Peng S., Zhang J., Engineering Geology for Underground Rocks

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5.6 Water impact on sedimentary rock strength 125

The cohesion in the rock with pore pressure effect may be expressed

as (Zhang et al. 1997):

tanpCC



(5.19)

where C and C

are the cohesions without and with the effect of a pore

pressure of p.

Rock strength also depends on the water content in the rock. Table 5.8

presents the experimental results of water content’s influences on rock

strength, Young’s modulus, and Poisson’s ratio in China’s coal measures.

The results show that rock compressive strength decreases significantly as

the water content increases.

Table 5.8. Experimental results of water influence on rock mechanical properties

Lithology

Water

content

W (%)

Compressive

strength

UCS (MPa)

Young‘s

modulus

E (GPa)

Poisson’s

ratio

0 142.5 73.33 0.330

0.23 82 48.30 0.340

Fine-grained

quartz sandstone

0.35 44.56 23.28 0.286

0 110.52 41.04 0.032

0.06 89.92 41.69 0.161

Limestone

0.11 58.08 42.35 0.189

0 119.4

0.56 88.5

0.78 51.6

Muddy siltstone

0.84 68.3

Lab tests show that rock compressive strength decreases as the water

content increases and can be expresses as follows (Zhang et al. 1997):

bWUCSUCS



(5.20)

where W is the water content (%); b is experimental constant.

For the rocks in China’s coal measures, the following experimental cor-

relations were given to describe the water effect on rock uniaxial strength

(Peng and Meng 2002):

For fine-grained quartz sandstone and water content of less than 0.35%:

126 5 Sedimentary rock masses and discontinuities

WUCS

6.2774.143 

For muddy siltstone and water content of less than 0.84%:

WUCS

6.71121

For mudstone and water content of less than 5.13%:

WUCS

93.44.30 

For limestone and water content of less than 0.11%:

WUCS

33.4729.112 

References

Bandis SC, Barton NR, Christianson M (1985) Application of a new numerical

model of joint behavior to rock mechanics problems. Proc Int Symp on Fun-

damentals of Rock Joints, Bjorkliden

Bandis SC, Lumsden AC, Barton NR (1983) Fundamentals of rock joint deforma-

tion. Int J Rock Mech Min Sci Geomech Abstr 20:249-268

Goodman RE (1976) Method of geological engineering in discontinuous rocks.

West Publishing

Hudson JA, Harrison JP (1997) Engineering rock mechanics, an introduction to

the principles. Pergamon

Meng Z, Zhang J, Peng S (2006) Influence of sedimentary environments on me-

chanical properties of clastic rocks. Environ Geol 51:113-120

Meng Z, Peng S, Qu H (2000) Relationships between rock mechnaical properties

and rock contents and structure in coal roof and floor formations. Chinese J

Rock Mech Eng 19(2):136-139 (in Chinese)

Peng S (1998) Approach on mechanical properties of clastic rocks and rock micro-

structure. Proc. 5

Ann Conf of Young Scientists, China Coal Society. Coal

Industry Press (in Chinese)

Peng S, Meng Z (1999) Classification and its application of roof stability using

lithology and geological structure. Symp 46

Young scientists, China Sci

Tech Soc, China Sci Tech Press Beijing pp35-42 (in Chinese)

Peng S, Qu H, Luo L, Wang L, Duan Y (2000) Complete stress-strain and perme-

ability experiments. J China Coal Society 25(2):113-116 (in Chinese)

Peng S, Meng Z (2002) Theory and practice of mining engineering geology. Geo-

logical Press (in Chinese)

Yu X (1983) Analysis of underground engineering stability. Coal Industry Press

(in Chinese)

Zhang J, Bai M, Roegiers J-C, Liu T (1999) Determining stress-dependent perme-

ability in the laboratory. In: Amadei B, Kranz RL, Scott GA, Smeallie PH (ed)

Proc 37

US Rock Mech Symp Colorado. Rotterdam, Balkema pp 341-347

Zhang J, Zhang Y, Liu T (1997) Rock mass permeability and coal mine water in-

rush. Geological Publ House Beijing (in Chinese)

6 Double porosity poroelasticity and its finite

element solution

6.1 Introduction

Field observations have revealed a need for a better and more comprehen-

sive method for geomechanics modeling. For instance, in oil and gas

industry, the exploration and production of hydrocarbons now occur in

more difficult geological settings; such as in naturally fractured media, in

overpressured shale formations, in rubble zones of sub-salt 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 shale formations, not only does the state

of stress and pressure play a role, but also the properties and interactions

between the shale and the drilling fluid.

Most petroleum reservoirs are situated in fractured porous formations.

In fact, in order to drain such reservoirs, inclined and horizontal wells must

be drilled in such formations. Therefore, traditional borehole modeling

methods cannot completely satisfy this requirement. The double porosity

approach can be used to model boreholes drilled in the fractured porous

medium.

6.2 Double porosity poroelastic model

In the double porosity approach, a 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 a small volume but have the ability to trans-

mit a large portion of the total flow through the reservoir. In early ap-

proaches (Barenblatt et al. 1960, Warren and Root 1963, Kazemi 1969),

the reservoir was considered as two overlapping continua: matrix and frac-

128 6 Double porosity poroelasticity and its finite element solution

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

troducing source functions.

A typical dual-porosity formulation is usually written as (Barenblatt et

al. 1960):



*



*



)(

mafr

frfrfr

mafr

mamama

ncp

(6.1)

where the subscripts ‘ma’ and ‘fr’ represent matrix and fractures, respec-

tively; p

and p

are the fluid pressures for the matrix and fractures, re-

spectively; k

and k

are the permeabilities for the matrix and fractures,

respectively;

is the fluid dynamic viscosity, n

and n

are the porosities

for the matrix and fracture, respectively; c

, c

are the compressibilities

for the matrix and fractures, respectively; t is the time, * is the geometric

leakage factor. It can be seen that no solid deformation is considered in this

formulation.

Alternatively, Barenblatt et al. (1990) proposed a more complete dual-

porosity formulation than that expressed by Eq. 6.1 by considering the

cross-phase storativity interaction:



*





*





)(

mafr

frfm

frfrfr

mafr

mamf

mamama

ncp

(6.2)

where c

and c

are the cross-coefficients.

However, effects of stresses and deformations on both the matrix and

fractures were not considered in Eqs. 6.1 and 6.2. Then, a coupled flow-

deformation approach within double porosity poroelastic media was pre-

sented (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 (Bai et al. 1993, Bai et al.

1999a, and Zhang and Roegiers 2005, etc.).

The naturally fractured reservoir can be assumed to be an ideal double

porosity and double permeability model as shown in Fig. 6.1 (Zhang and

Roegiers 2005). In this model the system, including matrix blocks and

fractures, is considered as a continuous medium.

6.2 Double porosity poroelastic model 129

Fig. 6.1. Naturally fractured and idealized double porosity reservoirs.

The relationship between total stress (

) and effective stress (

c) was

given by Terzaghi (1943) and Biot (1941). For double porosity media, the

effective stresses can be expressed as:

ijfrfrfrijfrij

ijmamamaijmaij

GDVV



(6.3)

where p is the pressure;

is the effective stress coefficient; and,

is the

Kronecker delta.

For separate but overlapping porous media, the linear elastic constitutive

relationships between the effective stresses and strains are defined as

(Elsworth and Bai 1992, Bai et al. 1999b):

frijfrijklfrkl

maijmaijklmakl

(6.4)

where C

maijkl

and C

frijkl

are the compliance tensors for the rock matrix (sub-

script ma) and fracture systems (subscript fr), respectively. The detailed

expressions for the tensors are listed in Appendix 6.1.

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

given by:

frklmaklkl

HHH



(6.5)

Substituting the double effective law, i.e., Eq. 6.3 into Eq. 6.4 while

combining with Eq. 6.5 and noting

ijfrijmaij

VVV

, the following

equation can be obtained:

130 6 Double porosity poroelasticity and its finite element solution

)(

ijfrfrfrijklijmamamaijklklmfijklij

pCpCD



(6.6)

The combined elasticity matrix D

mfijkl

is defined explicitly in a three-

dimensional geometry for an isotropic medium as:

)(





frijklmaijklmfijkl

CCD

(6.7)

where the detailed expression for D

mfijkl

is given in Appendix 6.1.

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 plastic. The elastoplastic stress-strain can be

described by:

İı ' '

mfijkl

(6.8)

where

mfijklmfijkl

mfijkl

DDD 

mfijkl

is the plastic modulus tensor.

The total strain increment is assumed to be the sum of the elastic and

plastic components; i.e.

İİİ '' '

(6.9)

where

İ' ,

İ' and

İ' are the total, elastic and plastic strain increments,

respectively. For more details about perfect plasticity solution, refer to

Zhang (2002).

For the separate but overlapping model, the double effective stress law

(Eq. 6.3) needs to be considered, and the combined elasticity matrix D

mfijkl

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 (Bai et al. 1999a,

Zhang 2002):

frmafr

frijklmfijklfrkkfr

mamafr

maijklmfijklmakkma

ifrfrijklmfijklfr

imamaijklmfijklmakikmfmfjjimf

qpp

CDp

qpp

CDp

pCD

pCDuGuG













)(

,,,

(6.10)

where

is the relative compressibility representing the lumped deform-

ability of the fluid and the solid; u is the solid displacement;

is the total

body strain;

is the transfer coefficient (Warren and Root 1963); s is the

6.3 Finite element discretization of the poroelastic solution 131

fracture spacing; q

and q

are the applied fluid flux;

is Lamé's

constant for the combined double porosity medium; and,

)21)(1(



mfijklmf

6.3 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 non-homogeneous problems. The second

step 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 can be 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.

6.3.1 Shape function

Interpolation or shape functions are used to map the element displacements

and fluid pressures at the nodal points (Peng and Meng 1999).

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

Npp

(

6.11)

where N is the shape function for the fluid pressure and solid displace-

ment.

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

ijji

pNp

(6.12)

For the eight-point three-dimensional elements:

132 6 Double porosity poroelasticity and its finite element solution

ijji

pNp

(6.13)

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

for pressure can be given in the following forms:

)1)(1)(1(

]K[

















(6.14)

where [, K, and ] represent local coordinates; and –1d[d1, –1dKd1 and

-1d]d 1.

A similar expression for the approximation in mapping nodal displace-

ments can be described as:

Nuu

(6.15)

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

¦¦

yjjy

xjjx

uNuuNu

(6.16)

For eight-point three-dimensional elements:

6.3 Finite element discretization of the poroelastic solution 133

¦¦¦

zjjz

yjjy

xjjx

uNuuNuuNu

(6.17)

where u is the nodal displacement vector, For simplicity, the superscript

‘*’ indicates the 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:

Buİ

(6.18)

where 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 case. 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-displacement

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

as:

(6.19)

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

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

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-dimensional setting. In a generalized plane strain

formulation, it is assumed that boundary conditions in the form of surface

134 6 Double porosity poroelasticity and its finite element solution

tractions, pore pressure, displacements, and normal flux, do not change

along the z-direction. As a result, the displacements, stresses, strains, pore

pressure, and fracture pressure are only functions of x, y and time t (Bai et

al. 1999a). For the generalized plane strain formulation, B can be written

as:

000

(6.20)

6.3.2 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, Zhang and Roegiers 2005):

³³

::

dSd

NfmNpCDB

mNpCDBİDB

frfrmf

mamamf

(6.21)

where

represents the integral domain; C is the compliance matrix

defined in Appendix 6.1 with C

= C

maijkl

, C

= C

frijkl

; D

is the matrix

expression of D

mfijkl

; f is the vector of applied boundary tractions; S is the

domain surface on which the surface traction f is applied.

Substituting Eq. 6.18 and dividing through by 't, the momentum

balance equation (Eq. 6.21) in the finite element form can be expressed as: