02 spe 164657-ms advances on partial coupling in reservoir simulation natc 2013-libre

8/9/2019 02 SPE 164657-MS Advances on Partial Coupling in Reservoir Simulation NATC 2013-Libre

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Advances on Partial Coupling in Reservoir Simulation: A New Scheme ofHydromechanical CouplingCarlos Emmanuel Ribeiro Lautenschläger, Guilherme Lima Righetto, Nelson Inoue and Sergio Augusto Barreto daFontoura, SPE, ATHENA – Computational Geomechanics Group / GTEP – Group of Technology in PetroleumEngineering / PUC-Rio – Pontifical Catholic University of Rio de Janeiro

Copyright 2013, Society of Petroleum Engineers

This paper was prepared for presentation at the North Africa Technical Conference & Exhibition held in Cairo, Egypt, 15 –17 April 2013.

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 have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper 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 SPE copyright.

AbstractThis paper deals with the implementation and validation of a new hydromechanical partial coupling methodology conducted

between two commercial simulators of flow and stress. Such configuration is based on a coupling methodology developed bythe Computational Geomechanics Group – ATHENA/GTEP - PUC-Rio, based on the consistent inclusion of terms in flow

equation in order to approach the results of fully-coupled simulations. The IMEX® flow simulator was included in the

workflow of the coupling code in order to widen the application scope of the methodology developed. To include the new

option of flow simulator was required some implementation effort together with validation through simplified models. The

algorithms developed to guide the programming were defined after detailed study of numerical and computational functioning

of the flow software. The results obtained with the new simulator were compared with the pre-existing configuration(ECLIPSE flow simulator), considering one and two way partial coupling and fully coupled models. In the comparison

scenarios set out to validate the implementation, it was evaluated changes of average pore pressure in the reservoir,compaction and subsidence, as well pore pressure variations. Comparisons with the results of pre-existing configuration and

the full-coupling scheme demonstrated the success of the developed algorithm. The exchange of coupling parameters between

simulators, in the new configuration, has been implemented effectively. Parametric studies of the variables also demonstrated

the quality of the new configuration coupling. The rigorous choice of exchange parameters between flow and stress simulators

is crucial for obtaining reliable results.

IntroductionReservoir production causes changes in the stresses and strains within the reservoir and surrounding rocks. Such changes give

rise to the so-called geomechanical effects, namely the effects observed in the system due to the change in pore pressure,characteristic of the extraction and injection of fluids in porous media. In a recent paper, Herwanger & Koutsabeloulis (2011)

illustrate some of these effects: subsidence of the surface or seafloor, slipping among stratigraphic planes, reactivation of

faults, loss of seal integrity and compaction of the reservoir.

The numerical analyses that consider the geomechanical effects should consider the phenomena in a coupled way.According to Settari & Vikram (2008), coupled problems in geomechanics must take into account the interrelationship of

hydraulic, thermal and mechanical variables in the solution of differential equations involved in each particular problem. Ingeneral, the mechanical problem is usually addressed by the finite element method and the flow problem by the finite

difference method.

The conventional reservoir simulation solves the hydraulic problem involving flow of oil, gas and water through a porous

medium. In these simulations, the variation of the pore volume is determined based only in the changes of pore pressure due to

the activity of production and injection, and a predefined value of rock compressibility. According to Inoue & Fontoura

(2009a), in this type of simulation the total stresses are held constant, and there is no compatibility of displacements between

the boundaries of the reservoir and the surrounding rocks: overburden, sideburden and underburden. In fact, what is observed

in a field development is the variation of fluid pressure that results in variation of the rock stress state. These variations, in

turn, cause changes in porosity, which is reflected in the pressure field. This process of interaction between phenomena is what

characterizes the nature of the coupled problems in reservoir engineering. Inoue & Fontoura (2009b) state that in the

conventional reservoir simulation – where only the mass balance equations, equations of state and Darcy's law are considered


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– the change in porosity is dependent only on the variation of the pore pressure and rock compressibility.

Using the concepts of poroelasticity, fully coupled results may be obtained, where the continuity equation, the Darcy flow

equation, equilibrium equation, Terzaghi's principle for effective stresses, stress-strain relation and boundary conditions arehonored simultaneously. Nevertheless, fully coupled simulations do not consist in a trivial task in the case of multiphase flow,

requiring other means for geomechanical analysis involved in the hydrocarbons extraction. The full coupling is the most

rigorous scheme, whereby the flow variables and the displacement field are combined in a single set of equations. However, in

the literature there are alternative schemes such as partial coupling, which use flow and stress simulations separately, as can be

observed in many studies such as Settari & Mourits (1994), Mainguy & Longuemare (2002), Walters et al. (2002), Settari etal.(2005), Dean et al. (2006), Samier & De Gennaro (2007), Rutqvist et al.(2002, 2007, 2008), Segura & Carol (2008a, 2008b),

Herwanger & Koutsabeloulis (2011), Inoue & Fontoura (2009a, 2009b), Inoue et al. (2011a, 2011b), Settari (2012). In the

partial coupling, each simulator solves its system of equations independently, requiring an external coupling code for

exchanging information between simulators.

As noted, great research effort has been devoted to the coherent consideration of geomechanical effects in the flow

simulator. Inoue & Fontoura (2009b) present a robust and innovative approach to partial hydromechanical coupling, where thecoupling terms intend to honor the result of fully coupled simulations. Comparative analyses conducted by Inoue et al. (2011b)

showed that commercial reservoir simulators, which seek to take into account the geomechanical effects, present very different

results from those obtained using more robust methodologies as the fully coupling. In the study aforementioned those results

were compared to the partial coupling methodology developed by ATHENA/GTEP - PUC-Rio, and it was demonstrated that

proper choice of coupling parameter is crucial in obtaining results with high technical accuracy. In the present paper, a new

flow simulator was included in workflow of the coupling code in order to widen the application scope of the methodology

developed. Next, the theoretical concepts about the methodology developed are presented.

Theoretical AspectsFirstly, the definition of the different coupling schemes for solving a geomechanics reservoir analysis is presented. This type

of analysis, which involves stress and flow coupling, can be done using two different methodologies, which are: partial and

fully coupling. The first one can be divided in two main coupling schemes between a conventional reservoir simulator and a

stress analysis program: the two-way partial coupling and the one-way partial coupling.

In the two-way partial coupling scheme, the flow variables (pore pressure and saturation of the phases) and the stress

variables (displacement field, stress and strain state) are calculated separately and sequentially, by a conventional reservoir

simulator and a stress analysis program, respectively. The coupling parameters are exchanged at each time step until

convergence is reached. Usually, the pore pressures are used to verify the convergence of the solution during the iterations.Figure 1 (a) illustrates the flowchart of this methodology.

Another partial coupling scheme, called one-way, can be considered as a special case of the two-way partial coupling. The

conventional reservoir simulator sends the information (pore pressure and saturation) to the stress analysis program but thecalculated results (displacements, strains and stresses) are not sent back to the conventional reservoir simulator. Thus, in this

scheme, the geomechanical effect does not affect the responses calculated by reservoir simulator. Figure 1 (b) illustrates the

flowchart of the one-way partial coupling.

Figure 1. Types of partial coupling: (a) Two-way and (b) One-way (Inoue & Fontoura, 2009b).

In the fully coupled methodology (poroelasticity), the variables of flow and geomechanics are calculated simultaneously

through a system of equations in which pore pressure, saturation and displacement are unknowns, assuring an internal

consistency. The method is also called implicit coupling because the whole system has a single discretization and is solved

(a) (b)


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simultaneously (Inoue & Fontoura, 2009b). The theory of poroelasticity can be seen in the several works developed by Biot, e.

g., Biot (1941).

The main disadvantages of using the fully coupled scheme to solve reservoir geomechanics problem are:

Numerical difficulties in solving the coupling between the mechanical equilibrium equation and the flow equation.

The flow simulation is, in general, simplified (generally single-phase flow).

In complex reservoir geometry this scheme is highly time consuming due to the large size of the matrix generated.

Governing Equations

The governing equations were formulated using continuum mechanics, which commonly uses the macroscopic scale to

describe the continuous distribution of the constituents in the control space. In this paper, only the equations of the flow

problem and the stress analysis problem will be shown. For more details about the development of the formulation see Inoue

& Fontoura (2009b). Next, it will be presented the flow and geomechanical equations as well as the coupling methodology

developed by the ATHENA/GTEP – PUC-Rio.

Flow Equations

The flow equation can be obtained by combining the mass conservation equation and the Darcy’s law. The law of mass

conservation is a material-balance equation written for a component in a control volume. In hydrocarbon reservoirs, a porous

medium can contain one, two and three fluid phases. In the conventional reservoir simulation, the porosity is related to pore

pressure through the rock compressibility using a linear relation, as shown in Eq. 1. On the other hand, in the fully coupledscheme, the porosity equation is composed of four components that contribute to the fluid accumulation term, as shown in Eq.

2, considering an isotropic linear elastic material. The details of these components are shown in Tran et al. (2004) and

Zienkiewicz et al. (1999).

oro ppc 1 (1)

oovv

o ppQ

1

(2)

Therefore, the governing flow equation for the conventional reservoir simulation and the governing equation used in the

fully coupled scheme are given by Eq. 3 and Eq. 4, respectively:

0 0 2

0f r

p k

c c pt

(3)

0 0 2 vf Sp k

c c pt t

(4)

Geomechanics Equation

The formulation of the geomechanical problem takes into account the equilibrium equations, stress-strain-displacement

equations, rock-flow interaction and the boundary conditions. Therefore, the governing equation of the geomechanical

problem may be written as indicated in Eq. 5 (Inoue & Fontoura, 2009b).

2u u1 2

GG p

(5)

Flow Equation for the Partial Coupling

The methodology used herein for the coupling between flow and stress problem was described in Inoue & Fontoura

(2009a, 2009b). The coupling is achieved through a convenient approximation between of the flow equation of the

conventional reservoir simulation and the flow equation of the fully coupled scheme (see Figure 2). In this methodology, the

effect of solids compressibility is removed from the fully coupled scheme and the effect of volumetric strain of the porous

medium is added to conventional reservoir simulation.

The parameters responsible for the coupling, which honor the fully coupling equation, are the porosity, Eq. 2, and the

pseudo-compressibility, Eq. 6. These parameters are updated during each iteration through the stress analysis information.

)( 1

1

n

i

n

i

o

n

v

n

vp

ppc

(6)


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Figure 2. Methodology developed for partial hydromechanical coupling.

Furthermore, the partial coupling between the stress analysis program and the conventional reservoir simulator is reached

using a staggered procedure (Inoue & Fontoura, 2009b). Figure 3 illustrates a more detailed flowchart of one time step of the

staggered procedure.

In the beginning of the time step, a commercial reservoir simulator is called to solve the flow equations, providing, as

result, the pore pressure and saturation field. The variation of the pore pressure in the time step is used to calculate the nodal

forces through a finite element code. A finite element program is called to solve the stress problem, providing the

displacements and stress/strain state. The coupling program calculates the pseudo-compressibility and porosity from the strain

state.The unknown variables of the flow problem (pore pressure and saturation field) in this procedure are calculated using the

pseudo-compressibility and the porosity that are evaluated at the end of each iteration.

Figure 3. Flowchart of the partially coupled scheme.

Coupling ApproachThe main motivation for the development of a geomechanical coupling code between different commercial software is the

possibility of exploiting the full potential of each individual program. Such harnessing is directly reflected in the quality of theresults obtained by using this kind of system, as extensively proved in studies published in this subject.

In this study was aimed to introduce the reservoir simulator IMEX® in the workflow of the coupling code developed by the

Computational Geomechanics Group – ATHENA/GTEP - PUC-Rio, as an alternative to using the ECLIPSE® flow simulation

software. The implementation of the new configuration was conducted after studying the structure of each flow simulator, in

order to map the similarities and differences between the operation modes of programs, allowing the adaptation of existing

code to the inclusion of IMEX®. The implementation was validated using the model shown in Dean (2006).

Validation Model

A reservoir model was developed in IMEX® based on the reference work presented by Dean (2006), which consists of a

prismatic reservoir surrounded by adjacent rock with a higher stiffness value. Figure 4 (a) shows the geometry (top view, side

view and front view) of the three-dimensional model of reservoir and surrounding rocks. The reservoir has a vertical well

located in the center, producing single-phase fluid at a constant flow rate of 7949.36 m³/day (or 50,000 BBL/day). Thehydrostatic gradient used was equal to 9.88 kPa/m (or 0.437 psi/ft) and the vertical stress gradient equal to 22.32 kPa/m (or

0.9869 psi/ft), with the initial horizontal stress equal to half vertical stress.

ECLISPE or IMEX

ABAQUS ®


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The finite element mesh of reservoir, employed in stress simulator, is coincident with the finite difference grid of flow

simulator. It is worth mentioning that in the flow model, there is no grid adjacent to the reservoir, once the geomechanical

effects are introduced into the flow response by updating of coupling parameters in iterative simulations. The reservoir modelhas 605 elements/cells, and the complete model with surrounding rocks (stress simulator) has 5292 elements. Figure 4 (b)

presents an overview of the finite element mesh of the complete model.

Figure 4: Geometry of problem analyzed: (a) Top, front and side views of model (in meters) and (b) Tridimensional view of the model.

It was constructed a main data file containing a series of include files developed to arrange the input parameters andproperties. It was created an include file for each of the following contents: horizontal coordinates of the cells, vertical

coordinates of the cells, types of rocks applied to each cell, compressibility and reference pressures, listing of null cells

delimiting the geometry, reservoir permeability in three directions and porosity of each cell. The separation of these data is

important to the organized operation of coupling code, because the data to be updated by the stress simulator are contained in

these include files. Table 1 presents the properties used in the stress and flow simulations.

Table 1: Fluid flow and geomechanical properties.

PROPERTIES VALUESSI units Field units

Formation volume factor at 0.1013 MPa (14.7 psi) 1.0 1.0

Viscosity 0.001 Ns/m² 1 cp

Fluid density at 0.1013 MPa (14.7 psi) 10 kN/m³ 62.4 lbm/ft3

Fluid compressibility 4.35 x 10-4 MPa

-1 3 x 10

-6 psi

-1

Horizontal permeability 1 x 10-9

m² 100 md

Vertical permeability 1 x 10-10

m² 10 md

Porosity 0.25 0.25

Young’s modulus (reservoir) 0.689 GPa 1 x 104 psi

Young’s modulus (surrounding rock) 6.89 GPa 1 x 106 psi

Poisson’s ratio 0.25 0.25

For the subsequent verification of the new coupling configuration implemented, it was necessary initially to check the

reproducibility of the results using only ECLIPSE® and IMEX

® for the model adopted. This analysis aimed to ensure that the

feeding files to the coupling code were exactly the same in terms of initial and boundary conditions, such that there was no

interference on the results from the new configuration ABAQUS-IMEX. The graph of Figure 5 shows the distribution of

pressures along the greater length of the reservoir in the initial condition and after 1200 days of production, obtained by

simulation using ECLIPSE® and IMEX

®. The curves showed very similar behavior, indicating that the models developed in

ECLIPSE® and IMEX® are numerically similar.

(a) (b)


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Figure 5: Analysis of the results reproducibility performed in the Dean model using ECLIPSE ®

and IMEX ®

.

After validation the base model, the implementation of IMEX® in the geomechanical coupling code was carried out. This

step consisted in the development of a strategy for updating information between simulators, using IMEX® and ABAQUS®

restart. Some studies have been conducted to perform the implementation, and are briefly presented in the following section.

Implementation aspects

From the point of view of implementation, functioning of the coupling code depends basically on the collection and storageof information, which is derived from the input and output files from simulators of stress and flow. The code should

manipulate information during the exchange of parameters between models in order to take the geomechanical effect to the

response variables assessed. The structure of ABAQUS-ECLIPSE coupling, which takes into account addition and removal of

terms in the flow equation to approach the full coupling response, can be used for the new implementation, since it was found

the theoretical similarity between the finite difference solutions of ECLIPSE® and IMEX® (ECLIPSE, 2009; IMEX, 2009).

As it is a study of geomechanical character, special attention was paid to the evaluation of porosity in flow simulatorsassessed, since this is one of the parameters used in the iterative coupling analysis. It was found that both simulators address

the variation of porosity in the same manner. Eq. 1 represents the change in porosity adopted by the simulators in each time

step as a function of pore pressure variations and the compressibility of the rock.

It was observed that the porosity updating performed by IMEX® due to the change in pressure field, does not take into

account the variation in the rock compressibility in each time step, and is linked only to the reference pressure assigned in the

data input, regarding the initial porosity. In the iterative process, the porosity must be updated in the reading files of flow

simulator at each time step, due to the geomechanical response from the stress simulator. Therefore, the reference pressure

should also be updated in accordance with the porosity present in each reference cell. The pseudo-compressibility, coupling

parameter that carries the effect of the volumetric deformation of the rock, must be updated every time step in flow simulator,

replacing the values of compressibility initially assigned.

Each time step of the flow simulation is an iterative process in the two-way coupling scheme, with the need to restart thesimulation every time advance, marked by obtaining convergence of the pore pressure in the cells with the highest pressure

gradient. In both coupling configurations, using ECLIPSE® or IMEX® as reservoir simulator, it was used restart to carry out

the flow simulations, facilitating the resuming process after convergence. However, differences were observed between

software in this respect: the ECLIPSE® restart information is stored in file generated over the simulation, while in IMEX®, the

input data files suffered modification along simulation.

In the pre-existing implementation, using the ECLIPSE® as reservoir simulator, the pore pressure values from the stress

simulator were modified in the actual restart file generated during the simulation, serving as reference pressure for the

calculation of new porosities. In IMEX®, it was observed that the values of pore pressure should be rewritten in the include

file, since, even being resumed simulation, the model values continued being based on reference pressure indicated in this file.

The other values to be updated in IMEX®, from geomechanical simulation in ABAQUS

® – porosity and pseudo-

compressibility – have also been updated in include files, in this case as had been done in ABAQUS-ECLIPSE coupling. Still,

the values of the variables of interest that are sent from IMEX ® to ABAQUS® – oil, water and gas saturations, pressure in the

cell – can be obtained directly from the results files through strings reading functions in the developed C++ code.

The implementation of the iterative process ABAQUS-IMEX was developed and completed after checking the similarities

and differences between the flow simulators employed, performing the necessary changes in the code in terms of the update of

input and output files of ABAQUS® and IMEX

®. The results of the validation process are presented in the sequence.


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ResultsThe results of partial coupling ABAQUS-IMEX were compared with those obtained by the simulation fully coupled

(ABAQUS) and partially coupled ABAQUS-ECLIPSE, in one and two-way. The values for the fully coupled analyses arepresented as reference. Results of reservoir’s compaction, seafloor’s subsidence and pore pressure within the reservoir will be

shown. Besides that, it was evaluated a parametric study varying the production flow rate.

Pore pressure behavior

Figure 6 shows a comparison among the average pore pressure in the reservoir after 1200 days of production, for the threecoupling scenarios. Figure 6 (a) and 6 (b) present the one and two-way coupling results, respectively.

Figure 6: Average pore pressure in the reservoir (a) one-way coupling (b) two-way coupling.

It was observed that the two-way coupling ABAQUS-IMEX resulted in a decrease of average pore pressure in the reservoir

over time much closer to the fully coupled simulation compared to the one- way coupling. It is also notable that the curves

obtained with the ABAQUS-IMEX coupling match ABAQUS-ECLIPSE coupling results, indicating the good quality of

implementing performed.

To analyze the pore pressure behavior in the cells of the model, it was considered a trajectory at the center of the reservoirin the direction I (1 < I


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Figure 8: Pore pressure values along reservoir path [ I, 6, 3 ] (a) one-way coupling (b) two-way coupling.

It is observed that the trajectory of two-way coupling presents very good agreement with the full coupling, indicating that

good behavior observed for the average pressure can be expanded to the remainder of the reservoir. Due to the lower pressuredrop in the one-way coupling, the pore pressure trajectory presents less accurate than results of fully coupled scheme. It was

also observed, as well as the analysis of average pressure, which the curves obtained with the ABAQUS-IMEX coupling

match ABAQUS-ECLIPSE coupling results.

Compaction behavior

The compaction values were obtained from the top vertices of the reservoir’s center block, around the production well, asshown in Figure 9. Figure 10 shows a comparison among reservoir compaction over time for the three coupling scenarios.

Figure 10 (a) and 10 (b) present the one and two-way coupling results, respectively.

Figure 9: Top vertices of the reservoir’s center block for compaction measurement.

Figure 10: Compaction of the reservoir (a) one-way coupling (b) two-way coupling.

(a) (b)

(a) (b)


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The results of the three scenarios evaluated here showed to be satisfactory in terms of compaction, keeping in mind the

expected behavior for the one and two-way coupling compared to fully coupled scheme. The curves obtained with the

ABAQUS-IMEX scheme overlap the ABAQUS-ECLIPSE results.

Subsidence behavior

The seafloor subsidence values were obtained from the top vertices of the overburden’s center block, as shown in Figure

11. Figure 12 shows a comparison among seafloor subsidence over time for the three coupling scenarios. Figure 12 (a) and 12

(b) present the one and two-way coupling results, respectively.

Figure 11: Top vertices of the overburden’s center block for subsidence measurement.

Figure 12: Subsidence of the seafloor (a) one-way coupling (b) two-way coupling.

In terms of seafloor subsidence, the results were shown to be satisfactory when compared to fully coupled scheme, as

observed previously. It was also observed that the curves obtained with ABAQUS-IMEX overlap the ABAQUS-ECLIPSE

results.

Parametric studies

The new version of the coupling code was tested in two different values of production flow rate, in order to carry out a

parametric analysis. Flow rates used were production of 50,000 BBL/day and 25,000 BBL/day. The results observed in theABAQUS-IMEX and ABAQUS-ECLIPSE two-way coupling showed excellent adjustment to those obtained with the full

coupling for each case, when evaluated compaction, subsidence and pore pressure variations. To prove that the difference

between the results obtained from different flow rates studied is only due to their own variation, graphs of compaction and

average pressure normalized by the production flow were built. This comparison is able to measure the quality of

implementation, since the overlap of the normalized curves indicates that no noise programming interfered in obtaining such

results, once the models are running in elasticity.Figure 13 (a) presents the average pore pressure drop normalized by flow rate (psi per BBL/day), and Figure 13 (b) presents

the reservoir compaction normalized by flow rate (m per BBL/day). For both values of flow rate, it was evaluated ABAQUS-

ECLIPSE and ABAQUS-IMEX partially coupled schemes and fully coupled scheme (ABAQUS).

(a) (b)


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Figure 13: Parametric studies. (a) ratio between average pressure drop and different flow rates produced versus time of simulation (b)ratio between reservoir compaction and the different flow rates produced versus time simulation.

Figure 13 (a) shows that the compaction curve normalized by the production flow rate in the models simulated with thepartial coupling ABAQUS-IMEX and ABAQUS-ECLIPSE, fall on a single curve. The curve obtained is very close to the

normalized curves of full coupling, indicating the quality of implementation even for different scenarios of production, since it

has been proven the unique influence of production flow rate in the observed of displacement values. In Figure 13 (b) it is

observed that the curves with average pressure drop normalized by flow rate fall on a single curve for the two-way coupling

ABAQUS-IMEX and ABAQUS-ECLIPSE. Again, they exhibit behavior very close to full coupling.

ConclusionsAs conclusion of this study, after comparison with the results of the ABAQUS-ECLIPSE coupling and fully coupling, the use

of IMEX® as reservoir software, associated with the stress analysis software ABAQUS®, resulted in a combination technically

feasible for analysis of hydromechanical phenomena in one and two-way schemes. The methodology used in this work proved

to be capable of simulating coupled process in reservoir geomechanics, highlighting the importance to consider the effect ofsurrounding rocks during prediction of reservoir production. Furthermore, the use of one-way partial coupling scheme showed

results quite different when compared with the two-way partial coupling scheme, which was developed in a more rigorousway, approaching more accurately to the poroelastic solution.

AcknowledgmentsThe authors would like to thank SIMULIA, Schlumberger and CMG for providing the academic licenses of the softwares

ABAQUS® /CAE, ECLIPSE

® and IMEX

®, respectively. Thanks are also extended to Petrobras for the financial support.

Nomenclatures

porosity

0 initial porosity

Biot’s parameter

Q Biot’s parameter

(a) (b)


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p pore pressure

cr rock compressibility

v bulk volumetric strain

n

v initial bulk volumetric strain of time step

1

nv final bulk volumetric strain of time step

k absolute permeability

f c fluid compressibility

sc solid matrix compressibility

fluid viscosity

G shear modulus

Poisson’s ratio

u nodal displacement

pc pseudo-compressibility

n previous time step

1n current time step

n previous time step

S saturation of the phase

stresses

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