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Hydraulic Fracturing in Produced Water Reinjection
SUMMARY
Reinjection of produced water has become a viable method for disposal, for support and for drive. Characteristic elements of these injection operations include longterm injection with consequent stress changes due to poro- and thermoelastic effects. Dilute concentrations of entrained particles in the produced water add another level of complexity. These micron-sized particles can plug the formation during matrix injection. During injection above fracturing pressure, these fines and carried-over oil will alter the near-fracture permeability, will afford development of external filter cake on the fracture faces and can plug the fracture tips or reduce the fracture conductivity itself. Successful produced water injection operations usually entail the intentional or unintentional development of hydraulic fractures.
Success is measured on an economic basis and, as such, economic planning and performance evaluation require reliable predictions of fracture geometries and the capacity of fractures to accommodate fluid. The basic mechanisms for fracture growth during produced water injection, available in the public domain, are summarized. Hydraulic fracturing for, or as a result of, produced water reinjection is compared with hydraulic fracturing for stimulation. Finally, various public domain models for designing and evaluating produced water hydraulic fracturing are briefly summarized.
INTRODUCTION
Hydraulic fracturing simulators for stimulation have evolved substantially.
Recently, some effort has been devoted to modeling fracturing processes that occur during flooding and disposal. For example, in maturing, water-drive oil fields, progressively increasing volumes of oily water are produced and must be disposed of. Reinjection is one disposal protocol that can be cost effective and environmentally attractive.
1 Declining well injectivity, often due to particles in the injected water, is one of the major factors in increasing costs of reinjection operations. In order to maintain injectivity, it is commonly necessary to inject above fracturing pressure. Economic forecasting is contingent on the fracture geometries that are created. The intent of this paper is to indicate some of the key differences between hydraulic fracturing for stimulation and hydraulic fracturing as a means for and a consequence of injecting produced water, as are currently available in the public domain. Also, the public domain methodologies for assessing fracture geometry and pressure during produced water injection will be summarized.
It can be surprising to realize the potential reduction in injectivity that can result from pumping dilute concentrations of small solids and oil. Wennberg, 1998, 2
1 Paige, R. and Ferguson, M.: "Water Injection: Practical Experience and Future Potential," Offshore
Water and Environmental Management Seminar, London, March 29-30, 1993.
2 Wennberg, K.E.: "Particle Retention in Porous Media: Applications to Water Injectivity Decline,"
Ph.D. Thesis, Department of Petroleum Engineering and Applied Geophysics, The Norwegian
University of Science and Technology, Trondheim (February 1988).
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Page 3 described injection into unfractured, gravel packed injectors in an unconsolidated sand in the Gulf of Mexico (Figure 1). Despite high native permeability, initial injectivity was low and repeated stimulation treatments were performed. After each stimulation, injectivity increased dramatically but then declined progressively more rapidly. The half-life of some of these wells was approximately 50 days. This means that within 50 days, the injectivity had decreased by 50% - economically unsatisfactory. It was concluded that fines were the culprits. The injected seawater was deoxygenated, filtered to at least five microns and treated for bacteria as well as inhibited for scale. The solids content in the seawater at the wellhead ranged from less than 1 to 7 ppm. None of the particles was larger than 4 microns and the average diameter was 2 to 3 microns. Available models for understanding injectivity, even for such radial flow scenarios, are inadequate.
Modeling of hydraulic fractures resulting from injection is also difficult. van der
Zwaag and Øyno, 1996, 3 provided a field case that highlights the currently increasing perception that almost all successful injectors are knowingly or unknowingly hydraulically fractured. They described injection trials in the Ula field where the purpose of the injection was to supplement weak reservoir support.
Seawater and seawater-produced water mixtures have been pumped. At the time of their publication they indicated rates of 200,000 BLPD into seven injectors.
Additional information has been provided by Svendsen et al., 1991.
4 Typical injection water, reservoir and completion properties are provided in Table 1.
9000 4500
8000 Rate
Pressure
4000
7000
6000
3500
3000
5000
4000
3000
2000
2500
2000
1500
1000
1000 500
0
0 50 100 150 200 250 300 350 400
0
Figure 1.
Time (days)
Injection decline for Well A09 (matrix injection, unconsolidated, Gulf of Mexico). (from Wennberg, 1998 2 ).
3 van der Zwaag C. and Øyno, L.: "Comparison of Injectivity Prediction Models to Estimate Ula Field
Injector Performance for Produced Water Reinjection," Produced Water 2: Environmental Issues
and Mitigation Techniques, M. Reed and S. Johnsen (eds.), Plenum Press, New York, NY (1996).
4 Svendsen, A.P., Wright, M.S., Clifford, P.J. and Berry, P.J.: "Thermally Induced Fracturing of Ula
Water Injectors," Europec 90, The Hague, The Netherlands, October 22-24, 1990.
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Table 1. Typical Injection Water Properties 3
Property Seawater
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50%SW:
50% PW
0.6 - 13.0 Total Suspended Solids, TSS (mg/l), including oil droplets
Suspended Solids (mg/l)
Mean Particle Diameter (microns)
Density (kg/m 3 )
Viscosity (cP)
Reservoir Height (feet)
Hole radius (inches) r e
/r w
Perforations (spf)
Perforation diameter (inches)
0.1-4.6
3.0
1023
1.011
293
4.25
1800
4
0.5
2.6-4.6
N/A
1035
1.116
Reservoir Permeability (md) 173
Various injection scenarios were evaluated and it was eventually discerned that the only reason that injectivity had been maintained was because the reservoir had been thermally fractured. After 45 days of injection "fractures with 2.2 m full
height and 20 m to 34 m half length were measured." 3 There is some controversy
over these dimensions. This is addressed by van den Hoek et al, 2000.
5
DIFFERENCES
Table 1 suggests some of the differences between hydraulic fracturing for stimulation and hydraulic fracturing during water injection. Settari and Warren,
1994, 6 described modeling of waterflood-induced fractures and the features that distinguish this process from conventional hydraulic fracturing. First, there are basic philosophical differences. In produced water reinjection or waterflooding, injectivity can be maintained if fracturing occurs. However, the engineer must consider more than the immediate impact of stimulation. Production economics are an essential consideration. A fracture alters the displacement pattern and can potentially decrease (or increase) recovery. There are significant differences in the time scale of the operations and the injected fluid viscosity. In water injection, the efficiency can be close to zero. "As a result, waterflood fracturing is leak-off
dominated as opposed to stimulation fracturing which is leak-off controlled." 6
5 van den Hoek, P.J., Sommerauer, G., Nnabuihe, L. and Munro, D.: "Large-Scale Produced Water
Re-Injection Under Fracturing Conditions in Oman," ADIPEC, paper prepared for presentation at the
9th Abu Dhabi Intl. Pet. Exhib., Abu Dhabi, U.A.E., October 15-18, 2000.
6 Settari, A. and Warren, G.M.: "Simulation and Field Analysis of Waterflood Induced Fracturing," paper SPE/ISRM 28081, presented at Eurorock 94 - Rock Mechanics in Petroleum Engineering,
Delft, The Netherlands, August 29-31, 1994.
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Settari and Warren suggested that the following factors might be important for produced water reinjection or waterflooding situations.
Significant pressure and saturation gradients may exist around the well from previous field injection or production. Reservoir properties may not be constant.
There can be large-scale reservoir heterogeneity and consequently leakoff variation.
Other offset producers or injectors will affect fracture propagation.
There can be thermally altered stresses, and changes in the fluid properties.
The average reservoir pressure can change during the time-scale of the injection operations.
Some of the relevant differences are described below. It is essential to recognize that an important difference between hydraulic fracturing for stimulation and hydraulic fracturing during waterflooding is that, during stimulation activities, the fracture will propagate (much) faster than the leakoff fluid front.
Mechanical Properties
As in stimulation scenarios, fracture length is strongly dependent on the leakoff.
"However, net pressure in the fracture is affected by K
C
, fracture friction and the factors controlling fracture containment (height) such as the confining stress profile with depth and modulus contrast in the same manner as in conventional fracturing.
Therefore, once the p foc
is fixed, the mechanical properties do not change the fracture length match. However, they determine the net pressure which is evident
during pressure fall-off test (PFOT) analysis." 6
other researchers have established that, particularly for layered formations, different mechanical properties can yield entirely different fracture geometries for the same pressure.
Rate
The preceding is not necessarily true.
Modulus strongly impacts thermal stresses. Also, Gheissary et al., 1998, 54
Many stimulation hydraulic fracturing treatments are performed at injection rates ranging from 10 to 50 bpm. A paradigm shift is necessary when thinking of produced water reinjection - consider rates/volumes in terms of barrels per day rather than barrels per minute. Rates may be up to multiple tens of thousands of barrels per day (10,000 BLPD ~6.9 bpm). This implies that while the rates may be similar, the volumes injected and the time scale of the operations can be quite
different. In terms of rates, van den Hoek et al, 2000, 5
Oman of 15,000 to 20,000 m 3
/day (~65 to 87 bpm). Also consider the potential for periodic shutdowns that are inevitable in any long-term operation and the fact that the injection rate may vary in accordance with meeting voidage requirements.
Potentially, injection rates may also be lower than for some typical stimulation operations, target formations may have very high permeability, and the injected fluid viscosity will be low, leading to low efficiency fracturing operations.
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Time Scale
Most hydraulic fracturing stimulation operations are completed in a matter of hours.
A few, large, specialty stimulations have injected large volumes at high rates.
Produced water reinjection and waterflooding are ongoing operations that can last for years. The consequences include substantial possibilities for poroelastic and thermoelastic stress field alterations and interaction with remote producers and
injectors. Figure 2 shows a typical field situation (from Detienne et al., 1995).
Fluids
Stimulation treatments with no polymer in the base fluid are rare. Even slickwater treatments will have small polymer loading to minimize tubular friction. Produced water typically has dilute concentrations of solid particulate matter, droplets of oil
(since this water has come from the production stream), and carried-over production chemicals. Many operators no longer do extensive filtering on injection water as it is anticipated that hydraulic fracturing will occur and that fractures will be able to accommodate the particulate material. The particles can be organic
(bacteria, plankton, etc.) or inorganic (e.g., clay minerals, quartz, amorphous silica, feldspar, mica, carbonates, etc.). Additives to produced water injection streams will characteristically include biocides, scale inhibitors and sometimes drag reducers although the price of the latter can sometimes be prohibitive. The viscosity of heated water represents the viscosity, since the re-injected water will likely be hot.
The reverse will be true if seawater is injected.
2500 250
Hydraulic Fracture
2000
Rate
WHP 200
1500
1000
500
Radial
TIF
150
100
50
0
0
Figure 2.
20 40 60 80 100 120 140 160
0
Time (days)
Events during a cold water injection program showing thermal fracturing (after Detienne et al., 1995 47 ).
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Thermo- and Poroelastic Effects
Water injection is a long-term, low viscosity operation. There can be significant changes in the total stresses due to reservoir cooling (seawater), reservoir heating
(possibly produced water) and pore pressure changes with the substantial injection volumes. Perkins and Gonzalez, 1984, 7 1985, 8 provided a view of stress alteration due to cold water injection. "During ordinary hydraulic fracturing operations … leakoff is controlled so that injected fluid volumes will be minimized. As a result, pressure and temperature changes in the rock surrounding the fracture do not ordinarily have a very significant effect on the fracturing operation. Therefore, the primary concern has been the effect that temperature has on fracturing fluid
rheology and leakoff behavior." 8
"... in some cases injection of cold fluid can significantly reduce tangential earth stresses around an injection well. It follows that vertical hydraulic fractures can be initiated and propagated at lower pressures than would be expected for hydraulic fracturing of a nearby producing well. The injection well fracture, however, would tend to be confined to the low stress region that lies within the flooded zone surrounding the injection well. If the injection rate is sufficiently high, or if injected solids plug the face of the fracture, then the pressure within the fracture could rise, thus permitting the fracture to extend beyond the confines of the cooled region.
After breakout, the fracture extension pressure should approach (and probably exceed because of the increased pressure field surrounding an injection well) the fracture extension pressure of nearby producing wells. The thermoelastic effect could have significant impact on fracture confinement at bounding zones. For injection wells, impermeable layers could confine fractures in vertical extent partly
because the impermeable layers have not been cooled as much as the pay zone." 8
Similar considerations apply to competing poroelastic effects. Detailed considerations of poroelastic calculations are available in the literature (for example, Detournay et al., 1989 9 ). Their real significance may be in produced water reinjection. Stevens et al., 2000, 10 gave examples specifically relevant to produced water reinjection. "Cooling is principally due to convection, and since the rock heat capacity per unit reservoir volume is approximately twice that of the water, the thermal front advances at about one-third the rate of the water saturation front." These are two competing phenomena. Thermal changes in viscosity are also a factor.
7 Perkins, T.K. and Gonzalez, J.A.: "Changes in Earth Stresses Around a Wellbore Caused by Radially
Symmetrical Pressure and Temperature Gradients," SPEJ (April 1984) 129-140.
8 Perkins, T.K. and Gonzalez, J.A.: "The Effect of Thermoelastic Stresses on Injection Well
Fracturing," SPEJ (February 1985) 78-88.
9 Detournay, E., Cheng, A.H-D., Roegiers, J-C. and McLennan, J.D.: "Poroelasticity Considerations in
In Situ Stress Determination," Int. J. Rock Mech. Mining Sci. Geomech. Abstr., 26 (1989) 507-513.
10 Stevens, D.G., Murray, L.R. and Shah, P.C.: "Predicting Multiple Thermal Fractures in Horizontal
Injection Wells; Coupling of a Wellbore and a Reservoir Simulator," paper SPE 59354, presented at the 2000 SPE/DOE Improved Oil Recovery Symp., Tulsa, OK, April 3-5.
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Plugging
It is known that the solids in produced water can be injected. The literature indicates field operations where several fracture volume equivalents of solids contained in the injected water have been successfully pumped.
11,12 In these situations, the fracture volumes were inferred from ancillary testing procedures
(hydraulic impedance testing, falloff surveys ...). This brings to mind the dominant
13 summarized the question: "Where do the solids go?" van den Hoek et al., 1996, issue:
"An essential difference with simulation of conventional waterflood fracturing is that owing to fracture fill-up with injected solids the fracture conductivity cannot be assumed infinite any more. This relates to the important PWRI issue of where the injected solids go. Using our model, we show that the pressure drop over a finite conductivity fracture can lead to a significant increase in fracture volume without necessarily leading to a significantly higher pressure. Thus, a picture emerges in which the fracture conductivity 'adjusts' itself in order to accommodate injected solids. This picture allows the computation of well injectivity as a function of total injected water volume, solids loading, etc. This concept can also be used to qualitatively explain the PWRI field observation that injectivity appears to be
partially or fully reversible as a function of water quality." 13
Wennberg, 1998, 2 and Wennberg et al., 1995,
14 presented the most comprehensive evaluation of water injection damage mechanics to date. The formation adjacent to the hydraulic fracture will be damaged due to particulate injection. Various empirical measurements have been made to facilitate representing injectivity decline as a function of injected volumes; particularly for matrix injection. Some of the highlights of these efforts are summarized below.
Donaldson et al, 1977, 15 showed that particles initially pass through the larger openings in a core and are gradually stopped by a combination of sedimentation, direct interception and surface deposition. They found that the larger particles initiate cake formation. Davidson, 1979, 16 found that the velocity required to
11 Martins, J.P., Murray, L.R., Clifford, P.J.G., McLelland, G., Hanna, M.F. and Sharp, Jr., J.W.: "Long-
Term Performance of Injection Wells at Prudhoe Bay: The Observed Effects of Thermal Fracturing and Produced Water Re-Injection," paper SPE 28936 presented at the 1994 SPE Annual Tech. Conf.
Exhib., New Orleans, LA, September 25-28.
12 Paige, R.W., Murray, L.R., Martins, J.P. and Marsh, S.M.: "Optimizing Water Injection
Performance," paper SPE 29774, SPE Middle East Oil Show, Bahrain, 1994.
13 van den Hoek, P.J., Matsuura, T., de Kroon, M. and Gheissary, G.: "Simulation of Produced Water
Re-Injection Under Fracturing Conditions," paper SPE 36846, presented at the 1996 SPE European
Petroleum Conference, Milan, Italy, October 22-24.
14 Wennberg, K.E., Batrouni, G. and Hansen, A.: "Modeling Fines Mobilization, Migration and
Clogging," paper SPE 30111, presented at the 1995 European Formation Damage Conference, The
Hague, The Netherlands, May 15-16.
15 Donaldson, E.C., Baker, B.A. and Carroll, Jr., H.B.: "Particle Transport in Sandstone," paper SPE
6905, presented at the 1977 SPE Annual Tech. Conf. Exhib., Denver, CO, October 9-12.
16 Davidson, D.H.: "Invasion and Impairment of Formations by Particulates," paper SPE 8210, presented at the 1979 SPE Annual Tech. Conf. Exhib., Las Vegas, NV, September 23-26.
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Page 9 prevent particle deposition is inversely related to the particle size (for the systems evaluated at least). Core measurements by Todd et al., 1984, 17 showed that the overall damage is related to the mean pore throat size. Cores damaged with aluminum oxide particles (with diameters up to 3 microns) exhibited damage along their entire length and as the particle size increased the damage gradually shifted to the injection end and external cake. Vetter et al, 1987, 18 found that particles with sizes from 0.05 to 7 microns caused damage and that the larger particles caused a rapid permeability decline with a short damaged zone. Permeability reduction with smaller particles was more gradual.
In conjunction with experiments, various researchers have attempted to mathematically characterize the mechanics of how fluid loss of water with particulates will damage the surrounding media. Most of these efforts have been continuum models based on conservation principles. The basic mass conservation relationship for one-dimensional flow is: d dt
c
d dx
uc D dc dx
0 (1) where:
................................................................................................... porosity, c(x) ...................................................... volume fraction of solids in the liquid, x ............................................................................................. the position,
(x) ........... volume fraction of trapped particles with respect to the bulk volume,
~................................................................................... indicates averaging,
A .................................................................................. cross-sectional area, u ................................................................................................... velocity, t ..................................................................................................time, and,
D .................................................................................dispersion coefficient.
Assuming incompressible flow, neglecting diffusion, assuming that particle deposition is the only mechanism for changes in porosity and finally assuming that c << 1:
dc dt
d
dt
c
0
(2)
17 Todd, A.C. et al.: "The Application of Depth of Formation Damage Measurements in Predicting
Water Injectivity Decline," paper SPE 12498, presented at the Formation Damage Control Symp.,
Bakersfield, CA, February 13-14, 1984.
18 Vetter, O.J. et al.: "Particle Invasion into Porous Medium and Related Injectivity Problems," paper
SPE 16625, presented at the 1987 SPE Intl Symp. on Oilfield and Geothermal Chemistry, San
Antonio, TX, February 4-6, 1987.
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Iwasaki, 1937, 19 worked on deep-bed filtration in sands and indicated that: u
c x
t
u
c x
uc
u
c
x
(
0
b
) uc
0 where:
(3)
........................................................................ filtration coefficient (1/cm).
Many of these concepts have been applied to comprehending how fluid loss is a transient process in matrix produced water reinjection. For example, Barkman and
Davidson, 1972, 20 outlined four mechanisms where entrained fines in the injection stream could reduce injectivity (Figure 3). These included development of an internal filter cake, where the particles invade the formation and are ultimately retained, reducing permeability; the consequent development of an external filter cake (the wall-building analog); plugging of perforations or other completions hardware; and, progressive coverage of the injection interval due to wellbore fillup.
They derived expressions for the time, t
I/I
0
,
where the injectivity decline ratio, =
(I is the injectivity index), has been reduced from 1 to . = 1/2 refers to the half-life of the well (cylindrical reservoir). Barkman and Davidson outlined a method to determine the water quality ratio where the suspension was flowed through a filtration membrane at a constant pressure to build an external filter cake, giving a straight line on a plot of cumulative injected volume versus the square root of time. Eylander, 1988, 21 revised the Barkman and Davidson model on the basis of core flooding measurements. His relationships accounted for porosity of the filter cake. van Velzen and Leerlooijer, 1992, variation of particle concentration with position:
22 hypothesized on the dc dx
(4) where:
c in e x c in
................................................................................... inlet concentration.
All of these models account for external and internal cakes separately. Pang and
Sharma, 1994, 23 extended these relationships by considering mutually interactive
19 Iwasaki, T: "Some Notes on Sand Filtration," J. Am. Water Works Ass., 29, (1937) 1591-1602.
20 Barkman, J.H. and Davidson, D.H.: "Measuring Water Quality and Predicting Well Impairment," JPT
(July 1972) 865-873.
21 Eylander, J.G.R.: "Suspended Solids Specifications for Water Injection from Core-Flood Tests,"
SPERE (1988) 1287.
22 van Velzen, J.F.G. and Leerloijer, K.: "Impairment of a Water Injection Well by Suspended Solids:
Testing and Prediction, paper SPE 23822, presented at the 1992 SPE Intl. Symp. on Formation
Damage Control, Lafayette, LA, February 26-27.
23 Pang, S. and Sharma, M.M.: "A Model for Predicting Injectivity Decline in Water Injection Wells, paper SPE 28489, presented at the 1994 SPE Annual Tech. Conf. Exhib., New Orleans, LA,
September 25-28.
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Page 11 formation of internal and external filter cakes with flow. They introduced the transition time, t*, which they related as the time when the deposition mechanisms change from internal filtration to external filter cake buildup. It was postulated that internal cake forms first, that as more particles are trapped on the surface a point is reached where invasion is limited, and that this is the time, t*, when the initial layer of external cake is formed. Before the transition time, internal filtration is applied and after external filtration is used. Other relevant references on matrix damage (i.e., the mechanics of formation plugging when there is no hydraulic fracture or there is a hydraulic fracture that is not propagating) include Khatib and
Vitthal, 1991, 24 and Khatib, 1994.
25
Figure 3. Schematic of mechanisms for particulate damage.
This is only part of the problem. Presuming that there is a mathematical methodology for inferring the variation of particle concentration with time and depth into the formation, it is also necessary to infer the variation of permeability due to the particle concentration (and ideally, using this new permeability distribution to infer future variations in the concentration profile). Models for explicitly calculating permeability change are based on Darcy's law and are usually single phase. Some empirical (semi-logarithmic - Nelson, 1994, 26 or power law -
24 Khatib, Z.I. and Vitthal, S.: "The Use of the Effective-Medium Theory and a 3D Network Model to
Predict Matrix Damage in Sandstone Formations," SPE 19649, SPEPE (1991).
25 Khatib, Z.I.: "Prediction of Formation Damage Due to Suspended Solids: Modeling Approach of
Filter Cake Buildup in Injectors," paper SPE 28488, presented at the 1994 SPE Annual Tech. Conf.
Exhib., New Orleans, LA, September 25-28.
26 Nelson, P.: "Permeability-Porosity Relationships in Sedimentary Rocks," The Log Analyst (1994) 38-
62.
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Rumpf and Gupta, 1971 27 ) relationships are used to interrelate porosity and permeability; porosity being determined from the deposition modeling. Many models still fall back on Kozeny-Carman representations, which have never been particularly successful.
A new perspective on fines deposition in porous media has been demonstrated numerically (using a mesoscopic model) by Wennberg et al., 1996.
28 They alleged that two generic classes for deposition are permeability reduction in bands orthogonal to the mean flow direction (localizations) or in bands parallel to the mean flow direction (wormholes). Which type of damage forms depends on the local pore velocity. "The possibility of low-permeability bands and high-
permeability wormholes further complicates modeling." 28
This area is incredibly complicated. Wennberg 2 proposed engineering
simplifications and outlined possible scenarios for the evaluation of the filtration
coefficient. Wennberg 2 recognized that it was important to consider linear flow
situations as well as radial, recognizing the overwhelming number of injectors that are actually hydraulically fractured. Conceptually at least, at the fracture tip,
(infinite conductivity) the velocity will be higher and particles will be transported to the tip causing fracture tip plugging. Eventually, leakoff will stabilize along the length of the fracture. If the whole fracture has a finite conductivity due to accumulation of particles, the flow pattern can deviate considerably from the purely elliptical flow pattern around infinite conductivity fractures (refer, for example, to
Liao and Lee, 1994 29 ).
Unfortunately, it is difficult to strictly apply these matrix injection mechanisms to situations where it is known that hydraulic fracturing is occurring. "Field experience shows that wells have been able to inject produced water over several years, despite injecting volumes of contaminant that significantly exceed any calculated fracture volume. Thus, not all the material injected into the fracture remains there, although in some cases the extrusion of sludge on shutting in of wells indicates that at least some of the material remains in the fracture. Material that remains within the fracture may be deposited as a low permeability filter-cake, as a fracture tip plug, it may form bridges with the fracture. Material may also be transported into
the formation causing an impairment by relative permeability effects." 46
It is important to realize that injectivity of produced water under fracturing conditions is determined by entirely different mechanisms than injection of produced water under matrix conditions. Plugging of the rock matrix by internal
27 Rumpf, H and Gupta, A.R.: Chem. Ing. Tech., 43 (1971) 367.
28 Wennberg, K.E. Batrouni, G.G., Namsen, A. and Horsrud, P.: "Band Formation in Deposition of
Fines in Porous Media," Transport in Porous Media, 24, Kluwer Academic Publishers, The
Netherlands (1996) 247-273.
29 Liao, I. and Lee, W.J.: "New Solutions for Wells with Finite-Conductivity Fractures Including
Fracture Face Skin," paper SPE 28605, presented at the 1994 SPE Annual Tech. Conf. Exhib., New
Orleans, LA, September 25-28.
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Page 13 and external filter cake during produced water reinjection while fracturing do not by themselves cause significant injectivity decline.
Leakoff
One-dimensional (Carter) fluid leakoff representations are commonly applied for low-permeability, stimulation hydraulic fracturing.
V
( t ) 4 h t
0 dt x f
(
0 t ) t
C
T dx
( x )
2 C
T x f h t (5) where:
V
....................................................................................... leakoff volume,
C
T
............................................................................. total leakoff coefficient, t ......................................................................................................... time, h .................................................................................. total fracture height, x f
................................................................................... fracture half-length, x ............................................................................................. position, and
.................................................... first time of exposure to the injection fluid.
"It is well-known that [Equation (5)] only works properly if the fracture propagation rate is large compared to the leak-off diffusion rate. If this is not the case, the use of [Equation (5)] can lead to overestimation of fracture length. For example, in waterflooding under fracturing conditions, this overestimation may be up to two
In this case [Equation (5)] needs to be replaced by a
proper description of the reservoir fluid flow around the fracture." 30 Differences in leakoff between linear (Carter) leakoff in low permeability stimulation on one hand and pseudo-radial leakoff in high permeability waterflooding on the other hand are
demonstrated in Figure 4, from van den Hoek, 2000.
Previous Injection History
Previous injection may have caused substantial changes in the near-well/fracture saturations and pressures, impacting fluid loss. Previous injection will reduce fluid loss and can also cause changes in the rate at which the fracture propagates.
Settari and Warren 6 considered this in detail. Beyond changes in formation
pressure, it is certain that injection rates will vary and there will be injection plant downtime. In addition, many injectors are converted producers that were hydraulically fractured. These issues place additional demands on hydraulic fracturing simulators - including tracking formation pressure during shut-ins and multiple injection cycles. In addition, representing conductivity associated with previous fractures is an issue that needs to be addressed. To account for reopening
30 van den Hoek: "A Simple and Accurate Description of Non-linear Fluid Leak-off in High Permeability
Fracturing," paper SPE 63239, prepared for presentation at 2000 SPE Annual Tech. Conf. Exhib.,
Dallas, TX, October 1-4.
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Page 14
and residual conductivity before and during reopening, Settari and Warren 6
modified the permeability of the opened fracture using empirical information to account for roughness, tortuosity, turbulence and closure stress. This issue is becoming more and more important as finite conductivity below "closure stress" is acknowledged (Abou-Sayed, 1999 31 ). At the opposite end of the spectrum, rather than reopening, a fracture can recede if a high fluid loss regime is encountered.
100.00
10.00
Numerical
Gringarten
Carter
Pseudo-Radial
Settari
1.00
0.10
0.01
0.001
0.010
0.100
1.000
Dimensionless Time (t
D
)
10.000
100.000
1000.000
Figure 4. A comparison between the dimensionless leakoff rate
(Q lD
= Q l
/{2 kh p}) versus dimensionless time (t
D
=k/{ c}) for various leakoff methodologies. Carter indicates conventional onedimensional fluid loss. Gringarten indicates Gringarten et al.'s solution (1974) 32 for transient elliptical diffusivity for a stationary, infinite conductivity line fracture. Settari indicates an elliptical
leakoff formula from Settari, 1980.
56 Pseudo-radial is a late-time
approximation of the transient elliptical flow solution from Koning,
45 Numerical is a numerical model solution for fully transient
elliptical flow around a propagating hydraulic fracture for arbitrary
pump rates, as developed by van den Hoek, 2000.
after van den Hoek, 2000 30 ).
In reopening (either for a newly converted, fractured producer or for an injector being brought back on line) any pre-existing cake will impact new cake deposition.
Suppose that there is an initial skin, s is:
2 provided a good analytical analog for this for radial injection.
0
. The injectivity during startup or reinjection
31 Abou-Sayed, A.S.: personal communication, June 1999.
32 Gringarten, A.C., Ramey, H.J. and Raghavan, R.: "Unsteady-State Pressure Distributions Created by a Well with a Single Infinite Conductivity Fracture," SPEJ (August 1974) 347-360.
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I
0
p wf
I
I
0 q
p
R ln
r e ln
r e
/
/ r w r w
ln
r e
s
2
kh
0 s
/
0 r w
s
s
0
(6) where:
I
0
....................................................................................... initial injectivity,
I ...................................................................................... current injectivity, q ........................................................................................... injection rate, p wf
................................................................................... injection pressure, p
R
....................................................................... average reservoir pressure, k .............................................................................................permeability, h ................................................................................................. thickness, r e
......................................................................................... external radius, r w
........................................................................................ wellbore radius, s
0
........................................................................................ initial skin, and, s ............................................................................................. current skin.
As can be seen, the higher the initial skin, the less the impact of cake that is deposited later. However, if there is initial skin, the injection skin, s, already has a head start and will develop more quickly.
Completing and Bringing Wells on Line
Stevens et al., 2000, 33 used simulations to indicate the importance of how wells are brought on line, for explicitly controlling the initiation of hydraulic fractures when thermal effects have changed in-situ stresses. In long horizontal injectors, multiple fractures may be created or fracturing may be concentrated only at the heel of the well. These applications demand explicit coupling with wellbore simulators to forecast bottomhole temperature and pressure at the sand face.
Conformance and Sweep
Produced water reinjection usually cannot be viewed strictly as a disposal operation. It is often an economic component in providing pressure maintenance or drive, and this is not just in high permeability situations. For example, Ovens et al,
1997, 34 discussed water injection in the Dan Field. The reservoir is located in
Tertiary Danian and Cretaceous Maastrichtian chalk formations, which are characterized by high porosities (20-40%) but low matrix permeabilities (0.5 - 2
33 Stevens, D.G., Murray, L.R. and Shah, P.C.: "Predicting Multiple Thermal Fractures in Horizontal
Injection Wells; Coupling of a Wellbore and a Reservoir Simulator," paper SPE 59354, presented at the 2000 SPE/DOE Improved Oil Recovery Symp., Tulsa, OK, April 3-5.
34 Ovens, J.E.V., Larsen, F.P. and Cowie, D.R.: "Making Sense of Water injection Fractures in the Dan
Field," paper SPE 38928, presented at the 1997 SPE Annual Tech. Conf. Exhib., San Antonio, TX,
October 5-8.
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Page 16 md). The highly porous D1 zone overlies a tighter D2 followed by the higher porosity Maastrichtian units M1 through M4. "In the case of low permeability reservoirs, it is possible to create large fractures, the size and orientation of which can have a profound effect on sweep patterns, producer placement and reservoir
Firm evidence from the later drilling of production wells indicated that fracturing had taken place (during a water injection pilot program) in both the Maastrichtian and Danian units, highlighting the need for fracturing models that can account for vertical growth and varying material and fluid transport properties. An extensive field monitoring program was carried out in addition to the development of new fracture prediction models. "With water injection above fracturing pressure, the problem is compounded by long term injection causing changes in reservoir pressure and stress, which in turn couples back to the fracture growth criteria.
Some commercial PC scale packages for fracture growth include this effect, but since their software architecture is primarily aimed at fracture stimulation, in the case at hand the use of these packages becomes clumsy, since some of the matching variables, such as swept zone width, would have to be computed separately outside each history match run. For this reason, the field data were matched using simple models of fracture growth, elaborated as required to compute
the matching variables, such as swept zone width, measured in the field." 34
MODELS
It is evident from the foregoing discussion that there are some unique challenges for modeling hydraulic fracturing during water injection operations. Various models and modeling philosophies are available or have been used. These will be described subsequently, but first, it is necessary to summarize the basic physical mechanisms that need to be considered. First, recall the differences that are important between produced water reinjection under matrix conditions and under fracturing conditions.
Suppose that injection is into an intact wellbore. For the sake of argument, consider that flow is radial (ignoring anisotropy). Field experience, laboratory measurements and analytical and numerical modeling all indicate that there will be the development of internal and external filter cakes. This will cause progressive development of skin. This causes a progressive and often precipitous decline in the injectivity (the injection rate divided by the difference between the injection pressure and the average formation pressure). On the other hand, if there is a fracture present, internal and external filter cakes will develop along the surfaces and ahead of the fracture. This causes a progressive increase in the efficiency and can ultimately facilitate discrete additional fracture propagation until a new equilibrium situation is reached. In combination with damage to the fracture face and in the formation, mass balance requires contaminant deposition in the fracture.
Depending on the particulars, it is suspected that this leads to plugging of the tip of the fracture and/or reduction in conductivity of the fracture itself. It also needs to be remembered that there is a larger surface area for development of the cake and there are important velocity differences between the linear and radial cases.
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The applicability of many of the matrix impairment events that have been described in the literature is relatively secondary for fracturing. For produced water reinjection under fracturing conditions, the most relevant stage of matrix type impairment is where an external filter cake has already started to build up. The real unknowns at this stage are the external filter cake's permeability and the external filter cake's thickness. The permeability is needed to assess the amount of fluid leaving the fracture and the thickness is required for mass balance considerations in evaluating how much "fill" is in the fracture itself. As is known from the hydraulic fracturing and drilling literature, it is presumed that the characteristics of a filter cake developed under fracturing conditions are different from those for a cake forming during matrix injection because of the linear flow geometry and shear rate effects leading to an equilibrium cake. The message is that filtration mechanics developed for radial flow models should be cautiously applied for fracturing scenarios.
As a result, even the most rudimentary fracturing model must account for movement of fluids and particulates into the adjoining formations, development of filter cake, and tip plugging or fracture conductivity impairment. The models should ultimately also account for erosional features associated with dynamic fluid loss mechanisms. The large injection volumes and common temperature differences absolutely require consideration of poro- and thermoelastic effects. The entire perspective of the reservoir must be considered for sweep efficiency considerations and for interaction with offset injectors and producers. The model must be able to represent more than one-dimensional fluid flow in the reservoir and have the capability to model discrete, albeit long-term injection events.
The models available for representing produced water fracturing range from modified stimulation simulators, through analytical two-dimensional produced water models, pseudo-three dimensional produced water models, and coupled or partially coupled reservoir simulators.
Modified Hydraulic Fracturing Stimulation Simulators
With injection into higher permeability formations and increased considerations over flowback design and in-situ measurements, the stimulation community has itself made efforts to modify their simulators. In an upcoming paper, van den Hoek,
2000, 30 asserts that changes in representation of fluid loss relationships are
essential for modeling fracpacking and cuttings reinjection. "In high-permeability reservoirs, leak-off rate may be high enough compared to fracture propagation rate to the extent that using the 1D Carter model ... is not justified anymore. This is especially true for those cases where the reservoir flow contribution to total leak-off
is the controlling factor, as can be the case for fracpacking operations." 30
Settari, 1980, 56 had shown that the classical form of fluid loss coefficient is strictly
valid only when the permeability in the direction of the fracture is zero or when the
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Page 18 leakoff rate is small. Nghiem et al., 1982, 35 presented a fully implicit model, coupling reservoir flow, fluid loss and fracture growth for multiphase situations.
Settari, 1980, had previously developed a sequentially coupled model. Fan and
Economides, 1995, 36 described relationships for pressure-dependent leakoff. Their method was based on flow of a non-Newtonian fluid in a porous medium. Plahn, et al., 1995, 37 used a reservoir model to evaluate fracture morphology during closure.
The real value of that paper may be the change in modeling philosophy where reservoir mechanics are merged with fracture mechanics.
Yi and Peden, 1994, 38 demonstrated a model, with realistic components. They included non-Newtonian fluid flow in the invaded zone. A representation of Darcy’s law, using power law methodology, was presented. There was Newtonian fluid flow in the reservoir zone. Ahead of the filtrate, formation fluid is displaced farther into the reservoir. This described flow of filtrate (fluid that has passed through the cake). Finally, within the fracture itself, cake is developed. The pressure gradient through the filter cake is proportional to the leakoff velocity raised to the nth power. Concurrently, the filter cake grows due to deposition. If the concentration of the materials causing cake development is C s
, a pressure drop relationship could be expressed as: p f
p w
c
V l v n '
c
2 K
n v
C s
1
C s
3 n
n
1
n '
1
n ' c
c
8 k c c
n '
1
2
(7) where: p f
..................................................................... pressure at the fracture face, p w
.................................................................. pressure behind the filter cake,
c
...................................................................................... filter cake factor,
V l
........................................................................................ leakoff volume,
v n .................................................................................... conversion factor,
C s
.................................................. concentration of pseudo-solids in the cake,
c
................................................................................... filter cake porosity,
35 Nghiem, L.X., Forsyth, P.A. and Behie, A.: “A Fully Implicit Hydraulic Fracture Model,” paper SPE
10506 presented at the 1982 SPE Symp. Reservoir Simulation, New Orleans, LA, January 30-
February 3.
36 Fan, Y. and Economides, M.J.: “Fracture Dimensions in Frac&Pack Stimulation,” paper SPE 30469 presented at the 1996 SPE Annual Technical Conference and Exhibition, Dallas, TX, October 22-25.
37 Plahn, S.V., Nolte, K.G. and Miska, S.: "A Quantitative Investigation of the Fracture Pump-
In/Flowback Test," paper SPE 30504, presented at the 1995 SPE Annual Tech. Conf. Exhib., Dallas,
TX, October 22-25.
38 Yi, T. and Peden, J.M.: "A Comprehensive Model of Fluid Loss in Hydraulic Fracturing," SPEP&F
(November 1994) 267-272.
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Page 19 k c
...................................................................... filter cake permeability, and,
c
.............................................. corrected cake factor accounting for erosion.
The numerical procedures are as follows.
1.
Solve for the pressure profiles in the invaded and the reservoir zones.
Discretize with standard finite difference techniques.
2.
Iteratively couple this with the filter cake model in accordance to specific farfield pressure or flow boundaries.
3.
Knowing the pressure profiles, calculate the leakoff velocity distribution.
Mayerhofer et al., 1993, 39 also recognized the importance of more explicitly coupling reservoir flow with fracture geometry, particularly for analyzing fracture calibration (minifrac) tests. They stated that the concept of the leakoff coefficient does not discriminate the controlling phenomena and the nature of their inherent deviations from ideality. "Fluid loss occurs normal to the fracture face through the filter cake, and into an invaded zone which does not extend more than a few centimeters into the formation. Outside the filtrate invaded zone the pressure
perturbation may extend for a significant distance into the formation." 39 These
authors explicitly separated the pressure drop occurring in the reservoir and in the filter cake. "Traditionally, specific leakoff coefficients have been postulated for separate phenomena such as compressibility-controlled, viscosity-controlled etc. ...
Then the individual zones have been combined as conductances in series. A simple
(harmonic average) and some more complicated techniques have been used to calculate the combined leakoff coefficient. Instead [they] addresse[d] the individual pressure gradients in their correct relative contribution and the components are added as resistances in series. This approach is straight-forward, since the solutions are given by well-known filtration models, that have been used frequently in well testing applications."
Mayerhofer and Economides, 1993, 40 presented a model which decoupled the reservoir and filter cake behavior as flow in the formation from a transient infiniteconductivity fracture with a rate and time-dependent skin effect. The assumptions made included:
Piston like displacement of the reservoir fluid by the filtrate,
The filtrate-invaded zone was modeled as a steady-state, but varying fracture face skin effect, added to the varying filter cake resistance.
39 Mayerhofer, M.J., Ehlig-Economides, C.A. and Economides, M.J: “Pressure Transient Analysis of
Fracture Calibration Tests,” paper SPE 26527 presented at the 1993 SPE Annual Technical
Conference and Exhibition, Houston, TX, October 3-6.
40 Mayerhofer, M.J., and Economides, M.J.: "Fracture Injection Test Interpretation: Leakoff Coefficient vs. Permeability Estimation," paper SPE 28562, presented at the 1994 SPE Annual Tech. Conf.
Exhib., New Orleans, LA, September 25-28.
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The analysis did not account for pressure dependence of the fracture face skin, relevant if there is significant pressure drop across the cake.
A material balance and PKN geometry considerations characterized, leakoff during closure.
Valko et al., 1997, 41 developed a radial flow model to represent two-dimensional reservoir flow that is associated with high permeability fracturing (for stimulation/completion).
Using a planar three-dimensional fracturing simulator, Morales et al., 1986, 42 approximated growth of a waterflooding induced fracture, resulting from filtered seawater injection into an oil-bearing limestone reservoir. Since the injected fluid was filtered seawater, fines were probably restricted. "The fluid loss rate was assumed to be represented by classic leakoff theory by the combination of the fluid loss coefficients C c
and C v
for the reservoir fluid and the injected fluid respectively."
Height growth was represented. An important observation was the rapid loss in thermal barriers to vertical growth once those barriers were ultimately penetrated by the fracture.
Clifton and Wang, 1988, 43 summarized three-dimensional modeling concepts in
TerraFrac TM , the code used by Morales et al. - particularly fluid loss and thermal stress effects. Leakoff through the walls occurs at a rate determined by the difference between the pressure in the fracturing fluid and the remote pore pressure divided by the time elapsed since the local fracturing surface was first exposed to the fracturing fluid. The fluid loss coefficient is normalized by the difference between the minimum in-situ compressive stress and the in-situ pore pressure. "The assumed proportionality of the fluid loss rate to the pressure difference (p-p f
) is consistent with the solution for one-dimensional flow into a semi-infinite porous medium with far-field pore pressure p f p maintained at the injection plane.
and a constant pressure
reviewed existing stimulation models and how fluid loss was
represented. He concluded that "none of the efforts addressing non-linear fluid flow around a hydraulic fracture have resulted in a model that can be used for a fracture propagating at arbitrary but not necessarily constant, velocity, i.e., that can be used to describe the growth of a fracture that propagates through a multilayer reservoir, with stress contrasts (leading to (temporary) retardation/ acceleration of frac growth) and rock mechanical property contrasts, and that can
41 Valko, P. and Economides, M.J.: "Fluid Leak-off Delineation in High-Permeability Fracturing," paper
SPE 37403, presented at the 1997 SPE Production Operations Symp., Oklahoma City, OK.
42 Morales, R.H., Abou-Sayed, A.S., Jones, A.H. and Al-Saffar, A.: "Detection of a Formation Fracture in a Waterflooding Experiment," JPT (October 1986) 1113-1121.
43 Clifton, R.J. and Wang, J-J.: "Multiple Fluids, Proppant Transport, and Thermal Effects in Three-
Dimensional Simulation of Hydraulic Fracturing," paper SPE 18198, presented at the 1988 SPE
Annual Technical Conference and Exhibition, Houston, TX, October 2-5.
44 Kurashige, M.: "Transient Response of a Fluid-Saturated Poro-Elastic Layer Subjected to a Sudden
Fluid Pressure Rise," J. Applied Mech., 49 (September 1982) 492-496.
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Page 21
also be used to describe the fracture closure after shut-in." 30 A numerical solution
will be presented in that paper for the fully transient elliptical fluid flow equation around a propagating hydraulic fracture for arbitrary pump rates(s). In addition, a simple analytical formula for leak-off rate is presented that is shown to yield an excellent approximation of the numerical results, both during fracture growth and after shut-in.
Two-Dimensional Produced Water Simulators
Conventional hydraulic simulators are evolving to represent high permeability environments which are characteristic of many produced water reinjection zones.
These stimulation codes do not necessarily explicitly represent the influence of fines and associated plugging, and rarely represent the poro- and thermoelastic effects adequately. The waterflooding and produced water communities have modified available codes or have developed new models to represent these effects. As with stimulation hydraulic fracturing, the first models developed were two-dimensional, either constant height or radial. Some of these models are briefly described below.
Perkins and Gonzalez, 1985, 8 developed a model for calculating the thermo- and
poroelastic stresses that are induced within elliptically shaped regions of finite thickness around a fracture. Presuming constant viscosity injection into a line crack
(two-wing, constant height, vertical fracture), the flood front propagates elliptically.
A region of thermal alteration, with a reasonably sharp front chases the flood front
(also elliptical). They cited the thermoelastic changes in an infinitely long elliptical cross section and approximations were derived for finite thickness situations.
These authors used Lubinski's developments to include poroelastic effects.
Following the conventional Perkins and Kern methodology, the fracture is assumed to start and initially grow radially, after which continued growth is lateral and confined. The stresses affecting growth are impacted by thermoelastic and poroelastic stress changes. One particularly interesting concept is the potential for development of secondary fractures.
Koning's analytical model, (Koning, 1988) 45 calculates fracture length, bottomhole injection pressure and dimensions of the waterflood front for a user-specified injection volume, V inj
. The waterflooded formation layer (with thickness, h) is bounded by impermeable zones above and below. The fracture, with a half-length, x f
, is at the centre of a set of concentric ellipses representing a zone where the reservoir temperature has changed, a flooded zone, and an unflooded virgin zone
(oil zone). Each region is characterized with its own mobility. Adopting a Geertsma and deKlerk methodology, the length is determined from: p f rac
( q , t , x f
, h )
initial
P
( q , t , x f
, h )
T
( Q , x f
, h )
K
IC
x f
(8)
45 Koning, E.J.L.: "Waterflooding Under Fracturing Conditions," Ph.D. Thesis, Technical University of
Delft (1988).
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Page 22 p frac
...................................................... fluid pressure in the fracture (uniform,
(since the fracture has infinite conductivity), q ........................................................................................... injection rate,
Q = qt ................................................................................. injected volume, t ......................................................................................................... time, x f
................................................................................... fracture half-length, h ................................................................ total height of layer and fracture,
initial
...................................................... total minimum in-situ principal stress
........................................................................................ (before injection),
P
.............................................. poroelastic back stress on the fracture face,
T
................................... thermoelastic back stress on the fracture face, and,
K
IC
....................................................................... Mode I fracture toughness.
The pressure, p frac
, is based on a steady-state solution for a two-dimensional, infinite conductivity fracture, accounting for the individual zones with different mobilities. Using this late-time transient approximation implies that fracture propagation is very slow in comparison to the diffusion of the fluid. In situations where this is not the case (i.e., low permeability injection or stimulation) this approach is unacceptable. From this pressure profile, Koning derived an analytical expression for poroelastic effects and adopted the Perkins and Gonzalez relationships for thermal effects.
Ovens and Niko, 1996, 46 formulated a radial version of Koning's model. They combined the Barenblatt fracture growth criterion with changes in back stress to derive a formulation relating changes in length to changes in fracture pressure.
Presuming that superposition is appropriate, the state of stress near the fracture tip was determined from the summation of two stress fields. The first one relates to the deformed surface resulting from the pressure applied to the fracture. The second stress state was for a continuous body subjected to body forces, in this case the loads arising from the pore pressure or temperature fields acting within the rock. An oblate spheroidal coordinate system has been used in formulating the equations leading to the stress changes.
For internal pressure in the radial fracture, Ovens and Niko 46
development by Abou-Sayed giving:
K
I
2
R
( 1 A p
) p i
( A p p
)
(9)
46 Ovens, J. and Niko, H.: "A Screening Tool for Predicting Lateral and Vertical Extent of Waterflood
Fractures, paper SPE 36892, presented at the 1996 SPE European Petroleum Conference, Milan,
Italy, October 22-24.
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Page 23 where:
R ......................................................................................... fracture radius,
A p
.............................................................................. poroelastic parameter, p i
..................................................................................... pressure in zone i, p
.............................................................. far-field formation pressure, and,
....................................................................................... far-field stress.
This was used in conjunction with thermo- and poroelastic stresses and volumes for a damaged zone, a cooled zone and an invaded zone. These expressions were used to evaluate an analytical version of Sneddon's relationship for a penny-shaped crack.
R r w p f
( r , t )
R
2
n
( r , t ) rdr r
2
2
R K
IC
(10) where: p f
(r,t) ................................................................................ fracture pressure, r ........................................................................................... radial position, t ......................................................................................................... time,
n
................................................................................... normal stress, and,
K
IC
....................................................................... Mode I fracture toughness.
Damage was accounted for. Ovens and Niko indicated that a rigorous way to include the effect of fracture face skin or reduced conductivity was to change the pressure boundary condition that governs calculation of the state of stress associated with internal loading. Reduced fracture conductivity would alter the pressure and flux distribution over the fracture face and thus alter the poroelastic
back stress. More details on this can be found in van den Hoek, 1996.
"Internal damage in the formation must be governed by the way in which the oil and solids are deposited within the formation. It is most likely that the deposition extends some distance away from the fracture face, since near the fracture the flow velocity may be sufficiently high to cause stripping of any deposited oil/solids. Thus
dynamic filtration theory may be required to model this effect." 46
"At the present time, the new radial model only accounts for the effects of internal damage, i.e. damage extending into the formation. This damage is crudely represented by two parameters, the damage factor K
F
DAM
. The damage factor K
DAM
DAM
and the damage volume
simply scales the water relative permeability to produce an ellipsoidal zone of reduced permeability around the fracture. The
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Page 24 volume of this zone is governed by F
relative to the total injected volume."
Detienne et al., 1995,
DAM
, which simply scales the damage volume
47 presented a convenient, basic model that has worked effectively in history matching wellhead pressure and injection rates for long-term
(3 to 5 years) injection. "The algorithm is sufficiently simple to be implemented in
a conventional reservoir simulator."
thermally-induced fracturing (TIF).
47 They particularly emphasized the concepts of
48,49,50 "The reservoir stress near the well is reduced when the reservoir is cooled, and fracturing will occur if the reservoir stress falls below bottom hole flowing pressure. It is this same mechanism which can
sometimes be heard as you drop an ice cube into a cocktail at room temperature." 47
TIF improves fracturing. The dimension of the cooled zone that develops around the well impacts the lengths of the fractures. The methodology adopted is as follows
1.
Wellbore Temperature Profile: A bottomhole flowing temperature is first calculated. The well is divided into segments and the transient heat exchange solution of Poettmann et al, 51 is used.
2.
Perform Calculations For Radial Injection: Matrix injection is initially assumed, bottomhole pressures are calculated and tested against the pressures required to cause hydraulic fracturing. Three radial zones are assumed - a near-wellbore cooled and flooded zone, a flooded and warmed up zone, and virgin reservoir.
In each zone, pressure drop is determined. Skin is incorporated in the cooled/flooded zone. Fluid properties and permeabilities are specified for each zone. The cooled radius is calculated, the flooded radius is determined from mass balance considerations and the pressure drop is found by adding the pressure drops in the three discrete zones. Thermoelastic effects are
incorporated by using the results from Perkins and Gonzalez, 1985.
cooled zones are indicated not to occur until the fracture grows beyond the radial-flow cooled zone. Poroelasticity was represented with a global effect to accommodate the evolution of the average reservoir pressure and a local effect due to pressure change in the immediate vicinity of the well.
3.
Fractured Well: When the pressure becomes adequate to initiate/grow a fracture, an iterative procedure is used to find x f
. An equivalent radius is used to represent the fracture in calculating skin. The skin incorporates a fracture conductivity component which accounts for the width and permeability of the
47 Detienne, J-L., Creusot, M., Kessler, N., Sahuquet, B. and Bergerot, J-L.: "Thermally Induced
Fractures: A Field Proven Analytical Model," paper SPE 30777, presented at the 1995 SPE Annual
Technical Conference and Exhibition, Dallas, TX, October 22-25.
48 Svendsen, A.P., Wright, M.S., Clifford, P.J. and Berry, P.J.: "Thermally Induced Fracturing of Ula
Water Injectors," Europec 90, The Hague, The Netherlands, October 22-24, 1990.
49 Morales, R.H., Abou-Sayed, A.S. and Jones, A.H.: "Detection of Formation Fracture in a Waterflood
Experiment," paper SPE 13747, presented at the SPE Middle East Technical Conference and
Exhibition, Bahrain, March 11-14, 1985.
50 Simpson, A.J. and Paige, R.W.: "Advances in Forties Field Water Injection," SPE 23140 (19191).
51 Poettmann, F.H. et al., "Secondary and Tertiary Oil Recovery Processes," Interstate Oil Compact
Commission, Oklahoma City, OK.
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Page 25 fracture, the geometric skin due to the existence of the fracture itself (s skin that can be associated with external filter cake (s with damage to the fracture face. s fracture
= G + s geometry
+ s cake
+ s cake damage geometry
),
), and skin associated s geometry
ln( 1
x f
/ r w
), s cake
w cake k
2 x f k cake s damage
k k damage
1
2 x w f k damage damage
(11) where: s geometry
..................................................................... geometric fracture skin, s cake
....................................................................... skin due to external cake, s damage
......................................................... skin due to fracture face damage, s fracture
.............................................................................. total fracture skin,
G ..................... skin due to fracture relative conductivity, (G 0.69, for F
CD
>30), x f
................................................................................... fracture half-length, r w
........................................................................................ wellbore radius, w cake
........................................................................ external cake thickness, k .............................................................................. formation permeability, k cake
..................................................................... external cake permeability, w damage
................................ depth of fracture face damage (internal cake), and k damage
.................... permeability of fracture face damaged zone (internal cake).
4.
Wellhead Pressure: Bottomhole pressure is converted to surface pressure by accounting for the head and the friction.
van den Hoek et al., 1996, 13 described a numerical model that couples the reservoir
engineering and fracture mechanics aspects of produced water reinjection, PWRI.
It incorporates finite, non-uniform fracture conductivity, fracture growth, and evolving accumulation of filter cake. The consequences of internal filter cake are addressed, as are stress changes associated with poro- and themo-elastic effects.
The numerical and analytical models are appropriate for constant-height fractures which fully penetrate a permeable layer bounded by impermeable layers, although a 'square fracture' option allows a first-order estimate of radial fracture dimensions.
The model is an extension of Koning's model for waterflood-induced fracturing. The fracture is surrounded by four elliptical pressure/temperature regimes. These are:
an impaired zone where oil/solids have penetrated,
a cooled zone (presuming the injection fluid is cooler),
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a flooded zone where the injected fluid has warmed up, and,
April 15, 2020
Page 26
a virgin hydrocarbon-bearing zone.
Each zone is characterized by its own temperature, viscosities and relative permeabilities. The extent of each zone is determined from the injected volume, the heat capacities, etc. There is an external filter cake composed of oil and solids.
The internal and external filter cakes and the finite conductivity fracture were elements not included in Koning's model. For clean water injection, the propagation criterion was represented as:
p
g f
(
( x q f
,
, t , h ) x f
,
K
h
IC x
) f
initial
PT
( q , t , x f
, h )
(12) where:
p f
.................................................... difference between the fracture pressure and the reservoir pressure at the start of the injection, q ........................................................................... total water injection rate, t ......................................................................................................... time, x f
................................................................................... fracture half-length, h ...................................................................... reservoir and fracture height,
K
IC
....................................................................... Mode I fracture toughness,
initial
................................ initial total minimum in-situ stress (before injection),
PT
........ sum of the poro- and thermoelastic stresses on the fracture face, and, g(x f
,h)...................................................................... fracture geometry factor
(PKN vs. KGD) 52 [for a KGD fracture, g(x f
,h)=1].
The pore pressure profile around the fracture followed Muskat's
derivations, the poroelastic stresses followed Koning's 45
thermoelastic stress field was determined after Perkins and Gonzalez.
damaging mechanisms were represented by:
53 original
A damaged zone around the fracture that was characterized by a uniform
"permeability impairment factor." Damage can be modification of the relative permeability in addition to solids-based damage. The outer extent of this impaired zone (internal filter cake) is calculated from the volume of injected oil and solids that deeply penetrates into the formation.
52 van den Hoek, P.J.: "A New Model for Optimizing Design of Hydraulic Fractures and Simulation of
Drill Cuttings Re-Injection," SPE 26679, European Offshore Conference, Aberdeen, UK (1993).
53 Muskat, M.: "The Flow of Homogeneous Fluids Through Porous Media," McGraw-Hill (1946).
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An external filter cake with uniform permeability that accumulates on the face of the fracture. The external filter cake (forming on the face of the fracture) is assumed to have a uniform permeability. The thickness is assumed to be elliptically variable. This will provide a uniform pressure drop through the external filter cake along the length of the fracture if the fracture has infinite conductivity. Mass balance is used to determine the thickness of the external filter cake [d(x), d(0) = d
0
]. d ( x ) d
0
1 ( x / x f
) 2
(13) where: d(x) .................................................................. external filter cake thickness, d
0
..................................... external filter cake thickness at the fracture mouth, x .................................................................. position along the fracture, and, x f
................................................................................... fracture half-length.
Once external filter cake development is initiated, it is presumed that internal filter cake development away from the fracture face will cease. Mass balance would then suggest that accumulation of solids could lead to plugging of the fracture itself and invalidation of continuing assumptions of infinite fracture conductivity, with accompanying pressure drops. The elliptical symmetry is no longer valid and pressure drop is determined semi-analytically using appropriately truncated Fourier series. Both poroelastic effects and fracture pressure depend on the position along the fracture half-length. "One problem that arises in modeling finite conductivity PWRI fractures is that both the average fracture permeability and the permeability profile in the fracture are
unknown." 13 If solids accumulate within the fracture after the initiation of an
external cake (presuming dynamic stability with erosion), solids fill the fracture and flow can be concentrated into discrete channels, possibly visualized as
"wormholes" within a filter cake that entirely fills the fracture. Continued injection can plug wormholes, and cause supplementary increases in local fracture width because of the consequent reduction in fracture conductivity.
These authors were among the first to publicly consider two limiting cases caused by differing distributions of plugging material. These are:
1.
Assuming that uniform fracture permeability will provide an upper-bound estimate of injection pressure.
2.
Assuming an infinite conductivity fracture with a low permeability plug behind the tip will result in a lower-bound estimate of fracture pressure. Analytical models have been developed for this situation.
It is anticipated that the tip of the fracture will plug first and that fracture permeability will decrease towards the tip. There are two extremes that one can imagine for modeling - a uniform, but finite fracture conductivity and a tip-plugged
PWRI
Model Comparisons
April 15, 2020
Page 28 region with infinite conductivity behind it. In the first case, iterations can be carried out on the fracture width until mass balance will allow all injected solids to be accommodated. In the latter case, it is presumed that the fracture behind the tip plug is dominantly filled but does contain adequate, discrete channels to minimize pressure drop and validate infinite conductivity assumptions, except at the tip. For the same wellhead injection pressure, case (i) [finite conductivity] will result in fractures with a smaller volume than for case (ii) [tip plugging]. On the basis of equal injected solid volumes, a lower injection pressure will characterize case (ii).
"... the assumption of a uniform fracture permeability profile will lead to an upperbound estimate of injection pressure, whereas the assumption of an impermeable plug will lead to a lower bound estimate of injection pressure."
When a uniform, finite conductivity fracture was assumed, observations from various simulations were:
Injection water temperature has a significant impact on well injectivity.
Determinations of filter cake permeability are problematic.
Most simulation runs showed a gradual reduction in injectivity due to fracture plugging.
Injectivity decreases for higher TSS, filter cake permeability, modulus, and injection water temperature,
Fracture size increases for higher TSS, lower filter cake permeability and lower injection water temperature,
Young's modulus had little effect on dimensions but affected pressure substantially,
Initial injectivity is not strongly dependent on solids loading because initial growth of the fracture is relatively rapid. Initial injectivity depends on the filter cake permeability and the injection water temperature.
Where an impermeable tip plug was assumed, simulations suggested that:
After prolonged injection, the pressures can be considerably lower than those forecasted for a uniform, finite conductivity fracture.
Injectivity hardly decreases with time. "This seems to be more in line with BP's
PWRI experience, who have been injecting for over 10 years in their Prudhoe
Bay field without observing any appreciable decline in injectivity." 13
These authors suggested that field information might preferentially support the tipplugging scenario, with infinite conductivity behind the tip plug. Modeling demonstrated that for a very large range of plug permeabilities, the computed wellhead pressure was fairly insensitive to the plug permeability and length (except for very high plug permeabilities). The actual mechanics of the plug are uncertain but one can speculate that the plug will be filled and compressed progressively up
PWRI
Model Comparisons
April 15, 2020
Page 29 to a point where the wellhead pressure no longer decreases. "The above discussion suggests that the picture of a low-permeability tip plug is more realistic than the
picture of a uniform fracture permeability distribution." 13
Analytical representation of the tip-plugging scenario is possible by using infinite conductivity assumptions over the unplugged length. In the plugged portion, the pressure profile and poroelastic stress regime are not uniform. Formulations are provided.
p f
( q , t , x f
, h )
initial
PT
( q , t , x f
, h )
( 1
1
2
A p
)
1 q
2
k
1 h ln
x x
f f
p f iltercake
1
2
sin
1
x f x f
(14)
g ( x f
, h )
K
IC
x f where:
p f
.................................................... difference between the fracture pressure and the reservoir pressure at the start of injection, q ........................................................................... total water injection rate, t ......................................................................................................... time, x f
................................................................................... fracture half-length, h ...................................................................... reservoir and fracture height,
initial
................................ initial total minimum in-situ stress (before injection),
PT
............... sum of the poro- and thermoelastic stresses on the fracture face,
A p
................................................................................. poroelastic constant,
i
........................................................................ effective viscosity in zone i, k i
.................................................................... effective permeability in zone i x' f
............................................... half-length of unplugged part of the fracture,
p filtercake
.................................... pressure drop through the external filter cake,
K
IC
................................................................ Mode I fracture toughness, and, g(x f
, h).................................................................... fracture geometry factor.
Another significant determination by van den Hoek et al., 1996, 13 was that the
computed fracture length is quite insensitive to the degree of internal fracture
plugging. This is demonstrated in Appendix B of van den Hoek et al., 1996, 13
(Equations B7 and B13).
In 1998, Wennberg 2 presented a description of the WID simulator that incorporates
many of the plugging concepts described earlier. Current versions of the code are proprietary. Nevertheless, the philosophical methodology is as follows.
1.
Determine the concentration of deposited particles as a function of distance and time. This requires us to know the initial filtration coefficient as well as the
PWRI
Model Comparisons
April 15, 2020
Page 30 variation of the filtration coefficient with time. (find the specific deposit
"Even more difficult to predict is the evolution of [the filtration coefficient] as deposition proceeds." However, the formulas [discussing transition time] rely on parameters that are difficult to determine theoretically,
but they can be determined by careful experiments." 2
2.
"The next step in damage modeling is to predict how the permeability changes as a function of specific deposit ( ). In general, it is difficult to relate the permeability reduction directly to the specific deposit, since the deposit
morphology will have considerable impact on the permeability reduction." 2 The
deposited particle concentration must be converted to a permeability distribution
... from which the corresponding skin factor can be deduced (the version
described by Wennberg 2 used the Kozeny-Carman model). Sum the resistances
in the discretization and update suspension concentration using mass balance.
3.
Knowing the skin, determine the change in injectivity.
"The reliability of WID (and any other simulation effort) is dependent on a correct understanding and prediction of the deposition rate. Due to the complexity of the process, involving particle type (size, shape and mineralogy), pore space morphology, pore mineralogy, liquid type, etc., it is virtually impossible to develop a correlation that will give a correct prediction of
Pseudo-Three-Dimensional Produced Water Simulators
As in conventional hydraulic fracturing, the next step in development of simulators has been to extend geometry with pseudo-three-dimensional considerations.
Gheissary et al., 1998, 54
extended van den Hoek's injection methodology
represented. Previously, Koning 45
layered reservoirs, where pseudo-three-dimensional fracture growth was
and Ovens and Niko, 1996, 46 had developed
solutions for a contained, and a radial fracture in a vertically unbounded reservoir, respectively. Gheissary et al. presented a method that approximates the gradual transition from three-dimensional to two-dimensional elliptical symmetry, if there is no crossflow - "... a new 3-dimensional fracture growth model which permits the description of fully contained elliptical fractures within the injection layer; with the fracture length, the height upwards and the height downwards all potentially
growing at different rates." 54 "We have developed an analytical model because it
needs to be coupled with a fracture simulator. A numerical model would be very time consuming as the reservoir pressure field needs to be evaluated at a large
number of succeeding time steps." 54
Presuming slow fracture growth, change in geometry and development of the pressure field were decoupled, and the pressure field was modeled with a constant fracture length. Fracture friction was neglected (rates are low and the injected fluid is water). It was assumed that there is no vertical crossflow away from the
54 Gheissary, G., Fokker, P.A., Egberts, P.J.P., Floris, F.J.T., Sommerauer, G. and Kenter, C.J.:
"Simulation of Fractures Induced by Produced Water Re-Injection in a Multi-Layer Reservoir," paper
SPE 54375, presented at the 1998 SPE/ISRM Eurock '98, Trondheim, Norway, July 8-10.
PWRI
Model Comparisons
April 15, 2020
Page 31 wellbore. Close to the fracture, the pressure field was approximately that for a fracture in an unbounded reservoir, whereas farther away it is two-dimensional
(asymptotically). Assuming pseudo-steady state and constant fracture pressure, analytical solutions are possible and there is a gradual transition from the three- to the two-dimensional solutions. The transition is represented by a discontinuity in the space for a fracture that is vertically centered in the layer. "The 3D-solution is taken inside the ellipsoid that touches the layer boundaries. The remainder of the reservoir is approximated by a 2D ellipsoidal volume and the associated 2D solution. A volume equal to the volume of the 3D ellipsoid is excluded, and the 2D solution beyond the 3D-2D transition is determined by an equivalent rectangular fracture over the full layer height. The boundaries of the 3D ellipsoid and the 2D excluded ellipsoid do not coincide; a pressure solution is only formulated inside the
3D ellipsoid and outside the excluded 2D elliptical cylinder. This allows pseudo-
three-dimensional fracture propagation." 54
"In the previous 2D model, 13 the filter cake was assumed to be uniformly
distributed over the fracture wall, with a possible tip plug at the end of the fracture where no water could penetrate. However, this resulted in often very high simulated bottomhole pressures as the friction in the very narrow "sheet" of fluid would become excessive. This observation pointed us to introduce "channeling" as
a mechanism to release the pressure."
reduces the area available for fluid loss, filter cake thickness is reduced, and an
Gheissary et al. found that:
54 Channeling apparently reduces friction,
increased pressure may be required to obtain the same opening as with sheet flow.
With more channeling, the pressure reduces and the fracture length increases.
Simulations are very sensitive to variations in oil viscosity with depth and temperature.
Simulations are very sensitive to the variations in water relative permeability with depth.
The simulator is unable to duplicate published field observations where there have been very rapid changes in the injection behavior when the TSS is changed.
Figure 5 is an example simulation using this model for water disposal in Oman.
PWRI
Model Comparisons
Shale
Initially
Planned
Injection
0.7 psi / ft
April 15, 2020
Page 32
Injection
Width (mm)
0 - 14.94
14.94 - 29.87
29.87 - 44.81
44.81 - 59.74
59.74 - 74.68
74.68 - 89.61
89.61 - 104.5
104.5 - 119.5
119.5 - 134.4
134.4 - 149.4
Minimum Stress (MPa)
Figure 5. Application of Shell's in-house pseudo-three-dimensional code for modeling produced water disposal in Oman (after van den Hoek et al.,
Coupled Reservoir Modeling
As is evident, it is desirable to incorporate the fracture modeling with detailed reservoir simulation. This has been done at varying levels of sophistication. Ali et al, 1991, 55 described injection above parting pressure in the soft chalk of the Valhall field. Four different simulation models were used for evaluation purposes. The first model was used to investigate the effect of injection water temperature on sweep and injectivity and to predict performance for three different reservoir descriptions.
This three-dimensional, three-phase, dual porosity model (with thermal capabilities) used an effective wellbore radius approach to model the induced fracture. A second
55 Ali, N., Singh, P.K., Peng, C.P., Shiralkar, G.S., Moschovidis, Z. and Baack, W.L.: "Injection Above-
Parting-Pressure Waterflood Pilot, Valhall Field, Norway," paper SPE 22893, presented at the 1991
SPE Annual Tech. Conf. Exhib., Dallas, TX, October 6-9.
PWRI
Model Comparisons
April 15, 2020
Page 33 model was used to evaluate the impact of long fracture lengths and different fracture orientations on breakthrough performance. "This was a very rigorous simulation of the actual fracture, assuming a single porosity reservoir system. An
in-house implicit formulation of the black oil model was used." 55 A third model was
used to address the uncertainties of fracturing. It was a single layer gridded system with three producers and one injector (dual porosity). Fractures of different lengths (all infinite conductivity) were incorporated. A fourth model was run to evaluate thermo-poroelastic effects.
"An in-house analytical fracture simulator was used to predict the thermo-poroelastic effect…This model assumed uniform thickness and an infinite reservoir with a propagating Perkins-Kern type fracture. It accounted for the effects of relative permeability and different viscosities for oil and water at reservoir temperature and injection temperature. The impact of fracture face plugging by solids and fracture extension as a result of injection of "dirty" water is also evaluated. The model assumes that the injected water displaces the oil in piston-like manner creating a
flooded domain which is an elliptic cylinder confocal to the fracture length." 55
Settari and Warren, 1994, 6 stated that "A typical waterflood pilot injection well may
experience low rate water injection, below fracturing pressure, step rate tests
(SRTs), several high rate tests followed by falloffs and an extended high rate injection until breakthrough is achieved. Ideally, the model should match the entire sequence of events in order to have confidence in the predicted fracture geometry."
They recommended:
Matching the injection below fracture pressure to provide basic reservoir characteristics and to indicate "ambient" conditions.
Analysis of SRTs † or high rate injection tests to give permeability and opening/closure pressures. Matching falloffs can give information on fracture volume, and leakoff.
Long-term injection can provide information on stress changes in the reservoir, as can repeated SRTs.
These authors described the use of a partially coupled model and its application to both static (propped or acidized) or dynamic (waterflood) fractures. A crucial component is the representation of leakoff. Two methods were proposed:
"Parametric" leakoff model. This assumes one-dimensional, piston-like displacement and describes the invaded (waterflooded) region by average mobility, saturation, compressibility and temperature. The leakoff model can approximate the influence of other wells or boundaries and correct the leakoff for 2-dimensional flow.
† Step-rate tests .
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Model Comparisons
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Page 34
In the numerical leak-off model, one-dimensional finite difference solutions are used for two-phase flow, heat transfer and stresses (approximately).
With either model, each fracturing event is modeled independently and is then
"interfaced" to a reservoir model at the appropriate time."
There are two evaluation components; waterflood fracturing simulation and reservoir simulation. For the fracture representation, a one-dimensional leakoff assumption underestimates leakoff. In 1980, Settari, injection rate (Q
D
= Q iw
/(hC 2 ).
56 described a correction factor for the one-dimensional leakoff velocity as a function of a dimensionless u
2 D
u
(15)
R
2 D
( Q
D
) where:
Q iw
= Q i
/2 ........................................................the injection rate for one wing,
R
2D
............................................................. growth reduction factor based on numerical solutions for a limited range of Q
D
, u ................................................................................................... velocity, h ........................................................................... formation thickness, and,
C ........................................................................... overall leakoff coefficient.
Koning, 1988, 45 provided analytical relationships between half-length and rate, from
which Settari and Warren derived relationships for the growth reduction factor, in single or multi-layered reservoir situations. For a single layer, two classical limiting cases were presented. For a large dimensionless injection rate (high rate and/or small height and/or small fluid loss): x f
t kc
T
2 h q
i p and R
2 D
1
(16)
For a small dimensionless injection rate (low rate and/or large height and/or large fluid loss): x f
3
kt c
T e q i
2
kh
p and R
2 D
6
kh q i
p e 2 q i
kh
p (17) where: x f
................................................................................... fracture half-length,
.................................................................................................. viscosity,
56 Settari, A.: "Simulation of Hydraulic Fracturing Processes," SPEJ (December 1980) 487-500.
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Model Comparisons
April 15, 2020
Page 35 t ......................................................................................................... time,
................................................................................................... porosity, k .............................................................................................permeability, c
T
................................................................................. total compressibility, q i
........................................................................................... injection rate, h .......................................................................................... thickness, and,
p ................................................................................. pressure difference.
There is a significant increase in fluid loss for a small dimensionless injection rate.
A standard reference for this subject is the publication by Clifford et al., 1990.
57
"The injection of cool surface water into higher temperature reservoirs frequently leads to thermal fracturing of the injection well. The fracture becomes a major influence on injectivity and, in some cases, on sweep efficiency. Fracture growth is dominated by stress changes due to cooling, and differs in a number of ways from
the conventional hydraulic fracturing process." 57 These authors presented results of
thermal fracturing in three-dimensional reservoir structures with spatially-variable permeability. A finite difference reservoir simulator was coupled with a threedimensional boundary integral fracture mechanics calculation. It was shown that:
Thermal fractures could often occur in high permeability layers that cool more rapidly.
Vertical fracture growth was found to depend on rates, pressures, temperatures, the perforation interval and on the horizontal and vertical permeability distribution.
Most water injection wells in deep reservoirs are fractured.
"Simulation of thermally induced waterflood fracturing requires the coupling of fluid and heat flow in the fractured reservoir with thermo- and poro-elastic stress and fracture mechanics. If the fracture can be approximated as 2-dimensional, extending over the full vertical interval of the pay zone for its full length, then simple plane strain formulae may be used to describe the fracture mechanics for a linear elastic medium. Perkins and Gonzalez constructed a simplified analytic model of 2-dimensional thermally induced fracturing based on the assumption of elliptical flow fields around the fracture. Models which couple two-dimensional fracture
models into a reservoir grid have been developed both for hydraulic fractures
for waterflood-induced fracturing.
58
The Dikken and Niko model calculates the
57 Clifford, P.J., Berry, P.J. and Hongren, G.: "Modelling the Vertical Confinement of Injection Well
Thermal Fractures," paper SPE 20741, presented at the 1990 SPE Annual Tech. Conf. Exhib., New
Orleans, LA, September 23-26.
58 Dikken, B.J. and Niko, H.: "Waterflood-Induced Fractures: A Simulation Study of Their Propagation and Effects on Waterflood Sweep Efficiency," paper SPE 16551, presented at the Offshore Europe
87, Aberdeen, September 8-11, 1987.
PWRI
Model Comparisons
April 15, 2020
Page 36 thermo- and poro-elastic stresses from the Goodier displacement potential, using methods developed by Koning." 59
The assumption of large vertical extent is often inappropriate and variable formation properties and layering need to be represented. The elastic modulus is the most important factor in determining thermal stress effects, in conjunction with permeability. The simulator uses a three-dimensional simulator developed by Gu and Yew, 1988, 60 in combination with the fluid and heat flow calculation from a three-dimensional reservoir simulator.
61 The stresses are calculated following
Koning's methodology. "For a consolidated rock without a network of natural joints, it is a reasonable approximation to separate the fluid and rock mechanics elements of the problem, at each time step. The assumption is that rock porosity and
permeability are nearly independent of fluid pressure and rock stress." 57
"It is rare for assumptions such as one-dimensional, transient leakoff, which is standard in many hydraulic fracture simulations, to be valid at any stage in thermal fracture studies. It is likely that flow over a wide region of the reservoir, including neighboring wells, will influence and be influenced by fracture growth. A reservoir
57 simulator is therefore an essential component of the model."
"At each time step, a pressure calculation must be performed which incorporates flow both in the reservoir and the fracture. ... Options exist for treating the fracture either as having infinite conductivity or having a finite conductivity, determined by
"The waterflood fracture is assumed to advance through a series of quasiequilibrium states ... in which the stress intensity factor K
I fracture boundary is equal to a critical value K
IC
at every point on the
Clifford et al., showed two simulation examples that are informative. The first was for a layered reservoir with moderate vertical permeability. These are communicating sandstone layers of different permeability (refer to Table 2). There is high permeability in the upper part. "Early breakthrough has occurred in production wells in the Brent formation, and has been attributed to channeling of water through the high permeability streaks in the Etive, leaving the Rannoch sands poorly swept. In order to reduce the vertical sweep problem both injection and
production wells have sometimes been perforated only in the Rannoch sands." 57
Case 1 is perforated only in Layers 5 through 7 and a fracture initiates in Layer 5.
There is substantial vertical flow into the Etive even before fracturing occurs. The fracture grows into the Etive within 20 days. Once the fracture has penetrated the
59 Koning, E.J.L.: "Fracturing Water Injection Wells - Analytical Modelling of Fracture Propagation,"
SPE 14686 (1985).
60 Gu, H. and Yew, C.H.: "Finite Element Solution of a Boundary Integral Equation for Mode I
Embedded Three-Dimensional Fractures, Int. J. Num. Meth. Eng., 26 (1988) 1525-1540.
61 Barker, J.W. and Fayers, F.J.: "Factors Influencing Successful Numerical Simulation of Surfactant
Displacement in North Sea Fields," In Situ, 12 No. 4 (December 1988).
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Model Comparisons
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Page 37
Etive that zone takes most of the water and the perforation placement has not done any good. Case 2 has perforations extending into the Etive. The fracture is assumed to initiate in Layers 3 and 4. There is no growth beyond the Etive. This is advantageous because of the high stresses in the upper Rannoch. There is more cooling in the Etive.
Table 2. Vertical and Horizontal Permeabilities
Layer Name Thickness
(feet)
Horizontal
Permeability
(md)
Vertical
Permeability
(md)
1 Ness 57.7 1497 899
2
3
4
5
Ness
Etive
Etive
Rannoch
60.2
68.5
47.0
90.8
652
1397
1807
346
391
1076
1391
246
6 Rannoch 60.2 208 131
7 Rannoch 46.7 33 16
The second example was an artificial case; a layered reservoir with low vertical permeability (refer to Table 3).
It is undesirable to inject directly into the high permeability zone, since there would be poor waterflood performance. The desired rates cannot be achieved by only injecting into the lower permeability zones. "A realistic strategy for vertical fracture confinement must therefore include a non-perforated low permeability buffer zone
between the perforated interval and the high permeability layers." 57
"It is therefore necessary to start injecting at relatively low rates, with a bottomhole pressure just greater than the cooled region stress. The rate is then allowed to build up gradually at roughly constant pressure, as the extent of cooling in layer 2 increases, and the fracture advances into it. The fracture must follow rather than
Table 3. Vertical and Horizontal Permeabilities
Layer Thickness
(feet)
Horizontal
Permeability
(md)
Vertical
Permeability
(md)
5
6
7
1
2
3
4
36.7
23.3
22.6
15.4
20.7
54.1
56.8
2.7
98.7
106.2
39.0
850.2
452.9
3.5
.0027
.0987
0.1062
.0780
2.5506
4.529
0.0105
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Model Comparisons
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Page 38
“More recently, the BPOPE code ‡ has been developed, amongst other things, to allow multiple thermal fracture generation and propagation in horizontal or multilateral wells to be modeled. Following on from these developments our attention has now turned to the way we handle bottomhole temperature in the BPOPE simulator, given the importance of temperature on the correct prediction of thermal fractures. Typically bottomhole temperature is treated as a single, time
independent parameter by many reservoir simulators….” 10 Wellbore and reservoir
simulators have been coupled together. The wellbore simulator was expanded to allow multiple entry points.
THE CHALLENGES
An evaluation of produced water models and generic model categories shows that each has advantages and limitations. Advantages of some relate to their simplicity, recognizing that it is particularly difficult to resolve all appropriate input parameters. Others more reliably incorporate the physical mechanisms of fluid loss and plugging while still others are capable of accounting for interactions with offset producers and injectors. There is no ideal model at the present time - just as the diversity of presentations at this workshop demonstrates that there is no ideal model for stimulation hydraulic fracturing. One of the upcoming challenges includes representing and comprehending the physics of particulate transport and plugging more effectively - “Where do the solids go?” Other challenges relate to the development of additional concepts for pseudo-three-dimensional modeling that approximates height growth without the burden of full-scale reservoir simulation.
Another challenge lies with extending fluid flow from two to three dimensions in the reservoir, within the realms of convenient computing. This may in fact be the greatest challenge since vertical crossflow away from the wellbore is an enormous economic consideration for recognizing and controlling sweep efficiency.
In the long run, effective modeling of fracturing associated with water injection may have more of an economic impact on the petroleum industry than fine tuning models for stimulation prediction.
‡
That has evolved from the code presented by Clifford, et al., 1990.
PWRI
Model Comparisons
REFERENCES
April 15, 2020
Page 39
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B.
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April 15, 2020
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van den Hoek, P.J., Matsuura, T., de Kroon, M. and Gheissary, G.:
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Wennberg, K.E., Batrouni, G. and Hansen, A.: "Modeling Fines Mobilization,
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Donaldson, E.C., Baker, B.A. and Carroll, Jr., H.B.: "Particle Transport in
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Vetter, O.J. et al.: "Particle Invasion into Porous Medium and Related
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Iwasaki, T: "Some Notes on Sand Filtration," J. Am. Water Works Ass., 29,
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Eylander, J.G.R.: "Suspended Solids Specifications for Water Injection from
Core-Flood Tests," SPERE (1988) 1287.
V.
van Velzen, J.F.G. and Leerloijer, K.: "Impairment of a Water Injection Well by Suspended Solids: Testing and Prediction, paper SPE 23822, presented at the 1992 SPE Intl. Symp. on Formation Damage Control, Lafayette, LA,
February 26-27.
W.
Pang, S. and Sharma, M.M.: "A Model for Predicting Injectivity Decline in
Water Injection Wells, paper SPE 28489, presented at the 1994 SPE Annual
Tech. Conf. Exhib., New Orleans, LA, September 25-28.
X.
Khatib, Z.I. and Vitthal, S.: "The Use of the Effective-Medium Theory and a
3D Network Model to Predict Matrix Damage in Sandstone Formations," SPE
19649, SPEPE (1991).
Y.
Khatib, Z.I.: "Prediction of Formation Damage Due to Suspended Solids:
Modeling Approach of Filter Cake Buildup in Injectors," paper SPE 28488,
PWRI
Model Comparisons
April 15, 2020
Page 41 presented at the 1994 SPE Annual Tech. Conf. Exhib., New Orleans, LA,
September 25-28.
Z.
Nelson, P.: "Permeability-Porosity Relationships in Sedimentary Rocks," The
Log Analyst (1994) 38-62.
AA.
Rumpf, H and Gupta, A.R.: Chem. Ing. Tech., 43 (1971) 367.
BB.
Wennberg, K.E. Batrouni, G.G., Namsen, A. and Horsrud, P.: "Band
Formation in Deposition of Fines in Porous Media," Transport in Porous Media,
24, Kluwer Academic Publishers, The Netherlands (1996) 247-273.
CC.
Liao, I. and Lee, W.J.: "New Solutions for Wells with Finite-Conductivity
Fractures Including Fracture Face Skin," paper SPE 28605 presented at the
1994 SPE Annual Tech. Conf. Exhib., New Orleans, LA, September 25-28.
DD.
van den Hoek: "A Simple and Accurate Description of Non-linear Fluid Leakoff in High Permeability Fracturing," paper SPE 63239 prepared for presentation at 2000 SPE Annual Tech. Conf. Exhib., Dallas, TX, October 1-4.
EE.
Abou-Sayed, A.S.: personal communication, June 1999.
FF.
Gringarten, A.C., Ramey, H.J. and Raghavan, R.: "Unsteady-State Pressure
Distributions Created by a Well with a Single Infinite Conductivity Fracture,"
SPEJ (August 1974) 347-360.
GG.
Stevens, D.G., Murray, L.R. and Shah, P.C.: "Predicting Multiple Thermal
Fractures in Horizontal Injection Wells; Coupling of a Wellbore and a
Reservoir Simulator," paper SPE 59354 presented at the 2000 SPE/DOE
Improved Oil Recovery Symp., Tulsa, OK, April 3-5.
HH.
Ovens, J.E.V., Larsen, F.P. and Cowie, D.R.: "Making Sense of Water injection Fractures in the Dan Field," paper SPE 38928 presented at the 1997
SPE Annual Tech. Conf. Exhib., San Antonio, TX, October 5-8.
II.
Nghiem, L.X., Forsyth, P.A. and Behie, A.: “A Fully Implicit Hydraulic Fracture
Model,” paper SPE 10506 presented at the 1982 SPE Symp. Reservoir
Simulation, New Orleans, LA, January 30-February 3.
JJ.
Fan, Y. and Economides, M.J.: “Fracture Dimensions in Frac&Pack
Stimulation,” paper SPE 30469 presented at the 1996 SPE Annual Technical
Conference and Exhibition, Dallas, TX, October 22-25.
KK.
Plahn, S.V., Nolte, K.G. and Miska, S.: "A Quantitative Investigation of the
Fracture Pump-In/Flowback Test," paper SPE 30504 presented at the 1995
SPE Annual Tech. Conf. Exhib., Dallas, TX, October 22-25.
LL.
Yi, T. and Peden, J.M.: "A Comprehensive Model of Fluid Loss in Hydraulic
Fracturing," SPEP&F (November 1994) 267-272.
MM.
Mayerhofer, M.J., Ehlig-Economides, C.A. and Economides, M.J: “Pressure
Transient Analysis of Fracture Calibration Tests,” paper SPE 26527 presented at the 1993 SPE Annual Technical Conference and Exhibition, Houston, TX,
October 3-6.
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Model Comparisons
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Page 42
NN.
Mayerhofer, M.J., and Economides, M.J.: "Fracture Injection Test
Interpretation: Leakoff Coefficient vs. Permeability Estimation," paper SPE
28562 presented at the 1994 SPE Annual Tech. Conf. Exhib., New Orleans,
LA, September 25-28.
OO.
Valko, P. and Economides, M.J.: "Fluid Leak-off Delineation in High-
Permeability Fracturing," paper SPE 37403 presented at the 1997 SPE
Production Operations Symp., Oklahoma City, OK.
PP.
Morales, R.H., Abou-Sayed, A.S., Jones, A.H. and Al-Saffar, A.: "Detection of a Formation Fracture in a Waterflooding Experiment," JPT (October 1986)
1113-1121.
QQ.
Clifton, R.J. and Wang, J-J.: "Multiple Fluids, Proppant Transport, and
Thermal Effects in Three-Dimensional Simulation of Hydraulic Fracturing," paper SPE 18198 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, TX, October 2-5.
RR.
Kurashige, M.: "Transient Response of a Fluid-Saturated Poro-Elastic Layer
Subjected to a Sudden Fluid Pressure Rise," J. Applied Mech., 49 (September
1982) 492-496.
SS.
Koning, E.J.L.: "Waterflooding Under Fracturing Conditions," Ph.D. Thesis,
Technical University of Delft (1988).
TT.
Ovens, J. and Niko, H.: "A Screening Tool for Predicting Lateral and Vertical
Extent of Waterflood Fractures, paper SPE 36892 presented at the 1996 SPE
European Petroleum Conference, Milan, Italy, October 22-24.
UU.
Detienne, J-L., Creusot, M., Kessler, N., Sahuquet, B. and Bergerot, J-L.:
"Thermally Induced Fractures: A Field Proven Analytical Model," paper SPE
30777 presented at the 1995 SPE Annual Technical Conference and
Exhibition, Dallas, TX, October 22-25.
VV.
Svendsen, A.P., Wright, M.S., Clifford, P.J. and Berry, P.J.: "Thermally
Induced Fracturing of Ula Water Injectors," Europec 90, The Hague, The
Netherlands, October 22-24, 1990.
WW.
Morales, R.H., Abou-Sayed, A.S. and Jones, A.H.: "Detection of Formation
Fracture in a Waterflood Experiment," paper SPE 13747 presented at the SPE
Middle East Technical Conference and Exhibition, Bahrain, March 11-14,
1985.
XX.
Simpson, A.J. and Paige, R.W.: "Advances in Forties Field Water Injection,"
SPE 23140 (19191).
YY.
Poettmann, F.H. et al., "Secondary and Tertiary Oil Recovery Processes,"
Interstate Oil Compact Commission, Oklahoma City, OK.
ZZ.
van den Hoek, P.J.: "A New Model for Optimizing Design of Hydraulic
Fractures and Simulation of Drill Cuttings Re-Injection," SPE 26679,
European Offshore Conference, Aberdeen, UK (1993).
AAA.
Muskat, M.: "The Flow of Homogeneous Fluids Through Porous Media,"
McGraw-Hill (1946).
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Page 43
BBB.
Gheissary, G., Fokker, P.A., Egberts, P.J.P., Floris, F.J.T., Sommerauer, G. and Kenter, C.J.: "Simulation of Fractures Induced by Produced Water Re-
Injection in a Multi-Layer Reservoir," paper SPE 54375 presented at the 1998
SPE/ISRM Eurock '98, Trondheim, Norway, July 8-10.
CCC.
Ali, N., Singh, P.K., Peng, C.P., Shiralkar, G.S., Moschovidis, Z. and Baack,
W.L.: "Injection Above-Parting-Pressure Waterflood Pilot, Valhall Field,
Norway," paper SPE 22893 presented at the 1991 SPE Annual Tech. Conf.
Exhib., Dallas, TX, October 6-9.
DDD.
Settari, A.: "Simulation of Hydraulic Fracturing Processes," SPEJ (December
1980) 487-500.
EEE.
Clifford, P.J., Berry, P.J. and Hongren, G.: "Modelling the Vertical
Confinement of Injection Well Thermal Fractures," paper SPE 20741 presented at the 1990 SPE Annual Tech. Conf. Exhib., New Orleans, LA,
September 23-26.
FFF.
Dikken, B.J. and Niko, H.: "Waterflood-Induced Fractures: A Simulation
Study of Their Propagation and Effects on Waterflood Sweep Efficiency," paper SPE 16551 presented at the Offshore Europe 87, Aberdeen, September
8-11, 1987.
GGG.
Koning, E.J.L.: "Fracturing Water Injection Wells - Analytical Modelling of
Fracture Propagation," SPE 14686 (1985).
HHH.
Gu, H. and Yew, C.H.: "Finite Element Solution of a Boundary Integral
Equation for Mode I Embedded Three-Dimensional Fractures," Int. J. Num.
Meth. Eng., 26 (1988) 1525-1540.
III.
Barker, J.W. and Fayers, F.J.: "Factors Influencing Successful Numerical
Simulation of Surfactant Displacement in North Sea Fields," In Situ, 12 No. 4
(December 1988).
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APPENDIX A
BPOPE Model
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BPOPE Model
Basic Description of the Model 62,63
Planar fractures of arbitrary shape are modeled using a three-dimensional boundary element method. Fracture growth is governed by linear elastic fracture mechanics.
Pore pressure and thermal stress changes are coupled in this model using a threedimensional finite difference reservoir simulation of fluid and heat flow in the region around the well. At each time step during injection, the pressure, saturation and temperature are calculated in the gridblocks of the reservoir model, using the fracture as the fluid source term. At intervals, the stress state in the plane of the fracture is calculated and the fracture size is updated so that it is in equilibrium with the new stress field. The effect of face-plugging due to suspended solids is modeled as a static filtration process. Model constants can be determined from core flooding tests. This model for formation damage due to suspended solids was found reasonable for low solid content injection.
Fracture Model
A three-dimensional boundary element method is used to relate the fracture
opening and the pressure for planar fractures of arbitrary shape.
growth criterion is based on the computed stress intensity factor and the input fracture toughness. This fracture model is a true three-dimensional hydraulic fracturing model. The fracture model is coupled with a reservoir model to calculate temperature change (and thus thermal stress) and pore pressure change (and thus poro-elastic effects) on fracture growth.
Reservoir Model
The fracture model is interfaced with a reservoir model. The reservoir model is based on a three-dimensional finite difference method for solving temperature change and pore pressure change. Saturation changes and temperature effects on water relative permeability are considered in this model. Once the temperature and pore pressure changes are obtained from the reservoir model, the stress changes due to these changes can be obtained numerically by integrating a threedimensional integral.
62 This 3-D integral may be reduced to a 2-D integral through
integration by parts. This improves the numerical performance of this coupled model. Having computed the stress changes due to thermal and poroelastic effects from the reservoir model, new stresses are applied to the fracture model to update the fracture geometry.
62 P.J. Cliford, P.J. Berry and H. Gu, “Modeling the Vertical Confinement for Injection Well Thermal
Fractures,” SPE 20741 (1990).
63 J.P. Martins, L.R. Murray, P.J. Clifford, G. McLelland, M.F. Hanna and J.W. Sharp Jr., “Long-Term
Performance of Injection Wells at Prudhoe Bay: The Observed Effects of Thermal Fracturing and
Produced Water Re-Injection,” SPE 28936, presented at the SPE 69 th Annual Technical Conference and Exhibition held in New Orleans, LA, 25-28 September 1994.
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Recent additions allow representation of multiple fractures in a deviated or horizontal well, and couple the model with a thermal wellbore simulator for calculation of the injection temperature along the well. These features are important for optimization of the injection along the wellbore.
Representation of Injected Solids 64
Injection fines are represented by the gradual build-up of a thin layer of low permeability skin along the fracture face, either on the surface or internal. If this layer has a thickness d sk
and a permeability of k sk the pressure drop across it in that region will be:
on some part of the fracture face, where:
p sk
d sk k sk
Q
A
Q/A ................................. flow rate through unit area of the fracture face and,
........................................................................................ water viscosity.
The buildup of this skin (on any region of the fracture surface) is assumed to depend on the cumulative flux of injected water through that region of the fracture face. For relatively low concentrations of fine solids, it is assumed that the face plugging can be described by the following equation: d sk k sk
L
C k rock where:
C ...................................................................... dimensionless constant and,
L ................................................................. cumulative flux in units of length
(m 3 of injected water volume per m 2 of fracture area).
The constant C is determined from core-flood experiments, and depends on the water quality and the formation properties, most notably the permeability. A typical test involves injection of several thousand pore volumes of representative water into a core plug of one inch length. If the effective plug permeability is found to be reduced 50% for 500 pore volumes of injection, then one can determine the corresponding C. For seawater, C usually varies from 0.01 to 0.1.
64 P.J. Clifford, D.W. Mellor and T.J. Jones, “Water Quality Requirements for Fractured Injection
Wells,” SPE 21439, presented at the SPE Middle East Oil Show held in Bahrain, 16-19 November
1991.
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APPENDIX B
BP Multi-Lateral PWRI Spreadsheet Model
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BP Multi-Lateral PWRI Spreadsheet Model
Basic Description of the Model
Fluid flow in a deviated well is first modeled for both laminar and turbulent flow.
Various pressure losses such as frictional loss along the pipe and perforation friction are considered explicitly. Both matrix injection with formation damage and fracturing injection are considered. The fracturing model considers a fracture with a fixed fracture height. Stress changes due to temperature and pore pressure changes are considered. Formation damage due to suspended solids and oil in water is considered. Permeabilities in the vertical and horizontal directions do not need to be the same. The well can be vertical or deviated.
The model is capable of modeling produced water re-injection into multiple zones such as for multi-lateral wells. The input data include the initial in-situ stresses, original reservoir pressure and temperature, drainage radius, coefficient of linear thermal expansion, poro-elastic properties, well trajectory, water quality, absolute permeability and end point relative permeability, the k v
/k h
ratio, fracture toughness,
Young’s modulus, the Poisson’s ratio, etc. The calculated data include injection rate into each zone, injection pressure and injectivity, whether it is matrix injection in each zone or fracturing injection. If it is fracturing injection, the fracture length and width are calculated.
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APPENDIX C
Duke Engineering Model
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The GEOSIM Model From Duke Engineering
Basic Description of the Model 65,66
This model considers many of the differences between conventional hydraulic fracturing and long term, lower rate water injection. One of the major differences is leak-off and thus efficiency. Fluid efficiency for stimulation fracturing is much higher than the fluid efficiency for PWRI. Therefore, the conventional Carter leakoff model was revisited in the model and a two-dimensional leakoff model adopted.
This model shows that Carter’s leakoff model may underestimate leak-off by several orders of magnitude, especially for low injection rates. The model presents a mechanism to partially couple the fracturing model with reservoir simulation, where fracture dimensions are determined from the fracturing model and reservoir model is executed with the predetermined fracture. Modelling parameters can be adjusted in order for the two models to give the same injection pressure. The model is capable of considering variations in thermal stress, pore pressure and saturation in the water invaded zone. This model also considers effects of previous injection, and pre-existing propped/acid fractures. These features are important in analyzing step rate tests and fall-off tests after a period of injection.
Factors Considered
For accurate PWRI simulations, the following factors need to be considered: 67 a) Significant pressure and saturation gradients may exist around the well from previous, long-term injection or production. It cannot be assumed that the fracture will propagate through a reservoir with constant properties. b) Large scale reservoir heterogeneity will cause leakoff variation in the fracture path as the fracture can be a few thousands of feet in half length.
68 c) Long-term cold water injection can create a large cooled zone around the fracture, with thermally-altered fluid properties and stresses. d) Average reservoir pressure and stresses can change during the time of fracture propagation.
65 A. Settari and G.M. Warren, “Simulation and Field Analysis of Waterflood Induced Fracturing,”
SPE/ISRM 28081, Eurorock 94 – Rock Mechanics in Petroleum Engineering, Delft, The Netherlands,
August 29-31, 1994.
66 A. Settari, G.M. Warren, J. Jacquemont, P. Bieniawski, and M. Dussaud, “Brine Disposal into a Tight
Stress Sensitive Formation at Fracturing Conditions: Design and Field Experience,” SPE 38893, presented at the 1997 SPE Annual Technical Meeting, San Antonio, TX, October 5-8, 1997.
67 “Fracture Propagation, Filter Cake Build-up and Formation Plugging During PWRI,” PWRI News
Letter, Feature Article, Volume 1, No. 3, October 1999.
68 J.E.V Ovens, F.P. Larsen and D.R. Cowie, “Making Sense of Water Injection Fractures in the Dan
Field,” SPE 38928, presented at the 1997 SPE Annual Technical Conference and Exhibition held in
San Antonio, Texas, 5-8 October 1997.
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Page C-3 e) The leakoff zone around the fracture becomes large and has a saturation and temperature distribution, which is three-dimensional. f) Formation damage and filter cake buildup due to solids and oil need to be considered to study the effect of water quality on injection performance.
Fracture tip plugging and branching have been observed both in the laboratory
Methodology of Modelling
Two methods have been presented for modeling the change of physical parameters during injection (Figure 1). In the “parametric” or analytical leakoff model, the model assumes a one-dimensional piston-like displacement and describes the changes of physical parameters such as temperature, water saturation, relative permeability, etc, in the invaded region by their average. In the “numerical” leakoff model, the changes of physical parameters from each element of the fracture are computed by a one-dimensional (perpendicular to the fracture surface) finite difference model which solves simultaneously for 2-phase flow, heat transfer and for the stresses.
Capabilities of the Model
Two-Dimensional Leakoff Correction
A 1-D leakoff assumption underestimates the leak-off, especially at low injection rates. Settari (1980) 69 introduced a correction factor to the 1-D leakoff velocity, which is a function of the dimensionless injection rate. Based on the results obtained from the Koning’s (1988) 70 model, a relationship is given between the correction factor and the dimensionless injection rate. This relationship shows that the 2-D leakoff correction can be several orders of magnitude different (Figure 2).
This needs to be confirmed because this has a big effect on fracture size if this is true.
69 A. Settari, “Simulation of Hydraulic Fracturing Processes,” SPE Journal,
December 1980, pp. 487-500.
70
E.J.L. Koning, “Waterflooding Under Fracturing Conditions,” PhD thesis, Technical
University of Delft, 1988.
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Figure 1. Two methods in handling the water invaded zone.
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Figure 2. Leakoff correction factor due to 2-D flow.
Effects of Previous Injection
This is very important in studying falloff or step rate tests after a preceding period of injection. For example, previous injection leaves large pressure and saturation gradients in the fracture path, which will affect the leakoff rate. In the analytical model shown in Figure 1, since average values are used, this model does not account for pressure and saturation dissipation during shut-in. In the numerical model shown in Figure 1, leakoff, pressure, saturation and temperature are computed during shut-in periods and the solution therefore accounts for the dissipation process during shut-in.
Relative Permeability and Thermal Effects
Since water saturation is a function of position and, in many situations the average mobility has a minimum at an intermediate water saturation, effective mobility in the invaded zone may be considerably lower than the end-point values (see Figure
3). In the numerical model, since the saturation and relative permeability are modeling variables, this changing relative permeability can be accurately represented. The effective mobility in the invaded zone, which can be constructed by Welge’s tangent to the fractional flow curve, is at S w
= 0.5. Figure 3 shows that
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Page C-6 this value is much lower than the end-point values. It also should be noted that the mobility is temperature dependent.
Figure 3. Total mobility in the invaded zone as a function of water saturation and temperature.
Pre-existing Propped/Acid Fractures
“When the pressure in the pre-existing (static) fracture increases during high rate injection, the closure stress on the fracture decreases, causing some increase of its conductivity. When the pressure reaches the confining stress, the entire fracture will unload (starting from the wellbore due to pressure losses) becoming part open and part propped fracture. Eventually, the dynamic fracture will start to extend
from the tip of the static fracture.” 65
Pre-existing propped/acid fractures must be considered in modeling SRTs where pressure both below and at fracture conditions must be matched. The problem in modeling this process lies in how to model the combined conductivity of a static fracture and a dynamic fracture. Injection pressure can not be modeled by simply overlaying the static and dynamic fractures or just by considering either the static fracture or the dynamic fracture. A model is presented to represent the combined permeability (k f
) c
:
( k f
) c
k f exp(
0 .
6931 (
p )
2
/
2
) where k f
= w 2 /12 is the permeability of the dynamic fracture with w being the fracture width and is an input value of p at which the full permeability is reduced to one-half and p = p oc
(opening-closing pressure)- p combine the static and dynamic fractures. f
(computed pressure from the reservoir model). Figure 4 shows an example of using this approach to
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Figure 4. Simulated injection pressure with combination of static and dynamic fracture.
Field Study Methodologies
In the analysis of field data, each part of the history provides valuable information: a) Matching the injection below fracture pressure provides basic reservoir characterization and establishes the “ambient” conditions prior to fracture propagation. b) Analysis of SRTs or high rate injection tests gives reservoir permeability and fracture opening/closing pressure. If a falloff is recorded, its match will yield information about fracture volume, net pressure and leakoff rate. c) Long-term injection under fracturing conditions can provide data on stress changes in the reservoir (from the trend of the fracture pressure).
The proposed approach in analyzing field data can be broken into two phases – fracturing simulation and reservoir simulation. Basically, fracturing simulation determines fracture growth as a function of injection volume or time. Having developed a fracture growth versus time scenario with the fracturing simulator, the
“description” of fracture and its conductivity is interfaced with a reservoir model to test the fracture growth versus time validity by comparing the computed injection pressures from both simulators. Injection pressure above fracturing can be matched by adjusting in-situ stress and net pressure. Pressure falloff can be used to determine net pressure and closure time. These can be history matched by adjusting leak-off and fracture volume. Ideally, the injection pressures from both fracturing and reservoir simulations should be the same by adjusting the input physical parameters. These adjustments should be determined from matching field
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Page C-8 injection history data and/or test data; such as step rate tests, falloff tests, hydraulic impedance tests. Usually the only parameters which need to be adjusted are those associated with leakoff rate in the fracture model.
A Case Study
An example has been published, showing how the model was used to reproduce
step rate tests performed for well in Valhall field.
65 Reservoir modeling of the SRTs
is approached in a staged fashion. First, the pressure response due to injection below fracture was matched with a conventional reservoir model with a static fracture (if the well is so completed) by adjusting reservoir permeability and the static fracture conductivity. Matching the pre-fracturing data also provides for a point of departure between the observed and calculated pressure, which indicates when fracture starts.
A fracture description, generated with this matched permeability, is then interfaced to the reservoir model at the time of pressure departure. If the fracture extension is correct, the calculated injection pressure in the reservoir model will match the observed data. For example, if the calculated pressure is too low, then fracture growth rate needs to slow down in order to give a higher calculated pressure.
If a falloff test is available after the SRT, the pressure decline after fracture closure should also confirm the matched reservoir permeability.
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APPENDIX D
Diffract Model
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APPENDIX E
Hydfrac/Hydfrac V3 Models
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HydFrac
HydFrac is a numerical model based on a three-dimensional, two-phase thermal reservoir simulator. It incorporates fracture mechanics and formation plugging due to injected particles. “Special attention is paid to the analysis of fracture closure during injection shut-in and to the description of formation damage.” 71 The media are represented as heterogeneous, anisotropic and compressible and there is a thermo-poroelastic stress model.
After fracture initiation, propagation is described by a two-dimensional PKN model with a model for fracture plugging by particles. Internal and external filter cakes are considered.
Capabilities:
1.
Three-dimensional, two-phase, thermal reservoir simulator for anisotropic, compressible media.
2.
One injector and several producers can be modeled.
3.
Injection can be specified by rate or pressure, at the wellhead or as bottomhole conditions.
4.
Temperature can be applied as a bottomhole or wellhead condition.
5.
When pressure/temperature are specified at the wellhead, heat exchange and pressure drop in the injector wellbore are represented.
6.
The model computes pore pressure, saturation and temperature in the entire field.
7.
From the pressure and temperature distributions, in-situ stress variations are computed using a thermo-poro-mechanical stress model and are solved using Koning’s method.
8.
The displacement filed is represented as the gradient of a scalar function and the mechanical problem reduces to the Poisson’s equation. The temperature and pressure are the source terms of the equation with corresponding thermal expansion and poroelastic coefficients.
9.
Thermal or hydraulic fracturing can be represented.
10.The fracture can increase or decrease in length at any time step.
The Fracture Model:
A two-dimensional model was used. A PKN representation was selected. Each vertical plane in the fracture is therefore assumed to deform independently of the others. The fracture widths in vertical planes are coupled by the fluid flow and continuity equations and the width is a function of the local pressure.
71 Longuemare, P., Detienne, J-L., Lemonnier, P., Bouteca, M., and Onaisi, A.: “Numerical Modeling of
Fracture Propagation Induced by Water Injection/Reinjection,” SPE 68974, paper presented at SPE
European Formation Damage Conference, The Hague, The Netherlands, (May 21-22, 2001).
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Page E-3
The equation for the width of the fracture is based on Sneddon’s equation and the propagation criterion is stress-based and takes the following form: p f x
x f
3 x
x f
0
Mass Balance:
Once fracturing has been initiated, a fracture fluid flow model determines the fluid pressure profile in the fracture, accounting for friction, leakoff, changes in fracture volume and particle plugging. Solution is fully implicit.
Fracture Plugging:
Four mechanisms are cited:
1.
Deposition on grain surface
2.
Formation of mono- or multiparticle bridges
3.
Internal cake formation (solids and oil) as soon as “the non-percolation threshold has been reached near the fracture face.” It is represented as a progressive permeability reduction function of the cumulative fluid filtration per unit fracture surface with a dependence on equivalent particle concentrations (oil and water). The depth of the permeability reduction is user-specified.
4.
External filter cake and complimentary fracture filling. After internal plugging, particles accumulate on the surface of the fracture. This filter cake is assumed to be incompressible (i.e., constant permeability) but the thickness is allowed to increase. The thickness depends on the fracture evolution and the accumulated volume of particles in the fracture. The external cake is dynamic.
“For each fractured cell, the formation damage effect is represented by a modification of the transmissibility between the fracture and the reservoir. The damaged transmissibility is calculated using an equivalent fracture face permeability taking into account the pressure drop induced by internal and external plugging.” (Permeability in series.) “The damaged transmissibility is integrated in the coupled fluid flow description between fracture and reservoir, which is solved in an implicit manner. This description ensures a good representation of mass fluid balance in the fracture and the reservoir.”
This is a synopsis of the cited SPE paper. For additional information and examples, refer to the paper.
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APPENDIX F
MWFlood Model
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MWFlood
MWFlood is a pseudo-three-dimensional simulator for predicting the pressure and geometry of conventional hydraulic fractures associated with waterflooding. The program was specifically designed for evaluating the effects of injecting large fluid volumes over long periods and for fracture efficiencies approaching zero.
MWFlood has options for conventional (diffusion controlled) and steady-state (nondiffusion) fluid loss. "At early times, fluid loss from the fracture is generally diffusion controlled, but at large times the fluid loss is governed by steady-state or pseudosteady-state leakoff. The fluid loss option has a marked effect on fracture geometry with larger leakoff rates at later times as compared to diffusion alone."
Features
Prediction of hydraulic fracture geometries for waterflood applications
Determine water and thermal fronts
Includes general MFrac features with waterflood limitations
Thermal and water front tracking
Multi-layer thermal stress properties
Effect of thermal stress distribution on fracture propagation
Conventional and steady-state fluid loss
Low injection rates, large pumping times and volumes
Application to fracture efficiencies approaching "zero"
Filtration Law
The Filtration Law has two options. Conventional is the standard diffusion type fluid loss model as used in MFrac. The Steady-State option is useful for long injection times when the leakoff rate is no longer controlled by diffusion but rather by steady-state injection and production.
Conventional
The Conventional option is the standard type of fluid loss mechanism where the rate of fluid loss to the formation is governed by the total leakoff coefficient. This is referred to as diffusion type leakoff because the fluid loss mechanism is diffusioncontrolled.
Steady-State
This option should be used for long periods of waterflood injection. The steadystate equations are based on the assumption that the production rate is equal to the injection rate resulting in a steady-state pressure behavior of the reservoir.
72
Although steady-state fluid loss is not diffusion controlled at long injection periods, the leakoff velocity at early times is diffusion controlled (i.e. the leakoff velocity is inversely proportional to the square root of time). This option accounts for the fluid
72 Note that the MWFlood documentation erroneously calls this pseudo-steady state. It is uncertain whether the formulation reflects this error, although the following paragraph suggests it does not.
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Page F-3 loss behavior changing from the conventional diffusion leakoff to a steady-state fluid loss controlled mechanism.
The resulting leakoff velocity for steady-state behavior approaches an asymptotic value. This results in a constant leakoff velocity since as time increases the fracture length asymptotes to a constant value.
Thermal Stress
The in-situ stress in MWFlood can be modified in the vertical and lateral directions to account for the effects of thermoelastic stresses. The fracture fluid temperature is specified by the user.
The thermal and water fronts are calculated based on the rate of creation of energy and mass. The fluid ahead of the thermal front is assumed to be at the reservoir temperature and the fluid behind the thermal front is at the fluid temperature specified for the fracture.
The injection fluid temperature is also used to calculate the induced thermoelastic stresses.
Ahead of the thermal front, the stresses are equal to the initial formation stresses.
Behind the thermal front (and toward the wellbore), the modified stresses are seen by the fracture system.
The thermoelastic stresses perpendicular,
T
(fracture) are given by:
, to the major axes of the ellipse
1
E
T where:
T
.............................................................................. Thermoelastic Constant
................................................................. Coefficient of Thermal Expansion
T .......................... Temperature difference between fluid and reservoir (T
1
-T f
)
The thermoelastic stresses are determined for regions of elliptical cross sections and finite zone thickness (h) using the calculations by Perkins and Gonzalez. The thermal constant for stresses perpendicular to the major axes for (b
1
/h) < 0.01 is
1 b
1 b
/
1 a
1
/ a
1
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Page F-4 where b
1
is the minor semi-axis of the elliptical thermal front perpendicular to the fracture and a1 is the major semi-axis of the thermal front (in the direction of the fracture length).
The limiting thermal constant for large minor axis to thickness ratios is unity (i.e., for b
1
/h > 10; 1.0). This condition generally occurs at later periods. Figure 1 shows typical Thermal Constant values for various thermal front conditions.
Figure 1. Thermal constant versus elliptical shape.
The modified stress is the minimum horizontal stress behind the thermal front
(toward the wellbore). Ahead of the thermal front, the stresses are equal to the initial in-situ stresses.
The modified layer stress is equal to the initial layer stress, , plus the thermoelastic stress,
T
,
M
T where
T
is negative if the fluid injection temperature is less than the reservoir temperature.
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Thermal/Water Front
The Thermal/Water Front input data include the oil displacement factor, the waterfront aspect ratio, the porosity, formation thickness and an equivalent drainage radius. This data are used to calculate the thermal front, waterfront, ellipsoidal waterflood shape, oil displacement and leakoff characteristics. The input data are:
Injected Fluid
The injected fluid represents the properties of the fracturing fluid. For waterflood applications, the user should specify water.
In-situ Fluid
The In-situ Fluid is the formation fluid that occupies the pores. Typically, this fluid is oil.
Oil Displacement Factor
The Oil Displacement Factor is the fraction of oil that is displaced by the water. This factor is directly related to the irreducible oil saturation (i.e., Oil
Displacement Factor = 1- irreducible oil saturation).
Figure 2. Flood and thermal fronts calculated for a specific situation using
MWFlood.
Waterfront Aspect Ratio
This is the limiting aspect ratio of the minor to major axes of the ellipsoidal thermal and waterflood regions. The minor axis is perpendicular to the fracture plane and the major axis is in the fracture plane.
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At early times, this aspect ratio is very large since it represents the fracture length divided by the leakoff distance perpendicular to the fracture face.
Figure 3 illustrates that as time progresses this aspect ratio will decrease and asymptote to the user specified Waterfront Aspect Ratio.
Figure 3. Thermal/Water Front Aspect Ratio as a Function of Time.
Formation Porosity
The formation porosity is the equivalent value over the fracture height used for calculating the thermal and waterflood regions.
Net Formation Height
The Net Formation Height is used in the mass conservation equations to calculate the water and thermal fronts. This height is also used for calculating the leakoff velocity for the Steady-State Filtration Law option.
Equivalent Drainage Radius
The equivalent drainage radius is used to calculate the steady-state leakoff velocity. The drainage radius is only used if the Filtration Law option is set to
Steady-State.
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APPENDIX G
Perkins and Gonzalez Model
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Perkins and Gonzalez Model
Basic Description of the Model 8
This is one of the earliest fracturing models that consider thermal stress and pore pressure change during injection. The model considers thermal stress that would result from cooled regions with fixed thickness and elliptical cross section.
Thermoelastic stresses for a region with an elliptical cross-section and finite thickness are determined approximately with a numerical procedure. Empirical equations were developed to estimate the average interior thermal stresses in elliptical cooled regions of any height. Stress changes induced by pore pressure changes during fracturing are calculated using the same equations that were derived for thermal stresses. Since for linear elasticity, the form of the equations is the same, this is accomplished by replacing the linear thermal expansion coefficient with the coefficient of pore pressure expansion and temperature change with pore pressure change. The computed thermal stresses and stress changes due to pore pressure changes are coupled with closed-form solutions for a PKN hydraulic fracturing model to determine fracture dimensions – including length and width as functions of injection volume or time. Examples, using typical elastic and thermal properties, showed that injection of cool water can reduce in-situ stresses around injection wells substantially, causing them to fracture at pressures considerably lower than would be expected in the absence of the themoelastic effect. Thermal effects have been proved to be a very important factor in many water injection
,47 A mechanism is also presented in the model to study the effect of
water quality on injection performance.
Thermoelastic Stress and Stress Change Due to Poro-Elasticity
When water is injected during PWRI, a region of cooled rock forms around the injection well. This region grows as additional water is injected. At any time, its outer boundary is approximately described as an ellipse that is confocal with the line crack (2D fracture). Three zones with sharply defined boundaries are assumed
(see Figure 1):
1.
The cooled-and-flooded ellipse from the wellbore out,
2.
Followed by a flooded, but not cooled? ellipse (the same temperature as the virgin reservoir, but increased injection water saturation) and,
3.
The undisturbed virgin reservoir.
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Figure 1. Plan view showing a two-winged vertical fracture oriented perpendicular to the plane of minimum horizontal in-situ stress.
Thermoelastic stresses for regions of elliptical cross-section and finite thickness were determined approximately with a numerical procedures. The following empirical equations were developed to estimate the average interior thermal stresses in elliptical cooled regions of any height: where:
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1T
..................... thermal stress in the direction perpendicular to the fracture,
2T
............................. thermal stress in the direction parallel to the fracture,
E ..................................................................................... Young’s modulus,
......................................................................................... Poisson’s ratio,
................................................. linear thermal coefficient of expansion, and h .................................................................................. reservoir thickness.
If it is assumed that the porosity and permeability are independent of the stress level, the change of stress induced by pressure change can be computed in a similar manner to the change in stress that is induced by a temperature change, with the linear thermal expansion coefficient replaced by the linear coefficient of pore pressure expansion. However, it should be noted that the equations for thermal stresses are obtained numerically with the assumption that the temperature in the elliptical region is uniform. This may be a good assumption, based on the numerical results, if the heat transfer is dominated by convection.
Pressure in the elliptical region is not uniform and stress changes due to pore pressure changes may have to be computed from a reservoir model.
Size of the Injected and Cooled Regions
The injected (flooded) region is approximately elliptical in shape, in its plan view, and it is confocal with the fracture length (Figure 1). The size of the elliptic region, its major and minor semi-axes, can be determined from volume balance of the injected water. The cooled region is also approximated as elliptical in cross-section, and is also confocal with the fracture; the major and minor semi-axes of the cooled region are determined from an energy balance. Heat transfer and energy loss to the upper/lower bounding layers are not considered.
Pressure Equations
The bottom hole pressure in the wellbore is given as:
P iwf
p
R
p
1
p
2
p
3
p s
p f
p p where: p
R
...................................... reservoir fluid pressure far from the injection well,
p
1
........................... pressure rise at the elliptical boundary of the flood front,
p
2
............... pressure increase between the flood front and the hot/cold front ,
p
3
................... pressure increase between the hot/cold front and the fracture,
p
S
......................... pressure increase across skin damage at the fracture face,
p f
............ difference between the wellbore pressure and the average pressure at the face of the fracture, and
p p
.................... pressure drop through perforations connected to the fracture.
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Equations and/or descriptions on how to compute each of the above terms are
given by Perkins and Gonzales (1985) 74 .
Opening of Secondary Fractures
Because the cooled region is nearly circular in shape when the fracture length is short, the thermally-related reduction of the horizontal stresses is nearly uniform in all directions. As the fracture length becomes large, the cooled region becomes more elongated. As the cooled region elongates, the thermal stress reduction parallel to the fracture exceeds the thermal stress reduction perpendicular to the fracture. This tends to reduce the difference between stresses within the cooled region and it is possible at some point the stress parallel to the fracture becomes as large as the stress perpendicular to the fracture. When this happens, fractures may initiate along the original fracture surface and propagate in the direction perpendicular to the original fracture. Whether this will happen or not depends on the difference in the principal horizontal stresses that are initially present in the reservoir, the thermal coefficient of expansion, the temperature change and the elastic modulus. This process is depicted in Figure 2.
Example Problem
An example problem was provided by Perkins and Gonzales. It showed a BHIP
(after thermal stress reduction) that was well below the initial minimum horizontal stress.
Summary
1.
Thermal stresses resulting from a step change in temperature, across a region of elliptical cross-section and finite thickness are considered.
2.
Stress changes due to pore pressure changes are considered.
3.
These stress changes are coupled with the PKN model.
4.
Damage due to suspended solids is considered.
5.
An example using typical elastic and thermal properties of rocks shows that the injection of cool water can reduce earth stresses around injection wells substantially, causing them to fracture at pressures considerably lower than would be expected in the absence of the thermoelastic effect.
6.
Depending on the shape of the cooled region and the difference between the minimum and maximum in-situ horizontal earth stresses, fractures perpendicular to the main two-winged fracture could eventually open, thus creating a "jointed" fracture system.
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Figure 2. Plan view showing that the shape of the cooled region controls the ration of principal stresses within the cooled region.
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APPENDIX H
Predictif Model
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Predictif
Basic Description of the Model 73
This model first considers the wellbore temperature profile as water flows down the injection string from the surface to bottom-hole. A linear geothermal gradient is assumed in calculating the temperature distribution along the wellbore. Thermal stress and poroelastic effects are considered in the model using the solution given by Perkins and Gonzalez.
74
Two-dimensional hydraulic fracturing model such as
KGD is used in predicting fracture length and fracture width. Radial flow is considered before fracturing. Case studies are available to show the importance of thermally induced fracturing in water injection. Based on the model, the entire injection history can be divided into different regimes and the injection history over each regime can be matched with the model.
Capabilities of the Model
Wellbore Temperature Profile
The model first calculates the bottom-hole flowing temperature, using the surface temperature, the injection rate and the wellbore configuration. A linear geothermal gradient is assumed. The solution also assumes that the injection rate is constant.
To cope with rate-varying behavior, an effective injection time has been used. It was found that as long as the injection rate does not vary too abruptly, the algorithm gives satisfactory results. This may explain the initial reduction in injectivity that is shown in Figure 1. As the bottom-hole flowing temperature decreases sharply initially (Figure 2), the average viscosity of the injected water increases and thus the injectivity decreases.
73 J-L. Detienne, M. Creusot, N. Kessler and B. Sahuquet and J-L. Bergerot, “Thermally Induced
Fractures: A Field Proven Analytical Model,” SPE 30777, presented at the SPE Annual Technical
Conference & Exhibition held in Dallas, Texas, 22-25 October, 1995.
74 T.K. Perkins and J.A. Gonzalez, “The Effect of Thermoelastic Stresses on Injection Well Fracturing,”
SPE Journal, February 1985, pp. 78 – 88.
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Figure 1. Injectivity index and viscosity.
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Figure 2. Bottomhole flowing temperature.
Radial Injection
The model calculates the bottom-hole flowing pressure by first assuming radial, matrix injection. It then tests whether this assumption is acceptable using a fracture criterion. If radial injection is acceptable, the model concludes that radial injection prevails and the model proceeds to the next time step. If the fracturing criterion is satisfied, strictly radial injection does not occur and the model calculates bottom-hole flowing pressure assuming that the reservoir has been fractured.
A conventional three-zone model is used for computing the flowing pressure during radial injection. For injection of a cooler fluid, there are: (1) a cooled-and-flooded zone from the wellbore out, (2) followed by a flooded zone, but with reservoir temperature and (3) finally the undisturbed virgin reservoir (see Figure 3).
For the cooled-and-flooded zone, viscosity is determined from correlations at different salinities and temperatures. Relative permeability is entered as data. For the flooded zone at reservoir temperature, the viscosity also comes from correlations. Relative permeability to hot water is also entered as data. Finally, for the undisturbed virgin zone, the viscosity and relative permeability are entered as data.
Determining the extent (size) of the cooled-and-flooded zone can be found in many papers.
75
However, all solutions assume injection of water at a constant bottomhole flowing temperature. In reality, the bottom-hole injection temperature is changing, particularly in the early stages. The bottom-hole injection temperature as a function of injection time can be modeled. A concept based on average temperature is used in the model to include temperature changes.
75 R.H. Morales, A.S. Abou-Sayed, A.H. Jones and A. Al Saffar, “Detection of Formation Fracture in a
Waterflooding Experiment,” SPE 13747, presented at the SPE 1985 Middle East Oil Technical
Conference and Exhibition held in Bahrain, March 11-14, 1985.
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Figure 3. Temperature and water saturation profiles due to cold water injection as simplified three-zone model.
Thermal Stress and Poro-elastic Effects
It has been observed that injectivity which is lost on conversion from seawater to produced water can often be fully restored on conversion back to seawater.
Including thermal and poroelastic effects is important for correct analysis of such cases. Thermal stresses have been proven to be an important factor in modeling long term injection such as waterflood. Thermal stress due to temperature change during injection are estimated in the model according to the method proposed by
The stress change due to poro-elastic effects is divided into a global reservoir effect and a local well effect. Each can be computed analytically.
Fracture Injection
After a fracture has been initiated, the fracture length and width are determined using a two-dimensional hydraulic fracturing model. An equivalent radius is used to represent the fracture in the injectivity index equation. This equivalent radius is based on skin calculation, which consists of geometric skin due to the fracture, filter cake skin, and skin due to damage. Skin is converted into an equivalent well radius. This equivalent well radius is then used for injectivity index calculation.
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A Case Study
One case study has been published for an offshore oil field in West Africa. Ten wells have been injecting for a period of 3 to 5 years. Well head pressure has remained fairly constant at 100 to 120 bars. Typical initial injection rates were 200 to 800 m 3 /d. Typical increases in injectivity indices due to thermally induced fracturing are a factor of 1.5 to 2, bringing injection rates to the 1000 to 2000 m 3 ranges. On three wells, the injectivity indices increased by a factor 10.
Pressure Matching
/d
Wellhead pressure was held constant at 120 bars. After 40 days of injection, injection rate increased abruptly from 200 m 3 /d and reached 2000 m 3 /d after 120 days. The radial flow model can correctly matches the wellhead pressure during the first 30 days (see Figure 4). If thermally induced fracturing is assumed at the point when the injection rate increased abruptly, the match of the radial injection regime during the first 30 days is unchanged. Now however, well history between
30 days and 60 days is reproduced as shown in Figure 5.
From this exercise it is concluded that:
From 0 to 30 days, injection is in the radial injection regime.
From 30 to 60 days, injection is in thermally induced fracturing regime.
At around day 60, the injectivity again suddenly increased. It was impossible to match the well behavior beyond 60 days with any reasonable set of reservoir parameters. One explanation offered was vertical fracture growth. The wellhead pressure and injection rate can be matched beyond 60 days by increasing the fracture height and reservoir permeability. It was found that in order to match the injection history, the fracture height had to be increased drastically from some 20 meters to 120 meters and later back down to 80 meters. With this increase and decrease of fracture height, the entire injection history (over 1200 days) can be
matched by the model (Figure 6). But, as Detienne et al.
no limits to the possibility of matching when k and h are allowed to vary from one time step to the next. A true 3D hydraulic fracturing simulator may be required to simulate this sudden fracture height growth and constrain the otherwise arbitrary variation of fracture height.
One feature which may also explain the observed phenomena can be secondary fractures due to stress orientation change resulted from thermal stress because thermal stresses in the directions parallel and perpendicular to the fracture
direction are different, as indicated by Perkins and Gonzalez.
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Figure 4. Attempt to match assuming radial flow.
Figure 5. Attempt to match assuming TIF is confined vertically.
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Figure 6. Final history match assuming an unconfined fracture.
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APPENDIX I
The PWFRAC Model
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The PWFRAC Model
Basic Description of the Model
PWFRAC was developed for the PEA-23 project and is not in the open literature. All of the information presented here is from the feature article in the October 1999
PWRI Newsletter (Volume 1, No. 3).
Coupling of the pore fluid movement, pore pressure change, and stress changes associated with injection operations are incorporated in the model. Both internal formation damage and external cake are considered. When the open gap (the part of the fracture that is open between the filter cake on the fracture walls) does not extend to the tip of the fracture, the pressure-flow relationships along the open fracture gap satisfy the usual equations for viscous hydraulic flow between two surfaces. The pressure within the closed gap (designated as a tip plug) may have different pressure profiles, depending on the filter cake permeability. Opening of the fracture is computed from pressure along the fracture, based on poroelastic theory, resulting from Darcy flow in the formation. The fracture propagation criterion is based on a stress intensity factor. The filter cake buildup is linked to the amount of solid particles that are deposited by PW entering the formation at the fracture face. Erosion of particles, caused by shear stresses on the filter cake surface, and the pressure drop across the filter cake are also considered. The model also provides a detailed description of the near-tip region. The simulator provides the following predictions, as a function of time during the injection period:
The injection pressure that is required to maintain a given injection rate,
The length and width of the created fracture,
The filter cake thickness and the open gap along the length of the fracture,
The impaired permeability of the formation and the extent of this formation damage.
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Fracture Propagation, Filter Cake Build-up and Formation Plugging
During PWRI
Produced water (PW) injection offers an acceptable means of disposing of the produced water and may provide an opportunity for a water drive when applied in waterflooding. The required rate of produced water injection can be anticipated using the expected pore volume replacement ratio and water cut estimated from the production forecast. Fracturing is likely to occur during PW injection at voidage replacement rates.
The extent (size) of the induced fracturing will significantly impact this process.
Therefore, it is necessary for well injectivity planning and fracture sizing to have an accurate estimate of the pore pressure, the rock's mechanical properties, and the minimum in-situ stress in the injection horizon. This collective information can be used to estimate the required injection pressure and the number of injectors throughout the production period. . Well planning and design can also benefit from predictions concerning the histories of the injector performance and the length of the created fracture. As a result, the waterflood planning cycle efficiency would be increased.
It is generally accepted that PW injection will lead to plugging of fractures and damage of injection zone permeability. The engineering problem faced by the operator is reduced to establishing the balance between two competing mechanisms. The first mechanism is related to the well injectivity improvement that may result from any fracturing associated with produced water injection. The competing mechanism results from plugging of the near crack tip region and the impairment of reservoir performance (permeability) around the fracture caused by water contaminant invasion of the injection horizon.
In the present article, we will confine the discussion to fracture propagation and its impact on well injectivity, under conditions of produced water injection in permeable reservoirs. Results of such analyses, in conjunction with experimental determinations, can provide estimates of filter cake permeability [if history matching is done, indications can be provided by the model alone, but, uniqueness is not guaranteed] and thickness, as well as the magnitude of permeability impairment around the fracture and the extent of the impairment zone.
Injector Fracturing Concepts
The objective of this article is to illustrate how fractures propagate during produced water injection. The role of porous formation mechanics on the interaction between a permeability-damaged zone around the fracture and a plug at the fracture tip is investigated. The article discusses the four concepts listed below:
Concept 1 The application of fracture mechanics techniques to hydraulic fracturing during the initiation of clean water injection and continuing through the life of the reservoir. Fracture mechanics can be used to predict the relationships
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Concept 2 As injection proceeds, particles in the produced water are deposited in the injection formation horizon and a "damaged" zone forms around the hydraulic fracture surface. These deposits decrease the permeability of the zone and tend to increase the required injection pressure (for a fixed injection rate). Considerations must be given to the water quality (concentrations and characteristics of the damage-causing contaminants) and its relationship to formation damage.
Concept 3 A second characteristic of the continuing injection process is that a plug of produced water particles can collect at the tip of the hydraulic fracture. This plug restricts flow at the crack tip and also tends to increase the injection pressure that is required to dispose water at a given rate and also cause the fracture to propagate.
Concept 4 The two combined phenomena (indicated in 2 and 3 above) affect fracture propagation. Although both phenomena tend to increase the required injection pressure for a given injection rate, their influences on the local stress state
(and their impact on the criteria for crack propagation) are quite different.
Fracturing Propagation Models for PW Injection
Fracture propagation during produced water injection in a permeable reservoir presents an added dimension to the problems encountered in applying fracture mechanics concepts to rocks. Coupling of the pore fluid movement, pore pressure change, and the stress changes associated with an injection operation must be incorporated in any analysis. The impact of water contaminants on plugging of the fracture and/or the formation depends on the injected water quality. General understanding gained from past experience and published data, as well as from earlier JIP work, provides the following insights into well fracturing during constant rate injection of produced water:
1.
Fracture propagation during waterflooding using clean water requires injection pressures that are far in excess of those required during hydraulic fracturing conditions at a constant specified injection rate. A decrease in formation permeability or an increase in the injection rate will reduce the required pressure for fracture propagation at the same injection rate. If there is less fluid lost to the formation, pressure will develop in the fracture to facilitate propagation – this can occur if the rate is higher (poroelastic considerations, if the formation is less permeable and/or if cake develops along the fracture surface.
2.
The presence of contaminants in the injected water will have two competing effects (tip plugging versus fracture face impairment) depending on where these particles are deposited.
3.
Fracture extension during produced water injection operations may be significant without apparent changes in well injectivity, injection rate or
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4.
Fracture tip plugging by solids will lead to a higher required propagation pressure during waterflooding in comparison to the clean fracture tip case.
5.
Formation permeability damage around the fracture will tend to facilitate fracture propagation during constant rate injection operations. The extent of the permeability reduction and the magnitude of the damage in the permeability impairment zone will both significantly impact the injection operations and will, in general, lead to a decrease in the required fracture propagation pressure.
6.
Under constant injection rate, injector performance (for example, the injectivity index) can be maintained fairly constant, while both tip plugging and permeability damage occur. This can occur by fracture propagation without significant change in injection pressure. The two phenomena may actually be in balance.
7.
Fracture growth might allow the operator to inject lower quality produced water for a longer period of time without adversely impacting injector performance
(different water qualities significantly affect injectivity).
8.
Conclusion #7, above, is not true if vertical conformance or aerial sweep become negatively impacted by the extent of fracturing in the injector. Care should be taken when interpreting PW injector data to allow for estimation of the fracture size.
9.
Fracture propagation criteria for produced water injection can be best verified by comparison with available well performance records.
For predicting the behavior of a vertical fracture, propagating in a horizontal reservoir layer as a result of PW injection, a simulator must take into account the following phenomena:
1.
Loading of the formation caused by the water flow during leakoff,
2.
Permeability damage from produced water particles deposited in the formation during injection,
3.
Filter cake build up on the fracture surface and plugging at the fracture tip.
These features are required for the simulator to reliably compute a fracture's dimensions and a well's response during injection. The simulator predictions need to be consistent with a set of field observations. This is demonstrated in the current article by several illustrative examples. The simulator must provide the following predictions - as a function of time during the injection period:
1.
The injection pressure that is required to maintain a given injection rate,
2.
3.
The length and width of the created fracture,
The filter cake thickness and the open "gap" along the length of the fracture,
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4.
The impaired permeability of the formation and the extent of this formation damage.
To achieve the above, the simulator must represent the following six physical requirements and these must be satisfied simultaneously:
1.
The poroelastic solution for the opening of a fracture subjected to arbitrary stresses/body forces, resulting from Darcy flow in the formation.
2.
The fracture propagation criteria. These may be related to the stress intensity factor, K
I
, or other equivalent conditions.
3.
The flow of fluid leaking off inside the injection zone. This flow most likely satisfies Darcy's law. The gradient of the pore pressure at the fracture face
(within the damaged zone) is proportional to the rate of migration of the fluid that has leaked off (away from the fracture) and the impaired permeability.
4.
The pressure-flow relationships along the open fracture gap (the open fracture width) must satisfy the usual equations for viscous hydraulic flow. Both laminar and turbulent flow regimes must be considered.
5.
The filter cake buildup should be linked to the amount of solid particles that is deposited by water entering the formation at the fracture face. Erosion of particles, caused by the shear stress on the filter cake surface, and the pressure drop across the filter cake must also be accounted for.
6.
Alteration of the formation permeability, by the produced water particles that are deposited in the formation, must be taken into account.
General Observations
As the fluid flows in the pores, the pore pressure is increased and the effective compressive stress in the matrix is reduced. Note, however, that depending on the diffusivity and poroelastic characteristics of the formation, the total stress can increase due to poroelastic effects. The stress distribution in the matrix is altered, and there are associated displacements that tend to close the fracture. The fracture itself is responding to the local total stress field. The situation is more complicated when a filter cake and altered formation permeability are present.
When the open gap (the part of the fracture that is open between filter cake on the fracture walls) does not extend to the end of the fracture, the pressure within the closed gap (designated as a tip plug) could be constant and equal to the pressure at the tip of the open gap. The tip plug region may have other pressure profiles depending on the filter cake permeability. If the formation layers bounding the fractured horizontal layer are sufficiently "strong" (or the stresses in these bounding layers are large enough), the fracture can be contained within the injection horizon. In a containment situation, the bounding zones cause a stiffening influence to the fracture propagation within the injection layer - with a consequent increase in injection pressure (all other factors being equal).
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Filter Cake Behavior
A filter cake model for produced water injection depends on the particular volumetric fraction of the solids in the water entering a particular section of the fracture surface that is assumed to remain on the surface of the fracture (external cake). The remaining solids must pass along the fracture and may cause tip plugging, or, especially at early exposure times, form an internal cake. The solid material that collects on the fracture surface forms an external filter cake. The filter cake thickness should be determined in the simulator and with experiments and back-analysis, along with the permeability for the filter cake. Both of these quantities are needed for determining the pressure gradient in the filter cake as a result of fluid leakoff into the formation.
The solid material that is deposited on the fracture surface causes the effective fracture width to be reduced, leaving an open gap for flow along the fracture.
When the volume rate of flow into the fracture is specified, reduction in the width of this gap can lead to increased fluid velocity along certain regions of the fracture.
When the velocity in this effective fracture width is sufficiently high, material on the surface of the filter cake can be dislodged and swept into the flow along the gap. A criterion for "sweeping" material off the filter cake is required.
Formation Permeability Damage
When solids-laden produced water flows through a porous formation, some of the solids are deposited in the porous material. Some of the produced water particles become lodged in pores and do not move with the fluid. As flow continues, an equilibrium is reached between the concentration of particles trapped in the formation and those flowing with the fluid.
Consider the case where produced water, with a constant solids concentration, is entering a formation with no "fixed" (lodged, trapped…) particle concentration.
After some time, solids concentrations in the formation can be expected decrease away from the fracture face. With time, solids concentrations in the formation, at a fixed position, will approach equilibrium values.
Parameter Interrelationships During Fracture Propagation Scenarios
In order to gain some physical insight into the more general case of a porous material where there is a damaged zone, a rough approximation is considered here.
The results give a feeling for the influence of the actual damaged zone.
Consider that there is a porous skin (internal and external filter cakes) at the surface of the crack. Fluid pressure drops across this skin as the fluid flows from within the crack into the porous material (Figure 1). This skin represents the collection of particles from the produced water that has accumulated at and near the crack surface. Although the mechanism for the pressure drop across this skin depends on many factors, the pressure drop is taken here as a fraction,
( 0 1 ) , of the internal crack pressure, p i
, or the injection pressure.
Additionally, the volume rate of flow, Q, into the formation is reduced by the factor
( 1 ) , compared to the case with no porous skin.
PWRI
Model Comparisons
Undamaged Formation
Z
Crack
Plug
April 15, 2020
Page I-8
r
Damaged Zone
+ a r
a{1+
2}1/2
Figure 1. Schematic representation of a PWRI fracture.
(Use the symbol R on the figure like in the text instead of symbol r )
For a constant injection rate, there is a considerable increase in the stress intensity factor, K
I
as the skin factor increases ( .
For fixed values of , , K
Ic
and Q, the dependence of the crack radius, a, and the crack pressure, p i
, can be determined at incipient crack propagation. For example, consider the case of constant values for the injection rate (Q), the critical stress intensity factor (K
Ic
) and the permeability to viscosity ratio ( ). As increases, the crack length ( A) grows while the crack pressure ( B) decreases. As changes from 0 to 0.3 the crack length increases by about a factor of 3 while the crack pressure drops by about a factor of 2 (refer, for example, to Figures 2 and 3).
“These results are related to a penny-shaped crack model and I feel that the pressure decrease has more to do with this frac geometry than with plugging effects.” 76
76 J-L. Detienne, personal communication.
PWRI
Model Comparisons
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
Poisson's ratio = 0.1
Poisson's ratio = 0.2
Poisson's ratio = 0.3
April 15, 2020
Page I-9
Figure 2.
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Filter Cake Damage Parameter ( )
This figure shows a plot of A (dimensionless crack radius) versus for three values of Poisson's ratio. The plot illustrates the interplay between poroelastic and filter cake effects. The crack radius increases with increasing damage.
PWRI
Model Comparisons
25
20
April 15, 2020
Page I-10
Poisson's ratio = 0.1
Poisson's ratio = 0.2
Poisson's ratio = 0.3
15
10
5
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Filter Cake Damage Parameter ( )
Figure 3. This is an example plot of B versus for the indicated values of
Poisson's ratios. For the case of constant injection rate, Q, the curves show the relationship between the injection pressures and the amount of damage. According to this model the injection pressure, p i
, will decrease as the damage increases (due to damage and the geometry of the fracture). The pressure decrease may have more to do with the volume of the fluid lost than the extent of the damage.
The argument for this is that in conventional hydraulic fracturing simulations, excess pressure (for a contained two-dimensional fracture) decreases with increasing fluid loss. However, the increase in length shown in Figure 2 can be directly correlated with the plugging effects of the solids.
Porous Material Solution with Damaged Zone and Crack Tip Plug
The schematic in Figure 1 considers the following scenarios:
1.
A plug of length R - a occupies the crack tip(s). In this representation, the crack radius is R while the radius that is open for "fluid occupancy in the crack" is a.
There is no fluid flow into the injection zone from the crack surface between the radii a and R.
2.
A damaged zone of finite extent is introduced around the crack. The damaged zone is an ellipsoid occupying the region around the fracture. All quantities in the damaged zone are designated with subscripted plus signs. For example, P is the pore pressure in the damaged zone and
+
+
is the permeability to viscosity ratio in the damaged zone. The corresponding quantities outside of the
PWRI
Model Comparisons
April 15, 2020
Page I-11 damaged zone are designated with subscripted minus signs. The elastic properties, , and (Lamé’s constants and Poisson’s ratio) of the poroelastic material are the same in the damaged and undamaged zones.
From Figure 1, the relations between a, R, and may be written as
a / R and R
1
a 1 2 are factors for the sizes of the tip plug and damage zones.
1
2
2
1
is the elliptical coordinate of the damage zone.
1
0
0 1
is the permeability damage factor
This configuration can be used to predict fracture propagation parameters at a fixed injection rate, Q, and a fixed crack radius, R, (tip plugging and efficiency could be restricting growth) while is increased (a greater extent of the damage zone) and is decreased (longer plug at the crack tip). The figures in the following section illustrate the interaction between these plugging parameters (at the fracture tip and formation damage), with injection pressure and fracture length.
Results and Conclusions
For a fixed crack radius, R, and selected values of the dimensionless plug length, measured by ( 1 ) , the dependence of the crack pressure, p i
, on the permeability ratio, measured by , is plotted in Figure 4. The crack pressure is seen to rise as either the damaged zone permeability decreases or as the plug length grows.
These results make physical sense as permeability reduction and plug length growth both cause an impediment to fluid leakoff from the fracture into the injection zone.
Figures 4 and 5 show results for the case with no plug [(R-a)/R =0] and for different ratios of the (damaged region radius)/(crack length). These curves demonstrate that the injection pressure and the stress intensity factor (because of differing degrees of fluid lost to the formation) both increase with a reduction in the damaged zone permeability and an increase in the size of the damaged zone - as is intuitively expected.
PWRI
Model Comparisons
11500
11000
10500
10000
9500
9000
8500
(R - a)/R = 0
(R - a)/R = 0.048
(R - a)/R = 0.084
April 15, 2020
Page I-12
Figure 4.
8000
0.0
0.1
0.2
0.3
0.4
0.5
= 1 -
+
/
-
An example simulation of the variation in the pressure in the fracture with the magnitude of the reduction in permeability in the damaged zone. A larger value of implies that the permeability of the damaged zone is reduced. is equal to permeability divided by viscosity, the subscript "+" denotes the damaged zone, and the subscript "-" denotes the virgin, undamaged formation. As the reduction of permeability in the damaged zone increases, the pressure in the fracture increases. As the extent of tip plugging increases, the pressure in the fracture also increases because pressure is not transmitted to the fracture tip to allow propagation
(fixed crack radius).
PWRI
Model Comparisons
12500
12000
11500
11000
10500
10000
9500
(R-a)/R = 0
(R-a)/R = 0.048
(R-a)/R = 0.084
April 15, 2020
Page I-13
9000
0.0
0.1
0.2
0.3
0.4
0.5
= 1 -
+
/
-
Figure 5. An example simulation of the variation in the pressure in the fracture with the magnitude of the reduction in permeability in the damaged zone. This is similar to Figure 4, with the exception that the injection rate is 43 percent higher. More precisely, the ratio of the rate to the product of k/ and R is 43 percent higher. The consequence is an increase in the fracture pressure - caused by increased rate and/or decreased damage zone permeability, and/or increased viscosity and or an increase in the overall length of the fracture. A larger value of implies that the permeability of the damaged zone is reduced. is equal to permeability divided by viscosity, the subscript "+" denotes the damaged zone, and the subscript "-" denotes the virgin, undamaged formation. As the reduction of permeability in the damaged zone increases, the pressure in the fracture increases. As the extent of tip plugging increases, the pressure in the fracture also increases because pressure is not transmitted to the fracture tip to allow propagation.
Figure 6 is an alternate representation, demonstrating the increase in the injection pressure with an increase in the extent of the permeability-damaged zone.
PWRI
Model Comparisons
11500
11000
10500
10000
9500
9000
Damage Zone Radius/Crack Radius = 1.25
Damage Zone Radius/Crack Radius = 1.50
Damage Zone Radius/Crack Radius = 1.75
Damage Zone Radius/Crack Radius = 2.00
April 15, 2020
Page I-14
8500
8000
0.0
0.1
0.2
0.3
0.4
0.5
Figure 6.
= 1 -
+
/
-
This is a plot of the injection pressure with the degree of damage in the formation due to fluid loss. Moving along, the abscissa, it can be seen as the permeability of the damaged zone decreases, the pressure increases. In addition, looking at the three different curves, as the extent of the damaged zone increases, the injection pressure also increases.
Field experience has shown that, to maintain a fixed injection rate, the required injection pressure changes very slowly with time (very important). It is possible that an interplay between the competing influences on K
I
of the damaged zone and the plug accounts for this gradual injection pressure change.
In the following example, the nature of the damaged zone - plug length interplay indicates that the fracture can maintain a steady rate of growth while both the injection pressure and the injection rate remain nominally constant as damage
(measured either by the damaged zone permeability decrease or the damaged zone extent) increases. The fracture growth criterion remains constant while both the crack radius and the plug length increase.
Initially the crack has no damaged zone and no plug. In that situation, as either d = (damaged zone radius)/(crack radius) =
1
R
R a
1 2 , or,
= 1 - (permeability ratio) = 1
PWRI
Model Comparisons
April 15, 2020
Page I-15 are increased, the values of the crack radius, R, and the plug length, R-a, are found so that the initial value of the stress intensity factor, K
I
, occurs while the injection rate, Q, has a value of Q o
.
Consider Figure 7. In this figure, each curve is for a fixed, but different permeability in the damaged zone. An increase in means a reduction in the permeability in the damaged zone. The dimensionless damaged zone radius
1
R
R a
1 2 is the abscissa, the dimensionless plug length [(R-a)/R] is the ordinate and the parameter varied is which is indicative of the magnitude of the permeability reduction. For each curve in Figure 6, initially, there is a rapid rise increase in the length of the plugged zone at the tip with an increase in the extent of the dimensionless damaged zone (damaged zone radius)/(crack radius). The rate at which the tip plug grows decreases with increasing extent of the damaged zone up to a value of approximately 1.5, after which it levels off and then the dimensionless plugged length (plug length)/(crack radius) decreases slightly from its maximum value.
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
Figure 7.
0.00
1.0
1.2
1.4
1.6
1.8
2.0
Damaged Zone Radius/Crack Radius
This is a plot of the length of the plugged zone at each tip of the fracture (non-dimensionalized by the fracture radius) with the extent of damage in the formation due to fluid loss. Moving along, the abscissa, it can be seen that as the extent of the damaged zone increases, the plug length first increases rapidly up to reach a constant plug length-to-crack radius ratio. In addition, looking at the four different curves, as the permeability of the damaged zone decreases, the plug length to crack radius ratio also increases.
[supported by mass balance considerations].
PWRI
Model Comparisons
April 15, 2020
Page I-16
Figure 8 shows how the crack radius is altered by damage. Each curve is for a constant (but different) damaged zone permeability. As the size of the damaged zone increases, the radius of the crack grows monotonically. The rate of growth of the crack radius decreases with the size of the damaged zone.
3.50
3.00
2.50
2.00
1.50
1.00
1.0
1.2
1.4
1.6
1.8
2.0
Damaged Zone Radius/Crack Radius
Figure 8. This is a plot of the length of the fracture (non-dimensionalized by the original fracture radius) with the extent of damage in the formation due to fluid loss. Moving along, the abscissa, it can be seen as the extent of the damaged zone decreases, the radius increases.
In addition, looking at the three different curves, as the permeability of the damaged zone decreases, the crack radius also increases.
Figures 7 and 8 show the interplay between the plug length and crack radius when both the injection pressure and the injection flow rate are held constant. The figures show that a nearly constant value for (plug length)/(crack radius) is approached as damage continues to increase. The crack radius grows monotonically at a decreasing rate as damage continues to increase.
PWRI
Model Comparisons
APPENDIX J
Shell / Maersk Models
April 15, 2020
Page J-1
PWRI
Model Comparisons
April 15, 2020
Page J-2
Shell/Maersk Models
(Ovens and Niko, Ovens et. al)
Basic Description of the Model 68,77
The Barenblatt fracture growth criterion is combined with thermal and poroelastic effects and fracture toughness to yield a compact formulation, relating changes in fracture length to changes in fracture pressure. It is assumed that fractures grow with a constant height. Two dimensionless parameters are introduced: one relates the magnitude of in-situ stress changes due to thermal and poroelastic effects and one relates to toughness. The limitations of this model include:
Proximal producers are not considered. Coupling with a reservoir simulator is necessary for considering injector/producer interaction.
Flux of water exiting the fracture is uniformly distributed along the length of the fracture.
It is a two-dimensional, constant height model.
Case studies have been published for a number of wells in the Dan Field, in the
Danish sector of the North Sea - a low permeability chalk oil field. The reservoir has a porosity of 20 – 40% but low matrix permeability of 0.5 – 2 mD. Tectonic fractures are rarely observed except in the immediate vicinity of the main fault.
Several monitoring techniques were applied to evaluate fracture height, length, orientation and injector/producer interaction. The techniques included openhole and through-casing saturation logging, tracer injection, producer water cut monitoring and falloff surveys in injection wells.
“In general, the observed fracture wing areas are in line with those expected from the model. However, injectors with higher rates generally require higher permeability to match the field data. The physical origin of this effect could be the induction of micro-fractures near the plane of the main fracture, which enhances
the effective permeability seen by the fracture.” 68
This may be due to thermal stress effects because the minimum stress can change
orientation during injection. 74
The model is applied, in conjunction with fracture dimension monitoring techniques,
to Dan field water injection projects.
68 The following is a summary of the
monitoring techniques used in identifying fractures.
77 J. Ovens and H. Niko, “A New Model for Well Testing in Water Injection Wells Under Fracturing
Conditions,” SPE 26425, presented at the SPE 68 th Annual Technical Conference and Exhibition held in Houston, Texas, 3-6 October 1993.
PWRI
Model Comparisons
April 15, 2020
Page J-3
How to Determine the Orientation of the Fracture?
1. Injection of Radioactive Tracer
“a short lived radioactive -emitting tracer was injected into the water injectors
MFB-07m MFA-09A, MFB-05 and MFB-01, which surround the A-Flank west producer MFB-22. In addition, a suite of logs including -ray, was run in the MFB-
22 itself. The -ray logs were intended to identify the position of the induced fractures intersecting MFB-22 and to determine whether the fractures had multiple branches.”
If the injector and producer are both sub-vertical, this method of injection may not work because the fracture may not intersect with the producer.
2. Fractures Intercepted by New Wells
“In the summer of 1996, a new producer, well MD-3B, was drilled on the crest of the A-block. The well intercepted the fracture created by the injector ME-02, which had been injecting above fracture propagation pressure for some six months. The open-hole resistivity log from MD-3B clearly shows a waterflooded interval about
150 feet wide.”
The contrast between the injected water and formation water indicates where the fracture intersects with the producer. The sharp change could also indicate piston like displacement.
How To Determine the Fracture Height?
Temperature logs were used to infer the fracture height along the wellbore. This technique is questionable, especially after a large volume of water has been injected.
Fracture dimensions (length and height) are mainly inferred through pressure matching with numerical model.
Does fracture branch?
Three possible methods were suggested for detecting fracture splintering or branching – one is -ray tracer logging, another is saturation logging and the third is evaluation of the injection pressure signature. Multiple spikes and new spikes at a later time were suggested to be indications of multiple fractures. If saturation logging did not indicate swept zone from a single fracture and injection pressure showed a step-like increase, then this was suggested as an indication of multiple fracture growth. Periods of rapid pressure increase were inferred to be indications of multiple fracture growth. One needs to be cautious about this because periods of rapid pressure increase could be the indication of fracture tip plugging and sudden growth.
Fracture branching has been observed in the pilot injection well through logging the swept zone in a horizontal well about 1000 ft away. Two swept zones of about 50 feet wide and 100 feet apart were found.
PWRI
Model Comparisons
APPENDIX K
Shell 1 Model
April 15, 2020
Page K-1
PWRI
Model Comparisons
April 15, 2020
Page K-2
Shell 1
Background
The model is an extension of Koning’s model for waterflood-induced fracturing. The fracture is assumed to fully penetrate a permeable layer and is bounding above and below by impermeable material. The fracture is surrounded by four elliptically shaped zones that include:
1.
an impaired zone where oil and/or solids have penetrated,
2.
a cooled (or heated depending on the injected fluid) zone,
3.
a zone flooded by injected water that “has warmed up,” and,
4.
a virgin oil zone.
Each zone is characterized by its own temperature, saturant viscosities, and relative permeabilities. The extent of each zone is determined from mass balance as well as heat capacities of the water and the target formation (using the methods outlined in Koning’s thesis). The fracture face is covered with an external filter cake consisting of injected oil and solids that have not penetrated into the formation. Eventually, the fracture may be filled with solids (oil) that have not penetrated into the formation, leading to a finite fracture conductivity. This is a significant departure from Koning’s model.
Propagation
For clean water injection, the fracture is infinite conductivity, poro- and thermoelastic back stresses are applied and propagation is based on a critical stress intensity factor criterion. A geometry factor is included to account for whether the fracture has a KGD or PKN geometry. Poroelasticity is incorporated using analytical solutions for elliptical regimes. Thermoelasticity is based on the concepts of Perkins and Gonzalez.
Damage
The damage is represented as:
1.
A damage zone around the fracture which is characterized by a “uniform permeability impairment factor.” The boundary is calculated from the volume of injected oil (solids) that deeply penetrates (analogous to an internal filter cake). It is assumed that this is roughly equal to the extent of the residual oil saturation. A determination must be made of what percentage of the oil/solids deeply penetrates.
2.
An external filter cake on the fracture face with uniform permeability. The thickness of this filter cake is assumed to be elliptical. If the fracture conductivity is infinite this implies a uniform pressure drop over the entire surface. Thickness of the filter cake is calculated from the volume of injected solids/oil that remains in the fracture and the fracture surface areas.
PWRI
Model Comparisons
April 15, 2020
Page K-3
3.
Internal plugging of the fracture. When the external cake starts to form it is assumed that supplementary deposition of solids will be against the external filter cake and will progressively fill the fracture. Elliptical symmetry is lost but this is resolved mathematically in the model. Consequently, a finite conductivity fracture can result. It is visualized that wormholes will evolve.
“This picture allows one to calculate the fracture conductivity by requiring that at any moment in time, the fracture volume should be equal to the total volume of injected solids.”
Movement of fines towards the tip is envisioned and two extremes in fracture permeability are envisioned (one uniform permeability profile and one with an impermeable tip plug).
PWRI
Model Comparisons
APPENDIX L
Shell 2 Model
April 15, 2020
Page L-1
PWRI
Model Comparisons
April 15, 2020
Page L-2
Shell 2 Model
Introduction
This is a pseudo-three-dimensional fracture growth model which permits the description of elliptical fractures in a multi-layered reservoir. Symmetrical vertical growth is not a pre-requisite.
Principles
This model is analytical and it is coupled with a reservoir simulator. It is assumed that fracture growth and development of the pressure field can be decoupled. This permits modelling of the pressure field in the reservoir using a constant fracture length. The transient pressure is approximated by applying the Laplace equation with a moving boundary for the pressure disturbance.
Fracture friction (shear) is ignored although pressure drop along the length of the fracture can result due to plugging. It is assumed that when multiple layers are present that there is no crossflow in the reservoir.
Pressure Distribution
Pressure distribution is represented by coupling the three-dimensional solution for a radial fracture in an unbounded reservoir with a two-dimensional solution (far-field) for an elliptically symmetric, pseudo-steady state situation. There is a discontinuity at the transition between the two regimes. This transition concept also applies for multiple mobility zones. Thermo- and poroelastic effects are considered.
Damage
The external filter cake is represented as a zone of altered mobility. “In the previous 2D model, the filtercake was assumed to be uniformly distributed over the fracture wall, with a possible tip plug at the end of the fracture where no water could penetrate. However, this resulted in often very high simulated bottomhole pressure as the friction in the very narrow “sheet” of fluid would become excessive.
This observation pointed us to introduce “channeling” as a mechanism to release pressure.”
PWRI
Model Comparisons
APPENDIX M
TerraFrac
TM
April 15, 2020
Page M-1
PWRI
Model Comparisons
April 15, 2020
Page M-2
TerraFrac™
Basic Description of the Model 78
TerraFrac™ is a PC-based hydraulic fracturing simulator based on fully coupled three-dimensional elasticity and two-dimensional fluid flow between fracture surfaces. Fracture growth is governed by fracture mechanics. The fracture is subdivided into discrete triangular elements by an adaptive meshing generator and the governing equations for these elements are solved by an approach similar to finite element method. That is, the modal force and displacement are related by a stiffness matrix. These governing equations consists of:
1.
Elasticity equations that relate the pressure over the fracture to the fracture opening for an arbitrary shaped, planar fracture.
2.
Fluid flow equations that relate the flow of the slurry between the fracture surfaces to the pressure gradients in the fluid.
3.
A fracture criterion that relates the intensity of the stress state ahead of the fracture front to the critical stress intensity necessary for tensile fracture of the rock.
Thermoelastic and poroelastic effects are considered in the model. Elastic modulus contrast between layers and their effects on fracture growth are modeled. For long-term injection, fractures can cross many different zones and all or parts of the fracture can close during injection. The model can simulate fracture closure on part of the fracture and re-opening if pressure becomes high enough again during injection.
Adaptive Meshing Generator
In order to solve the governing equations numerically, the fracture must be subdivided into elements. For three-dimensional hydraulic fracture modeling, the fracture geometry must be determined during the simulation and can be very complex in shape. More importantly, the fracture constantly changes its shape during injection and therefore, the meshing generator must be adaptive.
The meshing generator implemented in TerraFrac™ is based on DeLaunay triangulation. It has proven to be robust and adaptive for continuing fracture shape evolution. The following are three examples generated by the meshing generator.
78 R.J. Clifton, “Three-Dimensional Fracture-Propagation Models,”, Chapter 5 in
Recent Advances in Hydraulic Fracturing, SPE Monograph Vol. 2, edited by J.L.
Gidley, S.A. Holditch, D.E. Nierode and R.W. Veatch, Jr.
PWRI
Model Comparisons
April 15, 2020
Page M-3
2 0
1 0
1 0
5 0
0
-5 0
0 1 0 2 0
Figure 1. A snap-shot mesh for a run-away.
2 0
1 0
1 0
0
H h e sL y
Figure 2.
-1 0
-1 0
-2 0
0 1 0 2 0 a tu eL
3 0 n th
4 0 5 0
A snap-shot mesh for an hour-glass shaped fracture due to either high stress or a large permeability pay zone.
PWRI
Model Comparisons
1 0
1 0
0
April 15, 2020
Page M-4
-1 0
-1 0
-1 0 0 1 0 2 0
Figure 3. A mesh for a non-symmetric fracture in dipping strata.
Examples
Several example simulations are shown below. These examples show the mechanisms such as stress, permeability, elastic modulus, fluid viscosity, etc., that can affect the fracture geometry. All of the mechanisms interact together to govern the fracture geometry. For example, the example shown for high stress barriers
(Figure 5) indicates that the fracture is contained by the high stress in the upper and lower zones. However, if the fluid viscosity were higher, the pressure drop from the wellbore to the fracture front in the lateral direction would be larger and the fracture could grow in the vertical direction, rather than laterally.
Contained Fracture Due To A Stress Barrier
This case is for a contained fracture in a three-layered formation with uniform properties except for the minimum in-situ stress, as shown in Figure 4. The large stresses in the bounding layers prevent out-of-zone growth. If the frictional loss along the fracture becomes large either (due to large fracture length or due to higher fluid viscosity), then out-of-zone growth may occur. Selected simulation using TerraFrac™ is shown in Figure 5.
PWRI
Model Comparisons
April 15, 2020
Page M-5
Stress Profile
Figure 4. Stress profile and schematic of the expected fracture shape.
Figure 5. r c je tio eS a ea W th
V lu e=2 0 0b
4 0
2 0
0
-2 0
-4 0
-6 0
-8 0
S le
P r e le
S d
S h
S e
-1 0
5 0 1 0 en th(ft)
1 0
Large stresses in the upper and lower shales prevent the fracture from growing substantially out of the sand.
PWRI
Model Comparisons
April 15, 2020
Page M-6
Circular Fracture
This is a classic simulation. There are even some closed-form solutions to restricted versions of this problem. The example shown here entails fracturing in a homogeneous, isotropic environment with only regular variation in the minimum horizontal stress (refer to Figure 6). Selected simulation using TerraFrac™ for this case is shown in Figure 7.
Stress Profile
Figure 6. Stress profile and schematic of the expected fracture shape.
4 0
3 0
2 0
1 0
0
.4
6
.4
0
.3
4
.3
8
.3
2
.2
6
.2
0
.2
4
.2
8
.1
2
.1
6
.1
9
.0
3
.0
7
.0
1
Figure 7.
-1 0
0 1 0 2 0 3 0 4 0 5 0 6 0
Fracture shape and fracture width contour plot for a “circular” fracture.
PWRI
Model Comparisons
April 15, 2020
Page M-7
Multi-Layered Formations with Different Stresses
This case shows a fracture in a layered environment, with different in-situ stress levels in each of the layers; as shown schematically in Figure 8. Selected simulation results using TerraFrac™ are shown in Figure 9.
Stress Profile
Figure 8. Stress profile and schematic of the fracture shape.
2 0
2 0
1 0
1 0
.6
0
.5
9
.5
8
.4
8
.4
7
.4
6
.3
5
.3
4
.2
3
.2
2
.2
1
.1
1
.1
0
.0
9
.0
8
0
Figure 9.
0
0 1 0 2 0 3 0
Fracture shape and width contour plots, at shut-in.
PWRI
Model Comparisons
April 15, 2020
Page M-8
Fracture Containment Due To A Modulus Barrier
This case demonstrates the non-symmetric features (inclined) and the to fundamentally incorporate elastic moduli contrasts between layers. The example shown here is a three-layer scenario. The layers dip at 45 o (for demonstration purposes only). The elastic moduli for the top and bottom layers are ten times the elastic modulus of the middle layer. Other properties for the three layers are the same. Figure 10 shows the fracture shape and width contours, at shut-in. As can be seen, fracture growth into the higher modulus layers is limited because of much smaller apertures and consequent larger pressure gradients.
1 0
1 0
0
.4
0
.3
0
.3
0
.3
1
.3
1
.2
1
.2
1
.2
1
.1
1
.1
1
.1
2
.1
2
.0
2
.0
2
.0
2
-1 0
-1 0
-1 0 0 1 0
Figure 10. An elevation view of the fracture shape and width contours for a formation dipping at 45 , showing the elastic modulus effect on fracture growth.
PWRI
Model Comparisons
April 15, 2020
Page M-9
A Leakoff Barrier
This example shows the effect of permeability and solid particles on fracture growth in drill cuttings re-injection. The fracture propagates rapidly upward due to lower in-situ stress at smaller elevations. As the fracture enters the upper zone, which large permeability, fluid leaks off into the formation and high concentration of solid particles prevents the fracture from growing up further, as shown in Figure 11. The same characteristic plugging effect can be anticipated for PWRI.
S lid c n yV lu
2 0
2 0
1 0
1 0
S lid
2p a g d r o p h r e es a t. S u lu ein ic te g h p r n e efr b c les n ee te eto ).
.5
0
.5
4
.4
7
.4
1
.3
5
.3
8
.2
2
.2
5
.1
9
.1
3
.0
6
.0
0
5 0
0
-5 0
0 1 0 2 0 en th(ft)
3 0
Figure 11. This case shows that the fracture is contained by a highly permeable sand during drill cuttings re-injection.
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Model Comparisons
April 15, 2020
Page M-10
Hourglass-Shaped Fractures
This case was a three-layered situation with larger in-situ stresses (or large permeability) in the perforated, middle layer. As the fracture enters the upper and lower layers, which have smaller in-situ stresses, the growth rate into these two layers becomes larger and the fracture is pinched in the middle layer. An hourglass-shaped fracture is developed, as shown in Figure 12.
F a tu eS p W th a tu w L w e sZ n )
2 0
1 0
1 0
0
S e sL y
W th
.2
3
.2
6
.2
0
.1
4
.1
8
.1
2
.1
6
.1
0
.1
4
.0
8
.0
2
.0
5
.0
9
.0
3
.0
7
-1 0
-1 0
-2 0
0 1 0 2 0 3 0 a tu g
4 0 5 0
Figure 12. Hourglass shaped fracture and the fracture width contours.
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Non-Symmetric Fractures
This example shows a non-symmetric fracture growing from an inclined wellbore and in three dipping layers, with larger in-situ stresses (or large permeability) in the perforated, middle layer. As the fracture enters the upper and lower layers, which have smaller in-situ stresses, the growth rate into these two layers becomes larger and the fracture is pinched in the middle layer, as shown in Figure 13.
F a
F r aD v
D p L y s
2 0 llb e
1 0
1 0
0 ip in L y s
W th
.2
8
.2
1
.2
4
.2
7
.1
0
.1
3
.1
6
.1
9
.1
1
.1
4
.0
7
.0
0
.0
3
.0
6
.0
9
-1 0
-1 0
-1 0 0 1 0 2 0 rizo ta o in te(ft)
3 0
Figure 13. A non-symmetric fracture from an inclined wellbore in a formation with dipping layers.
PWRI
Model Comparisons
REFERENCES - TerraFrac
TM
1.
2.
April 15, 2020
Page M-12
Cliford, P.J., Berry, P.J. and Gu, H.: “Modeling the Vertical Confinement for Injection
Well Thermal Fractures,” SPE 20741 (1990).
Martins, J.P., Murray, L.R., Clifford, P.J., McLelland, G., Hanna M.F., and Sharp Jr,
J.W.: “Long-Term Performance of Injection Wells at Prudhoe Bay: The Observed
Effects of Thermal Fracturing and Produced Water Re-Injection,” paper SPE 28936, presented at the 1994 SPE (69 th ) Annual Technical Conference and Exhibition held in
New Orleans, LA, September 25-28.
3.
4.
5.
6.
7.
8.
Clifford, P.J., Mellor, D.W. and Jones, T.J.: “Water Quality Requirements for
Fractured Injection Wells,” paper SPE 21439, presented at the 1991 SPE Middle East
Oil Show held in Bahrain, November 16-19.
Settari, A. and Warren, G.M.: “Simulation and Field Analysis of Waterflood Induced
Fracturing,” paper SPE/ISRM 28081 presented at the 1994 Eurorock 94 – Rock
Mechanics in Petroleum Engineering, Delft, The Netherlands, August 29-31.
Settari, A., Warren, G.M., Jacquemont, J., Bieniawski, P. and Dussaud, M.: “Brine
Disposal into a Tight Stress Sensitive Formation at Fracturing Conditions: Design and
Field Experience,” paper SPE 38893 presented at the 1997 SPE Annual Technical
Meeting, San Antonio, TX, October 5-8.
“Fracture Propagation, Filter Cake Build-up and Formation Plugging During PWRI,”
PWRI News Letter, Feature Article, Volume 1, No. 3, October 1999.
Ovens, J. Larsen, F.P. and Cowie, D.R.: “Making Sense of Water Injection Fractures in the Dan Field,” paper SPE 38928 presented at the 1997 SPE Annual Technical
Conference and Exhibition held in San Antonio, Texas, October 5-8.
Settari, A.: “Simulation of Hydraulic Fracturing Processes,” SPE Journal, December
1980, pp. 487-500.
9.
Koning, E.J.L.: “Waterflooding Under Fracturing Conditions,” PhD thesis, Technical
University of Delft, 1988.
10.
Detienne, J-L., Creusot, M., Kessler, N., Sahuquet, B. and Bergerot, J-L.: “Thermally
Induced Fractures: A Field Proven Analytical Model,” paper SPE 30777 presented at the 1995 SPE Annual Technical Conference & Exhibition held in Dallas, Texas,
October 22-25.
11.
Perkins, T.K. and Gonzalez, J.A.: “The Effect of Thermoelastic Stresses on Injection
Well Fracturing,” SPE Journal, February 1985, pp. 78 – 88.
12.
Morales, R.H., Abou-Sayed, A.S., Jones, A.H. and Al Saffar, A.: “Detection of
Formation Fracture in a Waterflooding Experiment,” paper SPE 13747 presented at the SPE 1985 Middle East Oil Technical Conference and Exhibition held in Bahrain,
March 11-14.
13.
R.J. Clifton, “Three-Dimensional Fracture-Propagation Models,” Chapter 5 in Recent
Advances in Hydraulic Fracturing, SPE Monograph Vol. 2, edited by J.L. Gidley, S.A.
Holditch, D.E. Nierode and R.W. Veatch, Jr.
14.
J. Ovens and H. Niko, “A New Model for Well Testing in Water Injection Wells Under
Fracturing Conditions,” SPE 26425, presented at the SPE 68 th Annual Technical
Conference and Exhibition held in Houston, Texas, 3-6 October 1993.
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Model Comparisons
APPENDIX N
Visage
TM
System
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The VISAGE
TM
System
Basic Description of the Model
The VISAGE TM System provides for two software options for studying injection during reservoir simulations in both small scale (around wells) and large scale reservoir simulations. The first option is for partially coupled simulations, whereby
VISAGE TM when linked to ECLIPSE, VIP, ATHOS and FRONTSIM forms the SIM2VIS
System. The second option is to use a fully coupled stress/fluid/thermal multiphase flow module called VIRAGE.
SIM2VIS
The SIM2VIS system can be used to study the effect of thermal gradients on preferred waterflood directionality, stress magnitude and orientation. This simulator assesses the effect of porous media deformation on fluid flow characteristics and is linked directly to ECLIPSE, VIP, ATHOS and FRONTSIM. 3-D reservoir models with complex pre-defined distributions of faults and a large number of gridcells (>500,000) are readily accommodated. During waterflooding, faults and fractures may become conduits of flow or indeed transmissibility barriers if sealing occurs. The evolution of fractures is constantly traced with hydraulic parameters being updated, incorporating experimental data obtained from core samples to update permeabilities and rock fabric characteristics, as such fracturing develops. SIM2VIS accounts for changes in the effective stress state and rock fabric.
For example, in a SIM2VIS simulation a staggered solution scheme is adopted where ECLIPSE performs the fluid flow and temperature calculations, using permeability fields that have been determined from a non-linear stress analysis using VISAGE TM . ECLIPSE determines pore pressure and/or temperature distributions which are used in the stress calculations to determine equilibrium levels of effective stress. V.I.P.S. stated that “if hydraulic fracturing takes place, normal and shear plastic fault/fracture strains are determined and used to enhance or reduce levels of permeability in the reservoir. If thermal fracturing takes place
VISAGE TM will take account of thermal gradients in the reservoir in determining distributions of plastic strain.” The frequency of the non-linear stress calculations and the associated permeability enhancement calculations are at the discretion of the user when using the SIM2VIS system.
Stress Sensitive Reservoir Simulator
The VISAGE TM System also offers a Fully Coupled Stress Sensitive Multiphase Flow
Simulator, the VIRAGE module, for studies in a compressible non-linearly deforming porous media. The simulator is based on the finite element method and uses Galerkin-based numerical discretisation techniques to obtain fully coupled solutions to the mass balance, force equilibrium and plasticity equations of continuum mechanics. Incorporating experimental data from core samples the
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Page N-3 multiphase simulator updates the full permeability tensor as fault activation and/or fracture initiation develops during thermal injection. The thermal front is determined from coupled solutions of the advection diffusion equation governing thermal energy transport.
Porosity levels, together with system compressibilities may be continually updated in response to changes in volumetric stress and strain. Direct, symmetric and asymmetric solvers enable large problems, which may involve complex distributions of faults and fractures, to be solved effectively and efficiently. VIRAGE is also linked to ECLIPSE, FRONTSIM, ATHOS and VIP. Pre- and post-processors, FEMGEN and FEMVIEW, provide a wide range of mesh and visualisation techniques for complex, structured or unstructured model generation and interpretation of predicted results, including deformations, stress/strain distributions, levels of induced pressure and saturation and vector plots of all the velocity components of all fluid phases. Colour contour maps of updated permeability and porosity levels can easily be obtained.
Fault/Fracture Model
Fracture models in the VISAGE TM System are available for both 2-D and 3-D simulations and use the finite element method to determine changes in fault and fracture apertures. The approach incorporates constitutive models for the rock fabric operating under the fundamental principles of viscoplasticity. In this manner the ‘intact’ rock, fracture sets and faults can obey independently different constitutive laws. Upon fracturing and/or fault activation, the simulator calculates normal and shear strains for each fracture set or fault. Normal strains represent the potential for fault opening and permeability enhancement, whereas shear strains represent the potential for fault sealing. The viscoplastic approach to solving ‘intact’ material non-linearity and/or fracturing is based upon iterative procedures which determine successive solutions until any effective stresses that violate the imposed constitutive models are returned to the ‘yield’ surfaces within strict tolerances.
For soft sands a ‘damage’ mechanic theory with or without a ‘cap’ and based on multi-plane theories may be invoked to assess the potential for micro-fracture initiation and fracture evolution and propagation during thermal injection.
Permeability and porosity changes in the rock fabric can be ascertained, which in turn alter the pressure and/or thermal fields. Both the VIRAGE and the SIM2VIS
Systems have ‘access’ to these procedures in an attempt to assess the effects of fracturing in soft sands and resulting reservoir performance.
Other VISAGE documents are available where it unsuccessfully was attempted to use the code foe PWRI simulations. (These are provided in Appendix __.)
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Model Comparisons
APPENDIX O
WID Model
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WID
WID is a PC-based simulator developed at the University of Texas at Austin. It can accommodate layered reservoirs, horizontal wells, and constant injection pressure boundary conditions.
The principles are as follows:
1.
Determines the concentration of deposited particles around the injection well as a function of time and distance from the well. This is done by solving the filtration equations in that region. Information on the filtration coefficient is available in Pang and Sharma, 1994, 79 and Wennberg, 1998 80 .
2.
Calculate the altered permeability in the near-well zone due to retained particles.
3.
Determine how this near-well damage changes the injectivity of the well.
This depends on the formation parameters as well as the completion geometry.
4.
Calculate the transition time, i.e., the time where an external filter cake starts building on the wellbore. Before the transition time, only internal filtration is considered. After the transition time, only external damage is considered. The default porosity for the external cake is 0.25 and permeability is calculated from particle size and the Cozeny equation.
Other Features:
1.
WID 3.1 can represent a constant half-length fracture with a constant width.
The conductivity is calculated assuming parallel plates.
2.
Completion skin can be incorporated.
3.
Surface properties are specified and downhole pressure is calculated.
4.
Damage is calculated using particle deposition and the Cozeny equation
(refer to Sharma et al., 1997
81
). Changes in porosity and surface area are considered, as is the consequent tortuosity and reduction in permeability. A damage factor is specified as is a filtration coefficient, , (d /dt= vc where is the deposited concentration, v is the velocity, and c is the suspended concentration).
5.
The injectivity ratio is calculated and plotted. This is the injectivity divided by the initial injectivity. The half-life is indicated (the injectivity ratio has a value of 0.5).
79 Pang, S. and Sharma, M.M.: “A Model for Predicting Injectivity Decline in Water Injection Wells,”
SPE 28489, paper presented at the 69 th Annual Technical Conference and Exhibition, New Orleans, LA
(September 25-28, 1994).
80 Wennberg, K.E.: “Particle Retention in Porous Media: Applications to Water Injectivity Decline, PhD
Thesis, The Norwegian Institute of Science and Technology, Trondheim (February, 1998).
81 Sharma, M.M., Pang, S., Wennberg, K.E., and Morganthaler, L.: “Injectivity Decline in water
Injection Wells – An Offshore Gulf of Mexico Case Study,” SPE 38180, paper presented at SPE 1997
European Formation Damage Control Conference, The Hague, The Netherlands.
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6.
“For fractured completions, we neglect injectivity decline due to internal filtration, i.e., damage to the rock matrix. The injectivity curve, therefore, stays at a value of 1 until the transition time is reached. Then the injectivity starts to decrease because the deposited particle layer at the fracture surface decreases the fracture conductivity. Just before the fracture plugs completely, the injectivity declines very rapidly.”
7.
The simulator defines a one-dimensional grid in the nearwell zone in which the filtration equation is solved and particle deposition is determined.
8.
The transition time is the time when the deposition mechanism changes from internal deposition to external cake build-up. “In practice, this transition will be gradual, but WID considers it to take place abruptly in each layer.
Different layers can have different transition times. …By default, the transition time is reached when the porosity in the first few layers of grains is reduced to that of the formation porosity times the filter cake porosity. This is the theoretical minimum value the formation porosity can achieve. All subsequent particles are trapped as an external cake.”
