1
P014 WAG STUDY – DOE-BASED SENSITIVITY
ANALYSIS AND MATRIX FRACTURE
INTERACTION UPSCALING
1
1
1,2
S. Fallah , S.A. Haghighat ,T.E.H. Esmaiel , and C.P.J.W. van Kruijsdijk
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1
Delft University of Technology, Department of Geotechnology, PO Box 5028, 2600 GA Delft, Netherlands
2
Kuwait Institute for Scientific Research, PO Box 24885, Safat, Kuwait 13109
Abstract
The fundamental aspects of WAG (Water Alternating Gas) injection are still not well
understood. This paper discusses two research topics into the WAG process being currently
studied at TUDelft. The first part of this study looks at the sensitivity of production to rock and
fluid properties on a pattern scale using tools derived from experimental design and response
surface modeling. The second aspect discussed involves the fine-scale interaction of the WAG
process in fractured media compared to the dual porosity formulation used in commercial
simulators.
The process of determining sensitivities in an organized manner is shown here with a limited
number of parameters. The proxy model generated for oil recovery, gas and water production
and NPV provide a fast and easy estimation that can facilitate Monte Carlo analysis. The
systematic approach provided in this work can be expanded to facilitate a general reservoir
model and additional rock and fluid properties as well as operational considerations.
The standard dual porosity formulation in commercial simulators is based on a continuous matrix
grid overlaid by a continuous fracture grid. The flow in each system is determined by standard
formulation. What is highlighted in this study is the transfer of fluid between the matrix and
fracture. The aim of this work is to determine how to up-scale the fine grid WAG process to be
implemented in a dual porosity model.
Introduction
The first WAG process reported in literature was in Canada 1957. As the process is approaching
half a century old, much of the fundamentals require more understanding through research. The
majority of published literature discussing field cases does not provide details of the simulation
model used or the decision analysis by management. The sensitivity of the WAG process and
applying it to new formation types such as fractured media must be studied.
Two research paths are discussed in this paper, one study looks at the systematic approach to
determining the sensitivity of the WAG process to rock and fluid parameters. A D-optimal
experimental design and response surface based proxy model are the statistical tools used in this
approach. This study is done on a five spot sector model representing a sector location of an oil
field. These tools are an aid in developing and understanding a WAG pilot study for the field.
The results are instrumental in understanding how these parameters affect both the recovery and
the associated problems of early gas breakthrough.
Characterization and modeling of fractured reservoirs is one of the challenging aspects in the oil
industry. The approximations that fractured media can be represented by, such as an orthogonal
array of matrix blocks surrounded by fractures, were initially expressed by Warren and Root8
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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and improved by Kazemi et al9. If a single porosity formulation is used for simulating these
types of reservoirs, the properties of the matrix and the individual fractures must be represented
by separate grid blocks. This approach is (currently) too expensive. To solve this problem a
numerical approach that divides the reservoir into 2 sets of grid blocks, one for the matrix and
one for the fracture system, was introduced. This model that results is called a dual porosity
model.
In the second part of the study we consider the simulation of the WAG process in fractured
media. We performed 2 dimensional fine grid simulations of WAG injection in a system
consisting of a matrix block exposed to one fracture. Subsequently the same system was modeled
using a dual porosity formulation. By comparing the results of these two models, we have
derived some recommendations for getting accurate results from dual porosity simulations.
WAG
The tertiary recovery process known as WAG is a combination of the two secondary recovery
processes of water flooding and gas injection. The WAG process was proposed originally to aim
for the ideal system of oil recovery: improvements in macroscopic and microscopic sweep
efficiency at the same time. The water is used to control the mobility of the gas as can be seen in
equations 1 and 2. The cyclic nature of the WAG process causes an increase in water saturation
during the water injection half cycle and a decrease of water saturation during the gas injection
half cycle. This process of inducing cycles of imbibition and drainage causes the residual oil
saturation to WAG to be typically lower than that of water flooding and similar to those of gas
flooding.
fw =
fg =
kw µ w
k w µ w + ko µ o + k g µ g
kg µ g
k w µ w + ko µ o + k g µ g
(1)
(2)
Literature on the WAG process typically discusses two major management parameters that affect
the economics of a WAG project. These operational aspects are the half-cycle slug sizes and the
WAG ratio. The two major problems faced are early breakthrough and injectivity losses. In a
separate paper11 we consider the scope for smart well implementation in addressing these two
issues.
Reservoir Model
The reservoir model is 2,641 by 2,641 by 144 feet, represented by 19 x 19 grid blocks aerially
and 26 gridblocks in the vertical. A standard 5-spot pattern with a central injector and 4
producers is used with all sides bounded by no flow boundaries. The model is inspired by a
Middle East oil field. The reservoir model is implemented in a commercial reservoir simulator.
The PVT model consists of 7 pseudo components, and is used to represent an oil and gas phase
which are fully miscible under initial pressure conditions. A WAG ratio of 1:1 is used with 3
months per injection phase. As mentioned above we face two primary problems in WAG
processes: early breakthrough and loss of injectivity. The economic constraints placed on the
wells are a maximum water cut of 0.5 stb/stb and a maximum GOR of 5 Mscf/stb at which point
the well is shut-in. The wells are also tested every 100 days and can be reopened if the test
shows it can operate.
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Methodology
Many petroleum-engineering applications of design of experiment (DOE) have been reported in
literature1-3. This technique can be applied to estimate the sensitivity of reservoir behavior to
various factors. The obtained information can be used to optimize data acquisition, parameter
estimation, history matching and consequently assist in field development planning. Moreover,
the design of experiment framework reduces the number of costly and time-consuming reservoir
simulations especially in reservoirs with complex structure.
Design of experiment is defined as a structured and organized method, based on statistical
principles that can be used to identify the impact of different parameters affecting a process. The
objective of using DOE is to achieve the most reliable results with optimal use of time and
money. Experimental design, in fact, changes different parameters systematically and
simultaneously within a limited number of experiments to give an overall view of the process.
The first step to construct a design is to identify those factors that are expected to have a large
influence on the response. Afterwards, the factor ranges are usually scaled to lie between “-1”
and “1” to represent factor’s maximum (1), minimum(-1) and mean(0) value. Factor ranges
should be chosen carefully to avoid dominance of experimental error on response (small ranges)
and to decrease the possibility of construction of a complicated response model (large range)4 .
Then a particular design (classical or optimal) depending on time and computer power can be
constructed. The combination of factors derived from DOE is used to feed into a simulator or to
implement experiments. The response surface model (RSM) is finally used to fit the simulation
or experimental results to a model. Usually the model being fit is a polynomial function, which
acts as a substitute for reality.
The model is denoted as y=Xβ + ε where X is the design matrix with the row dimension equal to
the number of experiments and column rank equal to the number of terms in the model
(regressors)5. The design matrix depends on both regression model (linear, quadratic, cubic, etc)
and the design of experiment method. ‘y’ is the vector of simulation or experimental result. ‘ε’
denotes a random vector with distribution of N(0, σ2), which represents the error. ‘ β̂ ’, given
by βˆ = ( X ' X ) −1 X ' y is the least square estimate of β which delivers the best set of coefficients. It
has the covariance matrix of (X’X)-1σ2 where X’ is the transpose of the design matrix (X). To
acquire maximum information or correspondingly obtain higher quality of model, a series of all
possible combinations of factors (candidates) should be chosen such that the determinant of the
covariance matrix be minimized. This can be done by constructing an optimal design. There are
a number of criteria to construct optimal designs among which “D-optimal” design is the most
common and widely used.
For a full 3-level factorial design, 3K (K: number of factors) experiments is needed. Therefore as
the number of factors increases, conducting a full factorial design becomes less feasible. The Doptimal design procedure provides various options to select from a list of valid points (i.e. 3k)
those points that will extract the maximum amount of information from the experimental region,
given the respective model that one expects to fit to the data. It maximizes the determinant of
(X’X) or equivalently constructs a design, which provides as much orthogonality between the
columns of the design matrix as possible6. Obviously, if all regressors are orthogonal to each
other, one could extract the maximum information from the experimental region. The flexibility
of D-optimal design combined with its accuracy has made it quite popular in engineering
applications.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Single porosity model
A matrix block surrounded by one fracture is considered in this study. Water and gas are injected
from one side of the fracture and oil, water and gas are produced from the other side. By the
assumption that the length of the matrix slab is long enough and the flow in the matrix is
symmetric, a 2 dimensional fluid flow is modeled.
The 15 ft x1.5ft x 26ft matrix block, exposed to the one fracture, including 1536 (48 x 1x 32)
grid blocks was selected for this model. Each grid is 0.33 x 1.5 x 0.82 feet. The fracture and
matrix permeability are 10 Darcy and 10 mD respectively. The fracture width is 0.07 feet.
The water injector is located at the bottom of the fracture and the gas injector at the top. When
water is injected, oil is produced from the top and when the gas is injected oil is produced from
the bottom. Each WAG cycle takes 91.5 days and the whole process is modeled for 10 years.
The grid blocks have approximately the size of the core plugs that undergo detailed core analysis
in the laboratory. Therefore there would be no need to do up scaling and the solution of this
model represents the correct answer. The oil recovery from the matrix is the most important
result for us.
Because the fracture permeability is much higher than the matrix permeability there would be no
viscous forces in the fluid displacement in the matrix. The main processes by which oil is
produced are capillary imbibition (during water injection) and gravity drainage (during gas
injection). Figure.1 shows the oil saturation after 3 years.
Firstly it was interesting to see how sensitive the oil recovery is with respect to the number of
grid blocks. The block is simulated with different grid blocks as shown in figure 2. You can see
the different behavior of the oil production due to the effect of numerical dispersion and scale up.
As the number of grids decreasing, the initial oil rate also decreases, but the final production is
higher than the base case (fine-grid). When the grid blocks are so big (3*1*2 grids and 3*1*1
grids), both the initial rate and the final oil production are lower than the fine-grid results.
Results and discussion
In this study, 80 different combinations of 6 factors (table.1) on 3-level variations were
constructed using the D-optimal experimental design approach. A commercial simulator was
used to simulate the set of 80 simulations. Table.2 shows an example of design with
corresponding simulation responses. The regression analysis is then applied to fit the field oil
production total (FOPT) by a polynomial quadratic function to the factor setting specified by the
D-optimal design. Other responses like the total water (FWPT) and gas (FGPT) production also
were considered. The correlation coefficient between response surface model and simulator’s
FOPT, FGPT, FWPT are 0.96, 0.95, 0.61 respectively which indicates a satisfactory match for
oil and gas total production and a disappointing fit for total water production. The low
correlation coefficient and hence the inappropriateness of RSM approach for FWPT is attributed
to discontinuities in the response surface, since there are some sudden changes from low to high
water production among 80 runs caused by water breakthrough in some experiments. However,
the both pre-breakthrough and post-breakthrough cumulative water production were separately
fitted with high accuracy.
The response surface model is employed to analyze the sensitivity of cumulative oil production
to each individual factor and quantify their effect. The sensitivity of response to each factor is
defined as the partial derivative of the response with respect to that factor. The Pareto chart
(figure.3) then demonstrates the most significant terms affecting the cumulative oil production in
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this study. As can be seen from the chart, eight terms which consist of some single terms (e.g. oil
rel-perm curvature factor), some quadratic terms (e.g. correlation length squared) and some
interactions between factors (e.g. product of ‘kro.c’ and permeability multiplier (kh)) have the
largest effect on cumulative oil production. The interaction term of two factors implies that the
effect of the one factor is more considerable when the other factor is moving towards its extreme.
For example the cross term of ‘kro.c&kh’ is found to influence the oil production quite
significantly. This means that the sensitivity of oil production to ‘kro’ is high when the value of
horizontal permeability multiplier is high.
Figure.4 shows the influence of each individual factor on the total field oil production.
According to the graph, the cumulative oil production increases as the end-point of the gas
relative permeability decreases. A raise in ‘krg.e’ results in moving the gas-oil rel-perm curve
upwards which leads to an increase in gas relative permeability for each gas saturation. As a
result, the gas mobility increases. This leads to a faster gas breakthrough and consequently lower
oil production.
Variation of water relative permeability and water viscosity show a negligible effect on the total
oil production. Although the water mobility increases as the ‘krw.e’ increases and/or water
viscosity decreases, the oil production increases. This may be attributed to the fact that the loss
of water injectivity can probably be compensated by higher water mobility which results in
contacting more oil by water in the reservoir and helping maintain reservoir pressure.
As the oil relative permeability curvature factor becomes smaller, the FOPT increases sharply.
According to the 3-phase relative permeability model used by simulator7, the smaller curvature
factor shifts the oil relative permeability curve upward for both gas-oil and water-oil systems.
Thus, the oil rel-perm increases in both systems which results in an increase in 3-phase oil
relative permeability.
Increasing and decreasing the correlation length both leads to an increase in cumulative oil
production. In the 5-spot pattern, the permeability field with scaled correlation length of 0 has a
relatively higher permeable area near to the production well(s). Thus injected gas flows faster
through this area and not only lots of oil are bypassed but also gas reaches the production well
causing well shut-in and consequently less oil is produced. Shorter correlation length normally
indicates higher variation of permeability in the field. However, this fluctuation leads to
construct a permeability field with smaller higher permeable areas. Therefore the injection fluid
can sweep the oil exists in the higher permeable zone and also the oil around that. Thus the
breakthrough time for this model when the other factors are the same is delayed which implies
greater production for model with kcor=-1. On the other hand, the model with ‘kcor’=1 has been
constructed with a correlation length comparable with inter-well spacing. The permeability field
in this case shows almost a uniform distribution. As a result, there exists no considerable higher
permeable area to cause early breakthrough of injection fluid. This model potentially yields
larger oil production.
Two-dimensional slices (of the 6-dimensional design region) through the factor space can
visualize the response surface model that is computed by regression analysis using the design
matrix and cumulative oil production (figure.5). Factors which are not plotted in each surface
are adjusted to zero. The associated contour map of the interaction term (figure.6) can be used to
identify the values of factors which maximize the production.
The response surface model can also be used to assess the uncertainty in the reservoir. Having
used the RSM model as a substitute for simulator, one can construct the full factorial result (729
experiments) and hence run as many Monte Carlo simulations as needed by preparing a
distribution of factors beforehand. Of course, direct Monte Carlo simulation will not be
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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economically feasible by using a simulator. As a result, the need for response surface model
becomes crucial especially if the number of factor increases.
Scale up
How can the coarse single porosity model (4x1x 2) be adapted to yield the same results as the
fine grid simulation? For this aim some modification must be applied to the rock-fluid properties.
The parameters that play the dominant role in the upscaling are the relative permeability and
capillary pressures. All of these parameters were tested to see how the best match can be
obtained. Among the parameters, Pcwo (water –oil) and Pcgo(gas-oil) have the most impact on the
recovery . Different matches can be found by modifying PCow , PCog , and both of them. The best
match obtained by using modified PCow (figure 7). This shows that the role of water in displacing
oil is more than gas in the WAG process in the fracture media.
Dual porosity models
In this part the process is modeled by a commercial simulator under the dual porosity option.
According to the Kazemi9 definition the shape factor σ =.0091 ft-2 is calculated for this case. The
first simulation with dual porosity default parameters showed higher recovery compare to the
fine grid model. Also the other difference was the much lower matrix water saturation and higher
matrix gas saturation of the dual porosity model compare with the fine grid single porosity. This
shows that the stronger capillary imbibition force should be applied to the matrix block to give
the same saturation profile. So the first step for matching the curves was defining pseudo
capillary curves for water-oil and gas-oil.
Another point for this curve matching is that a single shape factor may not be sufficient to model
this case. As mentioned before when the water is injected, the oil production will be by water
imbibition and during the gas injection by gravity drainage. Typically gravity drainage is slower
than imbibition which is true in this case and it needs smaller value for σ. We determined the σg
by looking at the oil rate profile of the fine grid model. The oil rate during the water injection is
almost two times bigger than the gas injection so σg = .0045 was chosen.
The last step for matching the curves was done by using the relative permeability modification
which provided by Eclipse. Changing the shape of the recovery versus time can be done by
modifying the relative permeability curves directly. But the simpler way is using the a parameter
“m (w,g)” by which a quadratic modification is applied to the oil relative permeability. In this
case “mw”(modification factor for the oil in water relative permeability),“mg” (modification
factor for the oil in gas relative permeability) and both of them were tested. In all the cases a
good match (especially for the end part of the curve) can be achieved but modified curve by
“mw” gives the best answer (figure 8)
This matching process also was done for the same models with the different WAG cycles
(180days, 60 days) successfully. The matches for the long Wag cycle are more accurate and the
effect of the gas properties becomes less.
Summary and Conclusions
The design of experiment and associated response surface model was successful in identifying
the most sensitive factors in the model with Water Alternating Gas as the recovery technique.
The methodology not only reduced the number of runs, but also provided the maximum
information from the design space.
Eight terms were identified to have the highest influence on the oil production of which oil
relative permeability curvature factor had the largest effect. The sensitivity of cumulative oil
production to water rel-perm end point and water viscosity were found to be quite low. The
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sensitivity to correlation length was found too interesting since the recovery improved both by
decreasing and increasing of this factor.
Cumulative water and gas production were also considered as the responses which the RSM for
water had low accuracy due to the water breakthrough in some cases which makes erratic data
for response surface model. The sensitivity of gas production to factors was similar to that of the
total oil production.
Second study
-single porosity model of oil displacement (by WAG) in the matrix block is sensitive to the
number of grid blocks due to the numerical dispersion and upscaling. By decreasing the number
of grid blocks the oil production increases but for the large grids (few number of grids) it also
decreases.
-The simulation results of the coarse-grid single porosity model can be matched with the finegrid (single porosity) by modifying (increasing) the capillary pressure curves (Pcow, Pcog or both
of them). The best match can be achieved by the modified Pcow curves.
-Dual porosity model can represent the results of the fine grid model by defining pseudo
capillary pressure curves (for both oil-water and oil-gas), giving a proper value for the gravity
drainage shape factor (according to oil rate during water injection and gas injection) and
modifying the relative permeability curves of the oil in water and oil in gas by modification
factor “m”. This matching process is valid for different WAG cycles.
Reference:
1.
White, C.D, Royer, S.A: “Experimental Design as a Framework for Reservoir Studies,” paper SPE 79676 presented at the SPE
reservoir simulation symposium, Houston, Texas, U.S.A, 3-5 February 2003
2.
Van Elk, J.F., Guerrera, L., Vijayan, K. and Gupta, R. :”Improved Uncertainty Management in Field Development Studies through the
Application of the Experimental Design Method to the Multiple realisations Approach,” paper SPE 64462 presented at the SPE
Annual Technical Conference and Exhibition, Dallas, Texas, 1-4 October 2000
3.
kjφnsvik, D., T. Jacobsen, and A. Jones: ‘The Effects of Sedimentary Heterogeneities on Production from Shallow Marine ReservoirsWhat Really Matters?,” paper SPE 28445 presented at the European Petroleum Conference, London, 25-27 October 1994
4.
Atkinson, A. C., Donev, A. N., :Optimum Experimental Designs, Oxford University Press, New York (1992)
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Jeff, C.F., Hamada, M., :Experiments: Planning Analysing, and Parameter Design Optimisation, John Wiley and Sons, Inc., New York
(2000)
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StatSoft, Inc. (2004). Electronic Statistics Textbook. Tulsa, OK: StatSoft. WEB: http://www.statsoft.com/textbook/stathome.html
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Schlumberger, Eclipse, Reservoir Simulator, 2003
8.
Warren, J. E., and Root, P. J.: “The behavior of Naturally Fractured Reservoirs,” SPEJ (Sept. 1963) 245-.255, Trans. AIME, Vol. 228
9.
Kazemi, H., and Gilman, J. R.: “Analytical and Numerical Solution of Oil Recovery From fractured Reservoirs Using Empirical
Transfer Functions,” (SPE 19849) presented at the 64th Annual Technical Conference and Exhibition of the Society of Petroleum
Engineers held in San Antonio, TX, October 8-11, 1969.
10. Gurpinar,o, and Kossack, C.A : “Realistic Numerical Models for Fractured Reservoirs, ” (SPE 59041) presented at the 2000 SPE
International Petroleum Conference and Exhibition in Mexico held in Villahermosa, Mexico, 1–3 February 2000.
11. Esmaiel, T.E.H. , S. Fallah, and C.P.J.W. van Kruijsdijk, Gradient Based Optimization of the WAG Process with Smart Wells,
ECMOR, 2004, Cannes, France
Table 1. Factors ranges and scaling
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Table 2. An example of D-optimal design and associated responses
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Figure
1.
Oil
saturation
distribution in the
matrix block after
3 years
Figure 3. The
Pareto chart shows
the influence of
each term on the
cumulative
oil
production.
An
increase in terms
with red bar leads
to a reduction in
FOPT
Figure 5. Response surface shows the effect of oil relative
permeability curvature factor and correlation length on
cumulative oil production in one plot.
Figure 7. Oil
recovery match
from matrix
block of fine
grid and coarse
grid models
using modified
capillary curves
Figure
2.
Comparison
between oil
production
of the single
porosity
model with
different
number of
grid block
Figure 4.
Sensitivity of
each factor
on total oil
production.
The dash
lines
represent the
confidence
interval.
Figure 6. Contour map derived from response surface of figure 3.
Figure 8. Oil
recovery match
between fine grid
and dual porosity
model