A note on the influence of a constant velocity boundary condition

Transcription

1 WATER RESOURCES RESEARCH, VOL. 36, NO. 10, PAGES , OCTOBER 000 A note on the influence of a constant velocity boundary condition on flow and transport in heterogeneous formations S.C. Lessoff Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel P. Indelman Faculty of Civil Engineering, Technion, Technion City, Haifa, Israel G. Dagan Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel Abstract. Fluid flow and solute transport in an anisotropic, heterogeneous porous medium with mean flow normal to a constant-flux boundary are considered. The statistical moments of flow and transport variables are determined at second order in log conductivity fluctuation, and they are expressed in terms of the log conductivity variance and integral scales, the mean flow velocity, and the distance from the boundary. The variance of the longitudinal and transverse components of velocity as well as hydraulic head variance and the longitudinal macrodispersivity are analyzed for a bounded medium with axisymmetric, Gaussian log conductivity covariance structure. In this case, all of the moments can be solved by means of a single numerical quadrature. The constant flow boundary increases the variability of head and of flow transverse to the mean flow direction and causes a reduction of the macrodispersivity in a zone adjacento the boundary. Our results should be useful for the design and testing of numerical models and have important implications for surface infiltration of solutes. 1. Introduction specified fluid flux on the soil surface is a common boundary condition. Thus Zhang et al. [1998] and Zhang and Winter Most research on the influence of spatially variable hydraulic conductivity on flow and transport in porous media has been concerned with unbounded domains [see, e.g., Dagan, 1989; Gelhar, 1993; Cushman and Ginn, 1993; Neuman and Zhang, [1998] use a fixed flow upper boundary in their numerical analysis of one- and two-dimensional flow and transport in soil. The main interest of their studies as well as that of Zhang [1999] is the effect of internal fluid sources, transient flow, and 1990; Rubin, 1990]. This idealization simplifies computations the water table lower boundary which cause the mean hydrauconsiderably and is justified because the influence of boundaries is limited to zones adjacento them under deterministic boundary conditions. Application of stochastic models to bounded domains has generally been limited to constant head or no-flow boundaries, which are regarded as appropriate for lic head to be nonstationary. A fixed flux condition may have a significant impact on the subsurface zone close to the upper boundary. This layer is of paramount importance in soil physics because it is the primary zone of biological activity and transient flow. Another distincsimulation of aquifer flow [Mizell et al., 198; Naff and Vecchia, tive feature of surface infiltration problems is the presence of 1986; Rubin and Dagan, 1988, 1989; Oliver and Christakos, three-dimensional structures having marked anisotropic heter- 1995, 1996; Osnes, 1995, 1998; Zhang, 1998]. The specified ogeneity. As a first step toward a better understanding of the water flux boundary condition has received less attention. For impact of this type of boundary condition, we investigate satsaturated flow it has been adopted in a stochasticontext only urated flow and solute transport in a semi-infinite domain with by Bellin et al. [199] in Monte Carlo simulations of flow and constant flux applied on the upper boundary. Russo [1993] has transport in two-dimensional isotropic media. They used the found a marked similarity between transport in the unsaturfixed flow boundary condition to ensure a specified spatially ated and saturated regimes. averaged velocity in stochastic simulations of transport. The The hydraulic conductivity is modeled as a stationary, logmain purpose of Bellin et al. [199] was to analyze transport in normal space-random function of anisotropic axisymmetricothe central part of the domain where the presence of the variance. The axis of symmetry is the vertical one, z. Flow is characterized by the variances of the longitudinal and transboundary was not felt. Therefore, they did not invest effort into verse components of the velocity as well as the hydraulic head analyzing the zone neighboring the boundary. variance. Transport is characterized here by the macrodisper- Our interest in this topic stems from a different application, sion coefficient and the longitudinal second-order spatial monamely, flow and transport in the unsaturated zone. There ment of particle displacement. All these statistical moments Copyright 000 by the American Geophysical Union. Paper number 000WR /00/000WR are expressed as functions of z, the distance from the boundary. In order to derive simple analytical results we use a firstorder approximation in the hydraulic conductivity variance. 3095

2 3096 LESSOFF ET AL.: TECHNICAL NOTE. Mathematical Formulation of the Problem.1. General The pressure head H and the specific discharge q obey Darcy's law and the continuity equation q = -KVH V.q = 0 (x G D), (1) respectively. The flow domain D is the half-space z > 0, while the horizontal coordinates are x and y. With K(x, y, z) the hydraulic conductivity and Y = In K, elimination of q in (1) leads to VH + VYVH = 0 () to be solved with the boundary condition qzlz=o = -K(OH/Oz) z=0 = q0, (3) where q o stands for a constant and deterministic, given flux. 0 We assume that Y is normal and stationary, so that all its -k 'z(z, k) + 'z(z, k) = -kkos statistical moments are determined by the given mean, (Y) = In KG, and the two-point covariance, Cy(R) = where hatted variables stand for Fourier transforms, i.e., (Y'(x)Y'(x + R)) = trvpy(r), where KG is r geometric ' (z, k) = (w)-m/ J' -oo q' (r, z)e -ir'k dr with m = for mean and Y', trv, and Pv are the fluctuation, variance, and the two dimensions, k is the wave number vector and k = I kl. This autocorrelation function of Y, respectively, and angle brackets equation with boundary condition (7), i.e., :(0, k) = 0, and stand for ensemble averaging. the requirement of finiteness at z --> o can be solved by means To solve the flow problem, we expand H and q in perturba- of Green's function [Hildebrand, 1948] tion expansions in trv as follows: H = (H) + h + h + h +..., q = (q) + q, + q,()+ q,(3)+..., (4) Xzz(t) = Uzz(Ut', Ut") dt' dt", (8) f0t0 t where Uzz(Z, z) = (u(x, y, z )u(x, y, z)) stands for the autocovariance of the vertical component of velocity u for two points on a mean streamline. In turn, the longitudinal macrodispersion coefficient is given by axzz(t) Ozz(t) = 5 = Uzz(Ut', Ut) dt'. (9).. Derivation of the First-Order Solution We solve (7) in the half-space domain z > 0 by applying a Fourier transform (FT) in the x-y plane. This leads to the ordinary differential equation leading to G(, z, k) = -e-ke '(z, k), (10) -e-kz sin h(k )/k h(kz)/k z >- -< (11) with h and q' being the try-order fluctuations of head and water flux, respectively, while h (m) and q' (m) are O(tr ). Substituting (4) into (1)-(3) and ordering leads at zero order to O(H) V(H) = 0 (x G D), O = Ko -J (5) qo with the simple solution (H) = -Jz + constant. At first order we get for h (z=0) 'z = -JK k G(, z, k)f" (, k) d. (1) The main moment of interest is the vertical velocity covariance, U zz(z, z), which is needed for the computation of Xzz(8 ) and Dzz(9 ). The FT of Uzz(Z, z, r), where r = [(X1 -- X) q- (Yl -- y)] 1/, results from (1) as follows: azz(zl, z, k) = k 4 U OY' Oh Vh = -J (x G O_,) Oz, --= Jr' (z = 0). (6) ß G(,, z, k)g(, z, k) '½ l, :, k) d l d. At this order, (q) = {0, 0, qo} and q' = KG(JY' - Vh), where J = {0, 0, J}. Elimination of h in (6) leads to the following equation and boundary condition for q'z' Vq'z=JKGV Y ' (x G a), q'z = 0 (z = 0), (7) where V is the Laplace operator in the x-y plane. It is convenient to solve (7) and to use q : as the primary variable in order to determine the statistical moments of other variables and their dependence on z. The velocity is simply related to q by V = q/n, where the porosity n is assumed to be constant. With U = (V), at first order, the average velocity is Uz = U - q o/n = KoJ/n, and the fluctuation is u - q'/n. The transport of a passive solute is solved in a Lagrangean framework with neglect of pore-scale dispersion. By the wellknown procedure [e.g., Dagan, 1989] the variance of the longitudinal displacement of a particle originating at z = 0 for t = 0 is given by (13) In (13), Ov(z, z, k) is the FT of the stationary axisymmetric Cv(R) with R = {r, z - z}. In particular, the velocity variance is given by rru(z) - Uz (Z, z) - azz(z, z, k)k dk. (14) The variance, (14), grows from zero (at z -- 0) to its constant value in an unbounde domain (z --> o ). The latter is given in an analytical form by Russo [1995]. Another variable of interest is the head variogram '¾h(Z1, Z, r) = ([h(x, y, z ) - h(x, Y, z)])/ and, in particular, the head variance o' h (: ) =,Yh(:, oc, 0) -- tr (o ). The head variance for an infinite medium tr (o ) (1) is given by Dagan [1989, equation (3.7.16)]. With h(x, y, z) = [Oh(x, y, z')/oz'] dz' + h(x, y, o ) we get

3 LESSOFF ET AL.: TECHNICAL NOTE o-}/(o-,ij )... 5O-v/(o-yU ) o-u/(o-r U) Figure 1. Scaled head and velocity variances for an infinite anisotropic media with Gaussian log conductivity covariance. f 7h(Z1, Z, r) = Coh/Oz(Z t, z", O) dz' dz" -- C oh/oz( Zt, Ztt, F) dz' dz" IZl Iz Coh/Oz(Z1, Z, r) = JCy(Zl, z, r) - Cyqz,(Z, z, r) J 1 ør- K Cqz'(Zl' Z' r). (16) The FT of the cross-covariance Cyq; can be obtained from (1) as follows: ør- 5 C oh/oz( Zt, Ztt, O) dz' dz" ½yqz,(Z1, Z, k) = -JKG k G(, Z1, k)½y(% l, z, k) ds, -- O' (OO) ør- Ch(OO, oo, (15) The covariance of the vertical gradient of head C oh/o z can be expressed in terms of the moments of q and Y using Oh/Oz = -qz/'k G + JY'. This gives whereas &q;(z', z", k)/jk is given precisely by (13). (17) We will also use (15) in order to compute the variance of the component of velocity transverse to the mean flow direction f=l f=l/4 ' ' 1.6 f =1/ ";... Figure. The effect of a constant flow boundary on head variance normalized by the variance far from the boundary (the normalizing factor rrh(o )/(o'.ji ) = 0.03, 0.094, and 0.1 for f = 0.05, 0.5, and 1, respectively).

4 3098 LESSOFF ET AL.' TECHNICAL NOTE //j /./,.) ;'... f I ----f=l/4 I , z/i Figure 3. The effect of a constant flow boundary on longitudinal velocity variance normalized by the variance far from the boundary (the normalizing factor o- u (oo)/(u) = 0.53, 0.33, 0.19, and for f - 1, 1/, 1/4, and 1/0, respectively). o- (z). Our results depend on the first-order approximation v = -KG(Oh/Oy ) as follows [Dagan, 1989]: 0Th(g, Z, r) trv(z) = K lim r. (18) r 0 0 For z tending to infinity the transverse velocity variance of an infinite axisymmetric medium (see (0) below) is given by Russo [1995]. In section 3 we illustrate these general first-order results. 3. Illustration of Results 3.1. Infinite Medium We adopt a Gaussian log conductivity covariance of the scaling suggested by Gelhar and Axness [1983]. The log conductivity and its FT are given by Cr(z, r) = o'r exp T/'z 41v 1' T/-r ] OCz, a) = F} exp I 41f, (19) where I and I v are the horizontal and vertical integral scales, respectively, and f = IJI is the anisotropy ratio. The Gaussian Cv, (19), is very convenient because of the separation of z and r. The statistical moments of flow and transport in media with axisymmetric Gaussian Cv should be similar and representative of those moments in media with other Cv functions (e.g., exponential) for given o-v, I, and f [see, e.g., Zhang, 1998; Oliver and Christakos, 1995]. Since, far from the boundary, our results tend to those pertaining to an infinite domain, it is worthwhile to recapitulate the relationships developed previously in the literature, ap Figure 4. The effect of a constant flow boundary on scaled transverse velocity variance.

5 LESSOFF ET AL.: TECHNICAL NOTE f =1/4 / i I f =1/0 I///// ///. /////.. / ' ///'/' o o Figure 5. Scaled second moment of particle trajectories for three values off in an infinite medium (upper line of each pair) and for particles starting at a constant flow boundary (lower line of each pair). tu/iv plied to the Gaussian covariance (19). The longitudinal and transverse velocity variances were determined analytically by Russo [1995] for an exponential Cy and are reproduced here' O' u r}u f[fx/f-1(l+f)+(1-4f ) arccoth( _1)1 rr}u - 4(f - 1)5/ (f -, (0) l+(l+ff) arccoth( _1) 1 ductivity. It is worthwhile to emphasize that the velocity variances rh, and rrv do not depend on the particular shape of C., but only on the anisotropy ratio f. The head variance is given by Dagan [1989, equation (3.7.16)] for any medium of axisymmetric anisotropy. For C., (19), the result is j- =rr(l_f)[fx/l rr} g i arctan (x/lf _ -11. (1) rh, It is worthwhile to emphasize that the shape of C, affects the head variance rr, through a constant coefficient. Therefore the normalized head variance rr, (f)/[ rr, (1)] is the same for any axisymmetricovariance function. To illustrate the dependence of the variances (0) and (1) upon f, we have represented them in Figure 1 for 0 < f < 1. We consider anisotropy ratios f smaller than unity because our interest stems from upper soil related applications where field data indicates a layered structure of soils [e.g., Russo and Bouton, 199]. The results for anisotropic formations (Figure 1) include as a particular case those of isotropic formations (f = 1) in which rr /(rr.u ) = 8/15, rr,/(rr.u ) = 1/15, and O'h/(O' I S ) = /(3 rr). 3.. Effect of Boundary Condition on Flow Starting with rr (z) = h (Z, oo, 0) -- o' (oo), where rr,(o ) is given by (1), we express Th(Z, oo, 0) as a function of the moments of Y by combining (11), (15), (16), and (17). Integrating the resulting expression analytically over 1,, z', and z" gives the Fourier transformed variogram h(z 1, Z, k). Inverting the Fourier transform for r - 0 leads to an expression containing only one quadrature that was carried out nu- merically. To illustrate, we have represented the ratio rrh(z)/ rr,(o ) as a function of z/i in Figure for a few values of f. Head variability is increased near the constant flow boundary due to presence of steep head gradients in zones of low conductivity and shallower head gradients in zones of high con- It is seen that the "boundary layer," where o' h is increased, extends about two horizontal integral scales into the formation. This is similar to results for other boundary conditions [e.g., Rubin and Dagan, 1988, 1989]. We move now to the longitudinal velocity fluctuations whose variance (14) was computed with the aid of (13). Integrating twice analytically and inverting the Fourier transform leads to the computation of the variance in terms of a single quadrature that was carried out numerically. The normalized variance (z)/rr}(o ) (Figure 3) is quite insensitive to the anisotropy ratio, unlike rh, (ø ). The boundary layer where longitudinal velocity variance is inhibited has a thickness of around 1. This is somewhat smaller than the thickness of 31 determined by Bellin et al. [199, Figure 3a] in Monte Carlo simulations. However, the flow was two-dimensional in their case. Finally, the transverse velocity variance rr, (18) is represented in Figure 4. As expected from our discussion of the higher variability of the head along the boundary (Figure ), the transverse velocity fluctuations are larger than in an unbounded domain to compensate for the forcing effect of the imposed longitudinal velocity. At the boundary, rr, increases monotonically with f, but far from the boundary, reduction of the width of the contrasting permeability zones causes rr, to

6 3100 LESSOFF ET AL.: TECHNICAL NOTE Figure 6. Scaled macrodispersivity as a function of scaled travel time for three values of f in an infinite medium (upper line in each pair) and for transport starting at a constant flow boundary (lower line in each pair). decrease for f > 1/ (Figure 1). Thus the lines for f = 1 and f = 1/ cross as they approach the boundary in Figure 4. Bellin et al. [199] found that for rv > 1 the impact of nonlinear terms upon r u and O v becomesignificant. However, one may expect a lesser effect for three-dimensional flows investigated here. References Bellin, A., P. Salandin, and A. Rinaldo, Simulation of dispersion in heterogeneous porous formations: Statistics, first-order theories, convergence of computations, Water Resour. Res., 8, 11-7, Bresler, E., and G. Dagan, Solute dispersion in unsaturated heterogeneous soil at field scale, 1, Theory, Soil Sci. Soc. Am. J., 43, , Effect of Boundary Condition on Transport Cushman, J. H., and T. R. Ginn, Nonlocal dispersion in media with Our main aim is to analyze transport near a constant flux continuous evolving scales of heterogeneity, Transp. Porous Media, boundary. To this end, we illustrate, in Figure 5, the longitu- 13, , dinal particle displacement variance X (8) as a function of Dagan, G., Flow and Transport in Porous Formations, Springer-Verlag, New York, travel distance scaled by the vertical integral scale. The com- Gelhar, L. W., Stochastic Subsurface Hydrology, Prentice-Hall, Engleputation required again only one numerical quadrature. wood Cliffs, N.J., The constant flow condition significantly suppresses X for Gelhar, L. W., and C. I. Axness, Three-dimensional stochastic analysis a particle originating at the boundary as compared to its tra- of macrodispersion in aquifers, Water Resour. Res., 19, , jectory in an unbounded domain. Decreasing f enhances this Hildebrand, G. B., Advanced Calculus for Engineers, Prentice-Hall, boundary effect because as f -- 0, particles cannot circumvent Englewood Cliffs, N.J., regions of low permeability and flow becomes one-dimen- Mizell, S. A., A. L. Gutjahr, and L. W. Gelhar, Stochastic analysis of sional, thus maintaining the constant velocity from the bound- spatial variability in two-dimensional steady groundwater flow asary deep into the formation. suming stationary and nonstationary heads, Water Resour. Res., 18, The same effect is displayed by the "macrodispersivity" D , 198. (9), which was computed also in terms of one numerical Naff, R. L., and A. V. Vecchia, Stochastic analysis of threedimensional flow in a bounded domain, Water Resour. Res.,, quadrature. It is seen that for f - 1/0 the macrodispersivity , is practically zero for a travel distance of about 10Iv (Figure 6). Neuman, S. P., and Y. K. Zhang, A quasilinear theory of non-fickian This finding leads to interesting conclusions. Thus flow nor- and Fickian subsurface dispersion, 1, Theoretical analysis with apmal to a constant velocity boundary will be markedly different plication to isotropic media, Water Resour. Res., 6, , from flow normal to a constant head boundary. The suppres- Oliver, D. L., and G. Christakos, Diagrammatic solutions for hydraulic head moments in 1-D and -D bounded domains, Stochastic Hydrol. sion of longitudinal dispersion for a few vertical integral scales Hydraul., 9, 69-96, in saturated flow is in line with the simplified model of Bresler Oliver, D. L., and G. Christakos, Boundary condition sensitivity analand Dagan [1979] for unsaturated flow, which assumed that for ysis of the stochastic flow equation, Adv. Water Resour., 19, , anisotropic soils, transport takes place along vertical columns, the velocity being equal to the one imposed on the boundary. Osnes, H., Stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers, Water Resour. Res., 31, 981- At any rate, for processes occurring close to the boundary it is 990, important to impose the appropriate boundary condition in Osnes, H., Stochastic analysis of velocity spatial variability in bounded order to model correctly solute transport. rectangular heterogeneous aquifers, Adv. Water Resour., 1, 03-15, Rubin, Y., Stochastic modeling of macrodispersion in heterogeneous Acknowledgments. This research is part of a thesis to be submitted media, Water Resour. Res., 6, , in partial fulfillment of the requirements for a Ph.D. degree at the Tel Rubin, Y., and G. Dagan, Stochastic analysis of boundaries effects on Aviv University. We are grateful to A. Bonilla and the two anonymous head spatial variability in heterogeneous aquifers, 1, Constant head reviewers for their constructive comments. boundary, Water Resour. Res., 4, ,

7 LESSOFF ET AL.: TECHNICAL NOTE 3101 Rubin, Y., and G. Dagan, Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers,, Impervious boundary, Water Resour. Res., 5, , Russo, D., Stochastic modeling of macrodispersion for solute transport in a heterogeneous unsaturated porous formation, Water Resour. Res., 9, , Russo, D., On the velocity covariance and transport modeling in heterogeneous anisotropic porous formations, 1, Saturated flow, Water Resour. Res., 31, , Russo, D., and M. Bouton, Statistical analysis of spatial variability in unsaturated flow parameters, Water Resour. Res., 8, , 199. Zhang, D., Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media, Water Resour. Res., 34, , Zhang, D., Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media, Water Resour. Res., 35, , Zhang, D., and C. L. Winter, Nonstationary stochastic analysis of steady state flow through variably saturated, heterogeneous media, Water Resour. Res., 34, , Zhang, D., T. C. Wallstrom, and C. L. Winter, Stochastic analysis of steady-state unsaturated flow in heterogeneous media: Comparison of the Brooks-Corey and Gardner-Russo models, Water Resour. Res., 34, , G. Dagan and S.C. Lessoff, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. (dagan@eng.tau.ac.il; slessoff@eng.tau.ac.il) P. Indelman, Faculty of Civil Engineering, Technion, Technion City, Haifa 300, Israel. (Received September 30, 1999; revised May 18, 000; accepted May 18, 000.)