Stochastic Environmental Research and Risk Assessment 17 (2003) 370–383 Ó Springer-Verlag 2003 DOI 10.1007/s00477-003-0166-0
Flow and transport through two-dimensional isotropic media of binary conductivity distribution. Part 1: NUMERICAL methodology and semi-analytical solutions 370
A. Fiori, I. Jankovic, G. Dagan Abstract. Flow and transport take place in a heterogeneous medium made up from inclusions of conductivity K submerged in a matrix of conductivity K0 . We consider two-dimensional isotropic media, with circular inclusions of uniform radii, that are placed at random and without overlap in the matrix. The system is completely characterized by the conductivity contrast j ¼ K=K0 and by the volume fraction n. The flow is uniform in the mean, of velocity U=const. The derivation of the velocity field is achieved by a numerical method of high accuracy, based on analytical elements. Approximate analytical solutions are derived by a few methods: composite elements, effective medium, dilute systems and first-order approximation in logconductivity variance. The latter was employed by Rubin (1995), while the dilute system approximation was used by Eames and Bush (1999) and Dagan and Lessoff (2001). Transport is solved in a Lagrangean framework, with trajectories determined numerically from the velocity field, by particle tracking. Results for the velocity variance and for the longitudinal macrodispersivity, for a few values of j and n, are presented in Part 2. Keywords: Solute transport, Heterogeneous media, Composite media, Groundwater hydrology
1 Introduction Spreading of solutes in groundwater is dominated by advection by the spatially variable velocity field, which in turn is related to the spatial variability of the hydraulic conductivity K. To account for the complex and erratic behavior of K, it is common to model it as a space random function. As a consequence, all the related quantities (hydraulic head, velocity, concentration, etc.) are space random functions as well, and the stochastic approach to flow and transport in porous media is the appropriate mathematical framework to treat them. The stochastic method has been employed extensively in the last three decades in order to A. Fiori (&) Faculty of Engineering, Universita` di Roma Tre, Via Volterra 62, 00146 Rome, Italy E-mail:
[email protected] I. Jankovic Faculty of Engineering, University at Buffalo, Buffalo, NY14260-4400, USA G. Dagan Faculty of Engineering, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
capture the main features of the transport processes in natural formations. Most of the studies carried out in the past were devoted to formations characterized by a continuous distribution of the conductivity. Here we study transport in a type of formation which is sometimes encountered in nature (see e.g. Ritzi, 2000, Labolle and Fogg, 2001) in which the hydraulic conductivity distribution exhibits two dominant components. If the latter are widely separated, the K distribution can be approximated as a binary one, characterized by two distinct values K; K0 . Hence, we represent the porous formation as made up from lenses or inclusions of permeability K that are surrounded by a matrix of permeability K0 . The relevant parameter for flow and transport in such a formation is the conductivity contrast j ¼ K=K0 ; low values are typically associated with lenses of silt and clay in a sandy aquifer whereas high j inclusions may represent disconnected cracks. Another important parameter is the volume (or area) fraction n, defined as the volume occupied by the inclusions divided by that of the flow domain. The problem of solute transport in binary formations has been the subject of a few papers in the past, based on several different methodologies. Thus, Desbarats (1990) has solved the flow and transport equations numerically for a few particular configurations to illustrate the effect of such structures upon solute travel time distribution. In a pioneering study of a theoretical nature, Rubin (1995) has modeled the bimodal conductivity distribution with the aid of a random indicator function. Subsequently the flow and transport equations were solved by a firstorder approximation in the variance of Y ¼ ln K. Further theoretical studies addressed the problem of formations characterized by large j values. By modeling lenses as inclusions of regular shapes, the asymptotic macrodispersivity was derived semi-analytically by means of the ‘‘dilute’’ approximation (Eames and Bush, 1999; Dagan and Lessoff, 2001; Lessoff and Dagan, 2001). The latter is based on the assumption that for low volume fractions (i.e. for n 1) the inclusions are sparse, and velocity field interactions can be neglected. The approach was generalized by Dagan and Fiori (2003) for transient transport and for inclusions of finite density n, with the aid of a model of composite elements. In the present article we present the mathematical background for the solutions of flow and transport in binary formations obtained by means of high precision numerical simulations, based on the analytical elements methodology (Strack, 1989, Barnes and Jankovic, 1999). In this first stage, the analysis is limited to two-dimensional flow fields and statistically isotropic media. The paper describes also a few approximate solutions for binary systems available in the literature (Rubin, 1995; Eames and Bush, 1999; Dagan and Lessoff, 2001; Dagan and Fiori, 2003) as well as an additional one, the effective medium approximation, that is presented here for the first time. The approximate and numerical solutions are compared in Jankovic et al. (submitted; hereinafter denoted as Part 2). The plan of the set of papers is as follows. In Part 1 (this paper) the conceptual framework and the theoretical background for the numerical and approximate semi-analytical solutions are presented; the solutions available in the literature and the new one introduced in the present work are all set in the same general framework, in order to emphasize differences and common features; Part 2 is devoted to describing the implementation of the numerical approach, to the numerical results and to their comparison with the semi-analytical ones.
2 The random hydraulic conductivity field The two dimensional flow and transport domain of area X is considered as made up from a homogeneous porous matrix of conductivity K0 and a large number N
371
of disjoint inclusions of given shape and of conductivity K that are implanted at random. A major simplification, allowing for efficient numerical solutions and approximate semi-analytical ones, is to represent the inclusions by regular, well defined, shapes of area x. The volume fraction of inclusions is defined by n ¼ Nx=X. It is emphasized that the use of inclusions of simple shapes still permits one, by varying their volume fraction and dimension, to reproduce any given two-point correlation of K. For simplicity, we limit the present study to isotropic media for which the inclusions are circles of uniform radii R (Fig. 1). With xj the random centroid of the inclusions, the conductivity field is defined by 372
KðxÞ ¼ K
N X
" Iðx xj Þ þ K0 1
j¼1
N X
# Iðx xj Þ
ð1Þ
j¼1
where Iðx xj Þ is an indicator function equal to unity for x 2 xðxj Þ, the domain of the inclusion surrounding xj , and equal to zero otherwise. The medium is constructed by following two rules: the centers are located at random and there is no overlap between inclusions. In the numerical simulations the implementation scheme adopted here is quite simple: a first inclusion is placed at random in X. Hence, the univariate pdf of a center is given by f ðxÞ ¼ 1=X. The coordinates of the center of the following inclusions are also drawn at random, excluding however locations that lead to overlap. This procedure, as well as any other, limit the value of n from above (this problem is well known in the theory of packing of granular media, e.g. Cumberland and Crawford, 1987). We have limited the simulations to n 0:4, since the numerical algorithm arrived at a practical standstill at this value. Larger n can be achieved by periodic, non-random, setting or by using variable radii. These alternatives are not considered here. With the aid of the above pdf (1) it is easy to derive the exact relationships
hKðxÞi ¼ K n þ K0 ð1 nÞ;
r2K ¼ ðK K0 Þ2 nð1 nÞ
ð2Þ
Fig. 1. Definition sketch of the heterogeneous medium and of the flow domain. The mean streamlines and the mean velocity correspond to j ¼ K=K0 > 1
The derivation of the two-point covariance CK ðx yÞ ¼ hK 0 ðxÞK 0 ðyÞi, where K 0 ðxÞ ¼ KðxÞ hKðxÞi, follows from the definition (1) and from (2). The general relationship is
"* 2
CK ðx yÞ ¼ hK 0 ðxÞK 0 ðyÞi ¼ ðK K0 Þ
N X N X
#
+ Iðx xi ÞIðy xj Þ
n2
i¼1 j¼1
ð3Þ The detailed calculations in (3) are given in Appendix. The final result for the auto-correlation qK ðr=RÞ ¼ CK =r2K ; r ¼ jx yj, is given in Eq. (A4) and it does not depend on j. The auto-correlation (3) is represented in Fig. 2 for a few values of n and its behavior is discussed in Appendix. It is emphasized that the result (3) is underlain by the randomness of the centroids xi and xj , while obeying the non-overlap requirement. As mentioned above these two requirements can be obeyed for circles of uniform radii only for limited n < nmax . In Fig. 2 we compare the theoretical qK (3) with the simulated ones. The latter are obtained by spatial averaging of single realizations of the medium. However, due to the large number of inclusions (N ¼ 50; 000, see Part 2) it is believed that the exchange of the space and ensemble averaging is accurate. Examination of Fig. 2 reveals that nmax , defined as the value for which the simulated auto-correlation and the theoretical one are close, is nmax ’ 0:2. At the larger value of n ¼ 0:4 the discrepancy is attributed to the inability of the algorithm to create a distribution of genuine random inclusions centers. Nevertheless, volume fractions as large as n ¼ 0:2 are of definite interest in applications. At the dilute limit OðnÞ we obtain in (3) qK ¼ v, which was used in Dagan and Lessoff (2001). At this limit the integral scale is given by IK ¼ 8R=ð3pÞ.
Fig. 2. The conductivity auto-correlation for different values of the volume fraction n. The continuous lines represent qK (Eq. A1) and the auxiliary functions v (A2) and w (A3). The results based on the numerical simulations (NS) are explained in Part 2
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In principle, higher-order statistical moments of K can be obtained along these lines. However, the computations became quite complex and are not pursued here.
3 The flow and transport equations The equations of flow (Darcy’s law and continuity) are as follows V¼ 374
K rH; h
r V ¼ 0 ðx 2 XÞ
ð4Þ
where V is the random Eulerian velocity, H is the pressure head and h ¼ const is the effective porosity. The boundary condition on oX is of given uniform flow of constant velocity UðU; 0Þ, i.e. Vm =U:m, where m is a unit normal vector to oX. This condition ensures that hVi ¼ U. We denote by /ðxÞ ¼ K H=h the velocity potential, such that by (4) it satisfies r2 / ¼ 0 in X. The potential / is discontinuous at the interfaces ox between inclusions and the matrix, and the boundary conditions of velocity and head continuity are as follows
o/ðexÞ o/ðinÞ ¼ ; om om
/ðexÞ /ðinÞ ¼ K0 K
ð5Þ
where /ðinÞ is the potential inside inclusion x, /ðexÞ is the exterior one and o=om is a normal derivative. For the sake of convenience of the numerical solution (see next section) we extend the matrix of conductivity K0 to infinity, outside X, and replace the boundary condition on oX by o/=ox ¼ U1 for x ! 1. The domain X is selected as a circle of radius R0 R, such that N 1 (Fig. 1). Under these conditions the conglomerate of inclusions behaves in the mean like a homogeneous medium of an effective constant conductivity Kef , that is submerged in the matrix K0 . Furthermore, the mean velocity U inside X is constant (see next section) and the ratio U=U1 depends on Kef =K0 , which in turn depends on j and n. In order to solve the transport problem, we neglect at present the effect of pore-scale dispersion and molecular diffusion, which is justified by the high Peclet number characterizing aquifer flow and by the fact that we seek the spatial moments of plumes. Then, the concentration Cðx; tÞ of a passive solute satisfies
oC þ V rC ¼ 0 ðx 2 XÞ; ot
Cðx; 0Þ ¼ C0 ðx 2 VÞ
ð6Þ
where V is the domain of the initial plume of concentration C0 . In the Lagrangean representation (see e.g. Dagan, 1989), the exact solution of (6) is given by
C¼
Z
C0 ðaÞd½x Xðt; aÞda; V
dx ¼ V½X; dt
Xð0; bÞ ¼ a
ð7Þ
where X is the trajectory of a fluid particle originating at x ¼ a. For a large, ergodic, plume the spatial moments can be expressed with the aid of trajectories statistical moments hXi ¼ a þ Ut; X 0 ¼ X hXi,
Xij ðtÞ ¼ hXi0 ðt; aÞXj0 ðt; aÞi; . . . We shall concentrate here on the derivation of the longitudinal moment X11 moments solely. For large travel distance with respect to the heterogeneity scale, X tends to normality and is completely characterized by its mean and variance. The longitudinal macrodispersion coefficient and macrodispersivity are defined by DL ¼ ð1=2ÞdX11 =dt and aL ¼ DL =U, respectively, and similarly for the transverse one. In both numerical simulations and semi-analytical solutions, the trajectories are determined by a numerical particle tracking procedure. The difference between the two approaches manifests in the solution of the flow problem.
4 The solution of the flow problem (numerical methodology) The numerical methodology follows the analytical elements approach (see e.g. Strack, 1989) and the more recent work of Barnes and Jankovic (1999). A brief outline is presented herein. We represent the potential in a general manner as follows /ðxÞ ¼ Ux1 þ
N X
uð jÞ ðx; xð jÞ Þ
ð8Þ
j¼1
The perturbation potentials uð jÞ are attached to circular inclusions centered at xð jÞ and have different expressions uð jÞ ¼ uðin;jÞ for x 2 xð jÞ and uð jÞ ¼ uðex;jÞ for x 62 xð jÞ . The interior potential is a regular harmonic function, whereas ouðex;jÞ =ox1 ! 0 for jx xð jÞ j ! 1. The most general solution can be written in terms of the complex variable Z ¼ x1 þ ix2 as follows
uðin;jÞ ¼ Re
1 X
ð jÞ
aðmjÞ ðZ Z Þm ;
uðex;jÞ ¼ Re
m¼0
1 X m¼1
ð jÞ
ð jÞ
e aðmjÞ ðZ Z Þm
ð9Þ
ð jÞ
where Re stands for real part, am and e am are constant complex coefficients and ð jÞ their conjugates, respectively, while Z is the complex center coordinate. Eq. (9) expresses the solution by the common Taylor and McLaurin series, and similar though more complex solutions are available for the other 2D or 3D shapes (e.g. Fitts, 1991, for spheroids). Now, the flux continuity equation in (5) is satisfied exactly by the choice of the coefficients in (9), irrespective of their values or the level of truncation of the series. ð jÞ The coefficients am are determined from the head continuity boundary condition in (5). This condition yields ð jÞ
Z K0 K 1 þp / ðhÞdh K 2p p 6¼c Z K0 K 1 þp ¼2 / ðhÞeimh dh m ¼ 1; 2 . . . K0 þ K 2p p 6¼c
a0 ¼
ð10Þ
aðmjÞ
ð11Þ
where /6¼c ðhÞ is the discharge potential at angle h on the boundary of the examined inhomogeneity due to all other elements. It is seen that the coefficients ð jÞ am are equal to a constant times the coefficients of a Fourier expansion of /6¼c ðhÞ, ðkÞ i.e. they are linear functions of all am ; k 6¼ j. The integrals in these expressions are evaluated numerically following the discussion in Barnes and Jankovic (1999).
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Once these coefficients are determined, the velocity field is obtained by analytical differentiation of (9). In the numerical implementation, described in Part 2, the series (9) are truncated at m ¼ M 1, ensuring high accuracy even for large permeability contrasts, which is difficult to achieve by conventional numerical methods. However, these simulations are computer time demanding and were carried out as numerical experiments. Simple, approximate, solutions that can be used for general analysis and can be easily applied, are derived next. 376
5 The solution of the flow problem (analytical approximations) We are going to describe two approximate analytical solutions for finite j ¼ K=K0 and n values (the composite element and the equivalent medium), as well as the limiting cases n ! 0 (dilute system) and j ! 1 (first-order approximation). 5.1 The composite element approximation (CEA) The model of composite inclusions we are going to adopt was developed by Hashin and Shtrikman(1962) in a different context. Its numerous ramifications are reviewed in detail in Milton (2002); it has also been used by Dagan(1989, Sects. 2.7.3, 3.4.4). For the sake of completeness we reproduce here briefly the development of the model, along the lines of Dagan and Fiori(2003). The composite inclusion is made up from an interior circle of radius R and conductivity K, a circular shell of radius R r Re and conductivity K0 and a surrounding matrix r > Re of conductivity Kef (Fig. 3). The boundary condition is of uniform flow of velocity U in the matrix. The parameters R; Re ; K and K0 are given. The basic idea is to seek an exact solution of the flow problem with the potential / satisfying Laplace equation and the matching conditions (5) at the two interfaces r ¼ R and r ¼ Re , as well as the value of Kef , so that the composite inclusion does not perturb the uniform flow in the matrix. It turns out that this is possible and Kef =K0 is a function of j ¼ K=K0 and n ¼ R2 =R2e solely. It also follows that the medium can be filled with such noninteracting inclusions of different Re
Fig. 3. Definition sketch of composite elements medium (CEA): (a) a composite element and streamlines (j > 1Þ and (b) the heterogeneous medium
(Fig. 3b) and an exact solution of the flow problem is achieved for a dense collection of inclusions. Consequently Kef is the effective conductivity of such a medium and indeed most of the studies in the literature were devoted to deriving Kef . The main limitation of the model is that it precludes contacts (or closeness) between the internal cores of neighboring inclusions (the coating of conductivity K0 is always present), whereas in a completely random placement such events can occur. Besides we assume that the composite inclusions can be planted in a random fashion with no overlap and that they fill the space. The consequences of these assumptions upon the velocity field and on transport will be examined in Part 2. We have adopted the model (Dagan and Fiori, 2003) in order to investigate flow and transport in formations of finite volume fraction n. Unlike most of the previous studies, the determination of Kef is not of interest since it does not influence the velocity field. Although a completely dense system in which the inclusions occupy the entire space is topologically possible only if the inclusions are of different radii Re , we shall explore here the solution for identical elements as a prototype. While details can be found in the literature, we give here the main results for the sake of completeness. Thus, the velocity disturbance potential for a composite element is given by ðexÞ
u
bUx R2 nþ 2 for R < r < Re ; ðx x Þ ¼ r 1 bn ð jÞ
Ubðn 1Þ x uðinÞ x xð jÞ ¼ 1 bn r ¼ jx xð jÞ j;
ð12Þ
for r < R
bðK=K0 Þ ¼ ð1 jÞ=ð1 þ jÞ;
j ¼ K=K0
Differentiation of u (12) yields the velocity disturbance u. The velocity field for the entire medium is given by
VðxÞ ¼ U þ
N N X X u x xð jÞ ¼ U þ ru x xð jÞ j¼1
ð13Þ
j¼1
This was the starting point for computing the velocity covariance, the trajectories moments and the macrodispersion coefficient by Fiori and Dagan (2003) and the results will be compared with the numerical ones in Part 2 here.
5.2 The dilute system approximation (DSA) The model assumes that inclusions are sparse and non-interacting, such that the total velocity potential is the sum of the disturbances associated with each inclusion as if it were surrounded by an unbounded matrix of conductivity K0 . The disturbance velocity potential associated with an inclusion can be shown to be given by an expansion of (12) in n ¼ oð1Þ and retaining the zero-order term, i.e. UxR2 uðexÞ x xð jÞ ¼ bðK=K0 Þ 2 for r > R; r uðinÞ x xð jÞ ¼ UbðK=K0 Þx for r < R while the velocity field is given by summation (13, 14).
ð14Þ
377
The solution has been employed by Eames and Bush (1999), Dagan and Lessoff (2001) in order to determine the asymptotic macrodispersion coefficients and by Fiori and Dagan (2003) for transient transport. It is worthwhile to mention that the conductivity auto-correlation for the random setting of centers is given by (A4) with n ¼ 0, i.e. qK ¼ v.
378
5.3 The equivalent medium approximation (EMA) Another type of approach, which uses the dilute system solution as a first approximation, is that of an equivalent medium whose conductivity is determined with the aid of Maxwell model. It has been employed in the literature in the context of effective properties (see e.g. Milton, 2002, for a review). We apply the model here toward determining the velocity field in a binary system. The solution is identical to the one applying to a dilute system (14), except that nonlinear interactions between neighboring inclusions are taken into account indirectly by replacing the matrix of conductivity K0 by an embedding matrix of the effective conductivity of the heterogeneous medium, i.e. by substituting in (14) jef ¼ K=Kef and replacing bðK=K0 Þ by bðK=Kef Þ ¼ ð1 jef Þ=ð1 þ jef Þ. Thus, the approximation should be viewed as an extension of the dilute approximation for finite, but moderate, values of the volume fraction n. The effective conductivity is determined by Maxwell method as follows. The heterogeneous medium within the circle of radius R0 (Fig. 1) is replaced by a homogeneous one of conductivity Kef , submerged in the unbounded matrix of conductivity K0 . With Vx ! U1 at infinity, the exact solution for the disturbance potential is given by (14) with bðK=K0 Þ replaced by bðKef =K0 Þ and R by R0 , respectively. The constant total interior velocity inside the body of radius R0 is therefore given by Vx;in ¼ U1 þ ouin =ox ¼ U1 ½ð1 bðKef =K0 Þ. The mean velocity in the heterogeneous medium U is derived by using the dilute system approximation and by space averaging of V (13) to obtain after a few calculations Z 1 U¼ Vx dx ¼ U1 ½1 nbðK=K0 Þ ð15Þ X X It is emphasized that (15) stems from the interior potentials uðinÞ (14) solely since the contribution of the exterior ones is zero. By Maxwell method the interior velocity in the equivalent body is equated with U in (15) with the result
bðKef =K0 Þ ¼ nbðK=K0 Þ i.e.
Kef 1 þ j nð1 jÞ and ¼ K0 1 þ j þ nð1 jÞ
K 1 þ j þ nð1 jÞ jef ¼ ¼j Kef 1 þ j nð1 jÞ
ð16Þ
Hence, the velocity field in this approximation is given by (14, 13) with bðK=K0 Þ ¼ ðK0 KÞ=ðK þ K0 Þ replaced by bðK=Kef Þ (16). It is seen that this is a nonlinear approximation in n, supposed to account for the interaction between inclusions. The dilute limit is recovered for n ¼ 0 in (16), i.e. for Kef ¼ K0 .
5.4 First-order approximation in r2Y (FOA) An approximation that has been explored extensively in the literature pertains to weak heterogeneity: all flow and transport variables are expanded in a power series in the logconductivity variance r2Y and the leading order terms are retained. Under this in approximation the flow equation (4) is linearized and elimination of V leads to r2 H ¼ J rY
ð17Þ
In (17) J ¼ rhHi is constant (mean uniform flow) and Y ¼ ln K is the stationary logconductivity. After solving for the velocity field transport is also simplified: in (7) dX=dt ¼ VðhxiÞ. Thus, the trajectories X are obtained by integration along the mean streamlines. The simple expression of the asymptotic longitudinal macrodispersivity based on this approximation (Dagan, 1982, Gelhar and Axness, 1983) is aL ¼ r2Y IY , where IY is the logconductivity integral scale. For the binary system discussed here and with Y0 ¼ ln K0 and Y ¼ ln K we have, similarly to (1,2) hYðxÞi ¼ nY þ ð1 nÞY0 and r2Y ¼ nð1 nÞ½lnðK=K0 Þ2 . The integral scale IY ¼ IK , which depends on n, can be obtained from the integration of qK ðrÞ (Eq. (22), Fig. 2), which has to be performed numerically. At the dilute limit the closed form result is IY ¼ ð8RÞ=3p and aL ¼ n½lnðK=K0 Þ2 IY . The latter expression was used by Dagan and Lessoff (2001). As mentioned in the Introduction, Rubin (1995) has solved the flow problem for more general conductivity bimodal distributions under a first-order expansion and the expression of aL here is a particular case of his results.
5.5 Eulerian velocity covariance One of the statistical moments of primary interest is the velocity covariance Cu;ij ðx yÞ ¼ hui ðxÞuj ðyÞiði; j ¼ 1; 2Þ, where uðxÞ ¼ ru is the velocity fluctuation (i; j ¼ 1 and i; j ¼ 2 stand for x and y, respectively). In the numerical simulations the covariance was determined by space averaging in given realizations, i.e. by assuming ergodic behavior in the core of the circle of radius R0 (see Part 2). In the analytical approximations the starting point is (3) which leads to Cu;ij ðx yÞ ¼
N X N D E X ui x xðkÞ uj y yðlÞ
ð18Þ
k¼1 l¼1
where ensemble averaging is carried out over the random centroids coordinates. In the CEA (12) the only contribution in (18) stems from the diagonal terms since the velocity fields of different inclusions do not interact. By exchanging ensemble and space averages we therefore get (see for details Dagan and Fiori, 2003) N D E N Z X ðkÞ ðkÞ uj y x ¼ Cu;ij ðx yÞ ¼ ui x x ui ðx xÞuj ð y xÞdx X x k¼1
ð19Þ
379
The integration over a single composite element is carried out numerically in (19), after substituting (12). The variance (x ¼ y) can be determined analytically (Fiori and Dagan, 2003) and the result is
r2u1 3nð1 nÞð1 jÞ2 ¼ ; U 2 2ð1 þ j n þ jnÞ2
380
r2u2 nð1 nÞð1 jÞ2 ¼ U 2 2ð1 þ j n þ jnÞ2
ð20Þ
In the dilute system approximation the procedure is the same, except that integration in (19) is carried out over the entire space and (14) is used for the velocity u. It is emphasized that retaining only the diagonal term in the transition from (18) to (19) is an approximation since the velocity field associated with an inclusion overlaps with the ones of other inclusions. However the effect is of higher-order in n, and this approximation was adopted by Eames and Bush (1999), Dagan and Lessoff (2001) and Fiori and Dagan (2003). Again, analytical expressions can be obtained for velocity variances by expanding (20) in n and retaining the linear term, i.e. r2u1 =U 2 ¼ ð3n=2Þb2 ðjÞ and r2u2 =U 2 ¼ ðn=2Þb2 ðjÞ. The EMA approximation follows the same steps as the DSA, except the replacement of j ¼ K=K0 by j=jef ¼ K=Kef . Finally, the FOA is obtained from the previous by expanding the results in at first-order in ln j i.e. by replacing j by 1 þ ln j and retaining quadratic terms in ln j. These various approximations are compared with the numerical simulations in Part 2.
6 The solution of the transport problem In the present study we investigate the particle trajectories covariance Xij ðtÞ and the associated macrodispersivities. In both numerical simulations and semianalytical approach trajectories were determined numerically, by quadratures, with the aid of the Eulerian velocity fields by the definition Xðt; bÞ ¼ b þ Ut þ
Zt
u½Xðt 0 Þdt0
ð21Þ
0
were b is the initial particle coordinate, hXi ¼ b þ Ut; X 0 ¼ X hXi and Xij ðtÞ ¼ hXi0 ðt; bÞXj0 ðt; bÞi. In the numerical simulations a swarm of particles were injected along a line normal to the mean flow and their trajectories were determined with the aid of Eulerian velocity field by a particle tracking code (see Part 2). The trajectories variance X11 was subsequently derived by space averaging, while the longitudinal macrodispersivity aL ¼ ð1=2UÞdX11 =dt was determined by numerical differentiation. The same procedure was used for the analytical approximations of the velocity field, which are given in the preceding section. Due to the independence of the velocity fields of different inclusions mentioned above, it was sufficient to compute Xðt; b xÞ for an isolated inclusion (by one quadrature) and subsequently to determine X11 by space average over b for a fixed x ¼ 0 (see Dagan and Fiori, 2003). Thus, three quadratures were needed. The longitudinal variance X11 tends asymptotically to a Fickian, linear dependence on t and the longitudinal macrodispersivity aL tends to a constant. It could be determined directly by two
quadratures by the procedure forwarded by Eames and Bush (1999) and Lessoff and Dagan (2001). It implies taking b along a line far upstream of the inclusion and determining X1 ðt; bÞ for a sufficiently large t for which the residual X10 tends to a constant. The analysis of transport in a binary medium is the main objective of the present study. The results and their discussion are presented in Part 2.
7 Summary and conclusions In the present work, that constitute the first of a set of two papers, we have investigated numerical and approximate solutions for two-dimensional flow and transport in heterogeneous porous formations of a binary conductivity distribution. The type of formation considered is characterized by widely separated components of the conductivity distribution. In particular, we represent the porous formation as made up from lenses or inclusions of permeability K, located at random in the flow domain, that are surrounded by a matrix of permeability K0 ; the relevant parameters for flow and transport in such a formation is the conductivity contrast j ¼ K=K0 and the inclusions density n. The statistical structure of such formation can be adequately characterized by the autocorrelation function of the hydraulic conductivity. The problem is set first into a numerical methodology, based on the analytical elements approach (see e.g. Strack, 1989; Barnes and Jankovic, 1999). The results based on such approach are discussed in Part 2, and they serve as the basis for the comparison with a few approximate solutions available in the literature carried out in Part 2. Those solutions are presented here within a general framework, that helps in identifying the differences and the common features of the models. The approximate analytical solutions for flow and transport analyzed are derived by a few methods: composite elements, dilute systems and first-order approximation in logconductivity variance. The latter was employed by Rubin (1995), while the dilute system approximation was used by Eames and Bush (1999) and Dagan and Lessoff (2001). The solutions based on the composite elements methodology was developed in Dagan and Fiori (2003) and Fiori and Dagan (2003). Furthermore, a new approximate model, based on the effective medium approximation, is developed here for the first time. The proposed solution is identical to the one applying to a dilute system, except that nonlinear interactions between neighboring inclusions are taken into account indirectly by replacing the matrix of conductivity K0 by an embedding matrix of the effective conductivity of the heterogeneous medium. The effective conductivity of the binary formation is determined with the aid of Maxwell model (Milton, 2002). Thus, the approximation should be viewed as an extension of the dilute approximation for finite, but moderate, values of the volume fraction n. All the methods share the same solution at the limit of small heterogeneity and small inclusions density. When n 1, the solutions based on the composite elements approach, the effective medium approximation and the one for dilute systems are identical for any j. The main differences between models arise when the conductivity contrast j is large and the inclusions are dense. In fact, each methodology employs a different approximation to account for the increasing interaction between inclusions. Results for the velocity variance and for the lonigtudinal macrodispersivity, for a few values of j and n, are presented in Part 2. They are compared with the numerical results based on the analytical element approach.
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Appendix Derivation of qK ¼ CK =r2K Expanding the expression of the correlation function (3) yields + "* # N X 2 0 0 2 CK ðx yÞ ¼ hK ðxÞK ðyÞi ¼ ðK K0 Þ Iðx xi ÞIðy xj Þ n i;j¼1
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2* 3 + N N X X ¼ ðK K0 Þ2 4 Iðx xi ÞIðy xi Þ þ Iðx xi ÞIðy xj Þ n2 5 i¼1
i6¼j¼1
2 3 * + N X Nx vðx yÞ þ ð22Þ Iðx xi ÞIðy xj Þ n2 5 ¼ ðK K0 Þ2 4 X i6¼j¼1 " # Z Z ðN 1Þ2 2 2 ¼ ðK K0 Þ nvðx yÞ þ Iðx xi ÞIðy xj Þd xi d xj n X2 ¼ N ! 1 ¼ ðK K0 Þ2 Z Z n2 2 Iðx xi ÞIðy xj Þd nvðx yÞ þ 2 xi d xj n x ¼ nðK K0 Þ2 ½vðx yÞ nwðx yÞ P where Ni6¼j¼1 stands for summation of off-diagonal terms. In (22) the function v ¼ ðX=xÞhIðx xi ÞIðy xi Þi is given by
2 vðx yÞ ¼ ½cos1 ðr0 Þ r 0 ð1 r02 Þ1=2 ðr0 < 1Þ; v ¼ 0; ðr0 > 1Þ p
ð23Þ
with r 0 ¼ jx yj=ð2RÞ. The function v (23) stems from Cauchy algorithm: it is the normalized area of the overlap of two circles with centers at distance r (see e.g. Dagan, 1989). It represents the contribution of couple of points that lie entirely in the same inclusion. The function w (22) is defined by
1 wðx yÞ ¼ 1 2 x
Z Z
Iðx xi ÞIðy xj Þd xi d xj
ði 6¼ jÞ
ð24Þ
and it represents the contribution of the event of the couple of points lying within two different inclusions. The domain of integration in (24) is such as to exclude overlap between the two circles (the condition is j xi xj j > 2R). The integration is performed numerically. The function w is positive and has the following limits
wð0Þ ¼ 1 wðjx yj 4RÞ ¼ 0 Thus, the autocorrelation function is equal to
qK ðx yÞ ¼
vðx yÞ nwðx yÞ 1n
ð25Þ
and it does not depend on j ¼ K=K0 . The dilute limit is qK ðx yÞ ¼ vðx yÞ and was obtained before (Dagan and Lessoff, 2001). In Fig. 2 we have represented qK as function of r ¼ jx yj for a few values of n. The nonlinear (in n) interaction between inclusions, manifesting through w in (22), shows up in Fig. 2 for increasing values of the volume ratio n. The main effect of the increasing density n manifests through a negative correlation for intermediate values of the separation distance r=R. The negative part of qK , that is determined by the background conductivity, is a consequence of the non-overlap condition; the latter is introduced through the domain of integration of (24). Stated in words, the increasing density of the inclusions leaves a portion of the background around each inclusion that cannot be occupied by further inclusions because of the non-overlap condition. The latter circumstance leads to a negative qK . When n > nmax the negative portion of the conductivity correlation is dominant.
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