RESERVOIR ENGINEERING RESEARCH INSTITUTE

Research Program on Fractured Petroleum Reservoirs

DE-FG22-93BC14875

January 31,1996

Contract Date: September 30,1993

DOE Program Manager: Dr. Robert E. Lemmon

Principal Investigator: Dr. Abbas Firoozabadi

DOE/BC/l4875 =- 3 Vol. 4

4Q.95

October 1 - December 31,1995

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RESERVOIR ENGINEERING RESEARCH INSTITUTE

TOPIC 6c: WATER INJECTION IN FRAC'ITJIWDLAYERED POROUS MEDIA

Water Injection in Water-Wet Fractured Media: Experiments and Analysis

44.95

October 1 through December 3 1,1995

Mehran Pooladi-Darvish Abbas Firoozabadi

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

1

1 SUMMARY

A number of experiments have been performed to study water injection in fractured porous

media. These experiments reveal that: 1) the co-current imbibition may be the primary

flow process in water-wet fractured media, and 2) the imbibition may result in over 20 percent

recovery from very tight rock (Austin Chalk with IC,, of the order of 0.01 md) for an imbibition

period of about 2 months.

Theoretical considerations reveal that the exponential function of Aronofsky et al. [1958]

, does not describe the early-time, but may represent the late-time recovery.

2 INTRODUCTION

A large number of issues remain unresolved in the study of water injection of fractured

reservoirs. Much of the research work carried out in the past, center around the study of

counter-current imbibition for a single matrix block in a water-wet medium. There is no basis to

believe that counter-current imbibition is the proper imbibition mechanism for water injection

of fractured reservoirs. Co-current imbibition, which is of different recovery characteristics,

could be the dominant mechanism in some fractured reservoirs.

Fractured porous media comprise matrix blocks and fracture network. Interactions due to

1) reinfiltartion (especially for intermediate-wet, and mixed-wet systems), and 2) capillarity

(capillary pressure contrast between the matrix and the fracture) may also influence water

displacement efficiency. Therefore, the results from a single matrix may not fully apply to

fractured media.

A number of fractured reservoirs have a very low matrix block (kma of the order of 0.01

md). Gas-oil gravity drainage, due to tight matrix permeability and high capillary pressure

contrast between the matrix rock and the fracture [Firoozabadi 19931, may not be an efficient

recovery process for such reservoirs. Water injection, on the other hand, might be a viable

option. We are not aware of published work on the matrix recovery from very tight matrix

blocks.

2

To address the above and other issues of water injection in fractured reservoirs, we have

embarked on a comprehensive experimental and theoretical study. In this report, water in-

jection in water-wet fractured porous media will be discussed. The report is structured as

follows; after literature review and theoretical discussions, the data on the last experiment

for a three-stacked block will be presented. Analysis of the stacked block experiments for the

Austin Chalk and Berea will follow, and at the end conclusion will be drawn from the work.

3 LITERATURE REVIEW

Much work has been carried out in the study of water injection in fractured reservoirs. In

an early paper, Aronofsky et al. 119581 proposed an exponential function for oil production

due to imbibition from a matrix block immersed in water. Their solution was used in the

early reservoir studies of fractured reservoirs in Iraq [Freeman and Natanson 19591 and Iran

[Anderson et al. 19631. Aronofsky et al. [1958] accounted for the effect of a rising water level

in the fractures surrounding the matrix by performing a convolution, such that the time at

which oil production initiated from a matrix block started when the water in the fracturer

immersed the block. A more general convolution approach was used later by Parsons and

Chaney [1966] to calculate recovery from a tall block adjacent to a rising water level. They

assumed that imbibition occurred horizontally only; an element of rock in contact with water

in the fracture produced due to counter-current imbibition. The convolution on the fracture

water level was used later by Beckner et al. [1987] and Bech et al. f19911, who replaced the

empiric,d exponential function by the solution of a diffusion equation describing spontaneous

imbibition.

De S w a n [1978] used the exponential function of Aronofsky et al. for the transfer term in the Bnckley Leverett displacement in a fracture network surrounded by matrix rock. He as-

sumed the saturation continuity rather than the capillary continuity, and the fractional flow of

water to be equal to water saturation in the fracture, fwr = Swj. The latter assumption implies

unit mobility ratio and straight line relative permeabilities in the fracture. Chen and Liu [1982]

relaxed the assumption of equality of fractional flow of water and water saturation in the frac-

ture. Recently Kazemi et al. [1992] studied the same problem, and presented a similar solution

3

to that of de Swaan [1978] for the case of fwr = Swj. The authors presented the numerical

solution to the problem without the assumption of f w j = Swj, and showed good agreement

between the two solutions when a unit viscosity ratio was chosen. Kazemi et al. E19921 also

demonstrated that large errors incur with the exponential function at the early-time, and con-

cluded that the function should solely be considered as a curve fitting function. On the other

hand, Civan [1994] obtained a two-term exponential function for the matrix fracture imbibition

process, and claimed a physical meaning to the exponential solution. His solution simplifies to

the exponential function of Aronofsky et al. [1958]. Recently, Chen et al. [1995] calculated oil

production rate due to l-D counter-current imbibition by simulation, and observed that the plot

of recovery rate versus time on a log-log scale can be approximated by two straight lines. The

first straight line corresponds to the early-time behavior, indicating that oil production rate is inversely proportional to square root of time. Assuming this to be valid, Chen et al. [1995]

obtained the analytical solution for the saturation profiles and the imbibition rate.When the

nonlinearities were neglected, they showed that the late-time behavior could be described by

an exponentid solution similar to that of Aronofsky et al. [1958]. When the nonlinear problem

was considered, Chen et al. [1995] proposed a solution for the late-time behavior different from

the exponential solution, and concluded that the exponential function is only valid where the

Linear diffusion equation was considered. Their late-time solution indicated that recovery did

not approach one; it either approached zero or increased with time.

Nonlinearities in the mathematical description of the imbibition problem in a finite medium

restricts the development of an exact analytical solution. Experimental and numerical tech-

niques have been used to study the process. Laboratory scaling studies center around counter-

current imbibition. From the flow equations, Rapoport [1955] derived the scaling criteria for incompressible two phase flow in porous media. Mattax and Kyte [1962] modified Rapoport’s

results for l-D spontaneous imbibition, and verified their scaling group with the experimental

data. The study indicated that for scaling to be valid, the shape and the initial and bound-

ary conditions under the field conditions should be preserved in the laboratory. Additionally,

viscosity ratio and relative permeabilities should be identical between the model and the pro-

totype. Later efforts were directed toward relaxing these requirements. Kazemi et al. [1992]

incorporated the shape factor in the definition of dimensionless time in order to avoid the ne-

4

cessity of shape reproduction in the scaling studies. Zhang et al. [1995] introduced a modified

shape factor, and found that by using the geometric mean of oil and water viscosity different

viscosity ratios could be scaled. Marle [1981] considered a rising fracture water level problem,

and found that for every water level velocity in the fracture another experiment should be

performed. An equation indicating the same conclusion was previously obtained by Parsons

and Chaney [1966]. Marle [1981], similar to the previous investigators, found that the time

required to obtain a certain recovery in a counter-current process was proportional to square

of the length of the system.

In pardel to the experimental studies with immersion-type boundary conditions, rising

fracture water level experiments have been reported. Parsons and Chaney [1966] observed oil

production below as well as above the water level in the fracture. Hence co- and counter-

current regimes were both active. In contrast to the immersion-type experiments and for the

cases studied, recovery under a rising fracture water level did not depend on the size of the

block. Kleppe and Morse 119741 studied the effect of injection rate, i.e. fracture water level,

on a Berea sandstone. For the low injection rate experiment, maximum recovery was achieved

by the time water level in the fracture passed the top of the block. In the high injection rate

experiment, about 20% of the oil was recovered after water breakthrough from the fracture.

The block experiments with fracture water level rise and immersed boundary conditions

reveal certain differences. In the latter, the oil is forced to flow counter-current to water. In the

former, there is evidence that recovery is partly obtained by oil production above the water level

in the fracture. Bourbiaux and Kalaydjian [I9901 performed a detailed study to investigate

co-current and counter-current imbibition processes. The experiments with counter-current

imbibition showed slower recovery than those with predominantly co-current imbibition; the

half recovery time for co-current imbibition was 7.1 hrs, and for counter-current imbibition was

22.2 hTs for a particular case. As will be discussed later, the process of l-D counter-current

imbibition can be formulated as a nonlinear diffusion equation. The mathematical description

of co-current imbibition in terms of water saturation includes a convective term in addition

to the diffusion term. The convective term causes steepness in the saturation profile. Such a

difference was observed in the measured saturation profiles of Bourbiaux and Kalaydjian [1990].

5

Capillary driven oil production above the fracture water level can be considered as a 2-D,

co-current imbibition process. Under these conditions, when the pressure gradients in the

oil phase is neglected, the process can be described by a nonlinear 2-D diffusion equation

[Beckner et al. 19871. In the latter equation, the value of the diffusion coefficient is large, where

water is imbibed into the rock and water saturation is high. Hence, water is imbibed under

small saturation gradients. The diffusion coefficient in the counter-current imbibition process

however, has a bell-shaped behavior, vanishing at both saturation limits, unless the derivative

of the capillary pressure curve diverges faster than either of the relative permeabilities go to

zero. The low diffusion coefficient at high water saturation, i.e. at’the inlet, requires a high

saturation gradient there, reducing the imbibition rate.

In some rising fracture water level experiments, most of oil recovery is achieved by the time

the water level in the fracture passes an element of matrix. Based on this observation, Mattax

and Kyte [1962] introduced a critical rate, which corresponds to the velocity of water level

in the fracture at which the water level in the fracture is the same as the water level in the

matrix. Hence, for injection rates below critical rate the recovery is independent of the physics

of the imbibition process. In reality, water and oil in the matrix form a region of continuous

saturation change and no accurate water bevel in the matrix can be defined. Simulation studies

of Beckner et al. [1987] indicated that in some cases, the water in the matrix is imbibed above

the water level in the fracture. Recently, Bech et al. [1991] used the critical rate to refer to

the rate at which the water advance rate in the matrix equabs the water advance rate in the

fpurcture. Using the latter definition, and for sufficiently long matrix blocks, any finite advance

rate would be equal to the critical rate. This is because after a finite time, imbibition at the

lower portion of the matrix WU be complete, and any saturation contour in the matrix will

thereafter move with the same velocity as that of the water level in the fracture.

Blair E19641 studied the process of l-D counter-current imbibition using the classical equac

tions of two-phase flow. The author assumed a zero inlet capillary pressure corresponding to

maximum water saturation. For the water injection process, different studies have indicated

that water saturation builds up during a period of time which may or may not be significant

[Bentsen 1978, Shen and Ruth 19961. Blair [1964] showed that for water imbibition in a rock I

6

saturated with low viscosity oil (0.01- 1 cp) saturation profiles propagate in a frontal manner

through the rock. The profiles were more diffusive when oil viscosity was high (30 and 100 cp).

He found that, in the range studied, the lower the initial water saturation, the higher was the

imbibition rate. Kleppe and Morse [1974] used a 2-D, two-phase simulator to model their rising

water level imbibition experiments. The numerical study indicated that the behavior of a long

block was similar to a stack of two blocks of total equivalent height. Bech et al. [1991] performed

a fine-grid simulation and confirmed the observation obtained by Kleppe and Morse [1974] that

a system of stacked blocks behaved similar to a long vertical block with the equivalent height.

This conclusion may not be valid for all the cases. A condition can be thought of, where most

of the fractures are horizontal, and the block height is smaller than the block width. Under

these conditions, imbibition occurs mainly from the horizontal fractures, and their effect can

not be neglected.

The above brief review reveals that:

1. It is unresolved yet if the exponential expression of Aronofsky et al. [1958] provides a

proper recovery curve for a matrix block immersed in water.

2. There are different opinions on the behavior of a stack of blocks compared with a tall

block.

3. The differences between co- and counter-current imbibition, and their contribution under

rising fracture water level are not well understood. It is not clear if the scaling criteria

developed for counter-current imbibition are applicable to co-current imbibition.

In this work we attempt to elaborate on the differences between co- and counter-current

imbibition, discuss the conditions under which a stack of matrix blocks behaves differently

than a tall block, and show the physical significance of the exponential function as the late- +:-- mnl..t:nn ,F 1 TI :-h:h:+Ln 17v,..rr-:-mntQl thafi,a+:r3t -4A-nf l -e ---

6

saturated with low viscosity oil (0.01- 1 cp) saturation profiles propagate in a frontal manner

through the rock. The profiles were more diffusive when oil viscosity was high (30 and 100 cp).

He found that, in the range studied, the lower the initial water saturation, the higher was the

imbibition rate. Kleppe and Morse [1974] used a 2-D, two-phase simulator to model their rising

water level imbibition experiments. The numerical study indicated that the behavior of a long

block was similar to a stack of two blocks of total equivalent height. Bech et al. [1991] performed

a fine-grid simulation and confirmed the observation obtained by Kleppe and Morse [1974] that

a system of stacked blocks behaved similar to a long vertical block with the equivalent height.

This conclusion may not be valid for all the cases. A condition can be thought of, where most

of the fractures are horizontal, and the block height is smaller than the block width. Under

these conditions, imbibition occurs mainly from the horizontal fractures, and their effect can

not be neglected.

The above brief review reveals that:

1. It is unresolved yet if the exponential expression of Aronofsky et al. [1958] provides a

proper recovery curve for a matrix block immersed in water.

2. There are different opinions on the behavior of a stack of blocks compared with a tall

block.

3. The differences between co- and counter-current imbibition, and their contribution under

rising fracture water level are not well understood. It is not clear if the scaling criteria

developed for counter-current imbibition are applicable to co-current imbibition.

In this work we attempt to elaborate on the differences between co- and counter-current

imbibition, discuss the conditions under which a stack of matrix blocks behaves differently

than a tall block, and show the physical significance of the exponential function as the late-

time solution of l-D counter-current imbibition. Experimental and theoretical evidences are

considered to address the above issues.

7

4 THEORETICAL CONSIDERATIONS

The movement of water level in a fracture varies the boundary condition on the block.

While the matrix blocks immersed in water may continue oil production by counter-current

imbibition, most of the production from partially immersed blocks and the blocks above could

be by co-current imbibition. Different studies reviewed above, indicate that there may be

intrinsic differences between the performance of the two processes.

Imbibition, both co-current and counter-current, has been formulated as a diffusion process

[Mule 1981, Beckner et al. 1987, Bech et al. 1991, Dutra and Aziz 19921. In order to gain a

better understanding of the process, in the following, the behavior of a l-D diffusion equation

is mathematically examined.

4.1 l-D COUNTER-CURRENT IMBIBITION

In an incompressible, one dimensional system, where the gravity forces can be neglected, and

the total velocity of the two immiscible fluids is zero, the imbibition process can be described

by the following nonlinear diffusion equation [Marle 19811,

where,

Considering a cylindrical core of length 2L initially at connate water saturation, laterally

coated by an impermeable material, and immersed in water, the most commonly used initial

and boundary conditions are,

s w = S W C , t = O (3)

sw = 1 - so,, x = o t 2 o (4)

8

= 0. x = L t z o ( 5 )

The diffusion coefficient defined by Equation 2 is a strong function of saturation. However, if

we assume a constant diffusion coefficient, Equation 1 with the initial and boundary conditions

3 to 5 can be solved analytically [Carslaw and Jaeger 19591,

where,

We use the Heat Integral Method (HIM) [Goodman 19641 to solve a linear diffusion prob-

lem, and study the early- and the late-time behavior of l-D counter-current imbibition. The

advantage of using the HIM is that its application can be extended to solve nonlinear prob-

lems. The Heat Integral Method, one of the variations of the Method of Weighted Residuals

[Finlayson 19721, was recently used to solve diffusion dominated problems in thermal recovery

processes [Pooladi-Darvish et al. 19941. In this method, a trial function is assumed, which in

addition to satisfying the initial and boundary conditions, is forced to satisfy the integrated

form of the original differential equation. Hence, the method is an appropriate choice for

studying the problem of interest, since the total water imbibed into the system, which is an

integral characteristic of the system is obtained with’high accuracy. For increased accuracy of

the solution at early-time, the concept of penetration depth is introduced which corresponds

to the extent of the medium beyond which there is no effect of the diffusion process. For a lin-

ear diffusion problem, any change at the boundary is instantaneously transferred through the

whole domain, where beyond a certain limit the effect is negligible. The penetration depth is

9

analogous to radius of investigation in well-testing. Pooladi-Darvish et al. 119941 solved Equa-

tion l with initial and boundary conditions Equations 3 to 5 using HIM, where the diffusion

coefficient was assumed constant, and found the following solution,

6+d & A2 = - e 7 . 4

One can calculate water imbibition rate, using Equations 10 to 12,

10

The early-time production rate in the above finite system, is similar to that in a semi-infinite

medium. The exact solution for the semi-infinite problem is -&=, which is overestimated by

Equation 19 by 8.3%. Using Equations 19 and 20 the dimensionless cumulative volume of

water imbibed can be calculated as,

(22) 1 = 0.75 - - + QDed-ea t l y . t D 2 Q D (tD)Zate AD1 AD2

where the definition of QD is the same as recovery with respect to total recoverable oil in

place, and is the recovery at the end of the eraly time. By writing the R.H.S. of Equation 21 in conventional units one obtains,

Equation 23 implies that recovery during the early-time is proportional to square root of time,

and is inversely proportional to the length. In other words, the time corresponding to a specific

recovery is proportional to square of the length, which is in line with the conclusions obtained

from the previous scaling studies [Mattax and Kyte 1962, Marle 1981, Hamon and Vidal19863.

The recovery at the end of the early-time period from Equation 23 is 25% of the recoverable

oil, and is independent of the length. Therefore, Equation 22 can be written as,

(24) 1 R ( t ~ ) l ~ ~ ~ = f .- A e - X D l t D AD1 - L k e - A D 2 t D . AD2 t D 2

At the beginning of the late-time regime, the ratio of the first exponential function to the

second exponential term at the R.H.S. of Equation 24 is only 576, and decreases very rapidly

with time. Upon neglecting the first exponential function one obtains,

R(t&t, = 1 - A e - X D 2 t D . AD2 t D 2 R 1 (25)

where for a linear problem A2 and ADZ are constants and do not depend on the length of the

system. Equation 25 has the same form as Aronofsky et al.’s exponential function. One can

11

readily show from Equation 25 that, similar to the early-time behavior, the time corresponding

to a specific recovery is proportional to square of the length.

Figure 1. shows a comparison between the exact and the HIM solutions. Both the one-term

and two-terms exponential functions, Equations 24 and 25, are shown. Figure 1 indicates that the HIM solution slightly overestimates the exact solution, and that the one-term exponential

function introduces negligible error.

Equations 10 to 12 were obtained for a constant diffusion coefficient. If the HIM is used to

solve the nonlinear problem, a similar solution can be obtained, with A I , Az, Xol, ADZ functions

of the shape of the diffusion coefficient with respect to saturation. Of course, the solutions are

approximate and account, only partially, for saturation dependency of the diffusion coefficient.

However, if we assume that the form of the solutions does not change, the following caa be

mentioned.

1. The plot of recovery vs. ,,& during the early-time is a straight line, passing through

the origin (see Equation 23).

2. The data deviate from the straight line at a constant recovery, independent of the length

of the system.

3. The plot of (1 - A($)) data at the late-time vs. & on a semi-log scale is a straight line

with a negative slope, which may not pass through the point R(t = 0) = O(see Equation

25).

Figures 2 and 3 show the behavior of the exact solution of the linear problem at the early-

and late-times. Clearly, the above three conclusions can be verified. Also, the behavior of

the exponential function of Aronofsky et al. [1958] is shown, where an exponent of AD = 3

was chosen. Figures 2 and 3 indicate that the exponential solution of Aronofsky et al. [1958],

if chosen to match the overall behavior, under- and over-predicts the early- and late-time

solutions, respectively. The non-smoothness of the solutions at the late-time in Figure 3 is due

to the fact that the calculations were reported with three digits accuracy after the decimal

point. This was chosen to show that small errors in the late-time data are magnified when

12

plotted on a semi-log scale shown in Figure 3. This point should be considered when analyzing

the experimental data.

Figures 4 to 6 examine the validity of the early-time solutions for counter-current imbibition

experiments of Graham and Richardson [1960], Hamon and Vidal[1986], and Cuiec et al. [1994].

Clearly, all the experimental data follow a straight line, and deviate from the straight line,

independent of the length of the core, at 75 to 85% recovery. The physical properties of the

rock and fluids used are given in Table 1. The last data point reported was considered as

the ultimate recovery. In figure 4 the data corresponding to the 2.48 crn sample fall off the

main curve. This could be due to the errors in the oil production data of a very small sample.

Figure 2 is the corresponding plot with a constant diffusion coefficient.

Figures 7 to 9 show the plot of the above imbibition data on a semi-log scale. It can be

noted that in Figures 7 and 8 the late-time data can be approximated by a single straight

line. In Figure 8, the data of the 85 cm sample show a faster recovery than that predicted by

the straight line. Hamon and Vidal [1986] demonstrated that there was considerable effect of

gravity for the long samples. In Figure 9, the late-time data of Cuiec et al. I19941 can not be

approximated by a straight line. It can be observed, however, that the longer the sample the

sharper is the recovery. This is in line with the effect of gravity. Moreover, most of the data

in Figure 9 correspond to the last 2% of recovery. This increases the sensitivity of the analysis

to any error.

It is worth noting that the straight line passing through the late-time data of Figures 7 and

8, if extrapolated to zero time, predicts a non-zero recovery. If an exponential solution similar

to that of Aronofsky et al. [1958] is chosen such that it predicts a zero recovery at the initial

time, a behavior similar to that in Figures 2 and 3 will be observed; an exponent larger than

that obtained from the slope of the straight line will be needed to predict the overall behavior

of the system. The latter solution, then would under- and over-predict the early- and late-time

behavior, respectively.

It was pointed out that the dominant flow mechanism under rising water table conditions

may be appropriately described by 2-D co-current imbibition. When the oil phase pressure

13

gradient is neglected, the formulation of the problem leads to a nonlinear 2-D diffusion equation

[Beckner et al. 19871. Should the assumption be valid some of the conclusion obtained from the

study of the linear diffusion equation may be applicable to co-current imbibition. For example,

for linear as well as nonlinear diffusion processes, production rate during the early-time period

is inversely propdrtional to square root of time (see Equation 19) [Ames 1965, Crank 19751.

Superposition does not mathematically hold for nonlinear problems, however, similar to

Dutra and Aziz [1992], one may use superposition of l-D solutions for a 2-D nonlinear problem

to find an approximate solution. Under these assumptions, Equation 25 can be extended to

two dimensions as,

where it is assumed that the exponential function is valid for

(26)

all times. Parameters L, and

L, are half length of the system in the z and ,z directions, respectively.

4.2 MODELING OF RISING FRACTURE WATER LEVEL

Aronofsky et al. [1958] proposed Equation 27 for predicting oil production due to imbibition

from a matrix block immersed in water,

where,

Parameter X is a constant which determines the rate at which the recovery approaches its

maximum value. Previously, it was shown that the exponential function can explain the late-

time behavior of l-D counter-current imbibition. Similar to Parsons and Chaney [1966] one

can account for the effect of the rising water level in a fracture adjacent to a single matrix

block,

14

where Q(t ) is the cumulative oil produced at time t, A,, is the cross-sectional area of the

matrix perpendicular to the direction of water level rise, and Zf is the location of the water oil

contact in the fracture. Equation 30 assumes that imbibition occurs transverse to the direction

of water level rise in the fracture. Hence, the effect of horizontal fractures are neglected

and a stack of matrix blocks is assumed to behave similar to a tall matrix of equivalent

height. (We will modify Equation 30 to account for the effect of horizontal fractures later). In

previous studies using Equation 30, 2f was assumed known [Aronofsky et al. 1958, Freeman

and Natanson 19591. In a water injection process in a fractured reservoir, injection rate is

normdy known. Then, it is useful to calculate oil production from the matrix and water

level in the fracture simultaneously. By writing the material balance equation for water in the

fracture, one obtains,

where the left hand side is the total volume of water injected, and the right hand side represents

the volume of water in the fracture and the volume of water imbibed into the matrix. The

water flow in the fracture is assumed a piston-like displacement with zero residual saturation,

and the fracture porosity is assumed to be one. If X is known, Zf(t) is the only unknown in

Equation 31, and c m be solved for using a proper initial condition.

Differentiating Equation 31 with respect to t results in,

By eliminating the right hand integral between Equations 31 and 32, one can obtain,

For a constant injection rate, Equation 33 can be integrated analytically,

15

where,

and the initial condition was considered as,

2f(t = 0) = 0. (36)

Equation 34 is valid before breakthrough. During this interval, oil production rate is equal

to injection rate. After breakthrough, the water level stays at the top of the block, and the oil

production rate can be expressed as,

where tBT is the breakthrough t h e . When injection rate is not constant, Equation 32 can be

solved numerically, similar to that proposed by Kazemi et al. [1992]. Alternatively, Equation 33

can be solved numerically, for example using Runge-Kutta method. Note that Equations 32

and 33 are valid for an injection rate of more than the imbibition rate, i.e. water level is rising,

and the injection rate of less than imbibition rate, i.e. water level is dropping.

As mentioned earlier, Equation 31 assumes that imbibition occurs horizontally. Clearly,

at early periods and every time water level in the fracture passes the top face of a block, the

imbibition from the horizontal fractures becomes important. If we assume water in contact

with vertical fractures is imbibed horizontally only, and that from horizontal fractures is im-

bibed vertically only, multidimensional effects can approximately be taken into account, as in

Equation 26. For a single block imbibing from the bottom and side faces under a rising water

level condition, similar to Equation 30 one can write,

The algebra is similar to the above; water level in the fracture and oil production rate due to

imbibition are then obtained.

16

4.3 DEPENDENCE OF A ON SYSTEM PARAMETERS

The exponential function proposed by Aronofsky et d. [I9581 has been widely accepted

as an empirical equation matching oil production due to spontaneous imbibition with a step

change in boundary condition reasonably well. Some authors attribute a physical meaning

to the solution and others believe in lack of a physical meaning, as reviewed earlier. In the

previous section the physical significance of the exponential function was discussed as the late-

time solution of l-D counter-current imbibition, and its validity for some experimental data

was presented.

Mattax and Kyte [1962] combined the scaling group of Rapoport [1955] with Leverett’s J function, and derived a scaling group to account for the effect of sample size, permeability,

porosity, surface tension, and water viscosity. Subsequently, other authors showed the general

applicability of the above scaling group for counter-current imbibition, and offered modifica-

tions. In these studies it was shown that oil recovery from different experiments if plotted ver-

sus dimensionless time can be represented by a single curve. Assuming that Aronofsky et al.’s

exponential function is acceptable the above single curve can be approximated by,

R’(t) = Rb, (1 - e-ADtD) , (39)

where AD is a constant independent of length, shape, permeability, porosity, surface tension,

and water viscosity. Using the definition of t D given by Mattax and Kyte [1962], Equation 39

can be written in terms of t as in Equation 27, which yields,

For a multi-dimensional medium, Kazemi et al. [1992] replaced L2 by &,

where V is the volume of the imbibing matrix, Ai area of surface i open to flow, and dAi

is the distance from surface Ai to the center of the block; -& is the characteristic length of

the system and replaces L in Equation 40. Recently Zhang et al. E19951 introduced a similar

characteristic length L, defined as,

17

where X A ~ is the distance from the surface A; to the no-flow boundary. Hence, using either

Equations 41 or 42, for a vertical slab imbibing from the vertical faces, Equation 40 can be

written as,

If the slab is imbibing from the bottom face too, then using Equation 41, X can be written as,

It is worth noting that Equation 26 gave similar relations for A, where the underlying

assumptions were applicability of the late-time solution for all times, and the validity of the

performed convolution.

5 ANALYSIS OF EXPERIMENTAL DATA

We have conducted extensive experiments on water injection in a stack of blocks of the

Austin Chalk, and the Berea sandstone. The results from the experiments were presented by

Kwauk et al. [1995-a, 1995-b]. We have also conducted a new experiment during this quarter

using a stack of 3 blocks, each with a dimension of 14.71 x 14.71 x 60.48 cm. The setup for the

last test is similar to the one presented by Firoozabadi and Markeset [1984]. The measured

effective permeability of the fracture matrix system is 8.3 darcy, whereas the matrix block

permeability is around 650 md. The total fracture and matrix pore volume are about 8550,

and 150 cm3, respectively. The injection rate was around 330 cm3/hr. Breakthrough occurred

at t = 12 hours corresponding to 3920 cm3 oil production. Visual observations of flow in

the fractures surrounding the matrix blocks revealed that there was very little counter-current

imbibition. The production data and the position of the water level in the fracture for this

test (SB1) will be presented later. In the following, the experiments will be analyzed.

18

5.1 AUSTIN CHALK

The Austin ChaJk rock matrix used in our experiments has a permeability around 0.01 md.

The total pore volume of the six Austin Chalk matrix blocks stacked on the top of each other

was about 300 cm3. As a result of imbibition, about 20 to 25 percent of the oil was produced

over a period of two to four and half months.

We used the above model to calculate imbibition results and neglected the horizontal frac-

tures. Figures 10 to 12 show the comparison between the data and the model for Runs AC1

to AC3. Table 2 provides various data. Parameter X was used as a matching parameter and

was found to be 0.053 duy-l and 0.031 clay-' for Runs AC1 (and AC2) and AC3, respec-

tively. Application of Equation 27 implies that if the system was immersed in water from the

beginning 63% of the recoverable oil would have been produced in 19 and 32 days for Runs

AC1 (and AC2) and AC3, respectively. Water breakthrough in the above runs occurred within

nine hours from the beginning of the experiment. This indicates that if a higher injection rate

was used the performance of the system would not have changed. In other words the limiting

mechanism in the experiments performed on the Austin Chalk samples is imbibition in the

matrix.

In Figure 13 two injection schemes are compared for Run AC1. In one case, the injection

rate was held constant, and in the second, injection rate was reduced by a factor of 20 after

water breakthrough. No difference in oil production can be observed. This is because when

the blocks are immersed in water, the imbibition is a function of time only. In Figure 13 the

experimental data are shown for comparison. It can be noticed that, during the soak period

no oil is produced, and the experimental data fall below the other schemes. During this period

the produced oil due to imbibition is accumulated in the fracture and reduces the contact area

between the water and the matrix. This explains why after resuming injection, the two curves

do not fall on each other.

The fact that the blocks were immersed in water in a short time suggests that in Equation

38, t - r w t. Hence, the characteristic length of the matrix in different directions can be

grouped together, which is included in parameter A. In other words, accounting for imbibition

from the horizontal fracture would not change the behavior of the predictions. Of course,

19

this does not imply that the above experiments are independent of fracture intensity in the

vertical (or horizontal) direction. The characteristic length of a smaller block is less, as given

by Equations 26 or 43.

5.2 BEREA SANDSTONE

Figures 14 and 15 show the behavior of oil production, and the water level in the fracture

for Run SB1. The value of X was found by matching to be 0.45 h F 1 . This implies that if the system was immersed in the water from the beginning, 63% of the recoverable oil would

have been produced in about 2.2 hTs. Figure 14 indicates that the same volume of oil was

produced in about 9 hTs. Here, our lack of ability to provide the required water in a short

time has postponed oil production. In studying Figure 15 it should be noted that the capillary

pressure in the fracture, corresponds to about 25 cm, where the fracture capillary pressure

was found from Pc = 2crcosB/h (fracture aperture, h=150 pm, and cos0 = 1). This indicates

that small variations in fracture aperture could change the water level in the fracture by a few

centimeters, at least.

The value of X obtained from matching Run SBl, along with Equation 43 were used to ac-

count for the change in size and calculate the corresponding X for the stacked slab experiments

SS1 to SS3. This value was found to be 2.25 hr-l. Figures 16 to 18 show the comparison

between the prediction of the model with the experimental data. Although the predictions

are in general agreement with the experimental data, in all three Runs SS1 to SS3, observed

breakthrough times are less than the model predictions. This implies that the value of A for

Runs SS1 to SS3 is not as much as 5 times of Run SB1 as predicted from Equation 43. The

scaling rules developed for counter-current imbibition may not be, therefore, fully applicable

to experimental conditions. The mechanisms of co- and counter-current imbibition and their

differences are currently being studied. .

It was pointed out earlier that when the exponential function is forced to predict the overall

behavior of the imbibition process, overestimation of oil production at the late-time occurs.

This could explain partly the difference between the experimental data and the model results

with the exponential function.

20

Next, the effect of imbibition from the horizontal fractures was incorporated using Equa-

tion 38. When A,, A, and A, were scaled to the block size, very similar results to those in

Figures 14 to 18 were obtained. If the matrix blocks were closed on the vertical faces then Lz = half of block height (assuming imbibition occurring from top and bottom faces). However, with

the vertical faces of the block being open, and not covered with water the characteristic length

of the system in the z direction could be smaller. For L, = 10 cm the behavior of the system

is shown in Figures 19 and 20. The model predictions with A, and A, scaled for Runs SS1 to

SS3 are shown in Figures 21 to 23. A comparison between Figures 14 (and 15) and 19 (and

20) suggests limited effect of the horizontal fractures. This could be caused by the fact that

for the injection rate used, only a small fraction of oil was produced after the fracture water

level passed a block and covered its top face.

As mentioned earlier, in the experiments performed on the Berea Sandstone, imbibition

in the matrix was fast and oil production was controlled by availability of the injected water

to wet the rock surface. Figure 24 shows the validity of the above claim. In this figure, the

volume of produced oil for Runs SS1 to SS3 is plotted versus total volume of injected water.

It can be observed that the data, regardless of the differences in the injection rate for the most

part cover each other.

6 CONCLUSIONS

1. The experimental data reveal that water injection in water-wet fractured porous media

can be a very efficient recovery mechanism.

2. Imbibition from a stack of tight blocks (k,,=O.Ol md) could result in 20 to 25 percent

recovery in a period of about 2 months.

3. The observations on the Berea sandstone block revel that oil recovery is mainly due to

the co-current imbibition. Theoretical studies suggest basic differences between the co-

and counter-current imbibition.

4. The late-time behavior of l-D counter-current imbibition can be described by an expo-

nential function similar to that proposed by Aronofsky et al. [1958].

21

7 ACKNOWLEDGMENTS

The authors are greatful to Xianmin Kwauk and Rostam Jahanian for their help in the preparation

of the experiment.

8 NOMENCLATURE

Latin Letters

A

A

B D

Fs J 1;

L c P

Q R

R' s 2 f h

k

9

t

Cross Sectional Area, m2

Constant (see Equations 17 and 18)

Constant, s-l (see Equation 35)

Diffusion Coefficient, m2/s

Shape Factor, m-2 (see Equation 41)

Leverett 's Function

Half Length, m

Characteristic Length, m (see Equation 42)

Pressure, N/m2

Cumulative Production, m3

Recovery w.r.t. Maximum Recoverable Oil

Recovery w.r.t. Total Volume

Saturation

Height of Water Level, m

Fractional Flow

Fracture Aperture, pm

Permeability, pm2

Volumetric Rate, m3/s

time, s

Greek Letters

4 Porosity

x Exponent, s-l

P Viscosity, Nmls

U Surface Tension, N / m

9 Contact Angle, Rad.

7 Variable of Integration

Subscripts

BT

C

D

f early

2

1 ate

ma

0

or

4-

W

we

X

Y z

Breakthrough

Capillary

Dimensionless

Fracture

Early-Time

Injection

Late-Time

Matrix

Oil

Residual Oil Relative

Water

Connate Water

In the x Direction

In the y Direction

In the z Direction

Symbols

22

03 Corresponding to Infinite Time

23

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Table 1. Counter-Current Imbibition Data

b t h o r s I Length, cm I Permeability, md 1 1 Graham and Richardson I 12.5 1 236' I

Graham and Richardson 10.08 236

Graham and Richardson 7.56 236

Graham and Richardson 5.02 236 ~~ ~ I Graham and Richardson I 2.48 I 236 Hamon and Vidal 85.2 4070

Hamon and Vidal 40.0 4400

Hamon and Vidal 19.8 3200

I Cuiec et al. I 20 I 6.4 I I Cuiec et a ~ . I9 I 3.7 I

Cuiec et d. 5 1.25

"Assumed constant

27

Table 2. Relevant Data for Experiments Table 2. Relevant Data for Experiments

t D

.. ..

Figure 8. Hamon (A)-LateTime 1

4. .. . .. 0.. 9

& .

1 -

h - u w - * - I - * .

A1 -

. =

. ..

Figure 9. Culoc et al.- Late Time

l e

. 9

Figure 10. ACl , Injection Rate-2.2 cc/hr, nC6 70

8 0 -

5 0 - 8 :

H - I*: a - -

10 L Daa - Model

- Figure 12. AC3, Injection Ratez2.17 cc/hr, nCG+Crude

Qata Model -

Fuure 11. AC2, Injection Rab2.2 cclhr, nC6

60

Ma Model -

Figure 13. AC1, Different Injection Schemes

Wa - ModelcOnstantInjectbnRate 0 Model-Reducad 1n)ectnn Rate

0 0 200 400 6M) 8M) loo0 1200

Time, hr.

I

I

Figure 14. Oil Production vs. Time for Test S81, Injection Rate-330 cc/hr 5000,

Data Model

10 15 Time, hr.

20 25

Figure 15. Fracture Water Level vs. Time for Test SB1 , Injection Rate--30 cdhr 200

Data Model

5 10 Time, hr.

15 20 25

I .

Figure 16. Oil Produdion vs. Time lor Test SSl. Injection a t e 4 5 0 cc(hr L

5000 1

Figure 17. Oil Production vs. Time lor Test SS2, Injection Rated00 ccihr

-- w)(K I :

Data Model

Figure 18. Oil Produdion vs. Time for Test SS3, lnjedion Rate430 cdhr

D

Figure 19. Oil Production vs. Time for Test SB1, Injection Rate=330 cchr 5000

- . . * . . .d ..

0 5 10 15 20 25 Time, hr.

Figure 20. Fracture Water Level vs. Time for Test 331, Injection Rate=330 cdhr 200,

figwe 21.01 Production vs. Time far Test SS1; Injection Rate=150 cckr

- D a a Model -

0 20 40 80 80 100 120 140 160 180 Time, hr.

Figure 23. Oil Production vs. Time for Test SS3, injection Rate=530 cc/hr

Y

3 3 - n b - c 0

- 3

Figure 22. Oil Production vs. Time for Test SS2, Injmbn Rate=200 cchr r

0 40 60 80 100 120 Time, hr.

Figure 24. Oil Production vs. Water injected for Tests SS1, SS2. and SS3

ti t r

1

8