coupled geomechanics and reservoir simulation

SPE 77723

Coupled Geomechanics and Reservoir Simulation L. K. Thomas, L.Y. Chin, R. G. Pierson, and J. E. Sylte Phillips Petroleum Company

Copyright 2002, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 29 September–2 October 2002. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract An iterative procedure is presented in this paper to couple geomechanics and reservoir simulation models for the simulation of weaker rock formations with complex constitutive behavior. Parallel computing is employed in the coupled model to reduce run time in the compute intensive geomechanics model. The procedures developed here are general and can be applied to any reservoir simulation and geomechanics model. Field and example problems under a variety of exploitation scenarios are presented to demonstrate the utility and robust nature of the coupled model.

Introduction Conventional reservoir simulators calculate the effect of rock compaction on pore volume change through the concept of rock compressibility under a defined loading condition (hydrostatic or uniaxial strain). This approach is usually appropriate for reservoirs with competent rock. However, for weaker formations and complicated rock compaction behavior, coupled analysis of geomechanics and multi-phase fluid flow may be required for obtaining more accurate solutions from reservoir simulation (e.g., Refs1-7). Also, coupled analysis may be beneficial when key reservoir properties such as permeability are strongly influenced by the stress state and the loading conditions in the reservoir formation during fluid production.6,7 Reservoir simulation with coupled geomechanics for large-scale, full-field, 3D simulation problems can be quite time consuming. Consequently, computational efficiency and

convergence of numerical solutions are critical factors required to make coupled analysis economically and numerically feasible for practical field applications. In this paper, an iterative procedure for coupled analysis of geomechanics and multi-phase flow in reservoir simulation that takes advantage of parallel computing is proposed for large-scale, full-field, 3D problems. The proposed procedure is general and effective for handling reservoir rock with complicated constitutive behavior of rock compaction and permeability change while simulating various reservoir production scenarios. Descriptions of model formulations, constitutive equations, solutions procedures, and strategies for enhancement of computational efficiency are presented in the paper. Field and example problems are presented to demonstrate the capability of the developed procedure for iterative, coupled analysis. These coupled problems include the impact of stress and water saturation on compaction in water sensitive formations; and the effect of stress sensitive permeability on field productivity. Coupled Model Formulation An iterative, fully coupled procedure8 is used to integrate the reservoir simulation and geomechanics models in a generalized fashion. Discretized equations of these two models contain state variables that are shared by both models. These shared variables are porosity, pressure, and water saturation. A flowchart illustrating the iterative, coupled procedure is shown in Figure 1. The procedure consists of an initialization phase and a solution phase. At time zero, both reservoir and geomechanics models are set to their initial conditions. Based on the initial pressure and saturation distributions provided by the reservoir model, the geomechanics model solves its governing equations to establish the initial stress state in the reservoir, overburden, underburden, and sideburden under the prescribed initial conditions. Then, the geomechanics model sends initial porosity values and initial derivatives of porosity with respect to pressure and water saturation to the reservoir model. At this stage, the initialization phase is complete and the reservoir model is ready to start the first time-step calculation.

2 L. K. THOMAS, L.Y. CHIN, R.G. PIERSON, AND J. E. SYLTE SPE 77723

The reservoir model operates as the primary or controlling software in the coupled model. Pressures and water saturations for cases with water sensitive rock are passed to the geomechanics model each Newton iteration of each time step. The geomechanics model in turn iterates on its solution, taking as many Newton iterations as required, until convergence is reached. Porosities and porosity partial derivatives with respect to pressure and water saturation are then returned to the reservoir model for use on its next Newton iteration. This sequence is continued until convergence is reached based on criteria in the reservoir simulation model.

Description of the Models The governing equations for a black-oil reservoir model are presented here for simplicity, however, the coupling analysis described in this paper is general and can be applied to any reservoir model formulation. Based on the conservation of mass for the water, oil, and gas phases and Darcy’s Law, the following flow equations are obtained.9

( )[ ] ww

wwww q

BS

tzp +

∂∂=∇−∇∇ φγλ

(1)

( )

∇−∇+

∇−∇∇ zprzp gggsooo γλγλ

sgog

gs

o

o rqqBSr

BS

t++

+

∂∂= φ (2)

( )

∇−∇+

∇−∇∇ zpRzp ooosggg γλγλ

osgo

os

g

g qRqB

SRBS

t++

+

∂∂= φ

(3)

Additional equations include the constraint equation for the sum of saturations and capillary functions.

1=++ gow SSS (4)

)( wcwo SfP = (5)

(6) )( gcgo SfP

Finite difference methods are normally used to discretize the above equations. The reservoir simulator used in this paper is a general-purpose model for black oil and compositional applications.10 It contains compaction logic, which approximates results from the coupled reservoir and geomechanics model for specific cases. The governing equations for the geomechanics model are derived based on the balance of linear momentum and the effective stress law.11 The equations are stated as follows:

0b =+δ−σ⋅∇ ρ)'( p (7)

In order to derive the full set of governing equations, Eq. 7 must be supplemented with an appropriate constitutive equation for the solid porous matrix. In general, the constitutive equation is in the form:

),(' Κεσ f= (8)

Assuming the solid grain is incompressible, the solid strain rate is related to solid velocities as: (9) s

)(v∇=ε

These nonlinear constitutive equations (Eqs. 8 and 9) are usually appropriate to describe the nonlinear behavior of weak reservoir rocks. Details of the constitutive models used in this work are presented in reference 12. Data Exchange Interface Within a given time step and for each coupled iteration, pressure and water saturation values obtained from the reservoir model are directly passed to the geomechanics model through a data exchange interface. Porosity values and derivatives are passed from the geomechanics model to the reservoir model. Uniquely named disk files are used to exchange data between the two models. After each data file is written and closed, a second uniquely named file containing time, time step number, and iteration number is written. The existence of the second, small file signals that the model awaiting data may open and read the new data file. The mathematical system of the reservoir model is based on the Eulerian description with a fixed mesh configuration that is time-independent, while the mathematical system of the geomechanics model is based on the Lagrangian description with a deformable mesh configuration that changes with time. Therefore, the true porosity calculated from the geomechanics model cannot be directly passed to the reservoir model. To maintain numerical consistency with the definition of porosity used in the reservoir model, the true porosity of a given element has to be converted to the reservoir-simulation porosity defined by the constant bulk volume for the corresponding grid block in the reservoir model. The equations that relate the true porosity and the reservoir simulation porosity to the volumetric strain of a given element in the geomechanics model are

ve εφφ −−−=′ )1(1 0

(11)

)1(0 −+= ve εφφ (12)

SPE 77723 COUPLED GEOMECHANICS AND RESERVOIR SIMULATION 3

At time equal to zero, the initial reservoir-simulation porosity and the initial true porosity are the same. Note that εv is negative for compaction. The reservoir-simulation porosity for each grid block is obtained from the geomechanics model using Eq. 12 and is passed to the reservoir model. Parallel Computing Environment The coupled procedure presented in this paper is designed to run on a single PC, multiple personal computers with multiple CPUs, or on a cluster of PCs using the message-passing interface (MPI).13 Figure 2a illustrates the use of two nodes on the cluster to perform the coupled analysis with the controlling reservoir model on node one and the geomechanics model on node 2. These two models are run sequentially and therefore either or both can be run in parallel using multiple PCs. The two-headed arrow in Figure 2b indicates the data exchange and communication between the two nodes each coupled iteration. On field scale problems 70 to 80 percent of the run time is spent in the geomechanics model solving the system of linear equations. The compute intensive nature of this model is a function of several items. First, the finite element formulation of the geomechanics model has a 27 point stencil compared to the standard seven point stencil in the reservoir simulation model. The second factor is the number of unknowns in the geomechanics model, which is three at each node compared to one at each cell in an IMPES finite difference simulation model. The last item is the total number of grid blocks used in the geomechanics model is larger due to the overburden, underburden, and sideburden regions that are simulated in this model in addition to the reservoir grid blocks. To efficiently implement the parallel version of the solver, communication between CPU nodes must also be minimized to achieve the desired speed improvement. Due to the disparity in run times between the geomechanics model and the reservoir simulation model only the geomechanics model is run in parallel at this time which is illustrated in Figure 2b. The parallel version of the geomechanics model was developed with a parallel iterative linear solver based on the ILU preconditioned bi-conjugate gradient method. Domain decomposition with over-lapping boundaries was used to divide the model into multiple regions. Linear solver iterations are then conducted, for each Newton iteration, until convergence is reached. The number of iterations for the linear solver varied from 5-25 depending on the number of processors used and the difficulty of the problem. Likewise, the average number of Newton iterations required for the geomechanics model ranged from 1-5. Example Cases Five example cases are presented to illustrate the performance of the coupled reservoir and geomechanics models. The first

example is a single cell model, which is used to demonstrate the efficiency of the coupled model during depletion, repressurization by waterflooding, and blowdown. The second example is a field-scale, 3D, geomechanics problem that is run in parallel mode to demonstrate the efficiency of this model versus the number of processors used. The third example is a two-phase, multi-layered, five spot area of a field with varying layer properties that illustrates depletion followed by waterflooding. The effect of stress dependent permeability is also included in this example. A second 3D, five-spot example is used to illustrate the performance of the coupled model on a three-phase example. The last example is a large full-field model that includes historical production, gas injection, and water injection. A comparison between results from the coupled and stand alone simulation models is presented here for several of the example cases. Comparisons between coupled and uncoupled simulators for previous examples are given in reference 7.

Single Cell Example This example illustrates the coupled simulation of a two-phase, single cell model. The initial and boundary conditions for the geomechanics model are uniaxial, vertical load with rigid side and bottom boundaries. Thus, a direct comparison between the coupled reservoir and geomechanics models should give the same results as the “stand alone” reservoir model assuming that the same stress-strain data and hysteresis logic is used in both the geomechanics model and the reservoir model. The overburden pressure was held constant in this run at 9000 psia. The initial reservoir pressure is 7120 psia and the bubble point pressure is 2000 psia. Initial and connate water saturations were set equal to .15 and residual oil to water saturation was equal to .3. Normalized saturation squared curves10 were used to calculate relative permeabilities. Other pertinent data for this example are given in Table 1. The stress-strain curves for this case are a function of water saturation and are presented in Figure 3. This example was produced under primary depletion for the first 3000 days. Partial pressure maintenance was applied during the next 2000 days with limited water injection. At 5000 days massive water injection was initiated, repressuring the cell to approximately 6400 psia prior to blowdown at 7300 days. Plots of pressure, water saturation, and porosity versus time for this example as well as the stress-strain path taken during the run are presented in Figure 4, 5, 6, and 7 for both the coupled reservoir and geomechanics models and the “stand alone reservoir model”. Essentially identical results were obtained in both simulations. The number of time steps and iterations for both runs using a fixed 90 day time step were 84 and 86, respectively verifying the robust nature of the coupled model.


Geomechanics Example A field-scale, 3D, geomechanical problem was used to study the parallel computing method developed here. The problem is a rectangular, parallelepiped with a mesh configuration that is composed of 50x50x25 (62,500) hexahedral elements. The problem has a single point-load, which is a function of time, applied on the top surface of a corner element and the material modeled in the problem is treated as linear elastic. The performance of parallel computing for this problem is shown in Table 2. Speedup is the ratio of the run times between the parallel run and the single-node run. Efficiency is defined as the speedup in terms of the percentage of the number of nodes used. The results show that a computing efficiency of approximately fifty percent was accomplished when 16 nodes on the cluster were used. Five Spot Example -2Phase This example is used to illustrate the ability of the coupled model to simulate a 3D, two-phase, layered five-spot with stress dependent permeability and porosity that is a function of stress. Data for this example is essentially the same as that presented in reference 8, with the exception of the permeability versus stress data which is given in Table 3, absolute layer permeabilities that were set to 5, 10, 50, 10, 5, and the non-linear (plastic deformation) stress-strain relationships similar to those shown in Figure 3. The five-spot was produced under primary depletion for the first five years prior to the initiation of water injection. The producing well was assigned a maximum total liquid rate of 3000 STB/day and a minimum bottomhole pressure of 2000 psia. The maximum injection rate of each of the water injection wells was set equal to 1000 bbl/day with a maximum bottomhole injection pressure of 6000 psia. Coupled simulations for this example with and without stress dependent permeability were run. The coupled runs were made running the geomechanics model on one processor and on 16 processors in parallel with essentially identical results. Oil production rate, water cut, and average hydrocarbon pressure for the stress independent and dependent permeability runs are shown in Figures 8-10, respectively. Cumulative oil production for the stress dependent permeability case was approximately six percent less than the run with constant layer permeabilities at 20 years. In general there were insignificant differences between the stand alone and coupled runs presented for this example, however, this is not always the case as was illustrated in reference 7. Five Spot Example –3Phase The purpose of this example is to illustrate, under controlled conditions, ie. minimal changes in rate, the performance of the coupled model on a three-phase, multi-layered five-spot which

is set up to reflect the nature of the complex field example which is presented next.

The grid for the reservoir model was 11x11x5 and the grid for the geomechanics model was 11x11x15 with eight layers in the overburden and two in the underburden. The reservoir is initially undersaturated with an initial pressure of 7120 psia and a saturation pressure of 5562 psia. Porosity is constant in each layer and varies from 32 to 36 percent. Water dependent stress-strain curves similar to those in Figure 3 were used. A constant permeability of 40 md was used for all layers with kv/kh equal to .1. Layer thickness values are the same as those used in the previous example. The production well was produced at a constant liquid rate of 2500 STB/day throughout the run. The bubble point pressure was reached at approximately 5 years at which time the gas-oil ratio increased rapidly from its undersaturated value of 1530 SCF/STB to over 8000 SCF/STB at 9 years. Water injection at a rate of 3000 bbl/day per well was introduced at 9.3 years. Pmax was set to 6700 psia for the injectors. Plots of hydrocarbon average pressure and gas-oil ratio, for both stand alone and coupled runs (16 processors), are presented in Figures 11 and 12. Essentially identical results were obtained. The number of time steps and iterations for the stand alone and coupled runs were both 253 and 254, respectively. Ekofisk Example The Ekofisk problem was selected to illustrate the application of the coupled simulations to a large, complex full field example with a well documented14-17 compaction history. The field performance includes primary depletion, gas injection, water injection, and extensive infill drilling. The compaction mechanisms include pore collapse due to increasing effective stress levels and water weakening. Due to the complexity of the field and its history, the stand alone geomechanical and reservoir simulations themselves have been a challenge. The primary drive mechanisms during the depletion period include oil expansion, solution gas drive, compaction drive, and limited water influx. The field wide gas injection rates are shown in Figure 13. Oil rate from the start of production in 1971 through 2001 is presented in Figure 14. During this period the field average GOR increased from below 1500 SCF/STB at initial conditions, to in excess of 9000 SCF/STB prior to the start of water injection in 1987. The oil rates declined from a peak of 350,000 BOPD in 1976 to near 70,000 BOPD prior to the start of water injection, with reservoir pressures below 3500 psia (7120 psia initial) in large portions of the field. Full-scale water injection began in late 1987, Figure 15, with pressure stabilization achieved by late 1993. Despite stable to increasing reservoir pressures, surface subsidence continued at the 40 cm/year trend established during the depletion period


until mid 1998, primarily as a result of the water weakening phenomena. Due to further increases in reservoir pressure and the expenditure of the water weakening effect, surface subsidence is now at a relatively modest level approaching 10 cm/year. Production from the field has averaged around 300,000 BOPD the last few years due to a combination of an efficient waterflood, infill drilling, and installation of new production facilities, with a producing field GOR below 1100 SCF/STB, and water rates below 80,000 BWPD. The reservoir model for Ekofisk used in this study is a five-component, compositional model10 with a 67x41x14 layer grid. A cell size of approximately 450 feet was used in the central part of the field. The grid in the geomechanics model was 67x41x24 elements that consist of the reservoir section and the overburden, sideburden, and underburden regions. The mesh configuration and geological description of the geological section are identical to those of the reservoir model. The stress-strain relationships of the reservoir rock in Ekofisk are based on uniaxial data from the laboratory with adjustments for the natural fracture system (K0=.2) in this field and were modeled with a hypo-elastic/hypo-plastic constitutive model. Dry and fully water-weakened stress-strain curves for a range of porosity values were used to interpolate for strain knowing initial porosity, stress, and water saturation. The match of field GOR versus time for Ekofisk using the coupled model is presented in Figure 16. A similar match of water cut versus time was obtained. Maximum trapped gas saturations during re-pressuring that were used in the history matching process ranged from 10-18 percent and are consistent with values measured in the laboratory. This example was simulated using the coupled model with 16 processors for the geomechanics model and took 366 time steps and 661 iterations in the reservoir model for the approximately 30 year simulation. Thus, the coupled procedure performed the simulation in a robust and efficient manner. The total CPU and elapsed times for the run were 126 minutes and 144 minutes, respectively. These results indicate that the proposed procedure can be used for analyzing field-scale problems that integrate reservoir and geomechanics simulation. Discussion The example cases illustrate several important features of the coupled reservoir and geomechanics model presented here. First, the single cell example demonstrates that identical results should be obtained between a stand alone reservoir model and the coupled model if a consistent treatment of the rock stress-strain relationships is employed in the two models. The second example demonstrates the efficiency of the geomechanics model run in parallel mode. This efficiency starts at 100% for one node and decreases quadratically to approximately 50% with 16 nodes due to the communications

overhead between processors and the efficiency of the domain decomposition solution algorithm. The third and fourth examples illustrate the robust nature of the coupled model on two and three phase five-spot examples with variable layer properties subject to depletion and secondary recovery by waterflood. Both of these examples use stress-strain curves that were a function of water saturation. The last example demonstrates the successful application of the coupled model to a difficult full field simulation. An additional parallel run was made on this field to compare with scalar coupled results reported previously7. Here the run was made using bi-linear stress-strain curves up to 1988, which is prior to significant water injection. The elapsed times for the one node and sixteen node, coupled simulations for this case were 186 minutes and 36 minutes, respectively. Based on results from examples 3 and 4 where little difference was observed between forecasts from stand alone reservoir simulation and coupled runs, a feature was added to the coupled model to periodically skip calls to the geomechanics model for a number of time steps. An Ekofisk field case was then run where the geomechanics model was only called once a year during depletion up to mid-year 1988. After that time, the model was run in fully coupled mode. Essentially identical field results were obtained between this run and the fully coupled run. The ratio of the elapsed times for these two runs was .75. All run times reported in this paper were made on a Beowulf cluster consisting of 16 dual processor nodes. Each node is a dual PIII 1.13 Ghz rack mounted PC with 2 Gb memory. The ratio of run times running only one process on each of the sixteen nodes versus two processes on each of eight nodes is approximately .7. Conclusions

1. A fully coupled procedure has been developed for integrating reservoir simulation and geomechanics models using a parallel computing architecture based on the message-passing interface (MPI).

2. The procedure is general and can be used with any reservoir simulation model and geomechanics model.

3. The data exchange interface used in this development allows usage of the method on any PC with one or more CPUs, or a cluster of PCs.

4. The parallel computing method developed and implemented for the geomechanics model removes the major bottleneck in the coupled model.

5. Simulation results from the example cases included in this paper indicate that the procedure developed here can be used to effectively and efficiently analyze field-scale, 3D problems of fully coupled geomechanics and reservoir multi-phase flow.


Acknowledgements The authors would like to thank Phillips Petroleum Company Norway and their PL018 coventurers TotalFinaElf Exploration Norge A.S., Norsk Agip A/S, Norsk Hydro Produksjon a.s and Den norske stats oljeselskap a.s, for permission to publish this paper. Nomenclature Βl Formation volume factor of phase l K History dependent tensorial quantity K0 Ratio of horizontal to vertical effective stress pcgo Gas-oil capillary pressure pcwo Water-oil capillary pressure pl Pressure of phase l ql Volume of stock tank phase l produced per unit of reservoir volume per unit time rs Oil in gas phase Rs Solution gas-oil ratio Sl Saturation of phase l t Time vs

( ) Symmetric part of the solid velocity gradient z Depth measured positive downward Greek b Body force ε Solid strain εv Volumetric strain of an element

ε. Solid strain rate γl Weight density of phase l δ Kronecker delta λl Transmissibility of phase l φ Reservoir simulator porosity φo Initial porosity ρ Total mass density φ’ True porosity σ’ Solid effective stress Subscripts g Gas phase o Oil phase w Water phase References 1 Sulak, R. M., Thomas, L. K., and Boade, R. R.: “3D Reservoir

Simulation of Ekofisk Compaction Drive,” JPT, October 1991, pp. 1272-1278.

2 Cook, C.C. and Jewell, S.: “Simulation of a North Sea Field Experiencing Significant Compaction Drive,” SPERE, Feb. 1996, 48-53.

3 Settari, A. and Mourits, F.M.: “A Coupled Reservoir and Geomechanical Modeling System,” SPEJ, September, 1998, 219-226.

4 Gutierrez, M. and Lewis, R.W.: “The Role of Geomechanics in Reservoir Simulation,” SPE/ISRM 47392, Proc. Eurock 98, 2, Trondheim, Norway, July 8-10, 439-448.

5 Koutsabeloulis, N.C. and Hope, S.A.: ”Coupled Stress/Fluid/Thermal Multi-Phase Reservoir Simulation Studies Incorporating Rock Mechanics,” SPE/ISRM 47393, Proc. SPE/ISRM Eurock 98, 2, Trondheim, Norway, July 8-10, 449-454.

6 Settari, A. and Walters, D.A.: “Advances in Coupled Geomechanical and Reservoir Modeling With Applications to Reservoir Compaction,” SPE 51927, Proc. The 1999 SPE Reservoir Simulation Symposium, Houston, Texas, Feb. 14-17.

7 Chin, L.Y. and Thomas, L.K.: “Fully Coupled Analysis of Improved Oil Recovery by Reservoir Compaction,” SPE 56753, Proc. The 1999 SPE Annual Technical Conference and Exhibition, Houston, Texas, Oct. 3-6.

8 Chin, L.Y., Thomas, L.K., Sylte, J.E., and Pierson, R.G.: “Iterative Coupled Analysis of Geomechanics and Fluid Flow in Reservoir Simulation”, Oil and Gas Science and Technology – Rev. IFP, Vol. 57 (2002), No. 5, pp. 1-13.

9 Aziz, K. and Settari, A. (1979) Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, London and New York.

10 Coats, K.H., Thomas, L.K., and Pierson, R.G.: “Compositional and Black Oil Reservoir Simulation,” SPE Res. Eval..& Eng., August, 1998, 372-379.

11 Terzaghi, U. (1943) Theoretical Soil Mechanics, Wiley, New York.

12 Prevost, J.H. and Chin, L.Y.: “A Constitutive Model for Simulating Reservoir Compaction under a Constrained Stress Path that Leads to High Shear,” Proc. of the 8th Int. Conf. on Computer Methods and Advances in Geomechanics, Morgantown, West Virginia, May 22-28, 2139-2144.

13 Gropp, W., Lusk, E., and Skjellum A. (1999) Using MPI: Portable Parallel Programming with the Message Passing Interface, 2nd edition. MIT Press, Cambridge, Massachusetts.

14 Sylte, J. E., Thomas, L. K., Rhett, D. W., Bruning, D. D., and Nagel, N. B., "Water Induced Compaction in the Ekofisk Field," Paper 56426, presented at the 1999 SPE Annual Technical Conference, Houston, Texas, October 3-6, 1999.

15 Heugas, O. and Charlez, P.: “Mechanical Effect of the Water Injection on Ekofisk Chalk,” Third North Sea Chalk Symposium, Copenhagen, June, 1990.

16 Maury, V, Piau, J. M. and Halle, G.: “Subsidence Induced By Water Injection In Water Sensitive Reservoir Rocks: The Example of Ekofisk," Paper SPE 36890, presented at the 1996 European Petroleum Conference, Milan, Italy, 22-24 October, 1996.

17 Cook, C. C., Andersen, M. A., Halle, G., Gislefoss, E., and Bowen, G. R.: “Simulating the Effects of Water-Induced Compacton in a North Sea Reservoir”, Paper SPE 37992, presented at the SPE 14th Reservoir Simulation Symposium, Dallas, Tx., June, 1997.


Table 1 – Single Cell Example

Thickness = 230 ft ∆ X = ∆ Y = 2000ft Porosity = .40 Permeability = 30md Water compressibility = 3.5(10-6) psi-1 Oil compressibility = 10.0(10-6) psi-1 Water viscosity = .35 cps Oil viscosity = .5 cps Bwi = 1.0 RB/STB Bo (Psat) = 1.3 RB/STB

Rate Schedule (Pmin = 1000 psia, Pmax = 7500 psia)

Time, days Liq Rate, STB/day Water Inj, bbl/day 0-3000 3500 3000-5000 5350 3000 5000-7300 5350 8000 7300-7800 5350

Table 2 – Parallel Efficiency of Geomechanics Model No. of Nodes

Linear Solver

Elapsed Time (sec.)

Speedup Efficiency (%)

1 CG 1828.36 1 100 4 CG 537.16 3.40 85 8 CG 331.60 5.51 69 9 CG 308.11 5.93 66 16 CG 233.08 7.84 49 1 ILU-BiCG 1216.67 1 100 4 ILU-BiCG 375.50 3.24 81 8 ILU-BiCG 239.87 5.07 65 9 ILU-BiCG 207.17 5.87 63 16 ILU-BiCG 159.36 7.63 48

Table 3 – Permeability Factor vs Stress Stress Permeability Factor

2000 1.00 3000 0.78 4000 0.62 5000 0.51 6000 0.46 7000 0.44

ReservoirSimulator

GeomechanicsModel

p0 Sw0

ReservoirSimulator

GeomechanicsModel

pk Swk

Initializationt0=0

t=tndt=tn-tn-1n=1,2,3,..

φ0, (dφ/dp)0,(dφ/dSw)0

φk, (dφ/dp)k,(dφ/dSw)k

Newton loopConverged?

tn>=tmax

n=n+1

NoYes

Stop

Iteration kk=1,2,..

φn =φk

pn =pk

Swn=Swk

Reset k=1

Coupled Solution

NoYes

Figure 1 – Flow Chart of the Coupled Analysis

1 2

32

54

1

Reservoirsimulator

Reservoirsimulator

Geomechanicsmodel

Parallel Geomechanics model

a

b

Figure 2 - Schematic of the Computation Process for the

Iteratively Coupled Analysis

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 5 10 15

Axial Strain, %

Effe

ctiv

e A

xial

Str

ess,

psi

DryWet

Figure 3 Stress-Strain Curves

0

1000

2000

3000

4000

5000

6000

7000

8000

0 2000 4000 6000 8000Tim e, days

Pres

sure

, psi

a

Stand Alone

Coupled

Figure 4 – Single Cell Pressure vs Time


0

0.1

0.2

0.3

0.4

0.5

0.6

0 2000 4000 6000 8000Tim e, days

Wat

er s

atur

atio

n, fr

actio

n Stand Alone

Coupled

Figure 5 – Single Cell Water Saturation vs Time

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0 1000 2000 3000 4000 5000 6000 7000 8000

Tim e, days

Poro

sity

, fra

ctio

n

Stand Alone

Coupled

Figure 6 – Single Cell Porosity vs Time

0

1000

2000

3000

4000

50006000

7000

8000

9000

10000

0 5 10 15Axial Strain, percent

Effe

ctiv

e A

xial

Str

ess,

psi

Sw = 0 left

Sw = .15 m iddle

Sw = .325 rightGeom echanics Run

Coupled Run

Figure 7 – Single Cell Stress-Strain Path

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20

Time, years

Oil

Rat

e, S

TB/d

ay

K=f(stress)Base

Figure 8 – 5 Spot (2Phase) Oil Rate vs Time

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20

Time, years

Wat

er C

ut, P

erce

nt

K=f(stress)

Base

Figure 9 – 5 Spot (2Phase) Water Cut vs Time

0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15 20

Time, years

Hyd

roca

rbon

Ave

rage

Pre

ssur

e, p

sia

K=f(stress)Base

Figure 10 – 5 Spot (2Phase) HC Average Pressure vs Time


0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15 20

Time, years

Hyd

roca

rbon

Ave

rage

Pre

ssur

e, p

sia

Coupled

Stand Alone

Figure 11 – 5 Spot (3Phase) HC Average Pressure vs Time

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 5 10 15 20

Time, years

Gas

-Oil

Ratio

, SCF

/STB

CoupledStand Alone

Figure 12 – 5 Spot (3Phase) Gas-Oil Ratio vs Time

0

50000

100000

150000

200000

250000

300000

350000

400000

1970 1975 1980 1985 1990 1995 2000 2005

Tim e, years

Gas

Inje

ctio

n R

ate,

MC

F/da

y

Figure 13 – Ekofisk Gas Injection Rate vs Time

0

50000

100000

150000

200000

250000

300000

350000

1970 1975 1980 1985 1990 1995 2000 2005

Tim e, years

Oil

Rat

e, S

TB/d

ay

Figure 14 – Ekofisk Oil Rate vs Time

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1970 1975 1980 1985 1990 1995 2000 2005

Time, years

Wat

er In

ject

ion

Rat

e, b

bl/d

ay

Figure 15 – Ekofisk Water Injection Rate vs Time

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1970 1975 1980 1985 1990 1995 2000

Tim e, years

GO

R

Coupled Results

M easured Data

Figure 16 – Ekofisk GOR vs Time

coupled geomechanics and reservoir simulation

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