USMS020798 Miscible Displacement in Fractured Porous Media

Fernando Perez-Cardenas, Inst. Mexicano Del Petroleo;Juana Cruz-Hernandez, Inst. Mexicano Del Petroleo;Candelario Perez-Rosales, Apartado Postal 75-753

Copyright 1990 Society of Petroleum EngineersThis manuscript was provided to the Society of Petroleum Engineers for distributionand possible publication in an SPE journal. The material is subject to correctionby the author(s). Permission to copy is restricted to an abstract of not more than300 words. Write SPE Book Order Dept., Library Technician, P.O. Box 833836,Richardson, TX 75083-3836 U.S.A. Telex 730989 SPEDAL.

MAR 3 0 1990

SPEPUBLICATIONS

UNSOLICITED

MISCIBLE DISPLACEMENT INFRACTURED POROUS MEDIA

FERNANDO C. PEREZ -CARDENASJUANA CRUZ-HERNANDEZCANDELARIO PEREZ-ROSALES

INSTITUTO MEXICANO DEL PETROLEO

1990

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A B S T R ACT I

A theoretical and experimental study on themiscible displacement in fractured porous media ispresented. Thetheor,y is based on the ideas originallydeveloped by Coats and Smith in connection with mis-cible displacement in homogeneous media which containdead-end pores. The proposed model considers that fluiddisplacement takes place exclusively through the frac-tures by a convection-dispersion process, while thematrix blocks exchange matter with the fractures bymolecular diffusion. Four systems of contrasting geo-metries are analyzed. They consist of: infinite parallelplates, CUbes, spheres, and infinite parallel cylinders.It is found that the ~ehavior of these systems can bedescribed by the same mathematical formulation. Thissuggests that the proposed model is of a general type.To test the validity of the model, displacement runswere carried out with a core of Berea sandstone inser-ted within a hollow plastic cylinder. The annular space

,

formed between the core and the cylinder behaves as afracture of the system. Potasium chloride solutions ofdifferent concentration were used as the displacingand the displaced fluids. The experimental results arein accord with the theory.

I N T ROD U C T ION

In reservoir engineering, the study of miscibledisplacement in fractured porous media is importantbecause of its potential applications to enhanced oil

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recovery in naturally fractured reservoirs. The tHeoryconcerning fractured media presents some analyticalproblems that are not present in homogeneous systems.They are mainly due to the existence of factors suchas fluid channeling through the fractures and masstransfer between fractures and matrix.

The physical considerations adopted in this paperassume that fluid displacement takes place exclusivelythrough the fractures by a convection-dispersion pro-cess, and that the matrix blocks behave as stagnantelements which exchange fluids with the fractures bymolecular diffusion.

The mathematical model used to describe fluiddisplacement is the convection-dispersion equation ofCoats and Smith1 which applies to homogeneous porousmedia containing dead-end pores. The proposed modelestablishes a functional analogy between the matrixblocks of the fractured systems and the dead-spacesof the homogeneous media.

J

To make a theoretical study of the mass transferbetween fractures and matrix, four systems are analy-zed: (1) infinite parallel plates, (2) cubes, (3)spheres, and (4) infinite parallel cylinders. In spiteof their contrasting geometries, the behavior of thefour systems can be expressed by the same mathematicalformulation, which is similar to that proposed by Coatsand Smith1 for explaining mass exchange between flow-ing and stagnant pores.

Laboratory experiments were conducted to test thevalidity of the model. In this work, use was made of a

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simple technique to study the behavior of fracturedmedia. A core of Berea sandstone was inserted withina hollow plastic cylinder. Because of the irregularityof the core surface, there resulted a thin annularspace between the core and the cylinder. This annularspace actuated as a fracture of the system. Potasiumchloride solutions of different concentration wereused as the displacing and displaced fluids. The con-centration of the effluent was measured by electricalmeans. A good agreement between theory and experimentwas found.

FORMULATION

To establish the mathematical formulation, supposethat a linear fractured porous medium is completelysaturated with a fluid; then, at a given time, a secondfluid begins to be injected at one end. Assuming thatthe fluids are miscible, it is desired to find an ana-

,

lytical expression which gives concentration of theinjected fluid as a function of distance and time.

Laboratory and field. experiments indicate thatfracture permeability is much greater than matrix per-meability. Consequently, it will be considered thatfluid displacement takes place exclusively through thefractures by a convection-dispersion process, whilethe matrix blocks act as stagnant elements which ex-change matter with the fractures by molecular diffu-sion. By material balance considerations, it is foundthat the differential equation that describes the pro-

. 2cess 1.S

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~ ~c*= f + (1 _ f) 0

at Clt ~ 1)

where D is the dispersion coefficient, u is the ave-rage intersticial velocity, x is distance, t time,c fracture concentration of the injected fluid, cmatrix concentration of the injected fluid, and ffraction of pore space occupied by fractures. Thisequat'ion has the same form as that proposed by Coatsand Smith1 for the case of homogeneous media thatcontain dead-end pores; however, in the present case,c* represents the average concentration within theblocks which, for a block at a given time, is definedas

c"(t) = JJffc. (r.t)dV (2)where c'(r,t) is the concentration at a point withinthe block and V is the block pore volume.

Equation 1 has two unkowns: c and c*; hence,another equation is needed to obtain explicit solu-tions for these variables. This can be accomplishedby physical considerations, as follows: At the onsetof the experiment, the fracture concentration of theinjected fluid is zero. When the injection begins, thefluid advances preferentially through the fracturesand in a short time reaches a quasi-steady state, sothat the fracture concentration can be approximatedby a constant value which will be denoted by cq

The injected fluid penetrates the blocks exclu-sively by molecular diffusion; consequently, the con-centration within the blocks obeys a diffusion equation

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of the form.

. . . . . . .

r

( 3)

where D' is the molecular diffusion coefficient.

The initial and boundary conditions are

cl(within a block) = 0 forcl(on the surface) = cq for

t = 0

t ~ 0

In Appendix A, equation 3 is solved with condi-tions given by equations 4 and 5, for four differentsystems consisting of: (1) infinite parallel plates,(2) cubes, (3) spheres, and (4) infinite parallelcylinders. For the four cases it is found that

cq - c* = c~ exp(-kt) (6)where c~ and k are constants.

,

Deriving equation 6 with respect to time, yields

ac*- - - -at clk exp(-kt) = - k(cq q (7)

As stated above, cq is only an approximation ofthe actual fracture concentration, c, so that usingthis value instead of c , and defining K = (1 - f)k,qequation 7 reduces to

oc*(1 - f) -- = K(cch . . . . . . . (8)

where K is the so called mass transfer coefficient

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between matrix and fractures. I

Equation 8 was originally proposed by Coats andSmith1 for explaining mass transfer between flowingand stagnant (dead-end) pores of homogeneous media.Its validity has been established by some investiga-tors. 1,3,4 In this paper it is shown that the sameexpression can be applied to fractured media providedthat c~ represents the average block concentrationdefined by equation 2. Although here only four casesare analyzed, the results obtained suggest that equa-tion 8 is a general expression which applies to anyblock shape.

The analytical solution to the system formed byequation 1 and 8 is presented in Appendix B, whereuse is made of dimensionless variables. When theinjection of the displacing fluid is made in a conti-nuous form, and the appropiate initial and boundaryconditions are used, the solution for the fractureconcentration is

,

CO

fa (9)

An expression for the average block concentra-tion can be obtained in a similar way. In Appendix Cit is shown that

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m r~ exp( t n ) J b exp(Mxn ) ~ 21> = - ri 2 2J 2 (b+1-z )cos(ztn-NXn)

II 0 L(b+1) +z (1+z)

+z(b+2) sene ztn-NXn)] dz (10)

It is interesting to make a comparison between thebehavior of fracture concentration (given by equation9) and matrix concentration (given by equation 10).For illustrating purposes, consider a system of lengthL with the following characteristics: dimensionlessdispersion coefficient DD = 4, dimensionless masstransfer coefficient ED = 1, and volumetric fractionof fractures f = 0.05. Also consider that the obser-vations are made at the outlet of the sample, that isfor Xn = 1 (see Appendix B). The results obtained areshown in figure 1.

This figure shows that, at the beginning of therun, the fracture concentration increases much morerapidly than the matrix concentration. This means

,

that the injected fluid channels through the fractures.However, as time goes on, the displacing fluid beginsto penetrate the matrix, so that for long times thetwo concentrations tend to become equal.

The curve for the fracture concentration presentstwo well-defined regions. The large slope regioncorresponds to that part of the process which is do-minated by the displacement of fracture fluids, whilethe small slope region represents the part dominatedby the mass exchange between matrix and fractures.

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EXPERIMENTAL WORK I

The experimental study of fluid flow throughfractured porous media is not an easy problem, dueto the difficulty of having a precise control overthe geometrical properties of the fracture network.To deal with this type of problems, generally useis made of simple arrangements that in some way areequivalent to natural systems. In the present study,use was made of a system of simple geometry thatreproduces in a realistic way the displacement me-chanisms of fractured media.

Essentially, the system consists in a cylindri-cal sample of homogeneous porous rock inserted withina plastic container, as shown schematically in figure2. The rock sample, R, is fitted within the hollowcylinder, 0, of smooth walls. Due to the externalirregularities of the porous sample, a thin annularspace, F, is formed between the rock and the contai-ner. The fluids enter through A and leave at B. Whenfluids are flowing, they move much more rapidly throughthe annular space than through the rock, so that theannular space acts as a fracture of a fractured medium.By means of the pairs of electrodes I-I' and 0_0'electrical measurements can be made, from which it ispossible to determine the relative concentrations ofthe salt solutions used as displacing and displacedfluids.

The experimental runs were carried out with acleaned and stabilized Berea sandstone core insertedwithin a plastic cylinder. The characteristics of thesystem are indicated in table 1.

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I

The displacement runs were performed at constantflow rate, using two solutions of potssium chloride:15 g/cm3 for the displacing fluid and 3 g/cm3 for thedisplaced fluid.

The experimental results for a run are shown infigure 3. By fitting a theoretical curve to the expe-rimental points, it was found that DD = 0.20, KD = 0.36,f = 0.25. As it can be seen, there is a good agreementbetween theory and experiment.

It should be noted that the theoretical f is 4times greater than the measured value (compare valuesof f from figure 3 and table 1). This is probably dueto the fact that the fluids deep inside the sample donot participate in the movement of fluids, so that thepore volume associated with these fluids behaves as ifit were part of the solid phase.

CON C L U S ION S

1. Miscible displacement within fractured media occursin two steps. The first one, of short duration, isdominated by the movement of fracture fluids, andthe second step, of large duration, is associatedto fluid exchange between matrix and fractures.

2. The mathematical formulation which governs displace-ment processes in fractured media is similar tothat of homogeneous media.

3. Although in this work only four particular caseswere analyzed, it is considered that the formula-tion obtained is of a general type.

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4. The good agreement betwee~ theory and experimentconfirms the validity of ~he mathematical modelarrived at in this paper.

NOM ENe 1 A T U R E

c = fracture concentra~~onc' = matrix concentratic~

*c = average concentration within a blockcD = fracture dimension:ess concentrationc; = dimensionless avera5e concentration within

a blockCo = initial concentration of displacing fluidc q = quasi-steady fract~~e concentrationD = dispersion coefficient

n' z molecular diffusioL coefficientDn = dimensionless dispe~sion coefficient

f = fraction of pore space occupied by fracturesK = mass transfer coefficient between matrix and

fractures~ = dimensionless mass :ransfer coefficient

L = thickness of infin~~e plate; edge length ofcubic block

r = radial distanceR = radius of spherica: block; radius of infinite

cylindert = time~ = dimensionless time

u = average intersticial velocityV = block volume

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x = distance= dimensionless distance

REF ERE N C E S

1. Coats, K.H. and Smith, B.D.: "Dead-End Pore Volumeand Dispersion in Porous Media", Soc. Pet. Eng. J.(March 1964) pp. 73-84.

2. Perez-Rosales, C. and Perez-Cardenas, F.C.: "Disper-si6n de Trazadores en Medios Porosos Fracturados",Ingenieria Petrolera (Nov. 1985) pp. 19-25; Revistadel Instituto Mexicano del Petr6leo (April 1986)pp. 26-31.

3. Brigham, W.E.: "Mixing Equations in Short LaboratoryCores", Soc. Pet. Eng. J. (Feb. 1974) pp. 91-99.

4. Baker, L. E.: "Effects of Dispersion and Dead-EndPore Volume in Miscible Flooding", Soc. Pet. Eng.J. (June 1977) pp. 219-227.

5. Crank, J.: The Mathematics of Diffusion, Oxford atthe Clarendon Press, Second Edition (1975).

6. Churchill, R. V.: Modern Operational Mathematicsin Engineering, McGraw-Hill Book Company, Inc.,l~ew York (1944).

7. Perez-Cardenas, F. C.: Dispersi6n de Trazadoresen Medios Porosos Fracturados, Tesis Profesional,Facultad de Ciencias, National University of Mexico(1986).

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APPENDIX A

Solutions ~o equation 3 with conditions give~ byequations 4 and 5 are here presented, for four systemsof different geometries.

Infinite Parallel Plates

In this case, an infinite plate parallel to planeY-Z, of thickness L, is considered. In one c~mens~on,equation 3 takes the form

(A-1)

and the initial and boundary conditions are

c'(x,=;=O'-c' (O,t) = c' (L,t)

0< xo

(;"-2 )(A-3)

Using the ~ethod of separation of variaoles, onearrives at the solution

4 LCO 1 (2n+1) n"x (-:]' (2n"'1)2~t)J- - ----21sen L exp r

'IT n+ - Lt:::n=O

(A-4 )

and the average concentration given by equa~ion 2 re-duces to

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*c (t)L

= i fa c' (x, t)dx = Cq~ CD- _a2~ __1~-=-21/ L- (2n+1)

n=O

( _ D I (2n+ 1 ) 2 1r2t )

exp 2L

(A-5)

In figure A-1, graphs of -In Ec q - c*) /cqJversus time are presented, for four different cases.Curve A corresponds to a system of parallel plates withD I -5 2 I L= 1.35 X 10 cm /seg and = 1Q cm. It is seenthat, with the exception of small times, the behaviorcan be described by a straight line. Therefore, if kis the slope of the straight pa~~,

*c - cIn( 9 ) = kt + c.

c q

where d is a constant. Whence,

c - c~ = c' exp(- kt)q qwhere c' = c exp(- d)q q

Cubes

(A-6)

(A-7)

In this case, it suffices ~ith considering asingle cube. If it is assumed tt-at the diffusioncoefficient is isotropic, one has

(A-8)

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To avoid dividing by zero when using the methodof separation of variables, it is convenient to define

Q(x,y,z,t) = 1 - c' (x,y,z,t) . . . (A-9)

With this new variable, equation (A-8) takes the form

(A-10)

By considering a cube of edge L, and applying toc' the same conditions used in the previous case, bythe method of separation of variables it is found that

Q(x,y,z,t) = Qp(x,t)Qp(y,t)Qp(z,t)

= (1-c'(x,t))(1-c'(y,t))(1-c' (z,tp p p (A-11)

where the CIS are solutions of the form of equationpA-4.

From equations A-9 and A-11, we have that

c' (x,y,z,t) = [ 1-(1-C' (x,t)(1-c' (y,t(1-c' (z,tlp p P J (A-12)

Consequently, the average concentration within thecube is given by

*c (t)

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L L L- _1 ( ( ( c' (x,y,z,t)dxdydz- L 3 Jo Jo Jo

.dxdydz

L

= 1 - D. 1:,(1-c~( x. t) ) dXr (A-13)Curve B of figure A-1 illustrates the behavior of

a system with D l = 1.35 X 10-5 cm2/seg and L = 10 cm.As it is seen, its behavior is similar to the case ofparallel plates.

Spheres

Here, a sphere of radius R is considered. Sincethe ~iffusion is radial, the equation to be solved is

with the conditions

(A-14)

cl(r,O)=0c' (R,t) = cq

,

,

r

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v = c'r . . . {A-17)

equation A-14 transforms into

whose solution is

. (A-18)

c' (r, t)CD

+ 2R ~ .k1ln1rr L n

n=1

and

. (A-19)

c*(t)

Its behavior for D' = 1.35 X 1C-5 c~2 =eg andR = 5 cm is indicated by curve C of :igure ~-1.

Parallel Infinite Cylinders

In this case, a cylinder of rad~us B v:ll beconsidered. Due to the cylindrical s~e~I7. the dif-ferential ecuation is

.1. -1(rD' aC') = actr ar ar at

. (A-21)

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with the conditions

c' (r,O) = 0c'(R,t)=cq ,

r

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nn = D/uL

Kn = LK/u

cn = clcot\ itCn = c Ico

where L is the length of the sample and Co is theinitial concentration of the displacing fluid. Fromthe definition of dimensionless time, tn' it is seenthat this variable is equal to the number of porevolumes injected. With the above relationships, equa-tions 1 and 8 take the form

(B-1)

and

(B-2)

For a simi-infinite medium, the initial andboundary conditions are

cn(Xn,O) = 0 , Xn ? 0* 0 Xn ? 0cn(~'O) =

cn(cc,tn) = 0 , tn; 0cn(O,tn ) = 1 t n lO

Using the method of the complex inversion inte-gral of the Laplace transformation,6 the solution is

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. . . . . . . . . . (B-3)

where

M = 2~D (1 - Je'COS ~)N = ~ Jesen ~

D

ve = arctan U

U 1 + 411n~ + Knb + Kn(1 + ::)]= (1 + b) 2 +V = 4DDZ [f + b~ + z2 ](1 +b Kn= 1 + f

APPENDIX C

In this appendix a general expression for theaverage concentration, c;, within a block is obtained.The Laplace transform of equation B-2 is

(C-1)

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And making s = 1 + iz, one arrives at

-* KDcD = (1 + iz)(1 - f) +

(C-2)

". . . . . . . . (C-3)

Now, according to equation C.4 of reference 7, cDcan be expressed as

exp(MXn) ~On = 2 (cosNx..n - z senNx..n) - i (senNXn1 + z

+ z COSNx..n)]

By substituting equation C-3 into equation C-2,and separating the real and imaginary parts, yields

-* b exp(MxD).z2) {Eb z2)cosNxDcD = [(b + 1) 2 + z2J( 1 + 1 -+

- z(b + z)senNxDJ - i[(b + 1 - z2)senNXD

+ z(b + 2) COSN"DJ} (C-4)The inverse Laplace transform can be obtained by

means of the relationship6

co* exp( t D) fc

cD = tr (p coso

. (C-5)

(C-6)

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r

where p and q are the real and imaginary parts, res-pectively. Using the pertinent substitutions, onearrives at

Cf:)

i .b exp(Mxn ) r: 2r, 2 2J 2 L(b+ 1-z ) cos (ztn-NxD)o L(b+1) +z (1+z)+z(b+2)sen(z~-Nx:n)Jdz

81 Metric Conversion Factors

*in X 2.54 E+OO = cm

in2 X 6.451 6* E+OO = cm2

in3 X 1.638 706 E+01 = cm3

Ibm X 4.535 924 E-01 = kgmd X 9.869 233 E-04 = pm2

'* .Convers~on factor is exact.

TABLE 1 - CHARACTERISTICS OF SAMPLE

Code: DMF-1Diameter: 3.8 cmLength: 5.4 cmPrimary porosity: 0.20Secondary porosity: 0.03Fracture volumetric fraction: 0.06Permeability: 3.3 d

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Do= 4Ko= If =0.05Xo= I

O_...a....---a......---&....---L----Ir....-...a...---a......-.a...__a..---J

o 0.4 0.8 1.2 1.6 2.0PORE VOLUME INJECTED Ito

z:oi=; 0.8'