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Society of Petroleum Engineers

SPE 51920

Staggered In Time Coupling of Reservoir Flow Simulation and Geomechanical Defor-mation: Step 1 — One-Way CouplingSusan E. Minkoff, Charles M. Stone, J. Guadalupe Arguello, SPE, Sandia National Labs, and Steve Bryant, Joe Eaton,Malgorzata Peszynska, Mary Wheeler, SPE, The University of Texas at Austin

Copyright 1999, Society of Petroleum Engineers, Inc.

Thispaper was preparedfor presentationat the 1999SPE ReservoirSimulationSymposiumheld in Houston, Texas, 14–17February 1999.

This paperwas selected for presentationby an SPE ProgramCommittee following reviewof informationcontained in an abstract submitted by the author(s). Contents of the paper,as presented, have not beenreviewd by the Society of PetroleumEngineersand are subjectto correction by the author(s). The material, as presented,does not neccessarily reect anyposition of the Society of Petrloeum Engineers, its ofcers, or members. Papers presentedat SPE meetings are subject to publication review by Editorial Committees of the Societyof PetroleumEngineers. Electronic reproduction,distribution,or storageof any part of thispaperfor commercialpurposes withoutthe written consentof the Society of Petroleum En-gineers is prohibited. Permission to reproduce in print is restricted to an abstract of notmore than 300 words; illustrations may not be copied. The abstract must contain conspic-uous acknowledgementof 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.

AbstractAn isothermal, implicit, mixed nite element black oil reservoirsimulatorfrom the University of Texas is coupled to an explicit,

quasistatic, nonlinear nite element solid mechanics code fromSandia National Laboratories. Both codes are 3d and parallel.The former models (in a locally conservative manner) the owof oil, gas, and water uid phases in the reservoir while the lat-ter has been specialized to solve large-scalegeomechanics prob-lems involving signicant inelastic deformations. In this pa-per we illustrate a uni-directional coupling of the two codes inwhich ow simulation output (pore pressures) from a 10-yeartest case based on the Belridge Field in California drives thegeomechanics simulation for the same time period. The high-porosity, low-permeability Belridgediatomiteundergoessignif-icant compaction including 6 feet of vertical displacement at thetop of the reservoir.

IntroductionThere is widespread interest in the oil industry in modeling pro-duction in highly compactible weak formations. These simula-tions require an accurate representation of both uid ow andgeomechanical deformation to handleprimary and enhanced oilrecovery scenarios in a realistic manner. Writing a fully cou-pled simulator entails a great manpower investment. A more

exible and cost-effective alternative is to couple two separatesimulators that have been optimized for specic classes of ap-plications through a common high-level interface (see the pa-per by Settari and Mourits 1). In the current work we com-bine contributions from two projects of theUSDOE’s AdvancedComputational Technology Initiative (ACTI) program. The ul-timate goal of this study is to enable “2-way” staggered-in-timelinking of these two massively-parallel, three-dimensional sim-ulators. This technique will use periodic pore pressure outputfrom the ow simulator to drive the geomechanical deformationcode over that time interval. Resulting stresses are converted toporosity updates and fed back to the ow simulator for the nexttime step. As a rst step towards 2-way coupling, we demon-strate a simpler algorithm (“1-way coupling”) in which the owsimulator runs for the full time interval with xed porosity, out-putting pore pressures at specied times during the run. Sim-ulation of geomechanical deformation will cover the same to-tal time interval but will use the pore pressures as periodic in-puts to determine displacement and stress updates. The numer-ical example we describe in this paper is based on stratigraphicdata and material properties from the Belridge Field in Califor-nia which has been demonstrated to undergo considerable sub-sidence due to the extremely high-porosity diatomite reservoirrock.

Reservoir SimulationWe used an isothermal black oil reservoir simulatordeveloped atthe University of Texas to model the ow of three uid phases:oil, gas, and water. The black oil model code solves Darcy’s re-lation and the mass balance equations for the uid phase con-centrations, pressures, and velocities. This fully implicit, 3D,parallel model is one of several physical models currently im-plemented within the IPARS framework 2. The ow domain isparceled to different processors using domain decomposition,and the ow equations are discretized using an extended mixednite element method which maintains local conservation of mass. The poorly conditioned nonlinear system is solved via apreconditioned, parallel, GMRES iterative method.

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2 STAGGERED IN TIME COUPLING OF RESERVOIR FLOW SIMULATION AND GEOMECHANICS SPE 51920

Geomechanical DeformationThe geomechanics code (JAS3D) solves for nonlinear materialdeformations via explicit nite element technology developedover the last twenty years at Sandia National Laboratories. Thistechnologydoes not require the formationor factorizationof glob-al stiffness matrices but relies on iterative, nonlinear conjugate

gradient and self-adaptive, dynamic relaxation methods to re-duce nodal residuals and achieve global equilibrium. The lack ofa globalstiffness matrix allowsfor problemswith a large num-ber of elements to be solved efciently even on a workstation.

Numerical ExampleIn the ow simulation described here, we modeled a produc-tion scenario similar to the rst of four production scenarios thathave been implemented in the Belridge Field over the past 20years — a rectangular patch ( 2.5 acres) with four productionwells in the corners of the domain. In the eld, this productionscenario ran foreight years and led to substantial subsidence andwell-failure 3.

The two simulations covered the same areal extent but very

different depths: 688–2264 ft for ow versus 0–3800 ft for ge-omechanics. The grid spacing (in all three directions) also var-ied substantially between the two simulations. To capture ne-scale displacement and stress changes, we used 9700 elementsfor the ow simulation and nearly 10 times that many elementsfor the geomechanics simulation in the reservoir interval alone.Both simulations were run for 10 years.

Initial water pressure ranged from 331 psi at the top of thereservoir to 900 psi at the bottom. Permeabilities from .002–.2 mD are reasonable for this eld as are porosities (constant ineach vertical layer) ranging from 40–57%. Eight different setsof relative permeability and capillary pressure curves were usedto describe the layered ow simulation setup. The four vertical

wellswere completed to about 2/3’sof the total reservoir depth.Pore pressures were output every 90 days for input to the ge-omechanics code.

The geomechanical model included three layers of overbur-den, eight layers of diatomite and a layer of porcelanite, and an-other thick porcelanite layer constituting the underburden, for atotal of 13 stratigraphic layers. The three materials of the over-burdenand theporcelanitelayers were modeled using a Drucker-Prager plasticity model. The diatomite layers were modeled us-ing an extended Sandler-Rubin cap/plasticity model. The loadson the structure were the changes in pore pressure with time,as determined from the reservoir simulation, and gravity. Aninitial stress eld was also specied such that the total verticalstress at any point in the conguration was computed from theweight of the overlying material. The two principal horizontalcomponents of stress were taken to be 0.65 and 1.2 times thevertical values 3. No displacement was permitted normal to thevertical faces of the conguration, and all displacements at thebottom of the domain were xed.

Due to the extremely low permeability in the reservoir, thetotal production volume of hydrocarbons was low. Pressure de-creases migrate from the wells into the reservoir domain duringthe simulation. Moreover, surface subsidence and vertical dis-placement of the diatomitelayers increase with timeas showninFigure 1. Nearly a thirdof the total vertical displacement occurs

through compaction in diatomite layer J. This layer, which is un-dergoing large pore pressure change, is the weakest of the eightdiatomite layers. Ten years of primary production ow simula-tion results in vertical displacements as large as 6 ft.

ConclusionsIn this initial example, we demonstrate the capabilities of thetwo simulators for handling complex, three-dimensionaldata onthe eld scale for both ow simulation and geomechanical de-formation. The two simulation grids differed both in extent andspacing (an advantage of loosely coupled simulators). The nextstep includes addressing mass balance and convergence issuesthat arise in 2-way coupling where geomechanical porosity up-dates inuence the ow simulation.

AcknowledgementsThe authors thank Greg Deitrick of Shell E&P for his contin-ued contributionof expertise and support. Susan Minkoff, MikeStone, andLupe Arguelloare supportedby theU.S. Dept. ofEn-ergy under Contract DE-AC04-94AL85000. Susan Minkoff re-ceives funding for this work through DOE’s MICS Ofce. San-dia is a multiprogram laboratory operated by Sandia Corpora-tion, a Lockheed Martin Company. Mary Wheeler thanks theDOE (grant # 951522401), NGOTP, and sponsors of the Indus-trial Afliates Program at the Center for Subsurface Modeling.

References1. A. Settari and F. Mourits. Coupling of geomechanics and

reservoir simulation models. In Siriwardane and Zaman,editors, Computer Methods and Advances in Geomechan-ics , pages 2151–2158. Balkema, Rotterdam, 1994.

2. T. Arbogast, C. Dawson, M. Parashar, K. Sepehrnoori,P. Wang, M. Wheeler, and I. Yotov. A new generationEOS compositional reservoir simulator: Part I - formula-

tion and discretization. In Expanded Abstracts , number37979. Soc. Pet. Eng., 1997.3. K Hansen, M. Prats, and C. Chan. Modeling of reservoir

compaction and surface subsidence at SouthBelridge. SPE Production & Facilities , 10(3):134–143, 1995.

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Surface & Top of DiatomiteTop of Diatomite Layer JBottom of Diatomite Layer JBottom of Diatomite Layers

Figure1: Subsidence history fora xed locationand layerdepths of 0, 1185, 1327, and 1792 ft.