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Numerical simulation of water injection into layered fractured carbonate
reservoir analogs
Article in AAPG Bulletin · October 2006
DOI: 10.1306/05090605153
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GEOHORIZONS
AUTHORS
Numerical simulation of
water injection into layered
fractured carbonate
reservoir analogs
Mandefro Belayneh, Sebastian Geiger, and
Stephan K. Matthäi
ABSTRACT
Water flooding of fractured reservoirs is risky because water breakthrough can occur early, leading to a prohibitively high water cut.
In mixed or oil-wet carbonates, capillary drive is negligible or absent. For this scenario, we investigate fluid-pressure-driven displacement of oil by water in two-phase flow numerical models based on
naturally fractured limestone beds mapped along the British Channel coast. These reservoir analogs are represented by unstructured
finite-element grids with discrete representations of intersecting
fractures. We solve the governing equations for slightly compressible two-phase flow with our original control-volume finite-element
method. This permits the direct examination of displacement patterns in fractures and rock matrix.
We find that the irreducible saturation in the fractured carbonate is much higher than the value prescribed to the rock matrix.
The shape of water invasion fronts is highly sensitive to the viscosity ratio of oil and water. When the Brooks-Corey relative permeability model is applied to the rock matrix at a viscosity ratio of 1,
the total mobility, lt, is low at intermediate saturations. This stabilizes displacement fronts where a girdle of reduced lt develops,
but this effect disappears as the viscosity ratio increases.
For an idealized model with a water-wet matrix, we have also
evaluated the effect of countercurrent capillary-pressure–driven flow
across fracture-matrix interfaces. The rate of this countercurrent
imbibition scales with the specific fracture surface area and decays
exponentially as intermediate saturation zones develop adjacent
to the fractures. The resulting reduced lt feeds back into the fluidpressure-driven displacement process.
Copyright #2006. The American Association of Petroleum Geologists. All rights reserved.
Manuscript received September 30, 2005; provisional acceptance December 12, 2005; revised
manuscript received April 11, 2006; final acceptance May 9, 2006.
DOI:10.1306/05090605153
AAPG Bulletin, v. 90, no. 10 (October 2006), pp. 1473 – 1493
1473
Mandefro Belayneh ! Department of
Earth Science and Engineering, Imperial
College London, United Kingdom;
m.belayneh@imperial.ac.uk
Mandefro Belayneh is a research associate at
the Department of Earth Science and Engineering, Imperial College London, where he
obtained his M.Sc. degree and his Ph.D. in
structural geology. Prior to joining Imperial,
he had industrial experience in Ethiopia. His
research interests are studying the links between geological stresses, brittle failure, and
fluid flow in the Earth’s crust and their applications to fractured and faulted reservoirs.
Sebastian Geiger ! Department of Earth
Sciences, Swiss Federal Institute of Technology,
Zürich, Switzerland; present address: Institute
of Petroleum Engineering, Heriot-Watt University, Edinburgh, United Kingdom
Sebastian Geiger is a geoscience lecturer at the
Institute of Petroleum Engineering at HeriotWatt University and is currently an academic
visitor at the Department of Earth Science and
Engineering, Imperial College London. He
has been a postdoctoral researcher at the
Eidgenössische Technische Hochschule (ETH)
Zürich where he also received his Ph.D. in
2004. He holds an M.Sc. degree from Oregon
State University. His research interests are
the use of numerical simulations to study the
complex interplay of hydrodynamics and thermodynamics during multiphase flow in geologically complex systems.
Stephan K. Matthäi ! Department of Earth
Science and Engineering, Imperial College
London, United Kingdom
Stephan K. Matthäi is the Governor’s Lecturer
in Earth Science and Engineering at Imperial
College London. He received a Ph.D. from the
Australian National University and has postdoctoral experience from Cornell University,
Stanford University, and the Swiss Federal
Institute of Technology (ETH). He leads an industry consortium on enhanced oil recovery
from fractured reservoirs and has created the
Complex Systems Platform (CSP), an objectoriented numerical simulation application programming interface applicable to discrete
fracture modeling.
ACKNOWLEDGEMENTS
We thank J. R. Gilman for a careful, stimulating, and constructive review that has helped
improve the original manuscript. We also
benefited from the reviews of G. L. Prost,
E. A. Mancini, E. L. Cole, and an anonymous
reviewer. M. Belayneh thanks C. Thomas for
financial support during his Ph.D. at Imperial
College London, and S. Geiger thanks the
Swiss National Science Foundation Grants
SNF-20-59544.99 and 20002-100735/1 for
funding.
1474
Geohorizons
INTRODUCTION
The success of oil recovery from hydrocarbon reservoirs depends
on reservoir complexity. Only one-fourth to one-third of the oil in
place has been recovered from the fractured and faulted Iranian
carbonate reservoirs (A. M. Saidi, 2003, personal communication).
Globally, carbonate reservoirs account for approximately 65% of the
world’s remaining oil reserves, and 35% of the gas reserves in the
Middle East. Most of these reservoirs are fractured (Beydoun, 1998).
Fractures typically contribute less than a few percent to the total
pore volume of a reservoir (Dershowitz and Miller, 1984), but their
presence can give rise to highly localized flow (Nelson, 1985; Kazemi and Gilman, 1993; Sanderson and Zhang, 1999; Gentier et al.,
2000). Widespread difficulties in the history matching of fractured reservoirs suggest that current models lack the capability to
forecast production behavior (Casciano et al., 2004; Clifford et al.,
2005). More accurate predictions require a better understanding of
the multiphase behavior of such systems, emerging from the nonlinear interplay of basic flow processes in the complex flow geometry.
Previous numerical simulation studies of fluid flow in fractured
porous media fall into three categories: (1) single-phase flow models
of the fractures themselves (e.g., Zimmerman and Bodvarsson, 1996)
and of fractures and rock matrix (Matthäi et al., 1998; Taylor et al.,
1999; Bogdanov et al., 2003; Matthäi and Belayneh, 2004), (2) multiphase flow models with idealized fracture representations (KarimiFard and Firoozabadi, 2003), and (3) volume-averaged models like dual
porosity or permeability models (cf., Kazemi and Gilman, 1993). The
latter are widely used in reservoir engineering because complete descriptions of fracture patterns can only be obtained on the Earth’s
surface, and these models are the least computationally demanding.
In this article, we analyze the emergent behavior of fractured
rock in two-phase pressure-driven flow simulations based on discrete meter-scale representations of fracture patterns mapped on
wave platforms along the Bristol Channel coast, United Kingdom.
Computer-aided design (CAD) facilitates discrete fracture representation, making it possible to directly examine saturation fronts
and evaluate hypotheses about their nature (Yortsos, 2000). The governing equations are discretized on an unstructured finite-element–
finite-volume mesh that accurately represents fractures and matrix
(Geiger et al., 2004). Time is discretized by the implicit-pressure–
explicit-saturation approach (Aziz and Settari, 1979). These techniques are implemented in the numerical framework complex systems platform (CSP) (see Matthäi et al., 2001) and supported by an
algebraic multigrid solver well suited for large-scale computations
(algebraic multigrid method for systems, see Stüben, 1999). This
is important because models must be large enough for saturation
fronts to develop sufficiently removed from the prescribed boundary
conditions. Our second simulation goal is the evaluation of the hypothesis that, by analogy with porous media (Blunt et al., 1994),
viscosity contrasts between water and oil destabilize saturation
fronts in fractured porous media. A corresponding analysis of the
sensitivity of the flow patterns to the viscosity ratio employs the Brooks-Corey relative permeability model
(Brooks and Corey, 1964) for matrix and fractures and
is presented after the results on the development of
water invasion fronts at a ratio of 1.
GEOLOGY OF RESERVOIR ANALOG
The fractured carbonates that we use as a reservoir analog are exposed on the southern margin of the Bristol
Channel coast (Figure 1). They are lower Liassic limestone beds interbedded with shales that were deposited
during continued subsidence of the Permian– Triassic
basin. Basin subsidence is fingerprinted by vertical
joints preserved as calcite veins. Many joints are linked
across intervening shale beds by oblique fractures.
The fracture system that is investigated here is well
developed and postdates the aforementioned extension
fractures. It formed during basin inversion in the Late
Cretaceous to Tertiary, heralding the Alpine orogeny.
During this latter event, rollover folds were amplified, and
normal faults reactivated in reverse and strike-slip motion.
Fracture patterns were mapped in four limestone
beds, forming part of the gently north-dipping limb of
the Lilstock anticline and the hinge zone of a fold at
Kilve, approximately 2.5 km (1.5 mi) west of Lilstock.
Individual fractures are restricted to the 10–40-cm (4–
15-in.)-thick limestone beds and dip normal to bedding.
Despite their similar structural evolution, patterns in
Figure 1. Simplified structural map of Lilstock on the southern margin of the Bristol Channel coast, showing the Lilstock anticline
and the location of the studied limestone beds (modified from Rawnsley et al., 1998). Regional map shows the location of models
BED18 and KILVE1, which are not part of the stratigraphy at Lilstock.
Belayneh et al.
1475
Figure 2. Photographs of limestone pavements on Lilstock Beach capturing bed-to-bed variations in joint patterns: (a) bed 1 (cf., Figure 1)
represented by model BED1, (b) bed 2 by BED2 (Figure 7), (c) bed 3 by model BED3 (Figure 8), and (d) bed 4 by model BED4 (Figure 9).
adjacent carbonate horizons are often very different
(Figure 2) (Belayneh and Cosgrove, 2004). After marking the outcrops with a grid, fracture geometry was
captured by window samples and photographs. The
latter were rectified to enable the construction of undistorted two-dimensional joint trace maps. In the following, the fractured limestone beds that form the
basis of the numerical models are described in detail.
Model names are based on the nomenclature of the
beds. Figure 1 shows their location and stratigraphic
position, and Table 1 lists the applied model parameters.
Limestone Bed 1
Limestone bed 1 (Figure 2a) is located in the vicinity of an
east-west– and east-northeast–west-southwest–striking
normal fault (fault 5, Rawnsley et al., 1998) reactivated
in reverse motion during the Alpine orogeny. Bed 1 may
correlate with bed 105 of Whittaker and Green (1983)
or 1735 of Engelder and Peacock (2001). It is 10 cm
(4 in.) thick, situated on the gently dipping, northern
1476
Geohorizons
limb of the Lilstock anticline, and has a dip-dip direction of 05j/008j. Two joint sets in bed 1 exist: an early
set (J1) striking approximately east-west and a late
set (J2) of approximately north-south–trending short
joints that abut against J1 at approximately 90j to form
ladder patterns.
Limestone Bed 2
Limestone bed 2 (Figure 2b) is 16 cm (6 in.) thick and
has a dip-dip direction of 12j/004j. It correlates with
bed 109 of Whittaker and Green (1983) or 1848 of
Engelder and Peacock (2001). Based on abutting relationship, we distinguish six joint sets. The earliest
set, J1, of long straight joints strikes 125–130j. The second set, J2, strikes 110–115j (set 1, Loosveld and Franssen, 1992 or set J2 in figure 6a of Engelder and Peacock,
2001). Joints in J2 curve around as they approach J1. The
third joint set (J3) strikes 085–095j, approximately
subparallel to the Lilstock anticline axis (Figure 1) and
correlates with set 3/4 of Loosveld and Franssen (1992)
Table 1. Material Properties and Initial and Boundary Conditions Applied to Simulation Models
Model
GENERIC1
BED2
BED3
BED4
BED18
KILVE1
Material Properties
Units
X-dimension
m
50
4
14
3
6
8
Y-dimension
m
50
7
5
3
4
5.5
Matrix pore volume
m3
250
6.57
17.14
2.03
5.81
10.8
1.74
0.364
0.22
0.755
0.815
Joints volume
m3
Joint surface area
m2
665.2
782.87
64.19
202.95
351.13
Aperture range
m
0.001 – 0.01 0.003 – 0.005 0.003 – 0.005 0.003 – 0.005 0.003 – 0.005 0.003 – 0.005
Porosity
"
0.1
0.25
0.25
0.25
0.25
0.25
Specific joint
m2 m # 3
23.76
11.18
7.13
8.46
7.98
surface area
Matrix permeability
m2
1.00E – 13
1.00E – 12
1.00E – 12
1.00E – 15
1.00E – 15
3
#3
#1
1.00E – 09 1.00E – 09
1.00E – 09
1.00E – 09
1.00E – 09
1.00E – 09
Total system
m m Pa
compressibility
Brooks-Corey
"
2
2
2
2
2
2
parameter
Residual saturation
"
0.05
0.05
0.05
0.05
0.05
0.05
wetting phase
Residual saturation
"
0.14
0.14
0.14
0.14
0.14
0.14
nonwetting phase
Fluid Properties
Density oil
Density water
Viscosity oil
Viscosity water
Initial Values
Saturation oil
Saturation water
rp
kg m # 3
kg m # 3
Pa s
Pa s
"
"
Pa m # 1
800
1000
2.00E – 03
1.00E – 03
800
1000
1.00E – 03
1.00E – 03
800
1000
1.00E – 03
1.00E – 03
800
1000
1.00E – 03
1.00E – 03
800
1000
NA
1.60E – 03
0.95
0.05
9806.65
0.95
0.05
9806.65
0.95
0.05
9806.65
0.95
0.05
GENERIC2
1
1
0.25
4
0.01
0.25
4
1.00E – 13
1.00E – 09
2 to 3*
0
0.05
800
800
1000
1000
0.1 – 1.00e – 02 1.00E – 03
1.00E – 03
1.00E – 03
0.95
0.05
9806.65
0.95
0.05
0
*Fracture Brooks-Corey parameter = 0.2.
or J3 (figure 6a of Engelder and Peacock (2001). Set J3
abuts against J1 and J2, forming oblique ladder geometries. The fourth set, J4, is poorly developed and strikes
065 – 070j. J4 joints may correlate with J6 (figure 6a of
Engelder and Peacock, 2001). Sets 5 and 6 strike 335 –
345j and north-south (±10j), respectively. Overall,
the joint pattern in bed 2 is very similar to that produced
in analog experiments of noncylindrical folding (Rives
and Petit, 1990).
Limestone Bed 3
Limestone bed 3 (Figure 2c) has a thickness of 10 cm
(4 in.) and dips 08j/360j. It is separated from the
underlying bed 1848 of Engelder and Peacock (2001)
by 30-cm (12-in.)-thick shale. Based on abutting rela-
tionships, four joint sets are distinguished. The first,
J1, strikes northwest-southeast; the second, J2, strikes
north-south; the third, J3, strikes northwest-southeast;
and the fourth, J4, again strikes north-south, but it is
made up of short distinct joints. Arbitrarily oriented
joints are also common in bed 3.
Limestone Bed 4
Limestone bed 4 (Figure 2d) is 15 cm (6 in.) thick and
forms the uppermost layer of the gently dipping northern limb of the Lilstock anticline. It correlates with
bed 114 of Whittaker and Green (1983) and 1921 of
Engelder and Peacock (2001). Joint interaction criteria allow the distinction of three sequentially formed
sets. Sets 1 and 2 strike 320j and 300j, respectively, and
Belayneh et al.
1477
the former consists of long straight joints with wide
apertures. Joints of set 2 become parallel to set 1, where
they approach set 1. Set 3 comprises short joints with
variable orientations.
wet carbonates (Al-Hadhrami and Blunt, 2001). Thus,
Darcy’s law simplifies to
ui ¼ #k
kri
rp
mi
i 2 fo; wg
ð2Þ
Joint Patterns in a Fold Hinge
The total mobility lt is defined as
At Kilve, 2.5 km (1.5 mi) west of Lilstock, an anticline
with a subhorizontal axis trending 110j and a wavelength of more than 50 m (164 ft) is exposed over a
strike length of more than 80 m (262 ft). Early joints in
the hinge region are subparallel to the fold axis and interpreted as fold related. The second set terminates
against the long joints to form ladder patterns, which
possibly formed during uplift and exhumation. The fold
is cut from northeast to southwest by a strike-slip fault
with a sinistral displacement of about 30 m (100 ft). The
model based on this geometry is called KILVE1.
lt ¼ k
ui ¼ #k
kri
ðrpi þ gri rzÞ i 2 fo; wg
mi
fi ¼ k
rw
mw
1478
Geohorizons
kri
mi
þ kmro
ð4Þ
o
such that, in the absence of capillary and gravitational
forces,
ui ¼ fi ut
ð5Þ
The total fluid velocity ut is the sum of the phase
velocities
ut ¼ uo þ uw
ð6Þ
Assuming incompressibility, conservation of mass
implies
r ( u t ¼ qt
ð7Þ
where q t is a source or sink term, for example, caused
by the injection of water into the reservoir. An elliptic
pressure equation can be derived using equations 3 and
6 and inserting equation 2 into 7
ð1Þ
where ui is the fluid velocity of phase i; k is the permeability tensor; k ri is the relative permeability of phase i;
p is the fluid pressure; z is the depth from zero datum; and m, r, and g are the viscosity, fluid density,
and the gravitational acceleration vector, respectively.
Subscripts o and w denote the oil and water phases.
Because the fractured limestone beds are sandwiched
between impermeable shales and fractures are layer
bound, we restrict our simulations to the horizontal
plane in which gravitational forces are not active. In the
geometrically realistic models, we also neglect capillary forces because countercurrent imbibition (CCI) is
not an important recovery mechanism in mixed or oil-
ð3Þ
and the fractional flow function f for phase i as
NUMERICAL SIMULATION OF TWO-PHASE FLOW
We use our original implementation and adaptation to
complex fracture-matrix geometries of a dual-mesh
finite-element – finite-volume method (Baliga and Patankar, 1980; Huber and Helmig, 1999; Geiger et al.,
2004) to solve pressure and transport equations. The
pressure equation is solved with the finite-element
method. From the fluid-pressure gradient,rp, fluid velocities are computed via Darcy’s law and subsequently
employed in the solution of the hyperbolic transport
equation on the finite-volume mesh. Darcy’s law for
fluid phase i is given by
!
"
krw kro
þ
mw
mo
r ( ðlt rpÞ ¼ qt
ð8Þ
For slightly compressible fluids and transient flow,
equation 8 becomes
ct
@p
¼ r ( ðlt rpÞ # qt
@t
ð9Þ
where c t is the total system compressibility. Again,
assuming that capillary and gravitational forces are
absent, the conservation of fluid phase i simplifies to
f
@Si
þ r ( ð fi u t Þ ¼ q i
@t
ð10Þ
where S is the saturation (fluid volume fraction) of
phase i, and f is the porosity.
For a combination of fixed rate, fixed pressure, and
no-flow boundary conditions, the pressure field is computed by solving equation 9 using an implicit, Galerkin
finite-element formulation (Matthäi and Roberts, 1996;
Geiger et al., 2004). In contrast to computational methods that will only operate on grids with a fixed point
spacing, our finite-element method has the advantage
that inclined, complexly shaped, and large-aspect ratio
features such as fractures are adequately resolved without a prohibitively large number of cells. The saturationdependent total mobility, lt, does not vary spatially in
each finite element. This allows for the discrete representation of fracture-matrix interfaces. From the solution of the transient fluid-pressure field at a given time
step, the fluid velocity is computed at the center of
each finite element from Darcy’s law (equation 1) and
is used for the solution of the conservation equation
for the fluid phase i (equation 10). Advection is modeled with an explicit, i.e., backward in time, finitevolume algorithm that is second-order accurate in space.
It preserves steep gradients in saturation (Geiger et al.,
2004).
Fracture aperture is treated as equivalent to hydraulic aperture, and the cubic law for flow between
parallel plates is applied to obtain fracture transmissivity
(Witherspoon et al., 1980). Zimmerman and Bodvarsson
(1996) and Sisavath et al. (2003) found that this equivalent porous medium (EPM) approximation overestimates flow rates by an aperture relative to the surface
roughness-dependent factor of )2. However, this error
is less than the effect of the uncertainty in our outcropbased fracture aperture determinations and is therefore
not considered. The apertures assigned to our wellinterconnected fracture models translate into fracturematrix permeability contrasts of two to five orders
of magnitude. The lower end-member ratio (models
BED2–4) represents the case where fractures and matrix
carry the same fraction of the total cross sectional flow.
For the upper end-member ratio (model GENERIC2),
the matrix is largely stagnant (Matthäi and Belayneh,
2004).
The relative permeability is the key determinant
for the two-phase flow behavior of the models because
it controls the magnitude of the total mobility (equation 3) and the shape of the fractional flow function
(equation 4). We calculate the relative permeability
of the rock matrix using the Brooks and Corey (1964)
model with an exponent of 2. This value is frequently
applied to highly nonuniform materials (cf., Helmig,
1997), such as the limestone considered in this study
(Belayneh, 2003). As suggested by Valentine et al.
(2002), we also apply the Brooks-Corey model to the
fractures, albeit with an exponent between 0.2 and 1
and lower residual saturations than for the matrix
(Table 1). The rationale for this is as follows: The linear
relative permeability–saturation relation that is commonly used for fractures (e.g., Kazemi and Gilman, 1993)
was identified by Romm (1966) for water-kerosene
flow between impermeable parallel plates. It ignores
the effects of fracture surface roughness, which is significant as indicated by our field observations (Belayneh, 2003). A variable fracture aperture implies that
the nonwetting phase will preferentially occupy the
wider fracture segments, and because EPM permeability scales with the cube of aperture, a nonlinear relative
permeability saturation relationship is expected for
rough-walled fractures. The experiments conducted by
Neuweiler (1999) on synthetic rough-walled fractures
show that even at high flow rates, tortuous flow paths
result, and lt is reduced at intermediate saturations.
This behavior is captured by the Brooks-Corey model.
However, the viscous drag that flowing water should
exert on oil droplets in larger fractures is ignored. By
setting the residual saturation of oil in the fractures to
zero, we have partially addressed this deficiency, but it
is clear that relative permeability models for fractures
are a topic for future research.
MESH DISCRETIZATION OF FRACTURES AND
MODEL SETUP
For the finite-element meshing, we converted the joint
trace maps into CAD models. Each fracture trace was
divided into two parallel lines representing the fracture
walls separated by the average aperture. Then, unique
material properties were assigned to this void space. Prescribed aperture values do not represent in-situ conditions because aperture was altered during uplift and
exhumation. In addition, fractures were locally widened by chemical dissolution, which is especially prominent above the high-tide level (Figure 3). Subsurface
fracturing is commonly succeeded by mineral precipitation. In these cases, veins of euhedral calcite crystals
permit inference of in-situ aperture. Because the joints
described in this work are barren, precise conditions of
their formation have yet to be determined.
The CAD models of the fracture trace maps were
discretized with constrained conforming Delauney triangulations (Shewchuk, 2002) (see Figure 4). Matrix
Belayneh et al.
1479
are 0.14 and 0.05 for the rock matrix and zero for the
fractures, respectively. The total system compressibility of the model is 10 # 9 m3 m # 3 Pa # 1. The densities
of oil and water are 800 and 1000 kg m # 3, respectively.
We also generated a heuristic fractured reservoir model
(GENERIC1) with a well in its center. Model GENERIC2
consists of an oil-saturated block surrounded by stagnant fractures and was used to evaluate the rate of CCI.
SIMULATION RESULTS
Figure 3. Cliff at Lilstock Beach revealing a cross section
through limestone-shale stratigraphy and the localized effects of
carbonate dissolution on fracture aperture. Person, 1.84 m (6 ft)
tall, for scale.
blocks and fracture were tagged with region-identification points for the assignment of material properties after the meshing. To study the advance of saturation fronts, a hydrostatic, far-field, fluid-pressure
gradient (9.80665 " 103 Pa m # 1) was assigned parallel
to the model edges. This was accomplished by the application of uniform (Dirichlet) boundary pressures
on opposite sides (Figure 4). This leads to unrealistic
flow patterns near these boundaries because flow is
not naturally partitioned between fractures and matrix (see Matthäi and Belayneh, 2004). To circumvent
these limitations, the models were made large enough
to mimic the evolution of unbiased flow patterns in
their interior.
Fluid and material properties are listed in Table 1.
All models are initially oil saturated (S o = 0.95, S w =
0.05). Irreducible oil (S or) and water saturations (S wr)
1480
Geohorizons
The most prominent characteristics of the two-phase
flow in flat-lying beds crosscut by highly permeable
fractures are illustrated by an idealized 50 " 50-m
(164 " 164-ft) model (GENERIC1, Table 1). A circular well penetrates the rock matrix in the center of a
small hydrocarbon pool (Figure 5). Fracture aperture
has been deliberately exaggerated to 0.1 m (0.33 ft)
to show flow in the fractures, but fracture permeability was reduced correspondingly to the equivalent of
1-mm (0.04-in.)-wide fractures as predicted by the parallel plate model (1.0 " 10 # 9 m2 [1.07 " 10 # 10 ft2] =
1000 d). Matrix permeability is 1.0 " 10 # 13 m2 (1.07 "
10 # 14 ft2) (100 md).
The presence of the fractures leads to strong deviations from radial drawdown (cf., Matthäi et al.,
1998). Oil from the block around the well is recovered
quickly, but as soon as the oil near the tips of the surrounding fractures is depleted, the oil within them is
rapidly displaced by water. This water makes its way
to the well before the oil in the peripheral matrix blocks
is recovered, and the water cut at the well approaches
unity. Importantly, water coning occurs, although the
well does not intersect any of the fractures. Production from the central square area is controlled by the
inflow into the distant parts of the enclosing fractures.
This is a typical feature of two-phase flow in the
presence of well-interconnected fractures as judged
by the range of simulations that we have conducted
thus far.
Another important characteristic of fractured rock
with well-interconnected fractures is its short-term response to fluid-pressure changes. For fracture-matrix
contrasts in hydraulic diffusivity, k = k/(mfc t), greater
than two to three orders of magnitude, we observe
that the fractures transduce the pressure perturbation so fast that the slowly responding matrix blocks
stand out as isolated pressure highs. This is illustrated
with model BED18 (Figure 6). For its fracture geometry and a uniform total system compressibility,
Figure 4. Model configuration
and adaptively refined finiteelement mesh of a fractured
limestone bed; discretization,
placement of physical variables,
and boundary conditions.
c t of 10 # 9 m3 m # 3 Pa # 1, the characteristic distance of
pffiffiffiffi
pressure diffusion, L ¼ 2 kt , in 1 day is 22 m (72 ft)
for the rock matrix and 70 km (43 mi) for the fractures, respectively. Although the perturbation in the
fractures travels somewhat slower than indicated by
this simplistic calculation because fluid pressure is
buffered by the matrix blocks, the transient behavior
of model BED18 highlights how rapid drawdown pressure is transduced by the fracture network. Potentially,
such perturbations travel beyond the limits of the hydrocarbon pool. This is the case in model GENERIC1
(Figure 5). When this model is initialized with a steadystate pressure distribution matching the constant pumping rate, water breakthrough occurs significantly later
than when fluid pressure is uniform at the onset of the
calculation. The reason for this behavior is that before
a steady state is obtained, the flow response of the reservoir to the imposed pressure perturbation is dominated by the fractures. Thus, a larger proportion of fluid
from the water-saturated periphery of the model is
channeled into the well as compared with the steadystate model. It follows that ad-hoc water injection can
significantly damage a fractured reservoir if it fragments
the hydrocarbon pool. This is demonstrated by the experiments described below.
The Nature of Water Floods
Simulations conducted with model BED2 (Figure 7)
illustrate how oil is displaced from fractured rock caused
by the injection of water through the left model boundary. This experiment can be regarded as a proxy for
water injection from a horizontal well into a reservoir
layer.
Water invades the fractures orders of magnitude
faster than the rock matrix (Figure 7, see left model
margin at late time = 21.9 hr in c for comparison). The
flow-velocity histogram (Figure 7a) illustrates this by a
comparison of the velocity spectra for the matrix (gray)
and the fractures (white), respectively. The characteristic fracture velocity is three orders of magnitude greater
than the matrix velocity. However, this graph only
reflects the steady-state velocity distribution in a singlephase, water-saturated model. This ignores the dependence of total mobility, lt, on saturation. For the
Brooks-Corey (1964) relative permeability-saturation
relationship and the prescribed oil-water viscosity ratio of 1, lt has a minimum (= 0.3 lt(max)) at a water
saturation of about 0.5. Fluid-pressure contours after
5.2 hr (Figure 7b) are more closely spaced in the intermediate saturation zone directly behind the front,
Belayneh et al.
1481
Figure 5. Two-dimensional model GENERIC1 of water breakthrough into a production well in the center of a fracture-bounded
matrix block in an idealized fractured reservoir; constant well production rate 10 m3 day # 1 (353 ft3 day # 1), matrix k = 10 # 13 m2,
oil-water viscosity contrast = 2; exaggerated fracture aperture = 0.1 m [0.3 ft]. (a) Plan view of the pristine fractured reservoir horizon
(50 " 50 m; 164 " 164 ft) saturated with oil (red). (b) Saturation distribution after about 15% of the total hydrocarbon accumulation
has been produced; well now has water cut of 70%. Parts (c) and (d) are magnifications of (b) with velocity vectors indicating
the direction of fluid flow. The white box in (b) indicates the location of (d). Saturation is shown in rainbow coloring (oil = red,
water = blue).
fingerprinting an effective total mobility that is reduced
by a factor of 3–4 because the pressure gradient is inversely proportional to lt. A threefold lt reduction in
the fractures where S w ~ 0.5 also decreases the lt contrast between fractures and matrix. This enhances matrix flow near the front and, therefore, oil recovery. The
displacement front is fairly uniform, and a close examination of the flow patterns at the front reveals that
displacement occurs also in fractures parallel to the
front. This counteracts the formation of embayments
in the front and stabilizes its initial shape. Behind the
1482
Geohorizons
front, total mobility recovers where fractures become
fully water saturated (Figure 7c). The front therefore
stands out as a girdle of reduced total mobility, and the
final sweep efficiency is approximately 53%. This is
much higher than the figures for the Iranian fractured
oil reservoirs discussed in the introduction.
The gradual displacement of water by tortuous flow
through the rock matrix is fingerprinted by a decreasing
speed of the saturation front relative to the characteristic fracture flow velocity (compare Figure 7a with b, d).
The front initially moves with a speed of approximately
Figure 6. Fluid-pressure
response of model BED18
to a sudden fluid pressure drop of 10 MPa at
the right boundary. All
other sides are treated as
no-flow boundaries. The
actual fractures are too
narrow to be visualized
but they are overlain with
inverted grayscale encoded lines with a thickness that is proportional
to the flux, which increases
toward the right because
fluid is supplied from
adjacent matrix blocks.
These retain their original fluid pressure longer
than the highly permeable fractures.
one-half of the characteristic fracture velocity from the
histogram (Figure 7a), decreasing to approximately onefifth at water breakthrough on the right.
In summary, in this field-data-based model, oil is
displaced preferentially from the fracture network, including fractures oriented perpendicular to the flow.
However, there also is significant tortuous flow through
the rock matrix. This leads to a recovery of oil that is
orders of magnitude greater than the volume stored by
the fractures. The reduction of the total mobility of the
fractures at intermediate water saturation induces a
global total mobility minimum directly behind the saturation front. This reduction stabilizes the front. In a
following section of this article, this feedback is investigated for oil-water viscosity ratios greater than 1.
Dependence of Front Shape on Fracture Geometry
Snapshots of the saturation in model BED3 front at a
time, t, equal to 1, 2, 3, and 4 " 103 min are shown in
Figure 8. For the entire duration of the simulation, pressure contours are subparallel to the strike of the oblique
fractures. Initially, the front evolves into a convex shape
not influenced by the oblique pressure gradient but then
aligns itself with the contours. The total mobility reduction at the saturation front is not as pronounced
as in model BED2. The shape of the front, however,
is very stable throughout the simulation. Water breakthrough occurs just before 4000 min.
The pressure contours shown in Figure 9a–f are
very similar to those in Figure 8. The difference between the two models is that BED4 lacks fractures
aligned with the far-field fluid-pressure gradient. Therefore, the intact block of limestone on the right is swept
almost completely as the water front advances.
The saturation change in the model is accompanied by a decrease in the total velocity, mirroring a diminishing lt. Initially (Figure 9a), ut ranges from 1.21
to 2.83 " 10 # 1 m s # 1 (3.96 to 9.28 " 10 # 1 ft s # 1). Toward the end of the simulation, the maximum velocity
is only 4.05 " 10 # 2 m s # 1 (1.33 " 10 # 1 ft s # 1).
Effect of Viscosity Ratio on Shape of Saturation Fronts
Viscous fingering is a well-documented feature of twophase flow in heterogeneous porous media (e.g., Dullien, 1992; Blunt et al., 1994). If a viscosity contrast
exists between the two fluids, small differences in the
Belayneh et al.
1483
Figure 7. Water injection
into fractured limestone
(1) Dp = 106 Pa, matrix
porosity = 0.25, oil-water
viscosity ratio = 1, BrooksCorey parameter = 2,
Dp (mixed zone) = 4 " Dp.
(a) Velocity histogram
with separate curves for
fracture (white) and matrix
(gray). The vertical axis
gives the area fraction
that flows at the velocity
indicated on the logarithmic horizontal axis.
(b) – (d) Water injection
at left model boundary
leads to the formation of
a self-sharpening saturation front because total
mobility is reduced behind it by a factor of approximately 3 – 4. The
opposite is true if the viscosity ratio is increased
(see text). Shown are
fracture geometry (dark
lines), oil saturation (red =
0.95 > blue = 0.0), and
fluid-pressure contours
(red for high, blue for
low). The relative front
velocity decreases as oil
saturation is further reduced behind the front.
The irreducible oil saturation in this run was
0.42 (cf., Figure 13).
velocity of the saturation front get amplified, and viscous fingers develop. To evaluate whether this instability also has an influence on saturation fronts in fracture networks, we carried out simulations with model
KILVE1 (Figure 10) for a range of oil-water viscosity
ratios and a constant viscosity of water, mw of 10 # 3 Pa s.
Corresponding relative permeability and total mobility
curves are plotted in Figure 11. This graph shows that
the lt, minimum at intermediate saturations, disappears as the oil viscosity is increased, and lt becomes a
1484
Geohorizons
monotonously increasing function. Does this imply
that the stabilizing effect on the front shape seen in all
simulations with a viscosity ratio of 1 will disappear as
oil and water viscosities diverge?
Simulation results for model GENERIC1, as shown
for a viscosity contrast of 2 in Figure 5, indicate that
recovery is optimal when oil and water viscosities are
the same. It decreases rapidly as the ratio between oil
and water viscosity increases, supporting our hypothesis. However, because flow is attracted by the circular
Figure 7. Continued.
well and there are only a few fractures, this model does
not lend itself to investigate frontal instabilities. Because
all our models are two-dimensional, simulations cannot express fingering in the plane of the fractures. This
process could further accelerate water breakthrough and
is fingerprinted by selective tar staining of fractures in
sandstones of the Monterey Formation north of Los
Angeles, California.
Model KILVE1 contains two intersecting joint
sets. The first one is aligned with the long axis of the
model and delimits the second set of shorter joints
that are perpendicular to it. For the viscosity ratio of 1,
the displacement front is straight, and water-saturated
zones coalesce behind the front through the joints of
set 2 (Figure 12a, b). This front development is similar
to the other simulations.
For an oil-water viscosity ratio of 10 (Figure 12c, d),
total fluid velocities increase steadily from 7.75 " 10 # 4
to 5.42 " 10 # 3 m s # 1 (2.54 " 10 # 3 to 1.77 " 10 # 2 ft s # 1)
at 10,000 min to 1.01 " 10 # 3 to 7.06 " 10 # 3 m s # 1
(3.31" 10 # 3 to 2.31 " 10 # 2 ft s # 1) after 21,000 min.
In this second simulation, the saturation front is initially straight, but develops embayments with increasing displacement (Figure 12c). A comparison of the
spacing of the pressure contours in Figure 12a with c
and b with d shows that for the viscosity ratio of 10,
bulk lt is no longer reduced at intermediate saturations, and the front is much wider. A further pronounced difference of simulation 2 is that water saturation bridges in the short fractures are less likely to
develop as the front advances. We ascribe this feature
to the greater lt in water-saturated longitudinal as
compared with oil-saturated transverse fractures because it makes the water less prone to enter these. The
resulting behavior is comparable to fingering. The final
residual saturation in this second experiment is higher
than in the first one.
In summary, the best conditions for the water
flooding of fractures are a steady-state fluid-pressure
distribution and a viscosity ratio of 1. As the viscosity
Belayneh et al.
1485
Figure 8. Snapshots of model BED3 at
the time steps (a) t = 1000, (b) 2000,
(c) 3000 and (d) 3400 min. Water saturation is shown in gray scale (black = 1)
and tracks the fracture network. Stippled
contours represent isobars of fluid pressure.
of oil increases relative to that of water, its mobility
decreases, enhancing the fracture matrix lt contrast,
and it becomes more readily trapped in matrix blocks.
The reduced total mobility at intermediate saturations disappears, and embayments in the water front
are amplified.
At a viscosity ratio of 1 and provided that production is ramped up slowly to avoid the transient fracturedominated flow seen in model GENERIC1, the total
1486
Geohorizons
mobility reduction at and behind the saturation front
will actually postpone water breakthrough. This behavior might be exploitable during production.
Recovery History
Figure 13 shows the volume-integrated normalized
saturation in model BED2 as a function of time. After
water breakthrough at approximately 2900 min, the
Figure 9. Snapshots of model BED4 that contains regions that are not fully penetrated by fractures. Time t = (a) 200, (b) 1000,
(c) 2000, (d) 3000, (e) 4000, and (f ) 5000 min. Water saturation is shown in gray-scale (black = 1, white = 0) tracks the fracture
network. The contours represent isobars of fluid pressure.
fractures are swept clear from oil, and little further
saturation changes occur. The residual oil saturation is
0.42, three times the value we prescribed to the rock
matrix ( Table 1). The stippled curve in Figure 13 is
a sketch of the production history for a model homogenized to the effective permeability of BED2 using the procedure given in Matthäi and Belayneh (2004).
Water breakthrough only occurs at the very end of
the run as the sharp water front arrives at the right
boundary and the prescribed residual saturation is
reached. We explain this stark contrast in behavior
as follows. In the discrete fracture version of BED2,
S o in the matrix approaches intermediate values,
whereas S w in the fractures approaches unity as the
network is swept clear of oil. As k ro in the matrix
and k rw in the fractures evolve in opposite directions,
the final fracture-matrix contrast in the mobility of
the oil is orders of magnitude bigger than the initial
one. Even for models in which the rock matrix initially
carries most of the flow (Matthäi and Belayneh, 2004),
this implies that they will start to behave like the ones
where flow is fracture dominated. From a viewpoint
of production, the oil simply gets locked up into the
matrix blocks.
Capillary-Pressure Effects
Figure 14 shows the rate of countercurrent flow of
water into the rock matrix and oil into the fractures
for a square 1 " 1-m (3.3 " 3.3-ft) block of rock with
a surface area of 4 m2 (43 ft2) surrounded by waterfilled fractures. Depending on the Brooks-Corey parameter (exponents are 2 and 3) and entry pressure
(1000 Pa), the block releases oil at a rate of approximately 0.14 m3/day (4.94 ft3/day) once a relatively timeinvariant saturation gradient has established itself in the
matrix adjacent to the fracture. This flux is 1.5 orders
of magnitude less than the initial rate. Additional testing shows that the rate is relatively insensitive to S w in
the fracture, provided that the latter is high enough to
balance oil flow.
Belayneh et al.
1487
Figure 10. Photograph of Kilve anticline with prominent joint set parallel to its axis. This joint pattern from the hinge region is
represented in model KILVE1. The geological hammer at the center right of the photograph (black circled area) is 45 cm (17.7 in.)
long for scale.
Countercurrent imbibition has been studied extensively in physical experiments (e.g., Rangel-German
and Kovscek, 2003) and is typically accounted for
in reservoir-simulation models by transfer functions
(Kazemi and Gilman, 1993; Tavassoli et al., 2005). If
one could achieve a dynamic equilibrium where oil is
removed by production from the fractures at the
same rate as it drains into them from the matrix, fracture saturation would remain constant. To achieve
this, oil from a rock with similar properties as in our
experiment would have to be produced at a rate of
about 0.28 m3 m # 3 day # 1 (i.e., volume of fluid produced per unit volume rock per time). If the capillary
properties of the matrix were known from special
core analysis, the key constraint on the appropriate
production rate would be the fracture-matrix interface area per unit volume rock, A f (m2 m # 3). This
measure is considered by the shape factor in dualporosity models (e.g., Warren and Root, 1963; Kazemi
and Gilman, 1993) and is explicit in our experiments
( Table 1). However, because CCI can only occur
1488
Geohorizons
where water has entered the fractures, the reduced
number of water-infiltrated fractures behind the front
(cf., Figure 2d) implies that the shape factor would
overpredict the CCI rate for this model. In more general
terms, the shape-factor treatment should be applicable to real reservoirs only if
flow in the fractures is relatively uniform and time
invariant,
block size is relatively uniform (not the case in this
study),
block-to-block velocity variations are averaged out on
the larger scale,
an oil-water viscosity ratio near 1 precludes viscous
fingering in the fracture plane.
A deficiency of this argumentation is that it ignores the influence of CCI on fracture flow velocity. This feedback warrants further consideration because fracture velocity has a pronounced effect on
recovery.
Figure 11. Total mobility curves derived from the BrooksCorey model for different oil-water viscosity ratios ranging
from 1 to 100. For values near 1, lt has a distinct minimum
at intermediate saturations. This feature is lost at higher ratios, removing an associated stabilizing effect on saturation
fronts.
fracture-matrix oil mobility contrast diminishing the
recovery of oil. The magnitude of this effect is a function of the material properties. How accurately it is
captured by our models depends on the accuracy of
the applied relative permeability model. Any relative
permeability-saturation relationship, which causes a
significantly reduced total mobility at intermediate
water saturations, will lead to this behavior.
The high recoveries we obtain for an oil-water viscosity ratio near 1 may be misleading because we have
assigned uniform aperture values to fractures with the
same orientation. If this is changed, considering the
effects of stress on aperture (e.g., Zhang and Sanderson,
1996; Sanderson and Zhang, 1999), flow is likely to
become more localized. The worst case scenario for production is a fracture network at the percolation threshold
because flow will be extremely localized (Sanderson and
Zhang, 1999). Water breakthrough will occur early, and
the percolation path should stabilize if the invading fluid
has a lower viscosity than oil.
Clearly, this study is only a first step in the exploration of the indicated patterns. We are currently
conducting three-dimensional simulations incorporating gravitational and capillary effects into the models
to get a better understanding of their composite behavior (Matthäi et al., 2005). The presented models
are simplistic also because any variation in the properties of the rock matrix are ignored. Although we have
already conducted tests showing that random perturbations in matrix permeability and porosity do not
have a noticeable effect on the results presented herein, spatially correlated variations expected caused by
the presence of laminations, graded bedding, or other
sedimentary structures will have an effect (Grader
et al., 2000).
CONCLUSIONS
DISCUSSION
Our results indicate that, in the absence of CCI, a considerable fraction of the matrix oil can still be recovered by fluid-pressure-driven displacement in the rock
matrix. This recovery, however, depends greatly on
the saturation of oil that is maintained in the fractures.
As water saturation in the fractures increases and the
rock matrix adjacent to the fractures is desaturated,
the interior of the matrix blocks becomes shielded
by intermediate saturation zones. This enhances the
Field data-based discrete fracture models of oil
displacement by water in carbonate beds with welldeveloped layer-restricted fractures indicate that the
fractures impart a characteristic production behavior
on the fractured rock. Four important conclusions can
be drawn from this work:
1. The presence of fractures leads to early water breakthrough and a much higher residual saturation than
in the intact rock. Under the proviso that relative
permeability-saturation relationships in the matrix
Belayneh et al.
1489
Figure 12. Water injection front for model KILVE1 for two different oil-water viscosity ratios, 1 and 10. Snapshot taken at times
(a) 100, (b) 400, and (c) 10,000 and (d) 20,000 min for simulations at mo/mw = 1 and 10, respectively. Whereas the front is selfstabilizing in (a) and (b), it disintegrates at the higher viscosity ratio; see (c).
Figure 13. Normalized oil saturation in
model BED2 (see Figure 7) as a function
of time. Water breakthrough corresponds
to the kink in the saturation curve at approximately 2900 min. Note that the residual saturation in this run is about 0.42.
This is three times the value assigned to
the rock matrix in the model (S or = 0.14).
The stippled line marks the saturation
versus time curve for a version of model
BED2, which was homogenized to its effective permeability in the direction of
the fluid-pressure gradient.
1490
Geohorizons
Figure 14. Countercurrent oil-water exchange between large-aperture fractures
and surrounding rock matrix. (a) The fractures are initially water saturated, but
the water is gradually exchanged with
oil that drains from the water-wet matrix.
The final state, which is consistent with
experimental results, is an oil-saturated
fracture with a water-enriched halo.
(b) Countercurrent imbibition for a square
matrix block surrounded by water. The
imbibition rate (defined as the volume of
oil lost from the matrix per unit fracture
surface area per time) decays exponentially and varies as a function of the capillary entry pressure and the Brooks-Corey
parameter.
and the fractures are nonlinear with a total mobility minimum at intermediate saturation, tortuous
pressure-driven flow through the rock matrix suffices
to achieve a relatively high displacement efficiency
if a uniform far-field fluid-pressure gradient can be
imposed.
2. The viscosity ratio has a pronounced effect on sweep
efficiency and the shape of saturation fronts. For an
oil-water viscosity ratio of 1, the total mobility minimum at intermediate saturations has a stabilizing
effect on the saturation front. However, total mobility becomes a monotonously increasing function
as the viscosity contrast increases. This has important consequences for the oil recovery: As the total
mobility minimum at intermediate saturations disappears, recovery is strongly reduced, and water invasion will permanently damage the reservoir, im-
printing it with localized preferential flow paths for
water. The fracture-matrix oil mobility contrast increases unfavorably as fractures become water saturated. Then, it becomes more likely for the water
to bypass oil-saturated blocks with an intermediate
saturation halo.
3. In water-wet systems, CCI can mitigate against a
water saturation increase in the fractures. The capillary drainage speed scales with the specific fracture
surface area. This is an explicit parameter in our
models, so that under favorable circumstances, the
production rate matching capillary drainage capacity
can be computed.
4. The dependence of our results on assumptions about
the two-phase properties of fractures highlights the
need for better experimental data on fractures and
their in-situ aperture.
Belayneh et al.
1491
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