SPE 157147

Effect of Fracture Intensity and Longitudinal Dispersivity on Mass Transfer in Fractured Reservoirs A. Sharifi Haddad, SPE, H. Hassanzadeh, SPE, J. Abedi, SPE, Z. Chen, SPE, University of Calgary

Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPETT 2012 Energy Conference and Exhibition held in Port of Spain, Trinidad, 11–13 June 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract Accurate modeling of fluid transport in fractured reservoirs is challenging because of their heterogeneous nature. One of the

sources of heterogeneity is the fracture spacing. In this study, an analytical model which describes the effect of fracture

intensity on mass transfer during an advective-dispersive process in dual porosity systems is introduced.

The mass transfer process is modeled using different distributions of the fracture network that results in a various range of

matrix-fracture connectivity inside the reservoir and resembles the variable flow path for the mass transport. In this work

probable fracture network distributions in the field are tested and the effect of the matrix block size distributions and

longitudinal dispersivity inside the fracture network on the transport of the injected tracer in the reservoir is obtained. The

model consists of an infinite acting reservoir with planar matrix blocks and a radially divergent continuous injection system.

Results show that using the breakthrough time of the injected tracer, the stored mass inside the reservoir as a function of

fracture intensity and the dispersivity coefficient can be estimated. Analytical solutions are provided and can be used to study

the tracer transport in fractured reservoirs with variable fracture intensity.

Introduction Fractured reservoirs compromise about 30 percents of oil and gas resources around the world (Saidi, 1987). Current

simulators use the Warren and Root model (1963) to model the fluid dynamics in dual porosity reservoirs. In this model a

single rock matrix block size is assumed to represent the rock and fracture connectivity. Laboratory (Gwo et al., 1998; Hu

and Brusseau, 1995), field (Becker and Shapiro, 2000; Jardine et al., 1999), and modeling studies (Jelmert, 1995; Rasmusen

and Neretnieks, 1980; 1981; Sudicky and McLaren, 1992; Tang et al. 1981) have been conducted to analyze the behavior of

heterogeneous dual porosity reservoirs. In the field scale studies, as reported by many investigators (Haggerty and Gorelick,

1995; Jardine et al., 1999; Lee et al., 1992; Neretnieks, 1980) heterogeneity of geological formation affects the breakthrough

curves and the transport mechanisms in porous media due to the complexity of the flow path provided by micro and macro

fractures.

One of these sources which is not considered in the Warren and Root model is the fracture intensity in reservoirs. The degree

of fracture intensity determines the contact area between fractures and rock matrix blocks in the reservoirs. Consequently, it

affects the rate of the mass transfer inside a fractured reservoir. The effect of this heterogeneity has been investigated for the

pressure transient analysis (Belani and Jalali-Yazdi, 1988; Johns and Jalali-Yazdi, 1991). However, the mass transfer

problem still demands more investigations.

This study considers the effect of fracture intensity on the total accumulation of the injected tracer or catalyst in the fractured

reservoir, which can find applications in in-situ upgrading of heavy oil and geological characterization of naturally fractured

reservoirs.

Development of the model A model of tracer (catalyst) transport in single-phase divergent radial flow with incompressible fluid is assumed to represent

the fluid dynamics inside the fractured reservoir. No-flow boundaries at the top and bottom of the reservoir and slab shape

rock matrix blocks are considered. We further assume continuous injection through the whole pay zone. Physical properties

of fluid and porous media remain constant. The transport mechanisms in the fracture network are advection and dispersion

while the diffusive transport in matrix blocks is the only transport mechanism. Figure 1 shows the schematic of the dual

porosity reservoir considered in this study.

2 SPE 157147

Figure 1: Physical system for the study of the tracer or catalyst injection into the fractured reservoir.

Material balance for fractures and rock matrix blocks provides the governing equations of the tracer (catalyst) transport in

fractured reservoirs. Solutions of the partial differential equations can be found by applying the corresponding initial and

boundary conditions. These solutions are based on the assumption of a single rock matrix block size of the Warren and Root

model. The diffusion-dominated transport of the tracer in the matrix zone can be presented by:

2

m

2

mm

z

cD

t

c

(1)

In this equation, c represents the concentration of the tracer or catalyst in the porous media, Dm is the effective molecular

diffusion coefficient in the rock matrix zone, z is the coordinate in the vertical direction, and time is shown by t.

Concentration inside the rock matrix block is a function of the concentration inside the adjacent fracture. Therefore, the

fracture concentration is required to evaluate the accumulation of the tracer (catalyst) inside the rock matrix block. One can

write the partial differential equation for the fracture domain as given by:

r

crv

rr

crD

rrt

c

t

c

fffff

fmm

ff

~

(2)

Porosity of the rock matrix block and fracture zones are defined by m and f, respectively. Df is the dispersion coefficient

and v is the pore velocity inside the fracture. The one-dimensional transport of the tracer or catalyst in the radial system is a

function of r, which is the radial coordinate. Both terms on the right-hand side of equation (2) are velocity-dependent. The

first term is dependent on the velocity through the dispersion coefficient and the second term is directly related to the

velocity. The dispersion coefficient can be represented as (Fried and Combarnous, 1971):

ff vD (3)

Substitution for the velocity by the injection flow rate at the wellbore in the radial system and using the dispersion coefficient

given by equation (3) in equation (2) result in:

f

f

wff

f

wfmm

ff

22

~c

h

q

rrr

c

h

q

rrt

c

t

c

(4)

Introducing the dimensionless parameters into equations (1) and (4) gives:

D

mD

2

D

mD

2

t

c

z

c

(5)

RDD

Dm

Rf

m

D

Df

D

2

D

Df

2

DwD

Df 211

hzz

c

hr

c

rPe

r

c

rr

Pe

t

c

(6)

in which the dimensionless parameters are defined by:

wi

miDm

cc

ccc

(7.1)

wi

fi

fDcc

ccc

(7.2)

wr

rrD

(7.3)

w

Dr

zz

(7.4)

w

bR

r

hh

(7.5)

2

m

w

Dr

tDt

(7.6)

where hb is the rock matrix block height. Note that i and w stand for initial and wellbore, respectively. mfw 2/ DhqPe is

SPE 157147 3

called the Peclet number and shows the rate of advection of a constituent by the fluid flow to the rate of diffusion of that

constituent driven by the concentration gradient.

There are different methods to solve the coupled partial differential equations of mass transport. In this study the Laplace

transformation method is used. The real domain solution is then obtained using the Laplace numerical inversion methods.

Solutions of the tracer concentration in the rock matrix block and the fracture domains can be expressed as:

Df

R

DDm

2/cosh

coshc

hs

zsc (8)

3/2

D

2w

3/2

2w

Dw

Df)]([

)()2

(

)]([

)()2

(

))1(2

exp(

s

rsr

Ai

s

sfr

sAi

rr

c

(9)

where s is the Laplace domain variable, Ai is the Airy function, and β(s) is defined as:

w

R

Rf

m 2/tanh2

)(r

Pe

hsh

ss

s

(10)

Once the mass concentration inside the fracture as a function of time is found, the concentration inside the rock matrix blocks

can be calculated using equation (8). Then equation (8) can be integrated over the rock matrix block height to give the

average tracer concentration in the rock matrix blocks at the specific radius and time. This can be performed as given by:

R

0

DDm

Dm

)2/R(h

2~

h

dzc

c

(11)

All the solutions are required to be inverted into the real time. Stehfest (1970) introduced a numerical algorithm to invert the

Laplace domain results into the real domain, which has been widely used for this purpose. Taking the advantage of this

method the results obtained in the Laplace domain (i.e., equations (9) and (11)) can be inverted into the real domain.

The obtained solutions are based on the assumption of a single rock matrix block size, in order to apply the heterogeneity

caused by the fracture intensity in the fractured reservoirs (Dyer, 1983; Segall, 1981). Appropriate probability density

functions must be combined with the previously derived solutions. In this study the commonly used probability density

functions (Rodriguez, 2001; 2002; Johns and Jalali-Yazdi, 1991) have been employed. They can be defined as:

)exp()exp(

)exp()(

h

DDD

aaF

ahahf

(12)

bmhhf DDD )( (13)

h

DD1

1)(

Fhf

(14)

for exponential, linear, and rectangular probability density functions, respectively, where hD=l/lmax and, Fh=lmin/lmax, with l as

the characteristic length. For a probability density function one can show:

1)(

1

DDD

h

F

dhhf (15)

The parameter which represents the fracture intensity in the equation of concentration of the tracer or catalyst inside the rock

matrix blocks and fractures is the height of the rock matrix blocks. Therefore, incorporating the fracture intensity by the use

of the probability density function can be accomplished by performing the following integral:

D

1

DmDDDm

h

~)(~ dhchfcF

(16)

In the following section, the model described here is used to study the effect of dispersivity and fracture intensity on the

accumulation of the injected tracer or catalyst inside the reservoir.

Results and discussion The effect of the dispersivity and fracture intensity are discussed in this section. Figure 2 shows the effect of the fracture

intensity on the concentration of the tracer (catalyst) in the rock matrix blocks at rD=200 (corresponding to 10 meters far from

the injection point). As the fracture intensity increases (i.e., exponential pdf with a large positive exponential constant or

linear pdf with a negative slope) the average tracer concentration inside the rock matrix blocks increases. The Warren and

Root model with hR=6 corresponds to the highest number of fractures per unit volume of the reservoir rock. Therefore, the

average concentration of the injected tracer inside the rock matrix blocks in this case remains high as compared to other cases

with a lower number of fractures in the same volume of the reservoir rock. The exponential probability density function with

an exponential constant of a equals to 5 is the next realization with the highest number of fractures. The lowest average

concentration belongs to the Warrant and Root model with hR=60, which includes large rock matrix blocks and consequently

4 SPE 157147

the lowest number of fractures per unit volume of the rock. Other distributions have their corresponding curves located

between the two extreme cases of the Warrant and Root model.

Dimensionless time

1 10 100 1000

Dim

en

sio

nle

ss a

ve

rag

e m

atr

ix c

on

ce

ntr

atio

n

0.0

0.2

0.4

0.6

0.8

1.0Warren and Root model, hR=6

Exponential pdf a=5

Linear pdf m = -100/81

Rectangular pdf

Linear pdf m = 100/81

Exponential pdf a = -5

Warren and Root model, hR=60

m/

f=5

1 m

Pe=10000rD=200Large blocks

Small blocks

Figure 2: Effect of different fracture inensities on the average mass concentration inside the rock matrix blocks.

As the fracture spacing increases the contact surface available for the mass transfer between the fracture and rock matrix

blocks decreases; consequently, the tracer concentration inside the rock matrix blocks for the same period of time decreases.

Performing the integration over the whole domain (test volume) from the wellbore to the radius of investigation (e.g., the

monitoring well at rD=200) results in the accumulation of the tracer or catalyst inside the rock matrix blocks as given by:

T

r

r

Vc

drrhc

Mw

mm )2(~

w

(17)

Figure 3 describes the accumulation of the tracer or catalyst in the test volume for different fracture intensities. It can be seen

that for smaller blocks, the accumulation of the tracer or catalyst in the test volume is higher as compared to sparse fracture

systems.

Dimensionless time

0 1 2 3 4 5 6

Dim

ensio

nle

ss a

ccum

ula

tio

n o

f tr

ace

r in

sid

e t

he r

ese

rvoir

0.0

0.1

0.2

0.3

0.4

0.5

Warren and Root model, hR = 6

Exponential pdf a = 5

Linear pdf m = -100/81

Rectangular pdf

Linear pdf m = 100/81

Exponential pdf a = -5

Warren and Root model, hR=60

m/

f=5

1 m

Pe=10000rD=200

Figure 3: Effect of the fracture intensity on the accumulation of the tracer (catalyst) in the reservoir.

Figure 4 shows the tracer or catalyst accumulation inside the reservoir for different dispersivities. Since dispersion in the

radial direction spreads the tracer into a larger volume of the reservoir, it provides a larger number of matrix blocks with the

tracer and, therefore, enhances the mass transfer between the matrix blocks and fractures.

SPE 157147 5

Dimensionless time

0 1 2 3 4 5 6 7

Dim

ensio

nle

ss a

ccum

ula

tio

n o

f tr

acer

insid

e t

he r

eserv

oir

0.0

0.2

0.4

0.6

0.8

1.0

= 20 m

= 10 m

= 5 m

= 1 m

= 0.5 m

m/

f=5

hR

Pe=10000rD=200

Figure 4: Effect of dispersivity on the accumulation of tracer (catalyst) inside the reservoir, Warren and Root model.

Figures 5 and 6 are obtained for fracture intensity with the exponential probability density functions. Results show that as the

parameter “a” in the exponential pdf increases from negative to positive numbers (sparsely to intensely fractured rocks) the

accumulation of the tracer (catalyst) inside the reservoir increases. The results shown in Figure 5 reveal that the Peclet

number has a very notable effect on the accumulation of the tracer when the rock matrix blocks have different distributions.

Figure 6 shows that mass transfer between matrix and fracture is not sensitive to the dispersivity in sparsely fractured

systems. However, the mass transfer shows some sensitivity in the case of intensely fractured systems.

a

-20 -10 0 10 20

Dim

ensio

nle

ss a

ccum

ula

tion o

f tr

acer

insid

e t

he r

eserv

oir

0.0

0.1

0.2

0.3

0.4

0.5

0.6

small blockslarge blocks

Pe = 105

Pe = 104

Pe = 5 x103

Pe = 103

m/

f=5

1 m

tD=5

rD=200

Figure 5: Effect of Peclet number on the accumulation of tracer (catalyst) inside the heterogeneous reservoir with exponential probability

density function.

6 SPE 157147

a

-20 -10 0 10 20

Dim

ensio

nle

ss a

ccu

mula

tio

n o

f tr

ace

r in

sid

e t

he r

ese

rvoir

0.0

0.1

0.2

0.3

0.4

0.5

0.6

= 1 m

= 5 m

= 10 m

= 20 m

m/

f=5

Pe104

tD=5

rD=200

small blockslarge blocks

Figure 6: Effect of the dispersivity on the accumulation of tracer (catalyst) inside the heterogeneous reservoir with exponential probability density function.

The same observations made above can be confirmed for the linear and rectangular probability density functions. Figures 7

and 8 are related to the linear and rectangular (m=0) probability density functions, respectively. As the “m” value decreases

from positive to negative numbers (i.e., larger to smaller rock matrix blocks) again there is an increase in the accumulation of

the tracer (catalyst) inside the reservoir. Sensitivity of mass transfer between matrix and fracture to the dispersivity for such

heterogeneities is not significant.

m

-3 -2 -1 0 1 2 3

Dim

ensio

nle

ss a

ccu

mu

latio

n o

f tr

ace

r in

sid

e t

he r

ese

rvo

ir

0.0

0.1

0.2

0.3

0.4

0.5

0.6

m/

f=5

1m

tD=5

rD=200

Pe = 105

Pe = 104

Pe = 5 x103

Pe = 103

small blocks large blocks

Figure 7: Effect of Peclet number on the accumulation of tracer (catalyst) inside the heterogeneous reservoir with linear (rectangular m= 0) probability density function.

m

-3 -2 -1 0 1 2 3

Dim

ensio

nle

ss a

ccu

mula

tio

n o

f tr

ace

r in

sid

e t

he r

ese

rvoir

0.00

0.05

0.10

0.15

0.20

0.25

0.30

= 1 m

= 5 m

= 10 m

= 20 m

small blocks large blocks

m/

f=5

Pe104

tD=5

rD=200

Figure 8: Effect of the dispersivity on the accumulation of tracer (catalyst) inside the heterogeneous reservoir with linear (rectangular m= 0) probability density function.

SPE 157147 7

Summary and conclusions In this study it was shown that heterogeneity of a fractured reservoir has a profound effect on the rate of mass transfer

between the fracture network and rock matrix blocks. One of the heterogeneity sources is related to the variable fracture

intensity in porous media. The effect of fracture intensity was incorporated by developing the equations which included the

probability density functions for variable matrix rock blocks or fracture intensities. The results demonstrated that the smaller

the rock matrix blocks, the higher accumulation of the tracer (catalyst) inside the reservoir. In addition to the study of the

fracture intensity, the effect of dispersivity and the Peclet number are also investigated. The impact of the Peclet number was

more recognizable than dispersivity.

Acknowledgment The authors would like to thank the financial supports of the National Science and Engineering Research Council of Canada,

Computer Modeling Group Ltd., NSERC/AIEES/Foundation CMG, AITF, and the Alberta Energy Research Institute. The

first author is thankful for the help and comments of Peter P. Valkó, Texas A & M University and Danial Kaviani, University

of Calgary.

Nomenclature a = exponential probability constant [-]

b = intercept in the linear probability density function [-]

c = tracer concentration [ML-3]

c~ = average tracer concentration over the block height [ML-3]

c~ = average tracer concentration over the domain [ML-3]

c = tracer concentration in the Laplace domain [-]

D = molecular diffusion coefficient [L2 T-1]

f = function [-]

Fh = rock matrix block height ratio [-]

h = block height [L]

m = linear probability constant [-]

M = dimensionless accumulation of tracer [-]

Pe = Peclet number [-]

q = injection rate per repetitive element [L-3T-1]

r = radius from the wellbore [L]

s = Laplace variable [-]

t = time [T]

V = volume of the reservoir [L3]

z = vertical coordinate []

Greek

α = dispersivity [L]

= porosity, fraction [-]

Subscripts D = dimensionless

f = fracture

i = initial

m = matrix

max = maximum

min = minimum

R = ratio

T = test

w = wellbore

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8 SPE 157147

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