Double Porosity Finite Element Method for Borehole Modeling

8/6/2019 Double Porosity Finite Element Method for Borehole Modeling

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Rock Mech. Rock Engng. (2005) 38 (3), 217242

DOI 10.1007/s00603-005-0052-9

Double Porosity Finite Element Methodfor Borehole Modeling

By

J. Zhang1 and J.-C. Roegiers2

1 CIRES, The University of Colorado, Boulder, CO, U.S.A.2 Mewbourne School of Petroleum and Geological Engineering,

The University of Oklahoma, Norman, OK, U.S.A.

Received June 21, 2004; accepted January 24, 2005Published online March 15, 2005 # Springer-Verlag 2005

Summary

This paper considers the mechanical and hydraulic response around an arbitrary oriented boreholedrilled in a naturally fractured formation. The formation is treated as a double porosity mediumconsisting of the primary rock matrix as well as the fractured systems, which are each distinctlydifferent in porosity and permeability. The poro-mechanical formulations that couple matrix andfracture deformations as well as fluid flow aspects are presented. A double porosity and doublepermeability finite element solution for any directional borehole drilled in the fractured porousmedium is given. Compared with conventional single-porosity analyses, the proposed double-porosity solution has a larger pore pressure in the matrix and a smaller tensile stress in thenear-wellbore region. The effects of time, fracture, mud weight, and borehole inclination in thedouble-porosity solution are parametrically studied to develop a better understanding of physicalcharacteristics governing borehole problems.

Keywords: Double porosity, finite element method, inclined borehole, poro-mechanics, fractured

porous media.

1. Introduction

Field observations have revealed a need for a better and more comprehensive method

to model borehole stability, since the exploration and production of hydrocarbons now

occur in ever more difficult geological settings (Maury and Zurdo, 1996; Willson

et al., 1999); such as in naturally fractured media, in shaley formations, and at great

depths. In fractured porous formations, borehole instability has been of major concern

due to potential rock movements along fractures at the borehole wall. In the case of

shaley formations, not only does the state of stress play a role, but also the properties

and interactions between the shale and the drilling fluid. At great depths, the rock can


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become more ductile and behave as an elastoplastic material. In all those cases, the

design approach via conventional elasticity is not suitable.

Borehole modeling using linear elasticity was first used to predict the stability of a

vertical borehole subjected to a non-hydrostatic far-field stress and constant borehole

fluid pressure (Hubbert and Willis, 1957). Then, Goodman (1966) analyzed stress

distributions around circular openings and described weak plane effects using a

numerical method. The elastoplastic numerical analysis for underground openings

was given by Kovari (1977). Early analyses for a borehole of arbitrary trajectory in

linear elastic media were presented (Fairhurst, 1968; Bradley, 1979). Later on, a semi-

analytical model that took into account the influence of rock anisotropy on inclined

borehole stability was developed (Aadny, 1987). When the formation is saturated, the

effective stress field around a borehole is strongly modified by the pore pressure,

which is an important factor affecting borehole stability. Assuming a vertical borehole

with a plane strain deformation geometry, a poroelastic solution was developed in

the case of a non-hydrostatic stress field (Detournay and Cheng, 1988). For inclinedboreholes drilled in isotropic media a poroelastic analytical solution was derived

earlier by applying the generalized plane strain concept (Cui et al., 1997a). The

stability of boreholes in saturated rocks was also studied by using the finite element

method (Aoki et al., 1993; Charlez, 1999). A pseudo three-dimensional finite element

program for coupling anisotropic, nonlinear poroelasticity was formulated to simulate

inclined wellbore problem in porous media (Cui et al., 1997b).

However, most petroleum reservoirs are situated in fractured porous formations

(Pruess and Tsang, 1990). In fact, in order to drain the reservoir, inclined and hor-

izontal wells must be drilled in such fractured porous formations the pay zones.

Therefore, traditional borehole modeling methods cannot completely satisfy thisrequirement; and a new scheme needs to be developed to model boreholes drilled

in the fractured porous medium. This study considers such a medium as the combined

effects of solid rock deformation and fluid flow in both the rock matrix and fractures

by using a double porosity and double permeability geomechanical approach.

2. Double Porosity Poro-Mechanical Model

In the double porosity approach, the naturally fractured reservoir is classified as a

system containing two different physical domains. The primary rock matrix contains

a large volume of fluid but has a rather low permeability; while the fractures constitute asmall volume but have the ability to transmit a large portion of the total flow through

the reservoir. In early approaches (Barenblatt et al., 1960; Warren and Root, 1963;

Kazemi, 1969), the reservoir was considered as two overlapping continua: matrix and

fractures. Flow between the matrix and the fractures was accounted for by the introduc-

tion of source functions. However, the effects of stresses and deformations on both the

matrix and fractures were not considered. Then, a coupled flow-deformation approach

within double porosity poroelastic media was presented (Aifantis, 1997; Wilson and

Aifantis, 1982), and a constitutive model to define response of a fissured medium was

given (Elsworth and Bai, 1992). Later, a series of papers were published to study the

fluid flow and solute transport in multiple porosity media, such as Bai et al., 1993; Bai

and Roegiers, 1997; Bai et al., 1999a; Moutsopoulos et al., 2001 and Alboin et al., 2002.

218 J. Zhang and J.-C. Roegiers


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In this study, the naturally fractured reservoir is assumed to be an ideal double

porosity, double permeability model as shown in Fig. 1. In this model the system

including matrix blocks and fractures is considered as a continuous medium.The relationship between total stress (ij) and effective stress (

0ij) was given by

Terzaghi (1943) and Biot (1941). For double porosity media, the effective stresses can

be expressed as:

maij 0maij mapmaij;

frij 0

frij frpfrij;1

where subscripts ma and fr represent matrix and fractures, respectively; p is the

pressure; is the effective stress coefficient; and, ij is the Kronecker delta.For separate but overlapping porous media, the linear elastic constitutive relation-

ships among the effective stresses and strains are defined as (Elsworth and Bai, 1992;

Bai et al., 1999b):

"makl Cmaijkl0maij;

"frkl Cfrijkl0

frij;2

where Cmaijkl and Cfrijkl are the compliance tensors for the rock matrix (subscript ma)

and fracture systems (subscript fr), respectively. The detailed expressions for the

tensors are listed in Appendix I.

The total strain due to the elastic deformation in each of the systems is given by:

"kl "makl "frkl 3

Substituting the double effective law, i.e., Eq. (1) into Eq. (2) while combining

with Eq. (3) and noting maij frij ij, one obtains:

ij Dmfijkl"kl Cmaijklmapmaij Cfrijklfrpfrij: 4

The combined elasticity matrix Dmfijkl is defined explicitly in a three-dimensional

geometry for an isotropic medium as:

Dmfijkl Cmaijkl Cfrijkl1; 5

where the detailed expression for Dmfijkl is given in Appendix I.

In general, the stress-strain behavior of rocks is non-linear, especially in the case

where external loads exceed the elastic strength of material and the material becomes

Fig. 1. Naturally fractured and ideal double porosity reservoirs

Double Porosity Finite Element Method 219


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plastic. The elastoplastic stress-strain can be described by:

s Depmfijkl; 6

where Depmfijkl Dmfijkl D

pmfijkl, D

pmfijkl is the plastic modulus tensor.

The total strain increment is assumed to be the sum of the elastic and plasticcomponents; i.e.

e p 7

where , e and p are the total, elastic and plastic strain increments, respec-

tively. For more details about perfect plasticity solution, refer to Zhang (2002).

For the separate but overlapping model, the double effective stress law (Eq. (1))

needs to be considered, and the combined elasticity matrix Dmfijkl as well as other

related elastic constants need to be introduced. Then, the governing equations for solid

deformation and fluid phase in the dual-porosity poromechanical formation can be

written as (Zhang, 2002):Gmfui;jj mf Gmfuk;ki maDmfijklCmaijklpma;i frDmfijklCfrijklpfr;i 0;

kma

pma;kk maDmfijklCmaijkl

@"kk@t

ma@pma

@t !pfrpma qma;

kfr

pfr;kk frDmfijklCfrijkl

@"kk@t

fr@pfr@t

!pfrpma qfr;

8

where is the relative compressibility representing the lumped deformability of thefluid and the solid; u is the solid displacement; "kk is the total body strain; ! is thetransfer coefficient (Warren and Root, 1963); s is the fracture spacing; qma and qfrare the applied fluid flux; mf is Lamees constant for the combined double porositymedium; and, mf Dmfijkl=1 1 2.

3. Finite Element Discretization of the Poroelastic Solution

The first step in solving the coupled problem of fluid flow and solid deformations is to

discretize the problem domain by replacing it with a collection of nodes and elements

referred to as the finite element mesh. The values of the material properties are usually

assumed to be constant within each element but are allowed to vary from one element

to the next; making it possible to simulate nonhomogeneous problems. The secondstep in the finite element method is to derive an integral formulation for the governing

equations. This leads to a system of algebraic equations that can be solved for values

of the field variable at each node in each mesh. The method of weighted residuals is

used for the fluid flow and solid deformation modeling. The Galerkin method is then

used whereby the weighting function for a node is identical to the shape function used

to define the approximate solution.

3.1 Shape Function

Interpolation or shape functions are used to map the element displacements and fluid

pressures at the nodal points.



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For the fluid pressure approximation at phase i, one has:

pi Npi 9

where N is the shape function for the fluid pressure and solid displacement.

At the nodal level, for four-point two-dimensional elements:

pi X4j1

Njpij 10

For the eight-point three-dimensional elements:

pi X8j1

Njpij 11

For eight-point three-dimensional elements, the shape function vector for pressure can

be given in following forms:

N1 1

81 1 1

N2 1

81 1 1

N3 1

81 1 1

N4 1

81 1 1

N5

1

8 1 1 1

N6 1

81 1 1

N7 1

81 1 1

N8 1

81 1 1 ;

12

where , and represent local coordinates; and 1 1, 1 1 and1 1.

A similar expression for the approximation in mapping nodal displacements can bedescribed as:

u Nu: 13

At the nodal level, for four-point two-dimensional elements:

ux X4j1

Njuxj; u

y X4j1

Njuyj: 14

For eight-point three-dimensional elements:

ux X8j1

Njuxj; uy X8j1

Njuyj; uz X8j1

Njuzj; 15



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where u is the nodal displacement vector. For simplicity, the superscript indicatesthe finite element approximations that are omitted in the subsequent description.

Strains within a single element are related to nodal displacements through the

derivatives of the shape functions as:

Bu; 16where B is the strain-displacement matrix.

The generalized plane strain solutions maintain compatibility with the primary

unknown terms equivalent to the three-dimensional formulation, but geometrically they

are not related to the z-coordinate, similar to the two-dimensional cases. With reference to

the finite element formulation, the major differences among the generalized plane strain,

the plane strain and the three-dimensional situation are exhibited in the strain-displace-

ment matrix B. In a three-dimensional geometry the matrix B can be expressed as:

B

@@x

0 0

0 @@y 0

0 0 @@z@

@y@

@x 0

0 @@z@

@y@@z

0 @@x

2666666664

3777777775N; 17

It is well-known in plane strain problems, i.e., in an x-y plane, that the displace-

ment and the shear stresses are restricted along the z-direction. In the generalized

plane strain scenarios, however, these restrictions are removed. As a result, the number

of tensor components for stresses and strains are identical to that of a three-dimen-sional setting. In a general generalized plane strain formulation, it is assumed that

boundary conditions in the form of surface tractions, pore pressure, displacements,

and normal flux, do not change along the z-direction. As a result, the displacements,

stresses, strains and pore and fracture pressures are only functions of x, y and time t

(Bai et al., 1999a). For the generalized plane strain formulation, B can be written as:

B

@@x 0 0

0 @@y 0

0 0 0@

@y

@

@x0

0 0 @@y

0 0 @@x

266666664

377777775

N 18

3.2 Conservation Equations

The general force equilibrium equation, in terms of nodal variables for a double

porosity medium in generalized plane strain domain, is given by (Zhang, 2002):

BTDmfdma

BTDmfCmamNpma dfr

BTDmfCfrmNpfr d S

NfdS

19



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where represents the integral domain; C is the compliance matrix defined in

Appendix I with CmaCmaijkl, CfrCfrijkl; Dmf is the matrix expression of Dmfijkl; fis the vector of applied boundary tractions; S is the domain surface on which the sur-

face traction f is applied.

Substituting Eq. (16) and dividing through by t, the momentum balance equation

(Eq. (19)) in the finite element form can be expressed as:

BTDmfB@u

@td ma

BTDmfCmamN@pma

@td fr

BTDmfCfrmN@pfr

@td

S

N@f

@tdS 20

or,

K@u

@t R1

@pma@t

R2@pfr

@t

@F

@t; 21

where detailed expressions of the coefficients are given in Appendix II.Using the Gaussian quadrature method (Zienkiewicz, 1977), the double porosity

mass balance equations (last two equations in Eq. (8)) in the finite element forms can

be given for each system.

For the rock matrix system:

1

NTkmarN dpma ma

NTmDmfCmaB d@u

@t ma

NTN d@pma

@t

!

NTN dDp

NTN dqma 22

For the fracture system:1

NTkfrrN dpfr fr

NTmDmfCfrB d@u

@t fr

NTN d@pfr

@t

!

NTN dDp

NTN dqfr 23

where Dp pfr pma, mT (1 1 1 0 0 0); is the domain surface on which the fluid

flux q is applied; and Biots effective stress coefficients, , can be evaluated as:

ma 1 Ksk

Ks

fr 1 KskKfr

;24

where Ksk and Ksk are the bulk moduli of the skeleton for the matrix blocks and the

fractures, respectively; Ks and Kfr are the bulk moduli of the solid grains and fractures,

respectively; and, the relative compressibilities, , can be written as:

ma nma

Kf

ma nmaKs

fr nfr

sKn

fr nfrsKn

;25

in which Kf and Kn are the bulk modulus of the fluid and the normal stiffness of the

fractures, respectively; n is the porosity; and, s is the fracture spacing.



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Equations (22) and (23) can be written as the finite element forms:

M1@u

@t Q L1pma Qpfr N1

@pma@t

Qma 26

M2

@u

@t Qp

ma Q

L2

pfr

N2

@pfr

@t Q

fr; 27

where detailed expressions of the above coefficients are listed in Appendix II.

Equations (21), (26) and (27) represent a set of differential equations in time and

can be expressed in matrix form as follows:

0 0 0

0 Q L1 Q

0 Q Q L2

264

375

u

pma

pfr

264

375

K R1 R2

M1 N1 0

M2 0 N2

264

375 d

dt

u

pma

pfr

264

375

dFdt

Qma

Qfr

264

375:28

The discretization in space has been completed; Eq. (28) now represents a set ofdifferential equations in time.

3.3 Finite Element Discretization in Time

Using a fully implicit finite difference scheme in the time discretization domain, such that:

dutDt

dt

1

tutDt ut

dptDtma

dt

1

t

ptDtma ptma

dptDtfrdt

1

tptDtfr p

tfr

8>>>>>>>>>>>>>:

29

and substituting Eqs. (29) into Eq. (28), the finite element equations in the matrix form

for a double porosity poroelastic medium can be expressed as follows:

1

t

K R1 R2

M1 Q L1t N1 Qt

M2 Qt Q L2t N2

264

375

u

pma

pfr

264

375

tt

1

t

K R1 R2M1 N1 0

M2 0 N2

264 375 upmapfr

264 375t

F

t

Qma

Qfr

264 375tt

F

t

0

0

264 375t

: 30

There are 5 unknowns (ux, uy, uz, pma, pfr) and 5 equations per node; therefore,

displacements and pressures can be solved. In addition, strains and stresses can be

obtained through the following equation and Eq. (4).

"ij 1

2ui;j uj;i: 31

3.4 Stress Conversion for an Inclined Borehole

For an inclined borehole with its axis inclined with respect to the principal axes of the

far-field stresses (see Fig. 2), the following equations can be used to convert the global



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coordinate (far-field stress coordinate, x0, y0, z0) into the local coordinate (borehole

coordinate, x, y, z) system.

SxSy

SzSxySyzSxz

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

l2xx0 l2

xy0 l2

xz0

l2yx0 l2

yy0 l2

yz0

l2zx0 l2

zy0 l2

zz0

lxx0 lyx0 lxy0 lyy0 lxz0 lyz0

lyx0 lzx0 lzy0 lyy0 lzz0 lyz0

lzx0 lxx0 lzy0 lxy0 lzz0 lxz0

2666666437777775

Sx0

Sy0

Sz0

8

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Double Porosity Finite Element Method for Borehole Modeling