SPE 125760

Answers to Some Questions About the Coupling Between Fluid Flow and Rock Deformation in Oil Reservoirs Nelson Inoue, GTEP PUC-Rio, and Sergio A. B. da Fontoura, DEC PUC-Rio

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE/EAGE Reservoir Characterization and Simulation Conference held in Abu Dhabi, UAE, 19–21 October 2009. 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 been reviewed 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, its officers, 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.

Abstract In recent years operators have shown great interest in the coupling between the multiphase fluid flow and the rock deformation in oil reservoirs and surrounding rocks. Frequently, the geomechanical effects are approximated in a conventional reservoir simulation through only the rock compressibility. This means that the stresses in the reservoir and surrounding rocks may not be in equilibrium with the pore pressure, since the geomechanical behavior is not considered. The ideal solution for this coupled problem is to introduce the geomechanical effects through the stress analysis solution and to implement a scheme that assures that the governing laws of the flow simulation and stress analysis are obeyed simultaneously in each time step. It was developed one methodology to couple a conventional reservoir simulator (ECLIPSE) and a stress analysis program (Abaqus/CAE) that employs the pseudo-compressibility and the porosity as coupling parameters. It was implemented a C++ source code that manages all the work flow of the partial coupling in a fully automated manner. Two schemes of partial coupling were implemented: a) an iteratively two way coupled scheme and b) a one way coupled scheme. In this paper, it was examined the hydromechanical interaction results of these coupling schemes evaluating a problem of the soft reservoir and its surrounding stiff rock that was presented in others works. The results of average pore pressure in the reservoir, pore pressure field, subsidence and compaction were calculated. The results of the simulations using pseudo-compressibility as a coupling parameter in an iteratively two way coupled scheme are showed and compared with a fully coupled scheme. Introduction

The search for explanations for the large subsidence in Ekofisk has triggered the study of the effects of fluid production on the neighbor rock deformations. After that incident, suddenly the normal reservoir flow analysis and reservoir management activity had to incorporate means to take into account the in situ stresses and the stress – strain behavior of the adjacent rocks. Nowadays there is an on going effort amongst researchers to develop computational tools to carry out what has been known as the geomechanical analysis of reservoir fluid production. The suite of problems that may be addressed through a properly conducted geomechanical analysis includes the quantification of subsidence and reservoir compaction, the evaluation of seal integrity and the risk of fault reactivation as well as the prediction of long-term wellbore integrity.

The technical problem to be solved in a reservoir geomechanical analysis can be defined as follows. During fluid production and/or fluid injection, the reservoir rock tends to deform itself. However, these deformations cannot be modeled without obeying fundamental law of stress equilibrium and without honoring the displacement boundary conditions along the contact between the reservoir rock and the adjacent rocks. Unfortunately the so-called conventional reservoir simulators available in the market, and intensively used by the industry, were not developed with these concerns in mind. Ideally, the flow problem must be solved jointly with the stress-strain problem and therefore, the reservoir geomechanical analysis becomes a typical fully coupled hydro-mechanical problem. In general, fully coupled hydromechanical problems are solved numerically through discretization techniques such as finite element and the stress and flow problems are solved at the same time. The poroelastic problem as defined by Biot1 is at the basis of this formulation.

Gutierrez and Lewis2 present a fully coupled analysis in order to evaluate the importance of geomechanics in reservoir simulation. The formulation of the governing equations of a two phase fluid flow in a porous media and the formulation of the Biot’s theory for multiphase flow in a deformable porous media were presented in detail. The Biot`s equations for three-phase fluid flow (3D black-oil simulator) in deformable porous media were implemented in the finite element code CORES (Coupled Reservoir Simulator). Even though the fully coupled procedure is the correct one, it has a very high computational cost and that precludes its use in routine reservoir studies.

2 SPE 125760

An alternative procedure for the fully coupled scheme is to use schemes called partial (or explicit) or hybrid couplings. These types of coupling join different problems (flow and stress problems), different numerical methods (finite element method and finite difference method), distinct softwares and may join different domain discretizations (finite element mesh and finite difference grid). The advantage of these schemes is that the best software of each area may be used in the coupling procedure. However, it is necessary a follow a robust communication methodology between software in order to guarantee the equilibrium and convergence of the final solution of the two problems. The work of Settari and Mourits3, Mainguy and Longuemare4, Dean at al5 and Samier and De Gennaro6 are representative of the numerous papers published on this topic.

Settari and Mourits2 presented one of the first works about partial coupling of a commercial reservoir simulator (DRS-STEAM) and a 3-D stress code (FEM3D). The expression of the reservoir porosity (related to a fixed bulk volume) was derived using a stress analysis solution associated with the concept of true porosity. Although the deduction of the formulation is different from those frequently found in the literature, the deduced porosity expression is the same as the Biot’s theory of the poroelasticity. An iterative algorithm was proposed to the partially coupled scheme and the reservoir porosity calculated using the pore pressure variation and the mean normal stress variation is used as coupling parameter in the reservoir simulator.

Mainguy and Longuemare4 presented three formulations to correct the porosity equation that is used in the conventional reservoir simulation. The first equation is written in terms of pore pressure and volumetric strain, the second in terms of pore pressure and porosity and the third in terms of pore pressure and mean total stress. The porosity correction was deduced based on Biot’s theory of the poroelasticity, therefore the corrected porosity is the same as the one used by Settari and Mourits2 and others authors to consider the geomechanical effects in the reservoir simulation.

Dean et al.5 showed results of three methods for hydromechanical coupling, explicit coupling, iterative coupling and full coupling. For the explicit coupling, the reservoir simulator performs calculations for multiphase porous flow in each time step and performs geomechanical calculations for the displacements in selected time steps. In the trimesters when geomechanical calculations are not performed, the reservoir simulator calculates the pore volume through pore pressure change and rock compressibility. The three methods of coupling were implemented in the program ACRES (ARCOS’s Comprehensive Reservoir Simulator).

Samier and Gennaro6 proposed a new iterative partial coupling in which the reservoir simulator runs until the end of simulation and the pore pressures of N report times (times when the pore pressure and saturation data are printed in a output file) are used as pressure loads in the stress analysis simulation. Pore volume multipliers of N report times are calculated from the strain state and are used by reservoir simulation for a new iteration. ECLIPSE was the reservoir simulator used and the stress analysis was carried out by Abaqus.

This paper discusses the fundamentals of partial coupling schemes as replacements for the fully coupled procedure. In order to understand the differences and similarities between the schemes and the requirements for securing a good approximation of the correct solution we review the formulation of the flow equations as used in the conventional reservoir simulation and in the fully coupled scheme. A pseudo-compressibility calculated from response of the stress analysis is employed as coupling parameter, together with the porosity, also calculated from stress analysis. This combination of pseudo-compressibility and porosity guarantees a very robust coupling procedure and that the iteratively two-way coupled solution is very representative of the fully coupled solution. A simple example is used to compare the efficiency of the partial coupling scheme as compared to the fully coupled solution. We also indicate the influence of the discretization of the geometry on the final results. The widely used one-way coupled (Capasso and Mantica7 and Bostrom and Skomedal8) showed to produce results that may not be satisfactory from the practical point of view. Theoretical Formulation

Firstly we present the definition of the different coupling schemes for solving a geomechanical reservoir analysis. Types of coupling The two main schemes of the partial coupling between a conventional reservoir simulator and a stress analysis program found in the literature are the two-way coupling and the one-way coupling. Two-way coupling:

In this scheme, the flow variables (pore pressure and saturation of the phases) and the stress variables (displacement field, strain state and stress state) are calculated separately and sequentially, through a conventional reservoir simulator and through a stress analysis program, respectively. The coupling parameters are exchanged between simulators at each time step until convergence is reached. Pore pressures or stresses are used to verify the convergence of the solution during iterations. Figure 1 (a) illustrates the flow chart of the two-way coupling. One-way coupling:

This scheme can be considered as a special case of the two-way coupling. The conventional reservoir simulator sends the information (pore pressure and saturation) to the stress analysis program but the calculated results by stress analysis program are not sent back to the conventional reservoir simulator. In this scheme the geomechanical effect does not affect the responses calculated by reservoir simulator. Figure 1 (b) illustrates the flow chart of the one-way coupling.

SPE 125760 3

Figure 1: Types of partial coupling: (a) Two-way Coupling and (b) One-way Coupling.

Full Coupling

In this scheme the variables of flow and geomechanics are calculated simultaneously through a system of equations with pore pressure, saturation and displacement as unknowns, assuring an internal consistency. The method is also called implicit coupling because the whole system has a single discretization and is solved simultaneously9. The governing equations

Next we present the flow equations as seen by conventional reservoir simulator and by the fully coupled solution.

Flow Equation for the Conventional Reservoir Simulation The flow equation is obtained by combining the mass conservation equation with Darcy’s law. The law of mass

conservation is a material-balance equation written for a component in a control volume. In petroleum reservoirs, a porous medium can contain one, two and three fluid phases. For the sake of simplicity, the equations are developed considering single-phase fluid10 and figure 2 shows the finite-control volume.

Figure 2: Control volume for 1 D flow in rectangular coordinates (Ertekin et al.11).

The mass-conservation equation for 3D rectangular flow is expressed as equation (1). Darcy’s law is an empirical relationship between fluid flow rate through a porous media and potential gradient. For three-dimensional flow, the differential form of the Darcy’s law and of the potential gradient are written respectively as equations (2) and (3).

4 SPE 125760

lsclc

b

l

lzz

l

lyy

l

lxx q

BtVz

BuA

xx

Bu

Ay

xBuA

x−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=Δ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−Δ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−Δ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−φ

α (1)

Φ∇−=rr

μβ

ku c (2)

Zp ∇−∇=Φ∇rrr

γ (3)

Introducing the potential gradient equation into the Darcy’s law equation, results in:

( )Zpku c ∇−∇∇−=rrrr

γμ

β (4)

Finally, introducing the Darcy’s law equation into the mass-conservation equation yields the flow equation for single-phase flow, equation (5).

lsclc

b

ll

z

l

z

ll

y

l

y

ll

x

l

x qBt

VzZ

zp

BAk

zyZ

yp

BAk

yxZ

xp

BAk

x−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

∂∂ φ

αλ

μλ

μλ

μ (5)

Finite-Difference Approximation of the Spatial Derivative The flow equation (Eq. 5) is a second order, partial-differential equation (PDE) in space and first order in time. Frequently,

this equation cannot be solved analytically10 (exactly) due to its complexity. The solution of the PDE is obtained through numerical methods. The finite difference method (FDM) is the numerical method more used in the petroleum industry to obtain the numerical solution.

The finite difference equation in multiple dimensions is obtained applying the central-difference approximation two times to Eq. 5, resulting in

( ) ( ) ( ) ( )

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=+−−−+

−−−+−−−

+−

++++

+−

++++

+−

++++

−+

−+−+

lblsc

nkji

nkjilz

nkji

nkjilz

nkji

nkjily

nkji

nkjily

nkji

nkjilx

nkji

nkjilx

BtVqppTppT

ppTppTppTppT

ikjikji

kjikjikjikji

φ11,,

1,,

1,,

11,,

1,1,

1,,

1,,

1,1,

1,,1

1,,

1,,

1,,1

21,,21,,

,21,,21,,,21,,21

(6)

where,

kjill

xxlx xB

kATkji

,,21,,21

±⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

=± μ

, kjill

yyly yB

kAT

kji

,21,,21,

±⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

=± μ

and 21,,

21,,

±⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

=±

kjill

zzlz zB

kATkji μ

(7)

Finite-Difference Approximation of the Time Derivative The development of the right term of the Eq. 6 must be done carefully to guarantee the material balance of the problem.

Considering the formation volume factor (FVF) and porosity varying with the pore pressure we have that:

( )[ ]nnl

nln

l ppcBB

−+=

+

+

11

1 (8)

( )[ ]nnnn ppc −+= ++ 11 1 φφφ (9)

The finite difference approximation to the time derivative may be written as:

( )ni

ni

nl

lb pp

tBtV −

ΔΓ

≈⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ +

+1

1φ (10)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=Γ

+

+

nl

ln

nl

n

bnl B

cB

cV φφ φ

11 (11)

The final form of the finite difference equation for the conventional reservoir simulation is represented by Eq. (12).

SPE 125760 5

( ) ( ) ( ) ( )

( ) ( ) ( )ni

nin

l

ln

nl

nb

lscn

kjin

kjilzn

kjin

kjilz

nkji

nkjily

nkji

nkjily

nkji

nkjilx

nkji

nkjilx

ppB

cB

ct

VqppTppT

ppTppTppTppT

ikjikji

kjikjikjikji

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

Δ=+−−−+

−−−+−−−

+

+

+−

++++

+−

++++

+−

++++

−+

−+−+

11

11,,

1,,

1,,

11,,

1,1,

1,,

1,,

1,1,

1,,1

1,,

1,,

1,,1

21,,21,,

,21,,21,,,21,,21

φφ φ (12)

Flow Equation for the Full Coupling Scheme (Poroelasticity) Basically, the difference between the flow equation of the fully coupled scheme and conventional reservoir simulation is

the approach of the porosity equation. In the conventional reservoir simulation, the porosity is related to pore pressure through the rock compressibility using a linear relation (Eq. 9). Considering an isotropic linear elastic material in a fully coupled scheme, the approach of the porosity equation is composed of four components that contribute to the fluid accumulation term (Zienkiewicz et al.l1). 1) Component due to volumetric strain in the solid: vεΔ− ; 2) Component due to compression of the solid by pore pressure: ( ) SKpΔ−φ1 ; 3) Component due to compression of the solid by effective stress: ( )SvSD KpKK Δ+Δ− ε ; 4) Component due to volume change of the pore fluid: fKpΔφ ; The porosity equation is derived from the sum of the four components above:

( ) ( )nnnv

nv

nn ppQ

−+−+= +++ 111 1εεαφφ (13)

where Q is related to SK and fK through the expression

( )nS

nf

S

n

f

n

ccKKQ

φαφφαφ

−+=−

+=1 (14)

The parameter α is expressed in terms of DK and SK (Zienkiewicz et al.4).

S

D

KK

−=1α (15)

where: ( )ν2139

T

−==

EKD

Cmm is the bulk modulus associated with the tensor C .

Introducing the porosity equation (Eq. 13) into the accumulation term of the flow equation (first term of the right-hand side

of the Eq. 5), similar what was done in the Eq. 10, results in:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛++−

Δ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂ +

+

+

+

ni

nin

l

ln

nl

nv

nvn

l

b

lb pp

Bc

QBBtV

BtV 1

11

1

1 φεεαφ (16)

The final form of the flow equation, expressed in terms of finite differences for the fully coupled scheme is given by Eq. (17).

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛++−

Δ=+−−−+

−−−+−−−

+

+

+

+

+−

++++

+−

++++

+−

++++

−+

−+−+

ni

nin

l

ln

nl

nv

nvn

l

blsc

nkji

nkjilz

nkji

nkjilz

nkji

nkjily

nkji

nkjily

nkji

nkjilx

nkji

nkjilx

ppB

cQBBt

VqppTppT

ppTppTppTppT

ikjikji

kjikjikjikji

11

11

11,,

1,,

1,,

11,,

1,1,

1,,

1,,

1,1,

1,,1

1,,

1,,

1,,1

121,,21,,

,21,,21,,,21,,21

φεεα (17)

Stress Analysis Equations Based on the second Newton’s law, the time rate of change of the linear momentum of a deforming material region

embedded in Ω is equal to the sum of the forces applied to that region. These forces can be decomposed into a body force b and surface force t . The former is a volume force (gravity, for example), while the latter are exerted across the surface Ω∂ surrounding the material, see Eq. (18).

dVdSdVt ∫∫∫ ΩΩ∂Ω

+=∂∂ btv ρρ (18)

6 SPE 125760

Using Cauchy principle and divergence theorem in the Eq. 18, the equilibrium equation is obtained:

ub &&ρρσ =+⋅∇ (19)

In the static case, in which no quantities vary with time, the total linear momentum of the system remains constant resulting in the conservation of the linear momentum, the Eq. 19 reduces to

0=+⋅∇ bσ ρ (20)

The strain tensor ε can be written in terms of the displacement vector u as:

( )[ ]T

21 uu ∇+∇=ε (21)

The Terzaghi’s effective stress principle12,13 is written as:

pm-σσ α=' (22)

For multiple dimensions, the general stress-strain constitutive relationship is written as:

εCσ :'= (23)

Finite Element Discretization of the Stress Analysis Equations The boundary-value problem (BVP) of a linear elasticity problem is given by equilibrium equation, strain-displacement

relation, stress-strain relation and the boundary conditions. The continuum has a domain Ω and is enclosed by a boundary Γ (Figure 3) that may be divided into a boundary where displacements are prescribed and where surface forces are prescribed. Conditions on the boundary are written as: i) Essential (Dirichlet) boundary conditions:

uu = on uΓ (24)

ii) Natural (Neumann) boundary conditions:

nσt ⋅= on tΓ (25)

where: tu ΓΓ=Γ U .

Figure 3: Boundary conditions on Γ .

The partial differential equation form of the problem (Eq. 20), along with its boundary conditions, constitutes the strong form of the boundary-value problem. The dependent field variables (displacement u ) are strong continuity in strong form and the exact solution is very difficult to obtain for practical engineering problems. Numerical methods are used to achieve an approximate solution of the BVP, in stress analysis is widely employed the Finite Element Method (FEM). The effect of the flow problem is considered in the stress analysis introducing the Terzaghi’s effective stress principle (Eq. 22) into the Eq. 20 and the weak form of the BVP is written in the matrix form as (Zienkiewicz et al.11)

0u =−+ fQpKu (26)

where: Ω= ∫Ω

dTDBBK , Ω= ∫Ω

dpT mNBQ α , ( ) Ω= ∫Ω

dT

uu bNf ρ and uSNB = . For three dimensional problems, the matrix S

is defined as

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂∂∂

∂∂∂∂∂∂

∂∂∂∂

=

xzyz

xyz

yx

00

000

0000

S (27)

The variation of the displacement field due to variation of pore pressure field may be calculated using the Eq. 26. However the term Qp must be moved from the left to the right side of the equation. Rewriting the equation, results in

QpfKu −=− u (28)

SPE 125760 7

Flow Equation for the partially Coupled Scheme Figure 4 shows the assembly of the partial coupling equations, the flow equation (Eq. 12) is obtained from the conventional

reservoir simulation (finite difference approach) and the stress analysis equation (finite element approach) in terms of displacement and pore pressure is the same obtained from the fully coupled scheme (Eq. 26).

Figure 4: Assembly of the partial coupling equations.

The key issue of the partial coupling scheme is to make sure that during the solution of the flow problem using the

convetional reservoir simulator, the equation to be solved be as close as possible to the flow equation of the fully coupled scheme. Therefore, it is necessary to employ an external artifice to reformulate the flow equation. This may be achieved using the porosity and a pseudo-compressibility in each iteration that are both calculated using information from the stress analysis (from finite element program). The porosity is calculated through Eq. 13 and the pseudo-compressibility according with the Eq. 29.

( )ni

ni

o

nv

nv

pseudo ppc

−−

=+

+

1

1

φεε (29)

Partial Coupling using a Staggered Procedure

The partial coupling between the stress analysis program and the conventional reservoir simulator is implemented using a staggered procedure (Huang and Zienkiewicz14, and Radu and Charlier15). For any given time step, initially the flow simulation is carried out and a set of pressure and saturation is obtained. These two values are transferred to the finite element simulator and values of strain, stresses and displacements are obtained. Then, a new set of porosity and compressibilities is fed back in to the flow simulator. A new set of values of pressure and saturation are obtained. These are fed back to the stress simulator and a convergence criterion is evaluated. If this criterion is obeyed we proceed to a new time step. Figure 5 indicates the full procedure. Figure 6 illustrates a more detailed flowchart of one time step of the staggered procedure. In the beginning of the time step, the reservoir simulator (ECLIPSE, IMEX or any simulator) is called to resolve the flow equations, providing as result the pore pressure field and the saturation field. The variation of the pore pressure in the time step is used to calculate the nodal forces through a Finite Element Code. The nodal forces are written in the input file (*.inp) of the stress analysis (Abaqus or any other) program, using the keyword called CLOAD. The finite element program is called to resolve the stress problem, providing the displacements in each nodal point of the mesh and the stress state, evaluated in the integration points. The coupling program calculates the pseudo-compressibility field and the porosity field from the strain state. If an iterative scheme is used, the porosity field and the pseudo-compressibility field are written in the input file (*.data) of the flow simulator program and are employed to restart the reservoir simulation in the same time step, until reaching convergence. The keywords PORO and ROCK are used to introduce the pseudo-compressibility field and the porosity field, respectively. The convergence criteria adopted is the difference of average pore pressure between two consecutive iterations.

8 SPE 125760

Figure 5: A staggered procedure between the stress program and flow simulator

Figure 6: Flowchart of partial coupling between the programs: stress analysis and flow simulator.

The unknown variables of the flow problem (pore pressure and saturation field) in a staggered procedure are calculated

using the pseudo-compressibility and the porosity that are evaluated in the last iteration such as shown in the figure 7.

SPE 125760 9

Figure 7: Convergence behavior of a staggered procedure.

Case study: validation problem

This validation problem was initially analyzed by Gutierrez and Lewis2 and afterwards it was presented by Dean et al.5 and Samier and Gennaro6. The results obtained with the partial coupling program were compared with the results of the finite element program Abaqus that has a single phase poroelasticity model (tridimensional). The geometry of the problem consists of a soft reservoir and surrounding rocks (overburden, sideburden and underburden) as shown in the Figure 8. The finite difference grid and finite element meshes are made coincident. Therefore, the displacements field and pore pressures are calculated in the nodal points of the mesh in the fully coupled scheme while that in the partially coupled scheme the pore pressures are calculated at the center of the cells and displacements are calculated at the nodes of the elements.

The influence of the discretization of the model was analyzed using two degree of discretization for the same geometry. The first model uses the same discretization of papers mentioned above, a grid of 11x11x5 cells and a mesh of 21x21x12 elements, such as shown in the Figure 9 (a) that was called of coarse grid. The second model is more refined than the first one, with a grid of 21x21x12 and a mesh of 33x33x17 elements, such as shown in the Figure 9 (b) that was called of refined grid. The property values used in the fluid flow simulation and in the geomechanical analysis are given in the Tables 1(a) and 1(b), respectively.

Properties: Values: Properties: Values: Formation volume factor at 0.1013 MPa 1 Young’s modulus (surrounding rock) 6894.8 MPa Viscosity 1 cp Young’s modulus (reservoir) 68.95 MPa Fluid density at 0.1013 MPa 103 kg/m3 Poisson’s ratio 0.25 Fluid compressibility 4.35 10-4 MPa-1 (b) Horizontal permeability 9.86 10-14 m2 Vertical permeability 9.86 10-15 m2 Porosity 0.25

(a)

Table1: (a) Fluid flow properties and (b) Geomechanical properties. The vertical stress gradient in the z-direction is 22.62 MPa/m with initial vertical stress 0 MPa at the surface and the initial

horizontal stresses are equal to half of the vertical stresses. The essential boundary conditions at the side and bottom of the finite element mesh have zero normal displacement. A vertical well, with a wellbore radius of 7.62 x 10-2 m, producing at a rate of 9.2 x 10-2 m3/s, is completed in all layers in the center of the reservoir. The analysis was performed for a time period of 2000 days.

10 SPE 125760

Figure 8: Geometry of problem analyzed (units in meters).

(a)

(b)

Figure 9: (a) Coarse grid with a mesh 21x21x12 elements and (b) Refined grid with a mesh 33x33x17 elements.

Figures 10, 11, 12 and 13 show the results of pore pressure, average pore pressure, compaction and subsidence, respectively, considering a coarse grid (a) and a refined grid (b). In all figures, the results evaluated with an iteratively two-way coupled scheme are compared with the results of a fully coupled scheme (Abaqus). Comparing the results of coarse grid model and refined grid model, it is possible to verify that the two-way coupling results obtained with the refined grid are veru close to the full coupling results. However, the time to carry out one stress analysis run increased from 7 sec to 47 sec. When the precision of the solution is less important as is the case of a pre-analysis, it may be more advantageous to use a coarse grid, because the results obtained give a good order of magnitude and simulation time may be reduced. It is important to have in mind that much of the simulation time is spent in the stress analysis.

20

22

24

26

28

30

32

6000 7000 8000 9000 10000 11000 12000 13000X Direction (m)

Por

e P

ress

ure

(MP

a)

Fully Coupled Scheme

Two Way Coupled Scheme

100 days

500 days

1200 days

2000 days

(a)

20

22

24

26

28

30

32

6000 7000 8000 9000 10000 11000 12000 13000X Direction (m)

Por

e Pr

essu

re (M

Pa)

Fully Coupled Scheme

Two Way Coupled Scheme

100 days

500 days

1200 days

2000 days

(b)

Figure 10: Pore pressure variation along the X direction - (a) coarse grid and (b) refined grid.

SPE 125760 11

27

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0 400 800 1200 1600 2000

Time (days)

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rage

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Pa)

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Fully Coupled Scheme

(a)

27

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Pa)

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(b)

Figure11: Average Pore pressure along the time - (a) coarse grid and (b) refined grid.

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00 400 800 1200 1600 2000

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Com

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(m)

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(a)

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(b)

Figure 12: Compaction along the time - (a) coarse grid and (b) refined grid.

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Com

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(m)

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(a)

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(m)

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(b)

Figure 13: Subsidence along the time - (a) coarse grid and (b) refined grid.

Figures 14 (a), 14 (b), 15 (a) and 15 (b) show the results of pore pressure, average pore pressure, compaction and subsidence, respectively, obtained with the coarse grid (mesh) using a one-way coupled scheme and an iteratively two-way coupled scheme. The one-way coupling scheme is used more often in practice; however, great care should be exercized, because in this scheme, no variable is transfered from stress analysis program to the conventional reservoir simulator. Therefore, the mechanical behavior evaluated from the stress analysis does not affect the solution of the flow problem, only the stress analysis is performed from pore pressure variation. In the one-way coupling scheme, the reservoir is not affected by surrounding rocks and the pore volume variation is function only of the pore pressure change and rock compressibility. In this condition, the boundaries move freely, resulting in a more deformable reservoir. The pore pressure decreases more in a two-way coupled scheme, because the accumulation term supplies less energy to system (more rigid) whereas in a one-way coupled scheme, the system is more deformable, supplying more energy through the accumulation term. The one-way coupled scheme must be used when the reservoir rock is considerably rigid, i.e, the reservoir is less deformable for a given pressure

12 SPE 125760

variation. Thus, the porosity equation of the accumulation term that differentiates between the formulations has little effect on the problem.

20

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e Pr

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Pa)

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(a)

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(b)

Figure 14: (a) Pore pressure variation along the X direction and (b) Average Pore pressure versus time.

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Figure 15: (a) Compaction and (b) Subsidence versus time.

Conclusions Some topics related to the modeling of geomechanical reservoir analyses were discussed herein. Initially, the partial

coupling scheme was used as a substitute for the fully coupled solution. The partial coupling is only effective if the coupling parameters are well defined and transferred conveniently between the reservoir simulator and the stress analysis program. We defined a pseudo-compressibility that together with porosity are the elements that play the key role for the coupling between simulators. The pseudo-compressibility was defined to make sure that the flow equation solved in the reservoir simulator is the same as the flow equation solved during the fully coupled scheme. The tests carried out using the methodology proposed demonstrated the convergence of the iterative at each time step and the accuracy of the scheme. The results generated using the partial coupling are almost identical to the ones obtained using the fully coupled scheme.

The use of the so-called one-way coupling may conduct to wrong results. However, in cases of low compressibility rock rocks this coupling method may be used since accumulation term of the flow equation has negligible effect on the problem. The finite element mesh discretization in a coupled simulation is a question that needs special attention since much of the simulation time is spent in the stress analysis. High performance computional scheme must be used to solve system of linear equations in order to reduce the time required to perform geomechanical analysis of reservoirs in practical problems. Acknowledgments The authors would like to thank to SIMULIA and Schlumberger for providing the academic licenses of the softwares Abaqus/CAE and ECLIPSE, respectively. Nomenclature c component l ,, wo or g (components)

xA , yA , zA cross-sectional areas normal to x , y and z , respectively

lxu , lyu , lzu volumetric velocity components in x , y and z direction, respectively

SPE 125760 13

xΔ , yΔ , zΔ lengths along x , y and z direction, respectively φ porosity

lγ glρ= γ specific weight ρ density g acceleration of gravity

xk , yk , zk permeabilities in the direction of x , y and z axis, respectively

lc compressibility of phase l

φc porous compressibility 1+n current time step

n previous time step lΓ accumulation term coefficient for phase l

1+nvε final bulk volumetric strain of time step nvε initial bulk volumetric strain of time step

Q Biot`s parameter SK matrix bulk modulus

fK fluid bulk modulus α Biot`s parameter

DK drained bulk modulus m identity matrix E Young’s modulus ν Poisson’s ratio C drained tangent stiffness tensor u displacement vector u&& acceleration vector σ total stress tensor

'σ effective stress tensor pN shape function of pore pressure field uN shape function of displacement field

S strain matrix kjilxT

,,21±,

kjilyT,21, ±,

21,, ±kjilzT Transmissibilities of the porous media Reference 1 Biot, M. A.: “General Theory of Three-Dimensional Consolidation”, J. Appl. Phys., Vol. 12, 155-164 (1940). 2 Gutierrez ,M. and Lewis , R. W.: “The Role of Geomechanics in Reservoir Simulation”, paper SPE 47392 (1998). 3 Settari, A. and Mourits, M.: “Coupling of Geomechanics and Reservoir Simulation Models”, Computer Methods and Advances in

Geomechanics, Siriwardane & Zanan (Eds), Balkema , Rotterdam, 2151-2158 (1994). 4 Mainguy, M. and Longuemare, P.: “Coupling Fluid Flow and Rock Mechanics: Formulations of the Partial Coupling Between

Reservoir and Geomechanical Simulators”, Oil & Gas Science and Technology, Vol. 57, No.4, 355-367 (2002). 5 Dean, R. H., Gai, X., Stone, C. M. and Mikoff , S.: “A Comparison of Techniques for Coupling Porous Flow and Geomechanics”,

paper SPE 79709 (2006). 6 Samier, P. and De Gennaro, S.: “Practical Interactive Coupling of Geomechanics with Reservoir Simulation”, paper SPE 106188

(2007). 7 Capasso, G. and Mantica, S.: “Numerical Simulation of Compaction and Subsidence Using ABAQUS”, ABAQUS User’s Conference

(2006). 8 Bostrom, B. and Skomedal, E.: “Reservoir Geomechanics with ABAQUS”, ABAQUS User’s Conference (2004). 9 Settari, A. and Nghiem, L.: “New Iterative Coupling Between a Reservoir Simulator and a Geomechanics Module”, paper SPE 88989

(2004). 10 Ertekin, T., Anou-Kassem, J. H. and King, G. R..: “Basic Applied Reservoir Simulation”, SPE, Richardson, Texas (2001). 11 Zienkiewicz, O.C., Chan, A. H. C., Pastor, M., Schrefler, B. A. and Shiomi, T.: “Computational Geomechanics with Special Reference

to Earthquake Engineering”, John Wiley and Sons (1999). 12 Terzaghi, K. von and Rendulic, L.: “Die Wirksame Flächenporosität des Betons”, Z. öst. Ing.-u. Archit Ver., 86, No. 1/2, 1-9 (1934). 13 Terzagh, K. von: “The Shearing Resistance of Saturated Soils”, Proc. 1st ICSMFE, 1, 54-56 (1936). 14 Huang, M. and Zienkiewicz, O. C.: “New Unconditionally Stable Staggered Solution Procedures for Coupled Soil-Pore Fluid Dynamic

Problems”, Int. J. Numec. Meth. Engng. 43, 1029-1052 (1998). 15 Radu, J.-P. and Charlier, R.: “Modelling of the Hydromechanical Coupling for non Linear Problems: Fully Coupled and Staggered

Approaches” Computer Methods and Advances in Geomechanics, Balkema, Rotterdam, 783-788 (1994).