CHAPTER 11

TRANSPORT IN HETEROGENEOUS POROUS MEDIA

11.1 INTRODUCTION

Naturally fi ssured or fractured porous media are frequently encountered in subsur-face geological formations, including groundwater, geothermal, and petroleum res-ervoirs (Moench, 1984 ; Chang, 1993 ). * Most fractured formations are anisotropic and heterogeneous systems, characterized by a network of intersecting fractures partitioning the porous matrix into various regions, as depicted in Figure 11.1 (Civan and Rasmussen, 2005). For convenience in describing transport through such complex porous media, Barenblatt et al. (1960) introduced the double - porosity realization. In this approach, the fractures and porous matrix are envisioned as two separate but overlapping continua, interacting through the matrix – fracture interface (Moench, 1984 ; Zimmerman et al., 1993 ). Usually, the porous matrix, referred to as the primary continuum, is of a low - permeability and high - pore volume region, while the fracture, referred to as the secondary continuum, is of a high - permeability and

Porous Media Transport Phenomena, First Edition. Faruk Civan.© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

383

* Parts of this chapter have been reproduced with modifi cations from the following:

Civan, F. 1998c. Quadrature solution for waterfl ooding of naturally fractured reservoirs. SPE Reservoir Evaluation & Engineering Journal, 1(2), p. 141 – 147, © 1998 SPE, with permission from the Society of Petroleum Engineers;

Civan, F. and Rasmussen, M.L. 2001. Asymptotic analytical solutions for imbibition water fl oods in fractured reservoirs. SPE Journal, 6(2), pp. 171 – 181, © 2001, with permission from the Society of Petroleum Engineers;

Civan, F., Wang, W., and Gupta, A. 1999. Effect of wettability and matrix - to - fracture transfer on the waterfl ooding in fractured reservoirs. Paper SPE 52197, 1999 SPE Mid - Continent Operations Symposium (March 28 – 31, 1999), Oklahoma City, OK, © 1999 SPE, with permission from the Society of Petroleum Engineers;

Rasmussen, M.L. and Civan, F. 1998. Analytical solutions for water fl oods in fractured reservoirs obtained by an asymptotic approximation. SPE Journal, 3(3), 249 – 252, © 2001 SPE, with permission from the Society of Petroleum Engineers; and

Rasmussen, M.L. and Civan, F. 2003. Full, short, and long - time analytical solutions for hindered matrix - fracture transfer models of naturally fractured petroleum reservoirs. Paper SPE 80892, SPE Mid - Continent Operations Symposium (March 22 – 25, 2003), Oklahoma City, OK, © 2003 SPE, with permission from the Society of Petroleum Engineers.

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384 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

low - pore volume region in most fractured porous media (Moench, 1984 ; Chang, 1993 ; Zimmerman et al., 1993 ). The macroscopic description of transport in these two different continua requires different orders of magnitude representative elemen-tary volumes (REVs). As depicted in Figure 11.2 , the REV for the fracture network should be signifi cantly larger than that for the porous matrix (Moench, 1984 ). However, the majority of the double - porosity models consider the larger of the REVs to defi ne the volume averaged, that is, the effective properties of the various con-stituents and properties in both continua. For convenience in the mathematical modeling, Warren and Root (1963) resorted to a representation of naturally fractured porous media by an idealized and simplifi ed geometrical model, referred to as the sugar cube model, as depicted in Figure 11.1 .

Panfi lov (2000) distinguished two possible types of elementary fl ow patterns in a cell, referred to as the translation or fl ow through matrix and source fl ow or fl ow around matrix, respectively, as depicted in Figure 11.3 a,b. The translation fl ow occurs when the fracture and matrix media permeability are comparable with each other. Then, fl ows through both media are considered. Consequently, this approach requires two sets of similar equations for the naturally fractured porous media. The source fl ow occurs when the fracture medium permeability is signifi cantly greater

Figure 11.1 Representation of naturally fractured porous media (Civan and Rasmussen, 2005; © 2005 SPE, reproduced by permission of the Society of Petroleum Engineers).

Figure 11.2 Schematic naturally fractured porous media (prepared by the author with modifi cations after Moench, 1984 ).

REV for fracture network

REV for matrix

Impermeable layer

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11.2 TRANSPORT UNITS AND TRANSPORT IN HETEROGENEOUS POROUS MEDIA 385

than the matrix medium permeability. Then, the fracture medium forms the prefer-ential fl ow path, while the matrix medium acts as a source/sink for the fracture medium (Civan, 1998c ; Rasmussen and Civan, 1998 ; Civan and Rasmussen, 2001 ). As a result, equations for the fracture medium are suffi cient to describe the fl ow through naturally fractured porous media. Because the latter case prevails in most fractured reservoirs, the fracture – fl ow and matrix – source/sink formulation approach received the most attention. Ordinarily, the numerical solution of both the translation and source fl ow approaches requires the spatial discretization of both the fracture and matrix media, as demonstrated by Pruess and Narasimhan (1985) . However, for practical reasons, the source/sink fl ow approach based on the sugar cube model has been frequently solved by a semianalytical method. This is based on prescribing the interchange or transport between the matrix and fracture media by a suitable analyti-cal expression to be used as a source/sink function for the fracture fl ow equation, as described in the following sections.

In the following sections, a number of important topics are discussed, includ-ing transport units and transport in heterogeneous porous media, models for transport in fi ssured/fractured porous media, species transport in fractured porous media, immiscible displacement in naturally fractured porous media, and numerical solu-tions by the methods of weighted sums (quadrature) and fi nite difference.

11.2 TRANSPORT UNITS AND TRANSPORT IN HETEROGENEOUS POROUS MEDIA

11.2.1 Transport Units

The properties of heterogeneous porous media are spatially different. For convenient realizations and mathematical description of transport processes, heterogeneous porous media can be partitioned into various transport units, each involving transport of different types and orders of magnitude rates. Such models have been variably referred to by different names. For example, the frequently used models of macropores – micropores (Bai et al., 1995 ), double or dual porosity (Coats, 1989 ; Bai and Civan, 1998a,b ), matrix – fracture (Warren and Root, 1963 ; Rasmussen and

Figure 11.3 Flow patterns in a cell: (a) translation fl ow and (b) source fl ow (modifi ed after Panfi lov, 2000 ).

(a) (b)

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386 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

Civan, 1998c ; Civan and Rasmussen, 2001 ), and plugging – nonplugging paths (Gruesbeck and Collins, 1982 ; Civan and Nguyen, 2005 ) are based on a realization of heterogeneous porous media in two distinct scales of transport units. Different transport units are treated as mutually interacting continua (Valliappan et al., 1998 ) as schematically depicted in Figure 11.4 for a two - transport unit partitioning of heterogeneous porous media.

The defi nition of proper transport units in prescribed heterogeneous porous media depends on the specifi c applications and nature of porous media. For example, naturally fractured porous media can be analyzed by considering porous matrix interacting with fractures. Granular porous media usually display a bimodal pore size distribution and therefore, such media can be viewed as having the micropore and macropore regions interacting with each other by the exchange process. However, applying the Kozeny equation, the transport or fl ow units in heterogeneous porous media can be more adequately distinguished in terms of a quality index (Amaefule et al., 1993 ). The quality index is best defi ned by the mean hydraulic fl ow path diameter D h in porous media (Civan, 2007a ):

QI D Kh h≡ = 4 2τ φ , (11.1)

where τh is the tortuosity of preferential fl ow paths, K is permeability, and φ is porosity.

The volume fractions of the various transport units, denoted by f j : j = 1, 2, … , N , are a characteristic of heterogeneous porous media. By defi nition of fractions,

f j

j

N

=

−

∑ =1

1 0units

. . (11.2)

11.2.2 Sugar Cube Model of Naturally Fractured Porous Media

The sugar cube model is one of the most frequently used heterogeneous porous media models introduced by Barenblatt et al. (1960) and Warren and Root (1963) for the realization of naturally fractured porous media. As depicted in Figure 11.1 , the naturally fractured porous media are characterized in terms of a network of intersect-

Figure 11.4 Realization of heterogeneous porous media by interacting micropore and macropore regions (prepared by the author).

Inflow Outflow

Microporeregion

Macropore region

Exchange

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11.2 TRANSPORT UNITS AND TRANSPORT IN HETEROGENEOUS POROUS MEDIA 387

ing fractures, separating the porous media into a number of matrix blocks (Warren and Root, 1963 ; Chang, 1993 ).

Consider the representative elemental volume shown in Figure 11.5 a. D 1 and D 2 represent the fracture spacing. b 1 and b 2 denote the aperture widths for the frac-tures, and Γ 1 and Γ 2 denote the exchange rates between the matrix block and fracture in the x - and y - directions, respectively. Therefore, the following expressions for the representative elemental volume shown in Figure 11.5 a are given.

The cross - sectional area A and perimeter P of the matrix block are given, respectively, by

A l D D= =21 2 (11.3)

and

P l l D D2 1 2= + = + . (11.4)

Hence, the following expressions can be derived using Eqs. (11.3) and (11.4) :

lD D

l

D D= = +1 2 1 2

2 (11.5)

Thus,

lD D

D D=

+2 1 2

1 2

. (11.6)

The pore and bulk volumes are given, respectively, by

V D b b D bPV = +( ) +2 2 1 1 2 (11.7)

and

V D b D bBV = +( ) +( )1 1 2 2 . (11.8)

Hence, using Eqs. (11.7) and (11.8) , the fracture porosity is given by

φ fPV

BV

V

V

D b b D b

D b D b= = +( ) +

+( ) +( )2 2 1 1 2

1 1 2 2

. (11.9)

When b 1 << D 1 and b 2 << D 2 , Eq. (11.9) simplifi es as

φ fD b D b

D D

b

D

b

D≅ + = +2 1 1 2

1 2

1

1

2

2

. (11.10)

Figure 11.5 Representing (a) a rectangular block by (b) an idealized square block (prepared by the author).

(b)

b/2

Ideal square block

b/2

l

l

b/2 b/2

(a)

b2/2

Actualblock

b2/2

D2

D1

b1/2 b1/2

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388 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

If the same conditions are applied for b 1 = b 2 = b (Fig. 11.5 b), the result becomes (Valliappan et al., 1998 ):

φ fD D

D Db

D Db≅ +⎛

⎝⎜⎞⎠⎟ = +⎛

⎝⎜⎞⎠⎟

2 1

1 2 1 2

1 1. (11.11)

11.3 MODELS FOR TRANSPORT IN FISSURED/FRACTURED POROUS MEDIA

The commonly used models are reviewed in the following.

11.3.1 Analytical Matrix – Fracture Interchange Transfer Functions

Derivation of the matrix – fracture interchange transfer functions has occupied many researchers, including Warren and Root (1963) , Kazemi et al. (1976) , Moench (1984) , and Zimmerman et al. (1993) . As pointed out by Moench (1984) and Zimmerman et al. (1993) , the various approaches consider either pseudo - steady - state or transient - state conditions for describing the internal fl ow of the matrix medium. Moench (1984) claims that well test data support the presence of fl ows both at pseudo - steady - state or transient - state conditions. Most approaches, includ-ing those of Warren and Root (1963) , Kazemi et al. (1976) , and Lim and Aziz (1995) , have utilized constant fracture fl uid pressure along the matrix – fracture interface, referred to as the Dirichlet boundary condition. Moench (1984) considered the resistance to fl ow at the fracture – matrix interface owing to various reasons, including mineral deposition and alteration, and thus introduced the fracture skin concept. Consequently, Moench (1984) applied a Cauchy - type bound-ary condition. The mathematical fundamentals of these approaches are described in the following according to Moench (1984) , Chang (1993) , and Zimmerman et al. (1993) .

Referring to Figure 11.6 , describing a source fl ow, the fl ow inside a matrix block of a , b , and c dimensions in the x - , y - , and z - Cartesian coordinates is described by the equation of continuity and Darcy ’ s law. Assuming a single - phase, constant viscosity and a slightly compressible fl uid in a matrix block having constant poros-ity φm and anisotropic permeability K mx , K my , and K mz , and a constant total compress-ibility c m yields the following diffusion equation for the fl uid pressure:

φμm m

mm m mc

p

tp x y z V t

∂∂

= ∇⋅ ⋅∇⎛⎝⎜

⎞⎠⎟

∈ >10K , , , , , (11.12)

where Km and φm denote the permeability tensor and porosity of the matrix, and V m denotes the volume of the matrix. Initially, consider that the fl uid pressure through-out the matrix block is uniform:

p p x y z V tm i m= ∈ =, , , , .0 (11.13)

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11.3 MODELS FOR TRANSPORT IN FISSURED/FRACTURED POROUS MEDIA 389

Along the outer surfaces of the matrix block, consider the following Cauchy - type boundary condition, accounting for the fracture skin as illustrated in Figure 11.7 (Moench, 1984 ):

uK p

n

K p p

bA tn

mn m s m f

sm= − ∂

∂=

−⎛⎝⎜

⎞⎠⎟ >

μ μ, , ,0 (11.14)

where n denotes the outward normal direction at the matrix surface, K s and b s denote the permeability and thickness of the skin, p f is the fracture fl uid pressure assumed constant, and A m denotes the outer surface area of the matrix.

The total fl ow rate through the matrix surfaces is given by (Duguid and Lee, 1977 )

q V

K p

ndA t

m

mn m

Am

= − ∂∂

>∫10

μ, . (11.15)

As pointed out by Moench (1984) , the boundary condition given by Eq. (11.14) adequately justifi es the validity of neglecting the divergence of fl ow to drive

Figure 11.6 Rectangular parallelepiped shape matrix block (prepared by the author).

c

b

a

x

y

z

O

Figure 11.7 Matrix block with fracture skin (prepared by the author with modifi cations after Moench, 1984 ).

Skin bs

K f

Matrixblock

Km

Fracture

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390 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

the pseudo - steady - state model. K s → ∞ in the absence of the skin effect and, there-fore, Eq. (11.14) simplifi es to

p p A tm f m= >, , .0 (11.16)

In the following section, a number of special solutions to Eqs. (11.12) – (11.16) are presented, with modifi cations for the consistency with the rest of the presentation of this chapter.

11.3.2 Pseudo - Steady - State Condition and Constant Fracture Fluid Pressure over the Matrix Block: The Warren – Root Lump - Parameter Model

As described by Zimmerman et al. (1993) , the Warren and Root (1963) model treats the matrix blocks in a lump - parameter form. Thus, integrating Eq. (11.12) over the matrix block yields

φ μm m m

V

mn m

A

ct

p dVK p

ndA t t

m m

∂∂

= ∂∂

> >∫ ∫ , , .0 0 (11.17)

Next, defi ne the average fl uid pressure within the matrix block as

p Vp dVm

mm

Vm

= ∫1. (11.18)

Hence, invoking Eqs. (11.15) and (11.18) into Eq. (11.16) results in the fol-lowing equation:

φm mmc

dp

dtq= − , (11.19)

where q [ L 3 / T ] denotes the total volumetric fl ow rate of the fl uid across the matrix surfaces. Eq. (11.19) is often referred to as the pseudo - steady - state assumption (Dake, 1978 ). A simplistic approach similar to Kazemi et al. (1976) to estimating this term is to apply the right of Eq. (11.14) in Eq. (11.15) using the average fl uid pressure placed at the center of the matrix block, p pm m= , substituting K Ks mn≡ in the normal direction and taking the distance from the matrix center to the matrix surface for b s . This leads to the following expression:

qV

K A

a

K A

b

K A

cp p

m

mx x my y mz zm f= ( )

+ ( )+ ( )

⎡⎣⎢

⎤⎦⎥

−( )12

22

22

2μ μ μ/ / /, (11.20)

where A x , A y , and A z represent the surface areas of the matrix block normal to the x - , y - , and z - Cartesian directions, given by

A bc A ac A abx y z= = =, , . (11.21)

The volume of the block is

V abcm = . (11.22)

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11.3 MODELS FOR TRANSPORT IN FISSURED/FRACTURED POROUS MEDIA 391

Also defi ne a geometric average permeability for the anisotropic porous matrix as (Muskat, 1937 )

K K K Km mx my mz= ( )1 3. (11.23)

As a result, substituting Eqs. (11.21) – (11.23) into Eq. (11.20) yields the fol-lowing transfer function:

qK

p pmm f= −( )σ

μ, (11.24)

where the parameter σ is called the shape factor, given by

σ = + +⎛⎝⎜

⎞⎠⎟

42 2 2K

K

a

K

b

K

cm

mx my mz . (11.25)

Eq. (11.25) simplifi es to the shape factor of Kazemi et al. (1976) for an iso-tropic porous matrix. Eq. (11.25) implies that the shape factor σ has a reciprocal area dimension [ L − 2 ]. Invoking Eq. (11.24) into Eq. (11.19) yields

φ σμm m

m mm fc

dp

dt

Kp p= − −( ). (11.26)

A solution of Eq. (11.26) subject to the initial condition given by Eq. (11.13) yields a decay function for the matrix average fl uid pressure as

p p

p p

K t

cm i

f i

m

m m

−−

= − −⎛⎝⎜

⎞⎠⎟

1 exp .σμφ

(11.27)

Substituting Eq. (11.27) into Eq. (11.19) yields the following expression for the total volumetric fl ow rate of the fl uid across the matrix block surfaces per unit matrix block volume:

q cdp

dtp p

K K t

cm m

mf i

m m

m m

= − = − −( )⎛⎝⎜⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

φ σμ

σμφ

exp . (11.28)

11.3.3 Transient - State Condition and Constant Fracture Fluid Pressure over the Matrix Block

As demonstrated by Coats (1989) , Chang (1993) , and Lim and Aziz (1995) , an analytical solution for Eq. (11.12) , subject to Eqs. (11.13) and (11.16) , can be derived by applying the separation of variables method for the transient - state pres-sure distribution for the fl uid present in the matrix block. Then, integrating this analytical solution over the matrix block according to Eq. (11.18) yields (Lim and Aziz, 1995 )

p p

p pA B

t

cm i

f i m mnml

−−

= − ⎛⎝⎜⎞⎠⎟ −⎛

⎝⎜⎞⎠⎟

=

∞

=

∞

=

∞

∑∑∑18

2

3 2

000π

πμφ

exp ,, (11.29)

where

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392 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

Al m n

=+( ) +( ) +( )

1

2 1 2 1 2 12 2 2 (11.30)

and

BK

al

K

bm

K

cnmx my mz= +( ) + +( ) + +( )

2

2

2

2

2

22 1 2 1 2 1 . (11.31)

As a fi rst - order approximation to estimating the shape factor, Lim and Aziz (1995) considered only the fi rst terms in the summation series given in Eq. (11.29) , leading to

p p

p p

K

a

K

b

K

c

t

cm i

f i

mx my mz

m m

−−

≅ − ⎛⎝⎜⎞⎠⎟ − + +⎛

⎝⎜⎞⎠⎟1

82

3

2 2 2

2

ππμφ

exp⎡⎡⎣⎢

⎤⎦⎥. (11.32)

Then, substituting Eq. (11.32) into Eq. (11.26) yields the shape factor expres-sion of Lim and Aziz (1995) as

σ π= + +⎛⎝⎜

⎞⎠⎟

2

2 2 2K

K

a

K

b

K

cm

mx my mz . (11.33)

Consequently, the total volumetric fl ow across the matrix block surfaces can be obtained by substituting Eq. (11.33) into Eq. (11.28) . However, a direct substitu-tion of Eq. (11.29) into Eq. (11.26) yields a time - dependent expression for the shape factor as (Coats, 1989 ; Chang, 1993 )

σπ

πμφπμφ

=−⎛⎝⎜

⎞⎠⎟

−⎛⎝

=

∞

=

∞

=

∞

∑∑∑2

2

000

2K

AB Bt

c

A Bt

cm

m mnml

m m

exp

exp⎜⎜⎞⎠⎟

=

∞

=

∞

=

∞

∑∑∑nml 000

. (11.34)

11.3.4 Single - Phase Transient Pressure Model of de Swaan for Naturally Fractured Reservoirs

In the following, the formulation and solution of the model by de Swaan (1990) are presented in a manner consistent with the rest of the chapter.

Consider an elementary cell volume of the sugar cube model of naturally fractured porous media given in Figure 11.1 . Denote the representative elementary cell volume by V e , and the volumes of the fracture and matrix block by V f and V m , respectively. The permeability of the fracture and matrix block is represented by K f and K m , the porosities by φ f and φm, and the total compressibility by c f and c m , respectively.

The effective permeability of the fracture medium can be estimated by

K Kf f f= * ,τ (11.35)

where K f* is the intrinsic permeability and τ f is the tortuosity of the fractures sur-

rounding the matrix block in an elementary cell.

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11.3 MODELS FOR TRANSPORT IN FISSURED/FRACTURED POROUS MEDIA 393

The elementary cell volume and its total pore volume are given, respec-tively, by

V V Ve f m= + (11.36)

and

V V Vp f f m m= +φ φ . (11.37)

Note that the volume fraction of the interconnected fractures can be reduced for various reasons, including the presence of deposits inside the fractures and isola-tion and/or connecting to dead - end vugs.

Considering the fracture – fl ow and matrix – source/sink approach for modeling, the fracture medium pressure diffusivity equation is given by

φμf f

ff fc

p

tr K p t

∂∂

+ = ∇ >102 , , (11.38)

where r denotes the mass rate of fl uid lost from the fracture to matrix per unit fracture medium volume, given by

rV c

Vqm m m

fp p tf f= = ( )

φ. (11.39)

Applying Duhamel ’ s theorem, de Swaan (1990) corrected the matrix – fracture interface fl ow rate assuming constant fracture fl uid pressure p f = ct. to account for the variable fracture fl uid pressure p f = p f ( t ) as

q q tp

p p t p

ft

f f f= ( ) == −( ) ∂∂

∂∫ ττ

τct.

0

. (11.40)

Thus, combining Eqs. (11.38) – (11.40) yields the following fracture fl uid dif-fusivity equation:

φφ τ

ττ

μf ff m m m

fp

ft

f fcp

t

V c

Vq t

pK p t

f

∂∂

+ −( ) ∂∂

∂ = ∇ >=∫ ct.

0

210, . (11.41)

The initial condition for radial fl ow around a well is given by

p p r tf fi= ≤ < ∞ =, , .0 0 (11.42)

The boundary condition at the wellbore is given by

qB

A K

p

rA r h r r tw

w f

fw w w=

∂∂

= = >μ π, , , .2 0 (11.43)

The boundary condition at a suffi ciently long distance from the well, where the pressure remains undisturbed, is given by

p p r tf f= →∞ >∞, , .0 (11.44)

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394 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

As demonstrated by de Swaan (1990) , an analytical solution for Eqs. (11.41) – (11.44) can be derived by using the matrix – fracture function given by Eq. (11.28) based on the model of Warren and Root (1963) and then by applying a Laplace transformation.

11.4 SPECIES TRANSPORT IN FRACTURED POROUS MEDIA

The description of fl ow and species transport in fractured porous media has been of practical importance for various purposes, including environmental contaminant pollution modeling (Valliappan et al., 1998 ). In this chapter, the formulation by Valliappan et al. (1998) is presented with modifi cations for consistency with the rest of the material presented in this book. This approach considers the fractured porous medium to comprise the fracture network and the porous matrix as two separate continua interacting through a matrix – fracture transfer term.

The total mass and species – mass balance equations are given, respectively, by

∂∂( ) +∇⋅( ) =

tu mφρ ρ φ � (11.45)

and

ρ φ φρ φ∂∂

+ ⋅∇⎛⎝⎜

⎞⎠⎟ = ∇⋅ ⋅∇( ) + −( )w

tw w m w mA

A A A Au D � � , (11.46)

where �m and �mA denote the mass rate of fl uid and species A added per unit volume, respectively; D is the molecular diffusion coeffi cient tensor. Considering density variation by species concentration and pressure, and porosity variation by pressure, Eq. (11.45) can be expanded as

φ ρ ρ φ ρ φ∂∂+ ∂

∂+∇⋅( ) =

t tu m� . (11.47)

The volumetric fl ux of fl uid is given by Darcy ’ s law:

u K= − ⋅∇ρμ

Ψ, (11.48)

where Ψ denotes the fl ow potential defi ned by

Ψ = + −( )∫ dpg z z

p

p

o

oρ

. (11.49)

Next, consider that the fl uid density is a function of the pressure and concen-tration at isothermal conditions; that is,

ρ ρ= ( )p wA, . (11.50)

Hence, it follows that

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11.4 SPECIES TRANSPORT IN FRACTURED POROUS MEDIA 395

dp

dpw

dwA

Aρ ρ ρ= ∂∂

+ ∂∂

. (11.51)

A Taylor series expansion around a reference state, such as no species present and 1 - atm pressure conditions, yields

ρ ρ ρ ρ= + ∂∂

−( ) + ∂∂

−( ) +o

o

oA o

A Aop

p pw

w w higher-order terms. (11.52)

In addition, assuming a slightly compressible fl uid and porous matrix, the coeffi cients of compressibility can be defi ned by

cp

p =∂∂

1

ρρ

(11.53)

and

cp

φ φφ= ∂∂

1. (11.54)

Also, defi ne a coeffi cient of density variation by species concentration as

cw

wA

A =∂∂

1

ρρ

. (11.55)

The total compressibility coeffi cient is given by

c c ct p= +φ . (11.56)

Therefore, in view of Eqs. (11.48) – (11.56) , Eqs. (11.47) and (11.46) can be expressed, respectively, as

∇⋅ ⋅∇⎛⎝⎜

⎞⎠⎟= ∂

∂+ ∂

∂−ρ

μρφ ρφ φK Ψ c

p

tc

w

tmt w

AA

� (11.57)

and

∇⋅ ⋅∇( ) − ⋅∇ = ∂∂

− +φρ ρ ρφ φ φD uw ww

tm w mA A

AA A� � . (11.58)

Eq. (11.58) assumes that all the species A present in the solution are available for transport. This assumption is valid if the porous material is inert and therefore does not react with species A in the solution. However, most porous materials tend to interact with the fl uid media in various forms, including adsorption/desorption, absorption/desorption, deposition/dissolution, and chemical reaction. Such effects can be accounted for by a retardation factor (Neretnieks, 1980 ; Huyakorn et al., 1983 ), given by

R ks d= + −⎛⎝⎜

⎞⎠⎟

11 φφ

ρ , (11.59)

where ρs denotes the density of the solid material of the porous media and k d is a species distribution coeffi cient.

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396 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

The fl uid added to the fracture medium comes through the matrix – fracture interface from the matrix medium (Fig. 11.8 ). The mass rate of fl uid Γ transferred from the matrix - to - fracture media can be expressed by

Γ Ψ Ψ= = − = −( )φ φ ρ σμf f m m m

mm fm m

K� � , (11.60)

where σ denotes the shape factor. ρm is the fl uid density. The mass rate of species A ΓA added to the fracture medium (Fig. 11.8 ) is the

sum of the rate of mass diffusing from the matrix – fracture interface into the fracture medium, the species A carried by the fl uid transfer from matrix to fracture through the matrix – fracture interface, and the species A lost by chemical reactions within the fracture medium, expressed by

Γ ΓA f Af m m Am Af Af Am f f Afm D w w w w R w= = −( ) + + −( )[ ]−φ αφ θ θ λ φ� 1 , (11.61)

where α is an empirical coeffi cient. θ is a fl ow direction parameter, whose value is zero when the matrix – fracture interface fl uid fl ow is from the matrix into the fracture media, and unity otherwise. D is the molecular diffusion coeffi cient. λ is a rate coeffi cient for the fi rst - order decay reaction of species A .

Consequently, in view of the above - mentioned discussion, Eq. (11.58) should be modifi ed as

∇⋅ ⋅∇( ) − ⋅∇ = ∂∂

+ +φρ ρ ρφD uw w Rw

twA A

AA AΓ Γ. (11.62)

11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA

For example, an application of immiscible displacement in naturally fractured porous media is waterfl ooding. Incidentally, this is one of the economically viable tech-niques for recovery of additional oil following the primary recovery. However, the applications to naturally fractured reservoirs have certain challenges from the points of accurately describing the mechanism of oil recovery and fl ow of fl uids in naturally fractured formations and the effi cient numerical solution of the resulting equations. Parts of the following formulation of immiscible displacement in naturally fractured

Figure 11.8 Matrix – fracture interaction by advection and diffusion (prepared by the author).

PorousMatrix

Fracture Medium

Advection

Diffusion

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 397

porous media are presented from Civan (1998c) , Civan et al. (1999) , Civan and Rasmussen (2001) , and Rasmussen and Civan (1998) .

In general, the multiple - porosity modeling approaches proposed for naturally fractured reservoirs are highly computationally intensive or impractical for large - scale applications. Therefore, in search for a more practical approach, Kazemi et al. (1992) have adopted the single porosity with source modeling approach by deSwaan (1978) and have demonstrated that this approach is advantageous over the frequently used multiple - porosity approaches because only the solution of the fracture fl ow problem is required. Basically, deSwaan described the immiscible displacement process in fractured porous media by adding a source term to the conventional Buckley – Leverett equation to represent the matrix - to - fracture oil transport by an empirical function given by Aronofsky et al. (1958) . Kazemi et al. (1992) extended this function into a multiparameter empirical function. deSwaan ’ s approach is appli-cable when the fracture permeability is much greater than the matrix permeability, so that the fractures provide the preferential paths for fl ow and the matrix becomes the source of oil for the fractures. The resulting transport equation is an integro - differential equation.

Most researchers including Kazemi et al. (1992) neglected the capillary pres-sure effect in the modeling of fl ow through fractures. However, Gilman (1983) and Pruess and Tsang (1989) considered that the capillary pressure effect should be included. Civan (1993) extended the modeling approach by deSwaan (1978) to include the gravity and capillary pressure effects. Civan (1993, 1994a, 1998c) and Gupta and Civan (1994a) derived two - and three - exponent matrix - to - fracture trans-fer functions based on a proposed mechanism of oil transfer from matrix to fracture to incorporate into the deSwaan (1978) model. The analytical and numerical solu-tions of the model were obtained by various researchers.

11.5.1 Correlation of the Matrix - to - Fracture Oil Transfer

The permeability of the fracture system in most naturally fractured reservoirs is much greater than that of the porous matrix, and therefore the fractures form the preferential fl ow paths while the matrix acts as the source of oil for fractures (Civan, 1998c ). Then, modeling oil recovery by waterfl ooding of naturally fractured reser-voirs by the fracture porosity and matrix source approach provides a computationally convenient method (Aronofsky et al., 1958 ; deSwaan 1978 ). The accuracy of this method, however, depends on the implementation of a properly defi ned matrix - to - fracture transfer function in the Buckley – Leverett formulation. For this purpose, Aronofsky et al. used a one - parameter empirical function obtained by correlation of the cumulative matrix - to - fracture oil transfer given by

V t V eft( ) ,= −( )∞

−1 λ (11.63)

where λ is an empirical constant and V∞ is the volume of the movable oil initially available in the porous matrix, measured per unit bulk volume, although they recom-mended adding more exponential terms for better correlation of the experimental data. Hence, Kazemi et al. (1992) proposed an empirical function composed of an infi nite series of exponential terms, given by

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398 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

V t V a ef jt

j

j( ) .= −⎛

⎝⎜

⎞

⎠⎟∞

−

=

∞

∑11

λ (11.64)

Civan (1993, 1998c) and then Gupta and Civan (1994a) perceived naturally fractured porous media as having primary fracture porosity and secondary matrix porosity. The porous structure of the matrix consists of interconnected and dead - end pores (see Fig. 11.9 ). Because the fractures have relatively larger permeability than the matrix, the fl ow is considered to occur essentially through the fracture network, and the matrix feeds oil into the fractures owing to the imbibition of water and thus acts as a source. The imbibition of water into the matrix causes oil to discharge into the fractures. The oil existing in the dead - end pores passes into the interconnected pores and then to the matrix – fracture interface to accumulate over the fracture face, where it is entrained and removed by the fl uid system fl owing through the fracture. Civan (1993, 1998c) and then Gupta and Civan (1994a) derived two - and then three - exponential matrix - to - fracture transfer functions, respectively, based on the principle that dynamic processes occur at rates proportional to the governing driving forces. The proportionality factors are called the rate constants. Thus, for the present case, the oil transfers at various points of naturally fractured formations were assumed to occur at rates proportional to the oil available at those sites. Thus, Gupta and Civan (1994a) theoretically derived and verifi ed by experimental data that a maximum of three - exponential terms is suffi cient for accurate phenomenological representation of the oil transfer from matrix to fracture. However, the contribution of the dead - end pores varies for different types of porous media as indicated in the following example.

It is perceived that the oil expelled from the matrix by water imbibition accu-mulates over the fracture surface as oil droplets attach to the fracture surface, and then these droplets are entrained by the fl uid system fl owing through the fracture medium (Fig. 11.9 ). Thus, the transfer of oil from matrix to fracture is considered to occur through three irreversible rate processes in series as (Civan, 1993 ; Gupta and Civan, 1994a )

α β γV V V Vd n i f→ → → . (11.65)

Gupta and Civan (1994a) included the effect of the dead - end pores by expand-ing Civan ’ s (1993) rate equations as the following:

dV dt Vd d= −λ α3 , (11.66)

dV dt V Vn d n= −λ λα β3 1 , (11.67)

dV dt V Vi n i= −λ λβ γ1 2 , (11.68)

and

dV dt Vf i= λ γ2 , (11.69)

where V V V Vd n i f, , , and denote the cumulative volumes of oils remaining in the dead - end and interconnected pores of the matrix, accumulating over the fracture surface as oil droplets, and entrained by the fracture – medium fl uid, respectively, expressed per unit bulk volume of porous media (see Fig. 11.10 ). λ λ λ1 2 3, , and are empirically

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 399

determined rate constants, and α β γ, , and represent the orders of the governing transfer processes.

The underlying physics for Eqs. (11.66) – (11.69) is that the rate at which a dynamic process occurs is proportional to the pertinent driving forces, and the pro-portionality factor is the rate constant. Eq. (11.66) expresses that the rate of oil depletion in the dead - end pores is proportional to the oil remaining in the dead - end pores. Eq. (11.69) expresses the rate of entrainment of the oil droplets from the fracture surface by the fl uid system fl owing through the fracture as being propor-tional to the oil droplets available over the fracture surface. Eq. (11.68) expresses the accumulation rate of oil droplets over the fracture surface as being equal to the difference between the rate of oil expulsion from the matrix and the rate of oil entrainment by the fl uid fl owing through the fracture. Eq. (11.67) can be interpreted similarly to Eq. (11.68) .

A simultaneous solution of Eqs. (11.66) – (11.69) , subject to the initial condi-tions, given by

Figure 11.9 Imbibition - induced oil recovery from a porous media containing interconnected and dead - end pores and natural fractures (after Civan and Rasmussen, 2001 ; © 2001 SPE, reproduced with permission from the Society of Petroleum Engineers).

Oil Droplets AccumulatingOver the Fracture Ssurface

EffluxInflux

Oil

Oil

FractureVf

Water

Water

Matrix

InterconnectedPore

Dead-EndPore

Vi

Vn

Vd

λ1

λ2

λ3

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400 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

V f V V f V V V td d n n i f= = = = =°∞

°∞, , , , ,0 0 0 (11.70)

yields the following expression when α β γ= = = 1 (Gupta and Civan, 1994a ):

V t V a ef jt

j

j( ) = −⎛

⎝⎜

⎞

⎠⎟∞

−

=∑1

1

3λ . (11.71)

The parameters aj jand λ denote the empirically determined pre - exponential coeffi cients and the rate constants associated with the oil entrainment by the fracture fl uid, oil transfer from the interconnected pores to the fracture face, and discharge of oil from the dead - end pores to the interconnected pores, for j = 1, 2, 3, respec-tively (see the schematic shown in Figure 11.10 ). The expressions for the pre - exponential coeffi cients are given by Gupta and Civan ( 1994a ) for the respective processes as

a f fno

d12

2 1

3

1 3

0=−

⎡⎣⎢

⎤⎦⎥

−−

⎡⎣⎢

⎤⎦⎥{ }λ

λ λλ

λ λ, (11.72)

a f fno

do

21

2 1

3

2 3

= −−

⎡⎣⎢

⎤⎦⎥

−−

⎡⎣⎢

⎤⎦⎥{ }λ

λ λλ

λ λ, (11.73)

and

a fdo

31

1 3

2

2 3

=−

⎡⎣⎢

⎤⎦⎥ −⎡⎣⎢

⎤⎦⎥

λλ λ

λλ λ

, (11.74)

Figure 11.10 Representation of the Example A data by the one - and two - exponential transfer functions (after Civan and Rasmussen, 2001 ; © 2001 SPE, reproduced with permission from the Society of Petroleum Engineers).

00 1 2

Example A

Two-term fit

Data

One-term fit

tD3

0.2

0.4

0.6

Vf

V∞

0.8

1.0

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 401

where f fno

doand denote the volume fractions of the initially present movable oil

contained in the interconnected and dead - end pores of the matrix. It follows that

f fno

do+ = 1. (11.75)

It also follows from Eqs. (11.72) – (11.74) that (Civan and Rasmussen, 2001 )

a a a1 2 3 1+ + = , (11.76)

a a a1 1 2 2 3 3 0λ λ λ+ + = , (11.77)

a a a fno

1 12

2 22

3 32

1 2λ λ λ λ λ+ + = − , (11.78)

and

a a a f fno

do

1 13

2 23

3 33

1 2 1 2 1 2 3λ λ λ λ λ λ λ λ λ λ+ + = − + +( ) . (11.79)

These results will be used subsequently. Eqs. (11.76) and (11.77) are tanta-mount to the conditions V Vf f( ) and ( )0 0 0 0= =� applied to Eq. (11.64) .

The three - exponential matrix - to - fracture transfer function given by Eq. (11.71) contains several parameters, which are the characteristics of given rock and fl uid systems. These are the pre - exponential coeffi cients a a a1 2 3, , and (defi ned by Eqs. 11.72 – 11.74 ) in terms of the rate constants λ 1 , λ 2 , and λ 3 ; the movable oil initially present in the matrix V∞; the volume fractions of the initially present movable oil contained in the interconnected and dead - end pores of the matrices f fn d

° °and . Because there are nine unknown parameters related by four equations, namely, Eqs. (11.72) – (11.74) , 9 4 5− = parameters must be estimated by means of experimental data, using a least squares regression of Eq. (11.71) . However, because Eq. (11.71) is nonlinear, the parameter values cannot be determined uniquely. Therefore, some parameters should be directly measured. For example, the fraction of the dead - end and interconnected pores can be determined by a petrographical analysis of thin sections of porous rock. Uniqueness in the parameter values can also be achieved by enlarging the experimental database, similar to Ucan et al. (1997) , who developed a method of unique and simultaneous determination of relative permeability and capillary pressure curves from displacement data. For this purpose, waterfl ood oil recovery data obtained by injecting water into fractured cores along with the imbibi-tion oil recovery data obtained by exposing oil - saturated matrix blocks to water can be used together.

The matrix imbibition drive oil recovery experimental data by Guo et al. (1998) were considered by Civan and Rasmussen (2001) . They used 1.5 - in - diameter and 2 - in - long core plugs exposed to water and measured the cumulative volume of oil recovered as a function of time, expressed in percentage of the bulk volume of the cores, representing the matrix in naturally fractured reservoirs. A least squares, nonlinear regression of these data yielded the values of λ λ λ1 2 3, , , and fd

°. The remain-der of the parameters was calculated by Eqs. (11.72) – (11.74) using these values. The movable oil initially present in the matrix was determined by extrapolation, as the limit of the Guo et al. (1998) measured data. The location of the infl ection point of the plotted curves of the Guo et al. data was facilitated to aid in the regression process. The values of the parameters estimated using the Guo et al. (1998) data are presented in the following. The parameters of Example A are λ 1 = 0.247/day;

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402 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

λ 2 = 0.414/day; λ 3 = 0.640/day; a1, dimensionless = 3.886; a2, dimensionless = − 4.107; a3, dimensionless = 1.221; fd

o, fraction = 0.99; V∞, fraction = 0.471; φ f , fraction = 0.001; L = 1000 ft; vi = 168 ft/day; τ = 5.95 days; Swc, fraction = 0; and Sor, fraction = 0. The parameters of Example B are λ 1 = 0.0993/day; λ 2 = 1.83/day; λ 3 = 0.0060/day; a1, dimensionless = 0.471; a2, dimensionless = − 0.0175; a3, dimensionless = 0.557; fd

o, fraction = 0.522; V∞, fraction = 0.375; φ f , fraction = 0.001; L = 1000 ft; vi = 168 ft/day; τ = 5.95 days; Swc, fraction = 0; and Sor, fraction = 0.

Figures 11.10 and 11.11 show a comparison of the cumulative recovery pre-dicted by Eq. (11.71) , using one - , two - , and three - exponential terms, respectively, with the SP - 33 data extracted from Guo et al. (1998) . It is apparent that the three - exponent function fi ts the experimental data better than the others. These data are used in Example A by Civan and Rasmussen (2001) . Figure 11.12 shows that the three - exponential function, given by Eq. (11.71) , represents the SP - H9 data extracted from Guo et al. (1998) exactly because the number of parameters (fi ve) is equal to the number of data points. These data are used in Example B.

11.5.2 Formulation of the Fracture Flow Equation

Consider the natural fracture network shown in Figure 11.9 . Civan (1993, 1994d) has shown that a representative elemental volume averaging of the microscopic equation of continuity for a phase � leads to the following macroscopic, porous media equation of continuity:

Figure 11.11 Representation of the Example A data by the one - , two - , and three - exponential transfer functions (after Civan and Rasmussen, 2001 ; © 2001 SPE, reproduced with permission from the Society of Petroleum Engineers).

Vf

V∞

00 1 2

Example A

Three-term fit

tD3

0.2

0.4

0.6

0.8

1.0

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 403

Figure 11.12 Exact representation of the Example B data by the one - , two - , and three - exponential transfer functions (after Civan and Rasmussen, 2001 ; © 2001 SPE, reproduced with the permission of the Society of Petroleum Engineers).

00 5 10 15

Example B

Two-term fitThree-term fit

Data

One-term fit

tD

20

0.2

0.4

0.6

0.8

1.0V

f

V∞

∂∂( ) +∇⋅( ) −∇⋅ ⋅∇( ) + =

trε ρ ρ ε ρ� � � � � � � �u D 0, (11.80)

in which D� is an empirical hydraulic dispersion tensor. The third term can be inter-preted as a hydraulic dispersion due to spatial variations in porous media. The mass rate of phase � lost from the fracture to porous matrix is denoted by r�. The volume fraction, density, and volume fl ux of phase � are denoted by ε ρ� �, , and u�, respectively.

For the purposes of the analysis presented here, assume incompressible and immiscible fl uid phases in a constant porosity fracture medium and substitute

ε φ� �= f S . (11.81)

In this equation, φ f is the fracture porosity and S� is the saturation of phase � in the fracture. Also, defi ne the fractional volume, f�, of the fl owing phase � accord-ing to Civan (1994b) :

u u� �= f . (11.82)

In this equation, u is the total volumetric fl ux of the fl owing phase given by

u u=∑ �

�

. (11.83)

The volume rate of phase � lost from the fracture to porous matrix is given by

� � �q r= ρ . (11.84)

By substituting Eqs. (11.81) – (11.83) , Eq. (11.80) simplifi es to

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404 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

φ fS

tf q

∂∂

+ ⋅∇ + =�� ��u 0. (11.85)

The volumetric rate of oil transfer into the fracture is given by

� �q qdV

dtSo w

fw= − = ≤ ≤, . .0 1 0 (11.86)

Because the volume of water imbibed is equivalent to the volume of oil recov-ered, Eq. (11.86) also represents the rate of water imbibed by the matrix.

The derivation of Eq. (11.71) is based on the assumption that oil transferred into the fracture is rapidly carried away by the water fl owing through the fracture so that the fracture surface is always exposed to 100% water. Because the water saturation S w in a fracture varies, Eq. (11.71) should be used in Eq. (11.86) by a convolution according to deSwaan (1978) by applying Duhamel ’ s theorem:

� �q q tS

Sw

t

w= −( ) ∂∂

∂=∫ 1

0

ττ

τ. (11.87)

11.5.3 Exact Analytical Solution Using the Unit End - Point Mobility Approximation

The fractional fl ow function is given by (Luan, 1995 )

fMS

M Sw

w n

w n

=+ −

,

,( ),

1 1 (11.88)

where M is the end - point mobility ratio. Analytic solutions of the one - dimensional Buckley – Leverett fl ow problem

involving a one - parameter matrix - to - fracture transfer function have been presented by deSwaan (1978) , Davis and Hill (1982) , Kazemi et al. (1992) , and Luan and Kleppe (1992) . These analytical solutions have been possible after linearizing the governing integro - differential equation by invoking the unit end - point mobility ratio ( M = 1) assumption, which allows for approximating the fractional fl owing water f w of the fracture medium by the normalized water saturation S w , n according to

f SS S

S Sw w n

w wc

or wc

= ≡ −− −, ,

1 (11.89)

where Swc and Sor are the fracture medium connate water and the residual oil satura-tions, respectively, and Sw is the fracture medium water saturation.

Considering Eq. (11.63) and substituting Eq. (11.87) into Eq. (11.85) , deSwaan (1978) derived the following equation for the Buckley – Leverett immiscible dis-placement through a fracture network in porous media:

∂∂

+ ∂∂

+ ∂ −( )∂

=∞ − −( )∫S

t

u f

x

Ve

S tdw

f

w

f

t

t

w

φλ

φτ

ττλ τ

0

0, (11.90)

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 405

where u is the water injection fl ux into the porous medium, subject to the initial and boundary conditions given by

S S x L tw wc= ≤ ≤ =, ,0 0 (11.91)

and

f x tw = = >1 0 0 0. , , . (11.92)

deSwaan (1978) applied the unit end - point mobility approximation given by Eq. (11.89) and obtained an analytical solution given by

S x t tw n, , ,( ) = <0 α (11.93)

and

S x t e e I t d tw nt

o, , , .( ) = − −( )⎡⎣ ⎤⎦ ≥− −( ) −∫1 20

λ α τβ

λ α τ τ α (11.94)

Kazemi et al. (1989) derived an alternative form for Eq. (11.94) as

S x t e

e I t

e Iw n

to

o

, ,( ) =−( )⎡⎣ ⎤⎦

−( )⎡⎣ ⎤−

− −( )

− −( )+β

λ α

λ τ α

βλ α

βλ τ αλ

2

2 ⎦⎦

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪⎪

≥∫ d

tt

τα

α

, , (11.95)

where Io is modifi ed Bessel ’ s function of the zero type, and

α φ β λα φ= = ∞x u Vf f, . (11.96)

Luan and Kleppe (1992) have proven that Eq. (11.94) can be transformed into Eq. (11.95) , and therefore, they are identical.

11.5.4 Asymptotic Analytical Solutions Using the Unit End - Point Mobility Approximation

Civan and Rasmussen (2001) stated that the analytical solutions have resulted in complicated mathematical forms as demonstrated earlier, involving modifi ed Bessel functions and quadratures, and thus require tedious and frequently inaccurate pro-cedures to generate numerical values. Civan and Rasmussen (2001) presented a generalized methodology that obtains analytical solutions for imbibition waterfl oods in naturally fractured oil reservoirs undergoing multistep matrix - to - fracture transfer processes. The phenomenological representation of the oil transfer from matrix to fracture is based on a three - exponential matrix - to - fracture transfer function, the necessity for which is seen by the examination of the experimental data. The result-ing intego - differential equation is linearized by invoking the unit end - point mobility ratio assumption, converted to a fourth - order partial differential equation, and solved analytically by asymptotic means. It is shown that the asymptotic approximation approach signifi cantly reduces the complexity of the solution process and yields adequate solutions for a long - time evaluation of waterfl oods in naturally fractured reservoirs. The solution is not only computationally advantageous, but it also

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406 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

provides a physically meaningful interpretation of the propagation speed and dif-fusive spreading of the progressing wave front, which could not be readily obtained from the usual type of solution methods. Such analytical solutions are desired for convenient interpretation and correlation of laboratory core fl ood tests and verifi ca-tion of numerical solution schemes.

11.5.4.1 Formulation For the one - dimensional horizontal fl ow in fractured porous media considered here, the Civan et al. (1999) volumetric balance of fl owing water phase in fractures, obtained by combining Eqs. (11.64) and (11.85) – (11.87), can be written in partially dimensionless form as (Civan and Rasmussen, 2001 )

∂∂

+ ∂∂

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

∞ − − ′

=∑S

t

v

L

f

x

Va ew n

D

i w

D fj j

t t

j

t

j D D

D

, ( )τ τφ

λ τλ

1

3

0∫∫ ∂

∂ ′′ =

S

tdtw n

DD

, ,0 (11.97)

where vi is the fracture fl uid interstitial velocity given by

vu

S Si

f or wc

≡− −( )φ 1

. (11.98)

u is the fracture – fl uid volumetric fl ux, φ f is the fracture porosity, L is the reservoir length, and V∞ is the volume of the movable oil available in the matrix per unit bulk volume of porous media. Eq. (11.97) is an integro - differential equation.

The dimensionless distance and time are defi ned as

xx

LD ≡ (11.99)

and

tt

D ≡ τ, (11.100)

where τ is an appropriately selected characteristic timescale. As inferred by Eq. (11.97) , the characteristic timescale can be defi ned in a

variety of ways. The convection, matrix oil depletion, and rate process timescales can be defi ned alternatively as

τ =L

vi

(11.101)

or

τφλ

= =∞

f

j ja Vj, or 31 2, , (11.102)

or

τλ

= =11 2 3

j

j, , , .or (11.103)

Because the fl ow in fractures is convection dominated, Eq. (11.101) is selected for the characteristic time; that is, τ = L vi/ .

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 407

It is convenient to defi ne several more nondimensional parameters:

Λ jj

i

L

vj≡ =

λ, , ,1 2 3 (11.104)

and

KV La

v

V ajj

j j

f i

j

fj≡ = =∞ ∞λ

φ φΛ , , , .1 2 3 (11.105)

Further, defi ne the following nondimensional functions of x tD Dand :

C x t K e eS

tdt jj D D j

t t w n

DD

t

j D j D

D

( , ) , , ,,=∂∂ ′

′ =− + ′∫Λ Λ

0

1 2 3 (11.106)

When the above - mentioned nondimensional parameters and variables are used, then Eq. (11.97) can be written in the following simplifi ed nondimensional form:

∂∂

+ ∂∂

+ ==∑S

t

f

xC x tw n

D

w

Dj D D

j

, ( , ) .01

3

(11.107)

The initial and boundary conditions for either Eq. (11.97) or (11.107) are

S x tw n D D, ( , )≥ = =0 0 0 (11.108)

and

f x tw D D( , ) .= > =0 0 1 (11.109)

11.5.4.2 Small - Time Approximation For small times, the approximate ana-lytical solution is given by

S x t

x E

Ee

H t x

w n D DD E t x

D D

D D, ,

( )

,

( ) ≈ − − −( )⎡⎣⎢

⎤⎦⎥

−( )

− −( )11

11

2

2��

�α

ttD → 0,

(11.110)

where H t xD D( )− is the Heaviside unit step function ( H y( ) = 0 when y < 0, and H y( ) = 1 when y ≥ 0). The nondimensional diffusivity that governs the spreading out of the prevailing long - time bulk wave front is given by

ν β αβ

≡ −��

E

D1

3

( ), (11.111)

where

αφ

= ++ +

⎡

⎣⎢

⎤

⎦⎥

∞1 1 2

1 2 3 1 2

Λ ΛΛ Λ Λ Λ Λ

f Vno

f( ), (11.112)

βφ

= + ∞1V

f

, (11.113)

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408 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

�D ≡ Λ Λ Λ1 2 3, (11.114)

�E2 1 2 3≡ + +Λ Λ Λ , (11.115)

and

�E1 1 2 3 1 2≡ + +( ) .Λ Λ Λ Λ Λ (11.116)

Thus, initially, a discontinuity wave propagates to the right with the nondi-mensional speed vw = 1. The spatial variation between the injection face xD = 0 and the front location at x tD D= can be seen from a numerical example. The data used for the numerical examples, denoted as Examples A and B, are given in the above. These values will be explained in a later section. The small - time spatial variations with xD are shown in Figure 11.13 for tD = 0 01 0 02 0 03. , . , and . , according to Example B and Eq. (11.110) . For small times, Sw n, decreases from unity at the injection face to a minimum value and then increases to unity at the outward propagating discon-tinuity surface. After a long enough time, near the injection face, Sw n, begins to increase back toward unity, and the slope of the curve with respect to xD approaches its long - time value of zero.

11.5.4.3 Approximation for Large Time For large times, the approximate analytical solution is given as

S x t erfcb ax t

te erfc

b ax t

tw n D D

D D

D

ab x D D

D

D, ( , )

( ) ( )= −⎡⎣⎢

⎤⎦⎥+ +1

24 2 ⎡⎡

⎣⎢⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

, (11.117)

Figure 11.13 Short - time behavior of the saturation profi les at the dimensionless times tD = 0.01, 0.02, and 0.03 for Example B (after Civan and Rasmussen, 2001 ; © 2001 SPE, reproduced with permission from the Society of Petroleum Engineers).

Example B

00

0.5

1.0

0.01

tD = 0.01 0.02 0.03

0.02 0.03

xD

VwSw

,n

0.04

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11.5 IMMISCIBLE DISPLACEMENT IN NATURALLY FRACTURED POROUS MEDIA 409

where erfc y( )is the complementary error function with argument y . The parameters a and b are given by

a bv

≡ ≡ββ

and ,1

2 (11.118)

where ν is the effective diffusivity given by Eq. (11.111) . Letting Λ3 →∞ and then Λ2 →∞ yields the single - term transfer function result of Rasmussen and Civan (1998) .

When the argument of the second error function in Eq. (11.117) is very large, the value of the error function is exponentially small. Some computer software packages may set the error function erfc ( y ) identically equal to zero when y is large enough. This may cause some error in numerical calculations when x D is also large enough for the exponential multiplying term exp (4 ab 2 x D ) to compensate for the smallness of the error function. It is thus useful to use a two - term asymptotic expansion for erfc ( y ) when y is large and to express the second term in Eq. (11.117) as

e erfcb ax t

t

t

b ax t

t

b ax t

ab x D D

D

D

D D

D

D D

D4

2 2

2

12

( )

( )

( )

+⎡⎣⎢

⎤⎦⎥≈

+

−+

⎡

π

⎣⎣⎢⎤⎦⎥

− −⎡⎣⎢

⎤⎦⎥

exp( )

.b ax t

tD D

D

2 2 (11.119)

Therefore, we obtain the following solution at the bulk wave front x D = t D / a for large t D :

St

at

b t b tw n

DD

D D, , .⎛⎝⎜

⎞⎠⎟ = + −⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

1

21

1

21

1

8 2π (11.120)

For a comparison of results, Rasmussen and Civan (1998) used the values used previously by Kazemi et al. (1992) and Civan (1993) : v I = 168 ft/day, L = 1000 ft, λ = 0.1/day, ϕ = 0.001, R ∞ = 0.08, and S wc = S or = 0. These values yield τ = 5.95 days, Λ = 0.595, K = 47.6, a = 81, and b = 0.39. The profi les for S w , n as a function of nondimensional distance x D ≡ x / L are shown in Figure 11.14 for the nondimensional times of τ D ≡ t / τ = 5, 10, 15, and 20, which are relatively small. The approximate solution, according to Eq. (11.94) , compares well with the exact solution as com-puted by the formula of Kazemi et al. (1992) , and it improves as t D increases. (Note, for example, that t D = 5 represents a time t of about 30 days.) When S w , n is plotted for larger values of t D on a correspondingly expanded scale of x D , the difference between the approximate and exact solutions becomes harder to detect. The maximum error in S w , n , for a given value of x D near the central part of the wave, is about 3.5%. This error decreases as t D increases. This error is quite acceptable for fi eld applica-tions, considering that many reservoir parameters cannot be reckoned within this accuracy.

The long - time behaviors for Examples A and B are shown in Figure 11.15 for t D = 100. The dimensionless diffusivity for Example A is v = × −5 99 10 6. , whereas the value for Example B is much larger at v = × −1 15 10 4. . The dimensionless wave

c11.indd 409c11.indd 409 5/27/2011 12:35:46 PM5/27/2011 12:35:46 PM

410 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

speed of the bulk wave front, which is the propagation speed at approximately S w , n = 1/2, is vw = 0 00212. for Example A and v w = 0.00266 for Example B.

11.6 METHOD OF WEIGHTED SUM (QUADRATURE) NUMERICAL SOLUTIONS

Civan (1993, 1994a,c,d,e, 1998c) presented an effi cient solution for deSwaan ’ s formulation of waterfl ooding in naturally fractured porous media by the quadrature

Figure 11.14 Comparison of exact and approximate normalized water saturation as a function of nondimensional distance at various nondimensional times (after Rasmussen and Civan, 1998 ; © 1998 SPE, reproduced with permission from the Society of Petroleum Engineers).

00

0.2

0.4

0.6

0.8

1.0

0.05

10 15 20

Approximate Results

0.1 0.20.15 0.25 0.3 0.35

Nondimensional Distance, xD

Wat

er S

atur

atio

n, S

w,n

tD = 5

0.4

Figure 11.15 Long - time saturation profi les for Examples A and B for the dimensionless time of t D = 1000 (after Rasmussen and Civan, 1998 ; © 1998 SPE, reproduced with permission from the Society of Petroleum Engineers).

Vw Vw

00 0.1 0.2 0.3

Example A Example B

xD

Sw

,n

tD = 100

0.4 0.5

0.2

0.4

0.6

0.8

1.0

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11.6 METHOD OF WEIGHTED SUM (QUADRATURE) NUMERICAL SOLUTIONS 411

method. Comparisons of the analytic, fi nite difference, and quadrature solutions of the unit end - point mobility ratio problem reveal that the quadrature solutions are more accurate and are much more easily obtained than the fi nite difference solutions. The computational requirements of the nonlinear problem with capillary pressure included are comparable to that of the unit end - point mobility ratio problem. Civan (1998c) has shown that the fracture medium capillary pressure has a small effect, while the oil viscosity has a large effect on the oil recovery from naturally fractured reservoirs.

11.6.1 Formulation

Civan (1998c) considered a one - dimensional immisicible displacement of oil by water with a constant - rate water injection and constant fl uid and formation properties.

By substituting the interstitial fracture medium fl uid velocity defi ned by Eq. (11.98) , Eq. (11.85) can then be written for the fracture medium water phase as

∂∂

+ − −( ) ∂∂

− ∂∂

∂

= ≤ ≤ >

∞ − −( )∫S

tS S v

f

x

Ve

S

x L t

wwc or

w

f

t

t

w1

0 00

λφ τ

τλ τ

, , 00.

(11.121)

The conditions of the solution are considered as the following:

S S x L tw wc= ≤ ≤ =, ,0 0 (11.122)

and

f x tw = = >1 0 0, , . (11.123)

For computational convenience, Civan (1991, 1993, 1994a) also introduced a series of manipulations of Eq. (11.121) : (1) multiplying by e tλ , (2) differentiating with respect to time and dividing by e tλ , and (3) integrating with respect to time and then applying the initial condition given by Eq. (11.122) to obtain

∂∂

+ − −( ) ∂∂

+ ∂∂

∂⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ +⎛⎝⎜

⎞⎠⎟

∫∞

S

tS S

v

L

f

X

f

X

V

wwc or

w w

t

f

1

1

0

λ τ

λφ

SS Sw wc−( ) = 0,

(11.124)

in which the distance normalized with respect to the reservoir length L is given by

X x L= . (11.125)

The volumetric fl ux of the water phase through the fracture system is given by

u Kwr

wf w w

kp g zw= − ⋅ ∇ + ∇( )

μρ . (11.126)

Civan (1994a) defi ned a scalar fractional water, f w , according to

u uw wf= . (11.127)

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412 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

Thus, equating Eqs. (11.126) and (11.127) for a horizontal, one - dimensional fl ow and then integrating leads to the following expression for the water phase pressure:

p pu

K

f

kdxw w

w

f

w

rw

x

in= − ∫μ

0

. (11.128)

For convenience, the subscripts of S w and f w are dropped in the following. Function f denotes the fracture medium fractional water function (Marle, 1981 ) given by

f L S X= + ( )∂ ∂ϕ Ψ , (11.129)

in which

ϕ μ ρ ρ θ μ μ= + −( )⎡⎣⎢

⎤⎦⎥

+⎡⎣⎢

⎤⎦⎥−

o

r

fw o

w

r

o

rk

K

ug

k ko w o

sin1

(11.130)

and

Ψ = +⎡⎣⎢

⎤⎦⎥−

K

u

dp

dS k kf cf w

r

o

rw o

μ μ1

. (11.131)

where θ is the inclination angle; μ w and μ o are the viscosities, ρ w and ρ o are the densities, and krw and kro are the fracture – media fl uid relative permeabilities of the water and oil phases, respectively; K f is the fracture permeability; and g is the gravitational acceleration. The fracture media fl uid capillary pressure was repre-sented by a three - constant hyperbola given by Donaldson et al., (1991) :

pA BS

CScf =

++1

, (11.132)

where A , B , and C are some empirical constants. For convenience in numerical solution, Eq. (11.124) is decomposed into two

differential equations as

∂∂+ − −( ) ∂

∂+⎡

⎣⎢⎤⎦⎥

+ +⎛⎝⎜

⎞⎠⎟

−( ) =∞

S

tS S

v

L

f

Xy

VS S

wc or

fw wc

1

1 0

λ

λφ

(11.133)

and

∂∂= ∂∂

y

t

f

X, (11.134)

where y is a dummy variable. The initial and boundary conditions are given, respec-tively, by

S S y X twc= = ≤ ≤ =, , ,0 0 1 0 (11.135)

and

f X t= = >1 0 0, , . (11.136)

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11.6 METHOD OF WEIGHTED SUM (QUADRATURE) NUMERICAL SOLUTIONS 413

11.6.2 Quadrature Solution

The quadrature is an algebraic rule by which the numerical value of a linear opera-tion on a function at a given point is approximated as a weighted linear sum of the discrete function values at selected points around that point. The quadrature formula is given by (Civan, 1991, 1994a )

L f x w f x i ni ij j

j

n

( ){ } ≅ ( ) ==∑

1

1 2; , , , .… (11.137)

L is a linear operator such as a partial derivative or an integral of any order, or a composite of derivatives, integrals, function values, and constants. Variables x i ; i = 1, 2, … , n denote the location of discrete points, each of which are associated with a discrete function value of f ( x i ). Variable n is the number of discrete values considered for the quadrature approximation. w ij are the quadrature weights.

To determine the quadrature weights, a convenient function for the local rep-resentation of the solution of the problem is considered, generalized in the form of (Civan, 1994a, 1998c )

f x a B xv v

v

( ) = ( )=

∞

∑0

. (11.138)

Variables B v ( x ) are some monomials and a v are the associated weights. Then, invoking Eq. (11.138) , Eq. (11.137) leads to the following moment

equations:

B x w B x i n v nv j ij

j

n

v i( ) = ( ){ } = = −=∑

1

1 2 0 1 1L ; , , , and , , , .… … (11.139)

For a set of prescribed discrete point values, x i ; i = 1, 2, … , n , the set of linear equations given by Eq. (11.139) is solved simultaneously for the quadrature weights w ij .

For the application, consider a basic power series representation. Hence, Eq. (11.139) becomes

x w x i n v njv

ij

j

nv

i=∑ = { } = = −

1

1 2 0 1 1L ; , , , and , , , .… … (11.140)

In the applications, the quadrature weights for fi rst - order partial derivatives are required. Thus, the linear operator is

L ≡ ∂ ∂x . (11.141)

Applying Eq. (11.141) , Eq. (11.140) becomes

x w x x vx i n v njv

ij

j

nv

i iv

=

−∑ = ∂ ∂ { } = = = −1

1 1 2 0 1 1; , , , and , , , .… … (11.142)

Because the coeffi cients of Eq. (11.142) form a Vandermonde matrix, a unique solution for the quadrature weights w ij is obtained effi ciently using the method by Bj ö rck and Pereyra (1970) . The quadrature weights were obtained for 6, 11, and 21 equally spaced points and were used in the following.

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414 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

For numerical solution, replace the spatial derivatives in Eqs. (11.133) and (11.134) by the quadrature approximation to obtain the following ordinary differential equations:

dS

dtS S

v

Lw f y

VS Si

wc or ij j i

j

n

fi= − − −( ) +

⎛

⎝⎜

⎞

⎠⎟ − +

⎛⎝⎜

⎞⎠⎟

−=

∞∑1 11

λ λφ wwc( ) (11.143)

and

dy

dtw f n ti

ij j

j

n

= = >=∑

1

1 2 3 0, , , and .… (11.144)

Eqs. (11.143) and (11.144) are subject to the initial conditions that

S S y i n ti wc i= = = =, , , , and .0 1 2 0… (11.145)

The boundary value at the point i = 1, Eq. (11.136) , is

f i t1 1 1 0= = >, , . (11.146)

The water phase pressure is calculated by Eq. (11.128) as

p pu

Kc

f

ki nw w

w

fij

r jj

n

i in

w

= − ⎛⎝⎜

⎞⎠⎟

==∑μ

1

2 3: , , , ,… (11.147)

where c ij ’ s denote the integral quadrature weights. Then, the oil phase pressure is calculated by the fracture capillary pressure defi nition:

p p pcf o w= − . (11.148)

The numerical solution of the simultaneous differential equations was obtained using a variable - step Runge – Kutta – Fehlberg four (fi ve) method (Fehlberg, 1969 ). First, the solution of Eqs. (11.143) and (11.144) was carried out by approximating the fractional fl ow of water according to the unit end - point mobility ratio approach by deSwaan (1978) :

f S= . (11.149)

Second, the nonlinear fractional water function given by Eqs. (11.129) – (11.132) was used. By replacing the spatial derivative by the quadrature approxima-tion, Eq. (11.129) yields

fL

w Si ii

ij j

j

n

= +=∑ϕ Ψ

1

. (11.150)

Civan (1994a, 1998c) obtained the numerical solution of Eqs. (11.143) and (11.144) using Eq. (11.146) with Eqs. (11.149) or (11.150) for 6, 11, and 21 equally spaced grid points with the data given in the following: ϕ f = 0.001, k f = 1.0 darcy, L = 1000 ft, A = 1000 ft 2 , λ = 0.1/day, v = 168.4 ft/day, V ∞ = 0.08, S wc = 0.0, S or = 0.0, μ w = 1.0 cP, μ o = 1.0 or 3.0 cP, k rw = S , k ro = 1 − S , p cf = 2(1 − S)/(1 + 5S) psi, and θ = 0.0. Figure 11.16 shows a comparison of numerical solutions for f = s and f x= + ∂ ∂ϕ Ψ using 1 - and 3 - cP oil viscosities. It can be seen that the capillary pressure plays a small role, while the oil viscosity has a large effect on the displacement of oil by water in naturally fractured formations.

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11.7 FINITE DIFFERENCE NUMERICAL SOLUTION 415

The method presented here can be readily extended to multidimensional cases according to Civan (1991, 1994a) . However, in contrast to the one - dimensional example presented earlier, the velocity components depend on the multidimensional pressure and permeability distributions. Therefore, a simultaneous solution of the fracture medium fl uid pressure and saturation equations, similar to Kazemi et al. (1992) , is required.

11.7 FINITE DIFFERENCE NUMERICAL SOLUTION

Civan et al. (1999) investigated the effect of matrix - to - fracture transfer on the oil displacement by water imbibition in naturally fractured porous media in terms of the wettability effects and the governing rate processes. A mathematical model was developed by coupling the two - phase fl ow in the fracture network and in the porous matrix via an oil – water exchange function that incorporates the rates of transfer of oil from the dead - end pores to the network of pores and then to the network of fractures. The resulting integro - differential equation was solved numerically using the fi nite difference method. The parametric studies carried out utilizing this model indicated that the rate constants and the matrix wettability play important roles in obtaining an accurate description of the oil recovery during waterfl ooding in natu-rally fractured reservoirs.

Figure 11.16 Comparison of solutions for f = s and f = f ( P c ) (after Civan, 1998c ; © 1998 SPE, reproduced with permission from the Society of Petroleum Engineers).

0.1

0.2

0.3

0.4

0.5

Fra

ctur

e w

ater

sat

urat

ion,

s

0.6

f = s, 150 daysf = s, 300 daysf = s, 450 daysf = f (Pc), 1 cP oil,150 daysf = f (Pc), 1 cP oil,300 daysf = f (Pc), 1 cP oil,450 days

f = f (Pc), 3 cP oil,300 daysf = f (Pc), 3 cP oil,450 days

f = f (Pc), 3 cP oil,150 days

0.7

0.8

0.9

1.0

00 200 400

x, Distance(ft)

600 800 1000

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416 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

11.7.1 Formulation

Figure 11.17 presents a schematic of a network of fractures and pores in a naturally fractured reservoir. Assuming that the water and oil phases are incom-pressible, the water injection rate during waterfl ooding is constant, and the fl ow is horizontal and one dimensional, the macroscopic equation of continuity for the water phase in the REV of a network of fractures in a naturally fractured reservoir is given as

φ fw w

wS

tu

f

xq

∂∂

+ ∂∂

+ =� 0. (11.151)

�qw is the volumetric rate of water lost from the fracture to the porous matrix by imbibition. Neglecting the capillary pressure and gravity effects, the fractional fl ow of water for horizontal fl ow in the network of fractures can be expressed as

f

k

k

wro

o

w

rw

=+

1

1μ

μ . (11.152)

kro and krw are the fracture oil and water phase permeabilities, respectively. Assuming a linear dependency of the relative permeabilities of the fl uids in the

fracture on the normalized saturations, Eq. (11.152) can be simplifi ed as (Luan, 1995 )

fMS

M Sw

w

w

=+ −( )1 1

, (11.153)

where M and sw are the end - point mobility ratio and the normalized saturation given, respectively, by

Mk

krw o

ro w

=*

*

μμ

(11.154)

and

Figure 11.17 Schematic of a network of fractures and pores in a naturally fractured reservoir (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

WaterInjection

H

L

W

FractureRepresentativeElementary Volume Matrix

Oil and/or WaterProduction

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11.7 FINITE DIFFERENCE NUMERICAL SOLUTION 417

SS S

S Sw

w wc

wc or

= −− −1

. (11.155)

S wc and S or denote the connate water and residual oil, respectively. The Gupta and Civan (1994a) model, given by the following three -

exponent expression, was considered for the volume of oil transferred into the fracture network:

V V a e a e a eft t t= − − −( )∞

− − −1 1 2 31 2 3λ λ λ , (11.156)

where a 1 , a 2 , and a 3 , and λ 1 , λ 2 , and λ 3 are related by Eqs. (11.72) – (11.75) . λ 1 , λ 2 , and λ 3 can be expressed in a dimensionless form as

λ λ

σ θμ φ

iDi

s

w m

F ki=

⎡

⎣⎢

⎤

⎦⎥

=cos

; , , .1 2 3 (11.157)

Here, λ iD is the dimensionless rate constant; σ is the interfacial tension; θ is the contact angle; μ w is the viscosity of injected water; k is the absolute permeability of matrix; ϕ m is the porosity of matrix; and F s is the shape factor. The shape factor F s , which was proposed by Kazemi et al., (1992) , incorporates the variable shape of the matrix blocks as well as the imposed boundary conditions. F s is calculated by the following equation:

Fw

sf

= 42

. (11.158)

Here, w f is the mean fracture spacing of the system. By applying Eqs. (11.86) , (11.87) , and (11.156) , the volumetric fl ow rate of

the water lost by imbibition from the fractures to the matrix is obtained as follows:

�q Va e a e

a ew

t t

t=

++⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦∞

− −( ) − −( )

− −( )1 1 2 2

3 3

1 2

3

λ λλ

λ τ λ τ

λ τ ⎥⎥∂ ( )∂

∂∫ Sw

tττ

τ0

. (11.159)

Therefore, invoking Eqs. (11.155) and (11.159) into Eq. (11.151) leads to the following integro - differential equation for the water saturation in the fracture network:

∂∂

+ ∂∂

++

+∞

− −( ) − −( )

− −( )S

tv

f

x

V a e a e

a ew w

f

t t

tφλ λλ

λ τ λ τ

λ τ

1 1 2 2

3 3

1 2

3

⎡⎡

⎣⎢

⎤

⎦⎥∂ ( )∂

∂ =∫ Sw

tττ

τ 00

, (11.160)

in which

vu

S Sf wc or

=− −( )φ 1

. (11.161)

Here, v is the interstitial velocity of the fl uid in fracture. For convenience in the numerical solution, by applying the integration by parts

and noting that sw = 0 because s sw wc= at t = 0, Eq. (11.160) can be rearranged as

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418 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

∂∂

+ ∂∂

+ + +( ) −∞ ∞ − −( )S

tv

f

x

V Sa a a

Ra e S dw w w

f f

twφ

λ λ λφ

λ τλ τ1 1 2 2 3 3 1 1

2 1

00

2 22

3 32

00

2 3 0

t

tw

tw

tt

a e S d a e S d

∫

∫∫

⎡

⎣⎢⎢

+ +⎤

⎦⎥⎥=− −( ) − −( )λ τ λ τλ τ λ τ ..

(11.162)

Then, Eq. (11.162) can be solved to obtain the distribution of the water satura-tion in the fractures at different positions and times with the initial and boundary conditions given, respectively, by

S S x L tw wc= ≤ ≤ =, ,0 0 (11.163)

and

f x tw = = >1 0 0 0. , , . (11.164)

The cumulative oil volume produced at the outlet of the system is given by

Q Auf dto w

t

= ∫0

. (11.165)

The initial recoverable oil in the system is given by

V WHL S S S Sm wc or m f wc or f0 1 1= − −( ) + − −( )⎡⎣ ⎤⎦φ φ . (11.166)

The subscripts m and f denote the matrix and fracture, respectively. Therefore, the oil recovery factor can be calculated by the following equation:

RQ

Ve

o

o

= . (11.167)

11.7.2 Numerical Solutions

In Eq. (11.162) , there are three time integral terms that are of the same type but have different values of λ 1 , λ 2 , and λ 3 . Applying the trapezoidal rule, each of the integral terms can be discretized as

I e S dt

e S e S en tw

t

tw t

t twj t tj= = + +− −( ) − −( )

=( )− −( ) − −∫ λ τ λ λ λτ

0

00

22

Δ (( )

=

−

∑⎡

⎣⎢⎢

⎤

⎦⎥⎥

Swn

j

n

1

1

, (11.168)

in which Sw t=( ) =0 0 because S Sw wc= at t = 0. The Crank – Nicolson implicit fi nite difference of Eq. (11.162) (central in time

and backward in space) and using Eq. (11.168) leads to the following discretized equation:

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11.7 FINITE DIFFERENCE NUMERICAL SOLUTION 419

S S

t

v f f f f

x

Vaw

nwn

wn

wn

wn

wn

f

i i i i i i

+ + +∞−

+− + −⎛

⎝⎜⎞⎠⎟+− −

1 1 1

2 21 1

Δ Δ φ 11 1 2 2 3 31

2

1

3

1 12

2

22

λ λ λ

φ

λ λ

+ +( ) +( )

−

+

+

∞

=∑

a a S S

V

a I at

wn

wn

f

i i in

i

i i

Δee S S

at

t twij

wn

j

n

n j

i

− −( ) +

=

+ +⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

∑ λ

λ

1 1 1

1

2 22

22

Δee S S

at

t twij

wn

j

n

n j

i

− −( ) +

=

+ +⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

∑ λ

λ

2 1 1

1

3 32

22

Δee S St t

wij

wn

j

n

n j

i

− −( ) +

=

+ +⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪ ∑ λ3 1 1

1⎪⎪

⎫

⎬

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

= 0

(11.169)

Note that the I n value is calculated at the end of the previous time step. A Taylor series expansion truncated after the second term yields (Luan, 1995 )

f ff

tti

nin

i

n+ ≅ + ∂

∂⎛⎝⎜

⎞⎠⎟

1 Δ , (11.170)

in which, applying Eq. (11.153) ,

∂∂

⎛⎝⎜

⎞⎠⎟ =

+ −( )[ ]−+f

t

M

M S

S S

tw

i

n

win

win

win

1 12

1

Δ. (11.171)

Substituting Eq. (11.171) into Eq. (11.169) and solving for Swin+1 results in the

following equation used for the explicit numerical solution in the present study:

SS

t

v

x

f f f

MS

M S

win wi

nwin

win

win

win

win

+

− −+

= −− −

−+ −( )[ ]

1

1 11

2

2

1 1Δ Δ 22

1 1 2 2 3 3

2

2

2

Δt

R Sa a a

Ra I

win

f

f

i i i

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

− + +( )

+

∞

∞

φλ λ λ

φ

λ nn

i

i it t

wj

j

n

i

t a e Si n j

=

− −( )

==

∑

∑∑+⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥

+

1

3

2

11

3

1Δ λ λ⎥⎥⎥⎥

⎫

⎬

⎪⎪

⎭

⎪⎪

÷ ++ −[ ]

⎧⎨⎪

⎩⎪

+ + +∞

1

2 1 1

2

2

1 1 2 2

Δ Δ Δt

v

x

M

M S t

Ra a

win

f

( )

φλ λ aa

R ta a a

f3 3 1 1

22 2

23 3

2

4λ

φλ λ λ( ) − + +( )⎫⎬

⎭∞Δ .

(11.172)

Figure 11.18 shows the schematic of the grid system used. Eqs. (11.161) – (11.164) were solved according to the numerical scheme derived previously. The parameters used are listed in the following: S wc , fraction = 0.0; S or , fraction = 0.0;

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420 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

ϕ f , fraction = 0.001; ϕ m , fraction = 0.1; K = 1.18E - 13 m 2 ; μ w = 0.0009 Pa · s; R n , fraction = 0.08; R d , fraction = 0.02; v = 51.33 m/day; L = 304.8 m; W = 30.48 m; H = 3.05 m; M , dimensionless = 3.0; σ = 0.035 N/m; λ D 1 , dimensionless = 8.4; λ D 2 , dimensionless = 0.0807; λ D 3 , dimensionless = 0.0115; θ 1 = 0 ° ; θ 2 = 30 ° ; θ 3 = 60 ° ; w f 1 = 30.48 m; w f 2 = 60.96 m; and w f 3 = 152.4 m.

The calculations were conducted for fi ve different cases. First, only λ 1 was included in the calculation, which assumed that the fl ow rate from matrix to fracture is dominantly greater; that is λ 2 >> λ 1 and λ 2 >> λ 3 . Second, both λ 1 and λ 2 were included and λ 3 was neglected. The third case considered all three parameters, λ 1 , λ 2 , and λ 3 simultaneously. In the fourth case, the distributions of water saturation at 300 days were calculated for different wettabilities with contact angles of 0 ° , 30 ° , and 60 ° and by considering all three exponent functions. The fi fth case studied the effect of the fracture spacing on the distributions of water saturation in fracture. The calcula-tions were carried out for a fracture spacing of 30.48 m (100 ft), 60.96 m (200 ft), and 152.4 m (500 ft). Figures 11.19 – 11.23 show the simulation results for these fi ve cases, respectively. The oil recovery factor was calculated for the aforementioned cases of transfer functions, different wettability, and different fracture spacing. Figures 11.24 – 11.26 show the effects of these parameters on the oil recovery factor.

Figure 11.18 Schematic of the grid system used for the numerical solution (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

2

1

2

n

n + 1

Δt

Δx

n = 0i = 1

i, n + 1/2

i, n + 1i – 1, n + 1

i , ni – 1, n

i – 1 I – 1I – 2 I i

x

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11.7 FINITE DIFFERENCE NUMERICAL SOLUTION 421

Figure 11.19 Prediction of fracture water saturation using a single - exponent transfer function (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

00

0.1

0.2

0.3

Wat

er s

atur

atio

n (f

ract

ion)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100

150 days

300 days

450 days

200 300 400 500

Distance (ft)

600 700 800 900 1000

Figure 11.20 Prediction of fracture water saturation using the two - exponent transfer function (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

150 days

300 days

450 days

00

0.1

0.2

0.3

Wat

er s

atur

atio

n (f

ract

ion)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100 200 300 400 500

Distance (ft)

600 700 800 900 1000

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422 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

Figure 11.21 Prediction of fracture water saturation using the three - exponent transfer function (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

150 days

300 days

450 days

00

0.1

0.2

0.3

Wat

er s

atur

atio

n (f

ract

ion)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100 200 300 400 500

Distance (ft)

600 700 800 900 1000

Figure 11.22 Effect of wettability on fracture water saturation at 300 days using the three - exponent transfer function (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

Contact angle = 0°

Contact angle = 30°

Contact angle = 60°

00

0.1

0.2

0.3

Wat

er s

atur

atio

n (f

ract

ion)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100 200 300 400 500

Distance (ft)

600 700 800 900 1000

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11.7 FINITE DIFFERENCE NUMERICAL SOLUTION 423

Figure 11.23 Effect of fracture spacing on fracture water saturation at 300 days using the three - exponent transfer function (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission of the Society of Petroleum Engineers).

Fracture spacing = 100 ft

Fracture spacing = 200 ft

Fracture spacing = 300 ft

00

0.1

0.2

0.3

Wat

er s

atur

atio

n (f

ract

ion)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100 200 300 400 500

Distance (ft)

600 700 800 900 1000

Figure 11.24 Effect of the transfer function on the oil recovery factor (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

Single-exponent transfer functionTwo-exponent transfer functionThree-exponent transfer function

00

0.1

0.2

0.3

Oil

reco

very

fact

or (

frac

tion)

0.4

0.5

0.6

0.1 0.2 0.3 0.4Pore volume injected

0.5 0.6 0.7 0.8

c11.indd 423c11.indd 423 5/27/2011 12:35:48 PM5/27/2011 12:35:48 PM

424 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

Figure 11.25 Effect of wettability on the oil recovery factor (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

Contact angle = 0°Contact angle = 30°Contact angle = 60°

00

0.05

0.1

0.15

Oil

reco

very

fact

or (

frac

tion)

0.2

0.25

0.3

0.1 0.2 0.3 0.4Pore volume injected

0.5 0.6 0.7 0.8

Figure 11.26 Effect of fracture spacing on the oil recovery factor (after Civan et al., 1999 ; © 1999 SPE, reproduced with permission from the Society of Petroleum Engineers).

Fracture spacing = 100 ftFracture spacing = 200 ftFracture spacing = 500 ft

00

0.05

0.1

0.15

Oil

reco

very

fact

or (

frac

tion)

0.2

0.25

0.3

0.1 0.2 0.3 0.4

Pore volume injected

0.5 0.6 0.7 0.8

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11.8 EXERCISES 425

From these results, it was observed that λ 1 and λ 2 have signifi cant effects on the distributions of water saturation in fracture systems, but λ 3 has a much smaller effect compared with λ 1 and λ 2 . Wettability has signifi cant effects on the distribution of water saturation in the fracture. There is a smaller change of water saturation distribution when the contact angle changes from 0 ° to 30 ° , but a bigger change occurs when the contact angle changes from 30 ° to 60 ° . Fracture spacing is also important for the distribution of water saturation in fractures. The water saturation in fractures increases with the increase of the fracture spacing. The oil recovery rate appears to increase with a decrease in the number of exponents from three to two and then to one. Therefore, the single - exponent transfer function used by previous researchers may overpredict the recovery rate. The oil recovery rate decreases by increasing contact angles and fracture spacing.

11.8 EXERCISES

1. Determine the shape factor expression to replace Eq. (11.25) when the Cauchy - type boundary condition given by Eq. (11.14) is applied. (Hint: See Moench, 1984. )

2. Determine the shape factor expressions to replace Eqs. (11.33) and (11.34) when the Cauchy - type boundary condition given by Eq. (11.14) is applied. (Hint: See Moench, 1984. )

3. Determine the total volumetric fl ow across a matrix block using Eq. (11.34) for the shape factor.

4. Derive the expression for the shape factor for an infi nite, long, cylindrical - shaped matrix block of radius R using the formulation of Lim and Aziz (1995) given next and verify their solution:

φμm m

mm

mcp

tK

r rr

p

rr R t

∂∂

= ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟ ≤ ≤ >1 1

0 0, , , (11.173)

p p r R tm i= ≤ ≤ =, , ,0 0 (11.174)

and

p p r R tm f= = >, , .0 (11.175)

Show that

p p

p p R

K t

cm i

f i n

n m

m mn

−−

= − −⎛⎝⎜

⎞⎠⎟

=

∞

∑14 1

2 2

2

1α

αμφ

exp , (11.176)

where αn: n = 1, 2, ... are obtained as the roots of the Bessel function of the fi rst kind of order zero, given by J Ro nα( ) = 0. Using only the fi rst term in the summation, confi rm that the shape factor is given by σ =18 17 2. /L .

5. Derive the expression for the shape factor for a spherical - shaped matrix block of radius R using the formulation of Lim and Aziz (1995) given next and verify their solution:

φμm m

mm

mcp

tK

r rr

p

rr R t

∂∂

= ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟ ≤ ≤ >1 1

0 02

2 , , , (11.177)

p p r R tm i= ≤ ≤ =, , ,0 0 (11.178)

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426 CHAPTER 11 TRANSPORT IN HETEROGENEOUS POROUS MEDIA

and

p p r R tm f= = >, , .0 (11.179)

Show that

p p

p p n

n K t

c Rm i

f i

m

m mn

−−

= − −⎛⎝⎜

⎞⎠⎟

=

∞

∑16 1

2 2

2 2

21

ππμφ

exp . (11.180)

Using only the fi rst term in the summation, confi rm that the shape factor is given by σ = 25 67 2. /L .

6. Estimate how accurately a cylindrical block would approximate a matrix block formed in between two parallel fractures for the determination of the shape factor.

7. Estimate how accurately a spherical block would approximate a rectangular matrix block for the determination of the shape factor.

8. Applying the Laplace transformation, derive an analytical solution for the model of de Swaan (1990) given by Eqs. (11.41) – (11.44) .

9. Prepare the plots of the analytical solutions given by deSwaan (1978) and Kazemi et al. (1989) using q = 168 ft 3 /day; L = 1000 ft; λ = 0.01 and 0.1/day; R ∞ = 0.08; and ϕ f = 0.001 for 5, 10, 100, 300, 450, and 600 days as a function of distance. Compare the results.

10. Prepare plots of the short - and long - time saturation profi les at different times for Examples A and B mentioned in this chapter.

11. Consider a parallelepiped - shaped matrix block, which is separated from the surrounding matrix blocks by means of natural fractures. The equation describing the variation of the dimensionless average oil concentration with dimensionless time inside the matrix block is given by the following equation.

(a) How much dimensionless time does it take for the average oil concentration to decrease to one - half of its initial value in the matrix block?

(b) How much dimensionless time is required for the recovery factor value to attain a value of 0.75, calculated by the following equation?

RF T C T( ) ( )= −1 (11.181)

and

σ = − KL

dC dTC2 , (11.182)

where

σ: shape factor 0.5

: average permeability 100 md

: average di

==K

L mmension of matrix block 1.0 m

: dimensionless average concentr

=C aation of oil in the matrix block

: initial value of the dimensCo iionless average

concentration of oil in the matrix block 1.0

: d

=T iimensionless time

12. The following set of equations describes the pressure depression in a matrix block because of a decrease of pressure in the surrounding fracture medium.

c11.indd 426c11.indd 426 5/27/2011 12:35:49 PM5/27/2011 12:35:49 PM

(a) Write down the simplifi ed forms of these equations if steady - state condition and one - dimensional fl ow in the x - direction are considered.

(b) What is the general analytic solution of the one - dimensional model under steady - state conditions obtained without the application of the boundary conditions? Assume a slightly compressible fl uid and constant viscosity, and the permeability of the matrix is constant and isotropic.

(c) What is the specifi c analytic solution of the one - dimensional model under steady - state conditions obtained from the general solution after the application of the boundary conditions?

Consider (Rasmussen and Civan, 2003 )

φφ

ψ ψ∂∂

= ∇⋅ ⋅∇⎡⎣⎢

⎤⎦⎥t

DK

1K , (11.183)

subject to the conditions of solution, given by

− ⋅∇( ) ⋅ ≡ ⋅ =−⎡

⎣⎢⎤⎦⎥

K n u nψ ρψ ψ

Kb

sf

s

, (11.184)

where x = L / 2, K s = skin coeffi cient and

− ⋅∇( ) ⋅ = =K nψ 0 0, .x (11.185)

Express the pseudopressure function ψ in terms of the fl uid pressure p using the fol-lowing truncated Taylor series:

ψρμ

ψ ρμ

= ≅ ( ) + −( )∫ dp p p pb b

p

p

b

. (11.186)

13. Does the oil recovery become higher or lower for the waterfl ooding of a water - wet reservoir when the contact angle is higher?

14. Does the oil recovery become higher or lower for the waterfl ooding of a water - wet reservoir when the spacing between the natural fractures is larger?

11.8 EXERCISES 427

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