Concentration and temperature transients in heterogeneous porous media

Journal of Petroleum Science and Engineering 26 Ž2000. 221–233 www.elsevier.nlrlocaterjpetscieng

Concentration and temperature transients in heterogeneous porous media Part II: Radial transport I. Kocabas a , M.R. Islam b,) a

Department of Chemical and Petroleum Engineering, UAE UniÕersity, P.O. Box 17555, Al-Ain, United Arab Emirates b Faculty of Engineering, Dalhousie UniÕersity, Halifax, NS B3J 2X4, Canada Received 12 December 1998; accepted 15 December 1999

Abstract Being the most commonly used model of transport of solute and heat in porous media, analytical solutions of the convection–dispersion equation are of great importance for both interpretative and numerical model validation purposes. As in the linear case, the use of two different concentration variables namely resident and flux concentrations are equally common in radial transport; while the former is used in deriving the governing equations, the latter is the one measured in experiments. Therefore, description of transport processes in terms of dependent variables must be relevant to the way tracer experiments are to be performed. In addition, the common assumption of velocity and scale dependence of dispersion coefficient also leads to modification in governing differential equations whether they are expressed in resident or flux concentrations. This work presents a detailed classification of the solutions of convection–dispersion equation in radial coordinates based on velocity- andror scale-dependent forms of dispersion coefficient and also on the physical meanings of the solutions with respect to two different concentrations. The classification of solutions includes guidelines for selection of appropriate solution to be employed in field experiments and numerical validation as well. This classification has led to the development of several new solutions in this work that are of great importance for interpreting and designing field experiments. q 2000 Published by Elsevier Science B.V. Keywords: chemical transport; wellbore; miscible displacement; solutal diffusivity

1. Introduction Interpretation of concentrationrtemperature transients is of great importance in reservoir engineering ) Corresponding author. Tel.: q1-902-494-3980; fax: q1-902494-3108. E-mail address: [email protected] ŽM.R. Islam..

because almost all EOR processes involve injection of heat or some kind of a miscible slug into the reservoir. The purpose of any miscible fluid injection is to keep minimum smearing of the miscible slug or displacement front in case of heat injection. Hence, the process can achieve maximum displacement of hydrocarbon in place. Prior knowledge of transport parameter values is of great importance in the design

0920-4105r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 9 2 0 - 4 1 0 5 Ž 0 0 . 0 0 0 3 6 - X

222

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

and operation of a full-scale EOR process. Based on the knowledge of the transport parameters, performance of various schemes could be simulated prior to practical implementation. A significant number of these parameters are recovered from the interpretation of tracer return profiles. In fact, tracer tests have been an indispensable tool for studying the transport through porous media ŽThorbajarnarson and Mackay, 1997.. In general, both determination of tracer transport parameters and prediction of displacement performance of EOR fluid slugs involve solving the convection–dispersion equation. The analytical solutions of the radial convection–dispersion equation are valuable for both interpretation and numerical model validation purposes. Based on constant concentration ŽAvdonin, 1964; Tang and Babu, 1979; Chen, 1985; Ramirez et al., 1995. or constant flux ŽChen, 1987; Tang and Peaceman, 1987; Falade and Brigham, 1989. specification at the injection well, divergent ŽFalade et al., 1987. or convergent flow ŽFalade et al., 1987; Chen and Woodside, 1988; Moench, 1989., many approximate ŽHoopes and Harleman, 1967. and exact ŽAvdonin, ŽChen, 1986; Philip, 1994. analytical solutions have been developed for tracer flow in aquifers and heat transport during hot water injection in oil reservoirs. In an interpretation process, selection of a solution consistent with the actual conditions of the experiment could be achieved through the distinction of resident and flux concentrations ŽBrigham, 1974.. Such a distinction is necessary whenever a convective–dispersive transport is involved. The resident concentration C, is defined as the amount of material per unit volume of the system and mostly used for deriving the governing differential equations. In experiments, on the other hand, the measured quantity is mostly the flux concentration Cf defined as the ratio of the material flux to volumetric flux. The resident and flux concentrations are related by: uCf s uC y D= C

Ž 1.

Eq. Ž1. states that the flux concentration passing a cross-section is always greater than the resident concentration at that point. Using. Eq. Ž1., one can show that both resident and flux concentrations satisfy the same governing differential equation in linear coor-

dinates ŽPhilip, 1994.. In other words, both resident and flux concentration solutions are derived from the same partial differential equation. Hence, the boundary conditions determine whether the resultant solution is in C or Cf . While the solutions in resident and flux concentrations produce similar return profiles for small dispersivity systems, they differ significantly when the dispersive transport is considerable ŽKreft and Zuber, 1978.. Hence, the use of correct solutions becomes necessary for highly dispersive systems. Recognition of this distinction has led to a full classification of the solutions of the linear convection–dispersion equation. Such a distinction has also been recognized for the solutions of the radial convection–dispersion equation. However, a full classification of the solutions has not been performed yet. A major goal of this work is to perform a full classification of the solutions of the radial convection–dispersion equation. In addition, some new solutions have been derived during the course of this classification process. In fact, the most appropriate solutions to use in the interpretation of experiments are among these new solutions. An analysis of convection–dispersion equation in radial coordinates and its solutions in terms of the resident and flux concentrations is presented in the following.

2. Governing differential equations In order to study the governing differential equations and the physical meaning of the solutions in radial coordinates, we have to first specify the form of the dispersion coefficient D. Since each form of the parameter D leads to a different governing differential equation, each assumed form and hence each resultant solution has to be studied separately. There are four functional forms of the dispersion coefficient likely to occur in radial dispersive transport. Firstly, one is that D is independent of both velocity and scale and hence remains constant over the whole transport domain. Secondly, one may assume a linear velocity dependence of D over the whole domain. Alternatively, one can consider that D is linearly dependent on both velocity and scale. This also leads to a constant D, in radial divergent

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

flow. Finally, one can assume a linear dependence on velocity and an asymptotic linear dependence on scale. This in turn leads to separating the transport domain into three concentric regions. In the innermost region, D is assumed to be both velocity- and scale-independent. In the intermediate region, D is considered as linearly dependent on both velocity and scale. Finally, in the outer one, D is assumed to be dependent on velocity linearly, but independent of scale. Among these alternatives, we consider only two functional forms of D namely, D is constant over the whole domain and D is linearly dependent on the velocity. The reason for considering a constant D is that if one assumes the simple cases: D s a < u < and

a s br

us

1 E

2.1. Constant dispersion coefficient The transport in porous media is governed by the convection–dispersion equation derived in terms of resident concentration: EŽ f C . Et

Ž 2.

where S represents any source term that exists in the model. It may represent any of the following physical processes: radioactive decay, adsorption or matrix diffusion. Assuming matrix diffusion to be the dominant interaction, S is defined as: A m f Dm Vpf

²= Cm :G D

Ž 3.

Then, the equation of the transport within the matrix is: Dm= 2 Cm s

ECm Et

ž

Ž 4.

rf D

Er

/

1 E y r Er

Ž rf uC . y S s

EŽ f C . Et

Ž 5. Before we proceed further, let us define the following characteristic times each representing a specific process included in the modelling. r 02

td s

Ž 6.

D

pf hr 02

tc s

tt s

r

= . Ž f D= C y f uC . y S s

EC

r Er

qr2p h f

D would be constant over the whole domain. The other alternatives should be treated on a separate work. In the following, governing differential equations and their solutions of those two cases are presented.

Ss

In radial coordinates, Eq. Ž2. takes the form:

Ž 7.

q

t im s

where b is a constant. Since, in a radial flow

223

a2

Ž 8.

Dm

Ž Vpf rf A m . Dm

2

Ž 9.

where td , t c , t im , and tt are the characteristic times for dispersive and convective transports in the fracture, interaction between fracture and the matrix, and diffusive transport in the matrix, respectively. The ratio of first two of these characteristic times gives the well-known Peclet number that represents the ratio of the convective to dispersive transport in the fracture. Pe s

td

Ž 10 .

tc

Similarly, one can define two other Peclet numbers for interaction between two medium and diffusive transport in the matrix. Pt s

tt

Ž 11 .

td

Pim s

t im td

Ž 12 .

Let’s also define the following dimensionless variables: r rD s Ž 13 . r0 tD s

t td

Ž 14 .

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224

Substituting these variables into Eq. Ž5. results in the dimensionless resident concentration equation: E2 C Er D2

1 q rD

ž

1y

Pe 2

EC

/

Er D

EC y SD s

Et D

Ž 15 .

Similarly, substituting dimensionless variables into Eq. Ž1. gives: 2 r D EC Cf s C y Ž 16 . Pe Er D Using Eq. Ž16., one can perform a dependent variable transformation in Eq. Ž15. to obtain the dimensionless flux concentration equation: E 2 Cf Er D2

1 y rD

ž

1q

Pe 2

/

ECf Er D

y SfD s

ECf Et D

Ž 17 .

Eqs. 15 and 17 are the dimensionless governing different differential equations of resident and flux concentrations in radial coordinates for constant D. Rewriting the equations of transport in the matrix and the source in Eq. Ž15. in terms of dimensionless variables and parameters, one obtains: 1 ECm = D2 Cm s Ž 18 . Pim Et D SD s

(P (P

im t

1 Pim

²= D Cm :G D

Ž 19 .

Eq. Ž19. is evaluated at the outer boundary of the matrix, G D Ži.e. at the interface between the matrix and the fracture.. In solving Eqs. 15 and 18, one usually assumes that the concentrations at the interface are continuous. This assumption allows the following expression to be used for the source: t D EC SD s SuD Ž t D y t . dt Ž 20 . 0 Et

H

where SuD is the flux across the interface when Cm s 1 is the boundary condition instead of Cm s C in solving the matrix transport equation. 2.2. Velocity-dependent dispersion coefficient After many field observations, a number of authors have assumed a dispersion coefficient that depends on the velocity linearly. Dsa u

With this assumption, use of the following dimensionless variables is more convenient

Ž 21 .

rD s

tD s

r

Ž 22 .

a 1

t

Ž 23 .

2 r D tc

Notice that in this case, the characteristic time for dispersive transport and the Peclet number becomes: td s 2 r 0 D t c

Ž 24 .

Pe s 2 r 0 D

Ž 25 .

In terms of these new dimensionless variables, the following relations hold: E2 C Er D2

EC y Er D

EC y r D SD s r D

Ž 26 .

Et D

EC Cf s C y E 2 Cf Er D2

ž

Ž 27 .

Er D

y 1q

1 rD

/

ECf Er D

y r D SfD s r D

ECf Et D

Ž 28 .

where Eqs. 26 and 28 are the dimensionless governing differential equations of resident and flux concentrations, respectively, and Eq. Ž27. relates the two concentrations. Contrary to the linear form of the convection–dispersion equation, comparing Eq. Ž15. with Eq. Ž17. and Eq. Ž26. with Eq. Ž28. shows that resident and flux concentration variables satisfy different partial differential equations in radial coordinates. This difference is due to that velocity varies with the variable r. In such cases, the physical meaning of all solutions and the boundary conditions is inferred from the governing differential equations. The more appropriate flux concentration solutions for interpretation of experiments must either be derived by solving the appropriate equations namely Eq. Ž17. or Eq. Ž28. or substituting the resident concentration solutions in Eq. Ž16. or Eq. Ž27..

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3. Classification of solutions

C s d Ž t . at r D s r wD wthis workx

Based on the above discussion, we can classify the solutions according to its physical meaning. In other words, we specify whether the solution represents resident or flux concentration. In classifying the solutions of radial convection–dispersion equations, one again has to consider the velocity and scale dependence of the dispersion coefficient D. The initial and outer boundary conditions employed to derive the solutions were the same for all cases considered:

™ 0 as r ™ ` C Ž r ,0 . s 0 C Ž r ,t . ™ 0 as r ™ ` C Ž r D ,0 . s 0

C Ž r D ,t D . f

D

f

D

D

D

D

The earliest studies ŽAvdonin, 1964; Hoopes and Harleman, 1967. on radial convective–dispersive transport of heat or tracer assumed a constant D value. In classifying the solutions, we also start with constant D case. 3.1. Constant dispersion coefficient The dimensionless governing differential equation of resident concentration is Eq. Ž15. whose solutions are called as resident concentration solutions. The solutions of Eq. Ž15. in Laplace space and the inner boundary condition related to each solution and where that solution is first published are shown in the following. C s 1 at r D s r wD wAvdonin, 1959x Cs

1 s

rD

P e r4

ž / r wD

ž (S Ž1 q S ž (s Ž1 q S

. rD / uD . r wD /

K P e r4 K Pe r4

uD

Ž 29 .

C y Ž2 r D rPe .ŽECrEr D . s 1 at r D s r wD wthis workx Cs

Pe S

=

Ž (s Ž1q S Ž (s Ž1q S

uD

uD

. rD .

. rwD .

P e r4

Ž P e r4 . q1

K P e r4

225

Ž (s Ž1q S . r . Ž (s Ž1q S . r .

K Ž P e r4.q 1

uD

D

uD

wD

Ž 30 .

Cs

rD

P e r4

ž / r wD

ž (s Ž1 q S ž (s Ž1 q S

. rD / uD . r wD /

K Pe r4 K Pe r4

uD

Ž 31 .

C y Ž2 r D rPe .ŽECrEr D . s d Ž t . at r D s r wD wthis workx Cs

Pe

Ž (s Ž1q S

Ž (s Ž1q S

uD

uD

. rD .

. rwD .

P e r4

Ž P e r4 . q1

™

K P e r4

Ž (s Ž1q S . r . Ž (s Ž1q S . r . uD

K Ž P e r4.q 1

D

uD

wD

Ž 32 .

Notice that as r wD 0, the solution given by Eq. Ž29. reduces to the line source solution ŽTang and Babu, 1979.. Furthermore, as r wD 0, the solution given by Eq. Ž30. also reduces to the line source solution and hence becomes identical to the solution given by Eq. Ž29.. The same relation also holds for Eqs. 31 and 32. In addition, we may intuitively say that Eqs. 29 and 31 should be discarded as inappropriate to use in actual interpretation of return profiles because of the following. We usually inject a fluid of constant concentration into the system. This means that a constant ratio of tracer to volumetric flux enters the system, which corresponds to a constant flux concentration at the injection boundary. Boundary conditions employed in deriving these solutions imply a dimensionless flux concentration that is always greater than unity at the injection boundary. Of course, such a situation is unlikely to occur in actual experiments and hence material balance errors occur if these two solutions are used. Therefore, they should be considered as inappropriate to be employed in the interpretation of return profiles. One should also notice that those boundary conditions in question should be employed in deriving solutions to Eqs. 15 and 26.

™

3.2. Velocity-dependent dispersion coefficient Based on many field observations, later works ŽTang and Babu, 1979; Chen, 1985; Falade et al., 1987. on radial convection–dispersion equation involved a velocity-dependent dispersion coefficient D, given by Eq. Ž21.. Using this form of D leads to a dimensionless governing differential equation of resident concentration given by Eq. Ž26.. One can

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

226

solve Eq. Ž26. by first applying a Laplace transform and making a variable change of the form:

The solutions of Eq. Ž26. in Laplace space and the inner boundary condition related to each solution and where that solution is first published are shown in the following

y1

z D s r D q Ž 4 s Ž 1 q SuD . .

Ž 33 .

C s 1 at z D s z wD wTang and Babu, 1979; for S D s 0x Cs

1 s

exp

ž

z D y z wD 2

/

ž ( ž 2r3(s Ž1 q S

3r2 z D1r2 K 1r3 2r3 s Ž 1 q SuD . z D

z 1r2 wD

K 1r3

uD

.

3r2 z wD

/ /

Ž 34 .

C y ŽECrEz D . s 1 at z D s z wD wChen, 1987; Tang and Peaceman, 1987x

Cs

1 s

exp

ž

z D y z wD 2

/

z 1r2 wD K 1r3

2

ž 3 (s Ž1 q S

uD

3r2 . z wD

2

(s Ž1 q S . z / / q 2 z (s Ž1 q S . K

z D1r2 K 1r3

ž

3r2 D

uD

3

wD

uD

2r3

ž

2 3

(s Ž1 q S

uD

3r2 . z wD

/

Ž 35 . C s d Ž t . at z D s z wD wthis workx C s exp

ž

z D y z wD 2

Ž zD .

/Ž

1r2

z wD .

( Ž 2r3(s Ž 1 q S

K 1r3 Ž 2r3 s Ž 1 q SuD . z D3r2

1r2

K 1r3

Ž 36 .

3r2 uD . z wD

C y ŽECrEz D . s d Ž t . at z D s z wD wthis workx C s exp

ž

z D y z wD 2

/

ž (

z D1r2 K 1r3 2r3 s Ž 1 q SuD . z D3r2 z 1r2 wD K 1r3

ž 2r3(s Ž1 q S

uD

.

3r2 z wD

/ q 2 z (s q S wD

uD

/ ž (

3r2 . K 2r3 2r3 s Ž 1 q SuD . z wD

/ Ž 37 .

Using the same arguments on the boundary conditions, one could see that the solutions given by Eqs. 34 and 36 may also involve significant material balance errors and they should not be employed in actual interpretation practices.

4. Solutions for interpretation purposes In most lab or field experiments, the observed profiles are obtained by measuring the concentrations of the samples collected from the fluid coming out of the exit boundary. Hence, the return profiles are mostly plotted in flux concentration values.

Therefore, the solutions employed in interpretation of tracer return profiles must be in terms of flux concentration. Such solutions must either be derived by solving the corresponding equations namely, Eqs. 17 and 28 or by substituting the resident concentration solutions into Eqs. 16 and 27. The resident concentration solutions consistent with the actually possible boundary conditions of experiments are given by Eqs. 30 and 32 for constant D and Eqs. 35 and 37 for velocity-dependent D. Hence, the corresponding flux concentration solutions are obtained by using the relations given in Eqs. 16 and 27. We should mention that while it is possible to obtain the flux concentration solution by

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

solving Eq. Ž17. directly, the same was not presently possible for Eq. Ž28.. The Laplace space solutions Cf s 1 at r D s r wD wthis workx

Cf s

1

Ž pr4 .q1

rD

ž /

s

r wD

ž (s Ž1 q S ž (s Ž1 q S

for the flux concentration and where the solution is published first are shown in the following.

. rD / uD . r wD /

K Ž P r4.q1 K Ž P r4.q1

227

uD

Ž 38 .

Cf s d Ž t . at r D s r wD wthis workx

Cf s

Ž Pr4 .q1

rD

ž / r wD

ž (s Ž1 q S ž (s Ž1 q S

K Ž P r4.q1 K Ž P r4.q1

. rD / uD . r wD / uD

Ž 39 .

Cf s 1 at r D s r wD wChen, 1987x

Cf s

1 s

exp

ž

z D y z wD 2

/

ž ( ž 2r3(s Ž1 q S

ž ( ž 2r3(s Ž1 q S

z D1r2 K 1r3 2r3 s Ž 1 q SuD . z D3r2 q 2 z D 's K 2r3 2r3 s Ž 1 q SuD . z D3r2 z 1r2 wD K 1r3

uD

.

3r2 z wD

/ / q2 z

wD

's K 2r3

uD

.

/ /

3r2 z wD

Ž 40 .

Cf s d Ž t . at r D s r wD wthis workx

Cf s exp

=

ž

z D y z wD 2

/ ž (

ž (

z D1r2 K 1r3 2r3 s Ž 1 q SuD . z D3r2 q 2 z D 's K 2r3 2r3 s Ž 1 q SuD . z D3r2

ž (

/

(

'

ž (

/

3r2 3r2 z 1r2 wD K 1r3 2r3 s Ž 1 q S uD . z wD q 2 z wD s Ž 1 q S uD . K 2r3 2r3 s Ž 1 q S uD . z wD

/

The Cf solutions given by Eqs. 38 and 39 can also be obtained by solving Eq. Ž17. directly. However, solving Eq. Ž28. was not presently possible for us. Therefore, like other authors ŽChen and Woodside, 1988., we preferred employing Eq. Ž27., which relates C to Cf . The difficulty in obtaining a solution of Eq. Ž28. is embedded in the resultant Cf solution ŽEq. Ž40.. obtained by solving Eq. Ž26. and then using Eq. Ž27., since Eqs. 40 and 41 can only be expressed as a linear combination of two modified Bessel functions namely K 1r3 and K 2r3 . As an example, three solutions given by ŽEqs. 34, 35 and 38. with no source Ži.e. SuD s 0. are computed and plotted in Figs. 1–3. Fig. 1 shows that resident solutions given by Eqs. 34 and 35 practi-

Ž 41 .

/

cally yield the same return profile for small wellbore radius despite the significantly small Peclet number corresponding to a large dispersivity system. Nevertheless, this is expected because both solutions should converge to the line source solution as r wD decreases. Fig. 2 shows that for small Peclet numbers, if wellbore radius gets larger, those two solutions differ significantly. Fig. 3 shows that Eqs. 34 and 35 again practically yields the same return profile. This indicates that as Peclet number becomes larger, convective transport effectively dominates the large wellbore effect appearing in Fig. 2. Finally, in all three figures, the flux concentration solution, Eq. Ž38., yields much different results than the resident concentration solutions. This difference is significant

228

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Fig. 1. Normalized return profiles for resident and flux concentrations, for large dispersivity and small wellbore radius.

even in case of a large Peclet number as shown in Fig. 3.

5. Convergent radial dispersion In reservoir engineering practices, convergent radial dispersion is involved in single well tracer tests and sensible heat storage operations and so on. A number of authors ŽChen and Woodside, 1988;

Moench, 1989. have presented solutions for constant dispersivity and velocity-dependent dispersion coefficient. In addition, the classical work of Falade et al. Ž1987. on single well tracer tests in oil reservoirs must also be mentioned for this type of problem as well. In radial convergent dispersion problem, the distinction between C and Cf is also required. However, since at the observation well a zero gradient boundary condition is employed, C and Cf solutions

Fig. 2. Normalized resident and flux concentration return profiles for large dispersivity and large wellbore radius.

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229

Fig. 3. Normalized resident and flux concentration return profiles for small dispersivity and large wellbore radius.

become identical at this boundary Žof course here we assumed a negligible wellbore volume or zero mixing factor.. If we had measurements at points other than the production well, the distinction between the two concentration variables and hence derivation of appropriate new solutions would have been necessary. However, this is usually not the case in most field experiments. Therefore, we will not further pursue these solutions in this work. 6. Interpretation of tracer return profiles The interpretation of tracer return profiles usually consists of solving an inverse problem, where the

Cf s

1 s

Ž Pr4 .q1

ž (t s / ž (t s r /

Cf s

1 s

exp

ž

Ž Pr4 .q1

wD

z 0 D y z wD 2

ž( / ž (t s r /

K Ž P r4.q1 td s

d

d

controlling parameter values are recovered via matching the experimental profile to a theoretical curve. The matching of data to theoretical profile is usually performed by a computer program that employs a nonlinear regression procedure. The interpretation also includes determining the confidence intervals on the match and on the estimated values of the parameters as well. The theoretical functions in radial dispersion problems may have one to three parameters to be estimated. These parameters are best specified as the characteristic times representing various processes accounted by the model equations. The following are the model equations namely, Eqs. 38 and 40 in terms of the characteristic times ŽKocabas and Islam, 2000..

/

K Ž P r4.q1

d

Ž 42 .

wD

3r2 z D1r2 K 1r3 2r3 td s z 03r2 D q 2 z 0 D td s K 2r3 2r3 td s z 0 D

( ž ( / ž ( / ž 2r3(t s z / q 2 z (t s K Ž 2r3(t s z

z 1r2 wD K 1r3

d

3r2 wD

wD

d

2r3

d

3r2 wD

Ž 43 .

230

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

where z 0 D s r 0 D q Ž 4 sr Ž 2 t c r 0 D . .

y1

z wD s r wD q Ž 4 sr Ž 2 t c r wD . .

y1

Ž 44 . Ž 45 .

r 0 D s r 0 ra

Ž 46 .

r wD s r w ra

Ž 47 .

We first notice that no source term is included in these solutions. This is done for simplifying the computations. Secondly, for velocity-dependent dispersion case, td is given by Eq. Ž24.. Since Peclet number is defined by Eq. Ž25., in an interpretation or comparison of solutions, once the Peclet number, P, is specified, one can recover the value of dimensionless distance r 0D , from Eq. Ž25.. Application of the nonlinear regression to a simulated data with and without a random noise addition, and to areal field data will be presented elsewhere where heterogeneities will also be considered.

7. Conclusions and recommendations The analytical solutions of radial convection–dispersion equation are important for both interpretation and numerical model validation purposes in tracer testing, miscible displacement and so on. Based on the resident and flux concentration concepts, the most up-to-date classification of the solutions of radial convection–dispersion equation has been presented. We have shown that if dispersion coefficient depends on both scale and velocity, we can assume a constant D over the whole domain. Two new solutions to radial convection–dispersion equation have also been presented in this work. These new solutions are found to be more appropriate for return profile interpretation. Finally, an interpretation procedure has been outlined by using the appropriate flux concentration solutions. The regression parameters have been specified as characteristic times each representing a certain process included in the modelling. Computing these solutions including the matrixr fracture interaction for transport in heterogeneous

media or including linear adsorption or radioactive decay, and carrying out a sensitivity analysis will be the topic of our next work. Since numerical inversion still poses a serious problem in these type of problems ŽLarry, 1985; Moench, 1991., the most accurate and yet fast numerical inversion algorithm for our functions will also be explored in that work.

8. Nomenclature a Am B C Cm D Dm h Pe Pim Pt q r s S t td tm tt tw u Vmp Vfp d f h t l v

=

volume-to-area ratio of a matrix block interface area between mobile and immobile phases block geometry function mobile phase concentration immobile phase concentration longitudinal dispersion coefficient diffusion coefficient in immobile phase thickness of the reservoir longitudinal Peclet number of mobile phase Peclet number of immobile phase transverse Peclet number of mobile phase flow rate lspace variable along the flow direction Laplace transform variable amount of tracer generatedrlost per unit volume of the mobile phase per unit time time variable of the transport equations characteristic time for dispersion characteristic time for diffusion characteristic time for interaction between two phases breakthrough time of the convective front flow velocity pore volume of immobile phase pore volume of mobile phase Dirac delta function matrix porosity space variable of diffusive transport domain time convolution variable ratio of a block to the largest block size matrixrfracture pore volume ratio Žor Fourier transform parameter in Eqs. 17 and 17. gradient operator

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

231

Subscripts D dimensionless F or f flux R resident u for a unit concentration at the boundary o value at the outer boundary w value at the inner boundary Žwellbore.

Equating coefficients of Eq. ŽA4. to those of Eq. ŽA1., one obtains the following four conditions:

Superscripts y indicates Laplace transformation

Solving Eq. ŽA5. for a, b , c and p and then substituting them into Eq. ŽA2. and replacing y by Cf gives:

Appendix A. Derivation of flux concentration solutions

ž

Ž 1 y 2 a. s y 1 q

d2 C d r D2

q Ž 1 y 2 a. r D

2

/

r D2 c s r D2 ,

,

b c 2 s s Ž 1 q SuD . , and a2 y c 2 p 2 s 0

( = ž (s Ž 1 q S = ž (s Ž 1 q S = ž (s Ž 1 q S

dC d rD

Ž A5.

Ž Pr4 .q1

Cf s A Ž s Ž 1 q SuD . r D

Andrews states that given the following differential equation r D2

Pe

IŽ P r4.q1

uD

. rD / q B

uD

. rD /

uD

. rD /

Ž Pr4 .q1

K Ž P r4.q1

Ž A6.

The Laplace-transformed boundary conditions are:

q b 2 c 2 r D2 c q Ž a2 y c 2 p 2 . C s 0

Ž A1.

if b in Eq. ŽA1. is allowed to be pure imaginary, say b s i b , b ) 0 and p G 0, the general solution is: C s Ar Da Ip Ž b r Dc . q Br Da K p Ž b r Dc .

Ž A2.

In the following, the similarity between the Laplace-transformed governing differential equations and Eq. ŽA1. will be used to derive the selected relevant solutions, namely Eqs. 34 and 38.

Cf

™ 0 as r ™ `

Ž A7.

D

Cf s

1 s

at r D s r wD

Ž A8.

Imposing the boundary conditions given by Eqs. A7 and A8 reduces Eq. ŽA6. to Eq. Ž38., which is the desired solution of Eq. Ž17.. Note that this technique is also applicable to solve Eq. Ž15.. A.2. Variable dispersion case

A.1. Constant dispersion case

For the case of variable dispersion coefficient, Eq. holds. Taking Laplace transform of Eq. Ž26. and substituting Eq. ŽA3. for S D yields:

The Laplace transform of Eq. Ž20. is: S D s sCSuD

Ž A3.

Taking Laplace transform of Eq. Ž17., multiplying it by r D2 and replacing the term S D by Eq. ŽA3. yields: 2

r D2

d Cf d r D2

y r D Ž 1 q Per2 .

dCf d rD

y r D2 s Ž 1 q SuD . Cf s 0

Ž A4.

d2 C d r D2

dC y d rD

y r D s Ž 1 q SuD . C s 0

Ž A9.

Defining y1

z D s r D q Ž 4 s Ž 1 q SuD . .

Ž A10.

Ct s Cexp Ž yz D r2 .

Ž A11.

z t s Ž s Ž 1 q SuD . .

1r3

zD

Ž A12.

I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233

232

and performing these variable transformations in Eq. ŽA9. successively yields: d 2 Ct d z t2

y zt

dCt d zt

s0

Ž A13.

Eq. Ž13. is the standard Airy equation and its solutions is given in terms of Airy functions. Then the Airy functions are rewritten in terms of modified Bessel functions for computational purposes. Here we will use the similarity of Eqs. A1 and A13 to obtain the solutions directly in terms of modified Bessel functions. Multiplying Eq. ŽA13. by z t2 yields: z t2

d 2 Ct d z t2

y z t3

dCt d zt

b 2 c 2 s y1, and a 2 y c 2 p 2 s 0

Ž A15.

Solving the system of equations in Eq. ŽA15. for a, b , c and p and substituting them in Eq. ŽA2., we obtain: 2 3

z t3r2 q Bz t1r3 K 1r3

/

ž

2 3

z t3r2

/ Ž A16.

For simplicity, let’s apply the boundary conditions stated in Eqs. A7 and A8. These equations in terms of the new variables become: Ct

™ 0 as z ™ 0 t

Ct s

1 s 1

s s

1 s

exp Ž Ž z t y z wt . r Ž 2 b 1r3 . .

=

zt

ž / z wt

1r2

K 1r3 K 1r3

ž ž

2 3 2 3

z t3r2 3r2 z wt

/ /

Ž A19.

Substituting back the original variables into Eq. Ž19. results in Eq. Ž34..

Ž A14.

Ž 1 y 2 a . s 0, z t2 c s z t3 ,

ž

Cs

References s0

Equating the coefficients of Eqs. A1 and A14 results in:

Ct s Az t1r3 I1r3

Imposing Eqs. A17 and A18 on Eq. ŽA16. yields the solution:

Ž A17.

exp Ž yz wD r2 . exp Ž yz wt r Ž 2 b 1r3 . . at r wD

Ž A18.

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I. Kocabas, M.R. Islam r Journal of Petroleum Science and Engineering 26 (2000) 221–233 transform solution for aquifer tracer testing. Water Resour. Res. 25 Ž3., 439–447. Moench, A.F., 1991. Convergent radial dispersion: a note on the evaluation of Laplace transform solution. Water Resour. Res. 27 Ž2., 3261–3264. Philip, J.R., 1994. Some exact solutions of convection–diffusion equations. Water Resour. Res. 30 Ž12., 3545–3551. Ramirez, J.S., Samaniego, F.V., Rodriguez, F., Rivera, J.R., 1995. Tracer-test interpretation in naturally fractured reservoirs. SPE a28691, SPE Formation evaluation.

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