Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport

Transcription

1 COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol. 6, No., pp. 3-6 Commun. Comput. Phys. July 9 Laplae-Transform Finite Element Solution of Nonloal and Loalized Stohasti Moment Equations of Transport Eri Morales-Casique and Shlomo P. Neuman Department of Hydrology and Water Resoures, University of Arizona, Tuson, Arizona 857, USA. Reeived 6 November 7; Aepted (in revised version) 5 Otober 8 Available online 8 November 8 Abstrat. Morales-Casique et al. (Adv. Water Res., 9 (6), pp ) developed exat first and seond nonloal moment equations for advetive-dispersive transport in finite, randomly heterogeneous geologi media. The veloity and onentration in these equations are generally nonstationary due to trends in heterogeneity, onditioning on site data and the influene of foring terms. Morales-Casique et al. (Adv. Water Res., 9 (6), pp ) solved the Laplae transformed versions of these equations reursively to seond order in the standard deviation σ Y of (natural) log hydrauli ondutivity, and iteratively to higher-order, by finite elements followed by numerial inversion of the Laplae transform. They did the same for a spae-loalized version of the mean transport equation. Here we reount briefly their theory and algorithms; ompare the numerial performane of the Laplae-transform finite element sheme with that of a high-auray ULTIMATE-QUICKEST algorithm oupled with an alternating split operator approah; and review some omputational results due to Morales-Casique et al. (Adv. Water Res., 9 (6), pp ) to shed light on the auray and omputational effiieny of their reursive and iterative solutions in omparison to onditional Monte Carlo simulations in two spatial dimensions. AMS subjet lassifiations: 65Z5, 65Z5, 68U, 76S5 Key words: Geologi media, porous media, random heterogeneity, onditioning, stohasti transport, moment equations, nonloality, loalization, Laplae transform, finite elements, ultimatequikest. Introdution It has beome inreasingly ommon to desribe the spatial variability of geologi medium properties geostatistially and to analyze subsurfae fluid flow and solute transport in Corresponding author. addresses: emorales@hwr.arizona.edu (E. Morales-Casique), neuman@hwr. arizona.edu (S. P. Neuman) Global-Siene Press

2 3 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 suh media stohastially [, 44, 55]. The most ommon and straightforward method of stohasti analysis is omputational Monte Carlo simulation that produes a large number of equally likely results. These results are summarized statistially in terms of their sample averages, seond (variane-ovariane) or higher moments, or probability distributions. Results that honor measured values of medium properties and/or state variable are said to be onditioned on these data. One thus obtains (among others) onditional mean flow and transport variables that onstitute optimum unbiased preditors of these random quantities, and onditional seond moments that provide a measure of the assoiated predition errors. An alternative is to ompute these statistis diretly on the basis of orresponding onditional moment equations. Commonly, the moment equations are derived a priori in an approximate manner whih renders them loal in spae-time. We fous instead on exat forms of these equations, whih are generally nonloal in spaetime, and their subsequent solution by approximation. Exat spae-time nonloal first and seond onditional moment equations have been developed for steady state [4,4] and transient [48 5] flow in bounded saturated media as well as advetive [3, 39] and advetive-dispersive [56] transport in infinite domains; related but somewhat different unonditional formulations of nonloal transport in suh domains are found in [, 5, 3]. We onentrate here on exat first and seond nonloal moment equations for advetive-dispersive transport in bounded media developed reently by Morales-Casique et al. [37]. Morales-Casique et al. [38] solved Laplae transformed versions of these equations reursively to seond order in the standard deviation σ Y of (natural) log hydrauli ondutivity, and iteratively to higher-order, by finite elements followed by numerial inversion of the Laplae transform. They did the same for a spae-loalized version of the mean transport equation. Following a summary of their theory and algorithmi approah, we ompare below the numerial performane of the Laplae-transform finite element sheme with that of a high-auray ULTIMATE- QUICKEST algorithm oupled with an alternating split operator approah. We then review some omputational results due to Morales-Casique et al. [38] to shed light on the auray and omputational effiieny of their reursive and iterative solutions in omparison to onditional and unonditional Monte Carlo simulations in two spatial dimensions. Exat onditional moment equations for bounded media Morales-Casique et al. [37] express advetive-dispersive solute mass flux at some loal support sale ω, entered about point x in a Cartesian oordinate system, as J(x,t)=v(x,t)(x,t) D d (x,t), (.) where is onentration, v is veloity, D d is a onstant loal dispersion tensor and t is time. The veloity is given by Dary s law v(x,t) = K(x) h(x,t)/φ,

3 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp where K(x) is a random hydrauli ondutivity field, h(x,t) is hydrauli head and φ is a onstant porosity. It satisfies a stohasti flow equation v(x,t)= f(x,t) subjet to random initial and boundary onditions where f(x,t) is a random fluid soure (and/or aumulation term involving h/ t) normalized by φ. Lead multivariate statistis of log hydrauli ondutivity Y(x) = lnk(x) are inferred geostatistially from measured values of K(x), rendering it onditional on these data and thus (generally) nonstationary [, 44]. Lead joint statistis of v and f are obtained by solving the stohasti flow equation subjet to appropriate (generally random) initial and boundary onditions, onditioned on measurements of K(x) and/or h(x,t) and v(x,t) by forward and/or inverse solutions using Monte Carlo simulation or the solution of orresponding reursive onditional moments equations [3, 4, 8, 54]. The onentration of a non-reative solute in a domain Ω bounded by Γ is taken to be governed loally by the advetion-dispersion equation subjet to initial and boundary onditions (x,t) J(x,t)= g(x,t), x Ω (.) t (x,)=c (x), x Ω, (.3) (x,t)=c D (x,t), x Γ, (.4) D d (x,t) n(x)=w(x,t), x Γ, (.5) [v(x,t)(x,t) D d (x,t)] n(x)= P(x,t), x Γ 3, (.6) where g is a random soure of solute, C D is a random onentration presribed on boundary segment Γ, W is a random dispersive flux presribed normal to boundary segment Γ, P is a random advetive-dispersive flux presribed on boundary segment Γ 3, and n is an outward unit normal to any segment of Γ. Though theory does not require it, all foring terms g, C, C D, W, P are taken to be presribed in a manner that renders them statistially independent of v and eah other. All random funtions a(x,t) are deomposed as a(x,t)= a(x,t) a (x,t), a (x,t), where designates ensemble mean onditioned on measurements (as implied by the subsript; for simpliity and without loss of generality, foring terms remain unonditional) and primed quantities are zero-mean random flutuations about the mean. The former an be viewed as unbiased preditors of their random ounterparts, and the latter as the assoiated predition errors. Morales-Casique et al. [37] show that onditional mean transport is governed exatly by t (x,t) J(x,t) = g(x,t), x Ω (.7)

4 34 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 subjet to where (x,) = C (x), x Ω, (.8) (x,t) = C D (x,t), x Γ, (.9) D d (x,t) n(x)= W(x,t), x Γ, (.) [ v(x,t) (x,t) D d (x,t) Q (x,t)] n(x)= P(x,t), x Γ 3, (.) and J(x,t) = v(x,t) (x,t) D d (x,t) Q (x,t), (.) Q (x,t)= t Ω t Ω Γ 3 α (x,t,y,τ) y Q (y,τ)dτdy t γ (x,t,y,τ) (y,τ) dτdy t Ω t Γ 3 β (x,t,y,τ) y (y,τ) dτdy α (x,t,y,τ)q T (y,τ)n(y)dτdy β (x,t,y,τ) (y,τ) n(y)dτdy, (.3) α (x,t,y,τ)= G(x,t y,τ)v (x,t), β (x,t,y,τ)= G(x,t y,τ)v (x,t)v T (y,τ) γ (x,t,y,τ)= G(x,t y,τ)v (x,t) f (y,τ). The random Green s funtion G(x,t y,τ) satisfies a stohasti advetion-dispersion equation subjet to homogenous initial and boundary onditions. Morales-Casique et al. [37] present orresponding equations satisfied exatly by the onditional ovariane C (x,t,z,s)= (x,t) (z,s) of onentration and the onditional ovariane J (x,t)j T (z,s) of solute flux., 3 Reursive and iterative Laplae-transform finite element algorithms (FELT) Though the above onditional moment equations are exat, they annot be evaluated without a losure approximation. For steady state flow Morales-Casique et al. [37] derive reursive approximations in Laplae spae by expanding all ensemble moments to i th order in a small parameter, σ Y, representing the standard deviation of Y (x)=y(x) Y(x).

5 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp They define a deterministi differential operator ˆL () = λ [ v(x) () D d ], where tilde designates Laplae transformed quantities and the supersript indiates order zero in σ Y. They show that, to zero order, ĉ(x,λ) satisfies the boundary-value problem ˆL () ĉ(x,λ) () = ĝ(x,λ) C (x), x Ω, (3.) ĉ(x,λ) () = Ĉ D (x,λ), x Γ, (3.) D d ĉ(x,λ) () n(x)= Ŵ(x,λ), x Γ, (3.3) ] n(x)= ˆP(x,λ), x Γ 3, (3.4) [ v(x) () ĉ(x,λ) () D d ĉ(x,λ) () and to seond order the boundary-value problem ˆL () ĉ(x,λ) () = v(x) () ĉ(x,λ) () ˆQ () (x,λ), x Ω, (3.5) ĉ(x,λ) () =, x Γ, (3.6) D d ĉ(x,λ) () n(x)=, x Γ, (3.7) ] ĉ(x,λ) () D d ĉ(x,λ) () n(x) [ ] = v(x) () ĉ(x,λ) () ˆQ () (x,λ) n(x), x Γ 3. (3.8) [ v(x) () The omplete seond-order approximation is given by ĉ [] = ĉ () ĉ (). The authors provide orresponding seond order equations for the onditional ovariane of onentration predition errors, ˆQ () and solute mass flux. Eqs. (3.) and (3.5) are Laplae-transformed equivalents of a standard advetiondispersion problem but with nonstandard foring terms and variables (inluding parameters) that onstitute deterministi estimators of their random ounterparts. Eqs. (3.)- (3.8) an therefore be solved using existing numerial methods supplemented with routines to ompute the foring terms. Morales-Casique et al. [38] solve these reursive moment equations by Galerkin finite elements in a retangular domain Ω subjet to deterministi initial and boundary onditions, taking the veloity to be uniform in eah element and the loal dispersion-diffusion oeffiient to be a onstant deterministi salar, D d. They invert the results bak into the time domain using a numerial algorithm due to Crump [9] and De Hoog et al. [4]. The authors demonstrate that an improved solution is obtained upon adopting the iterative solution algorithm (Algorithm 3.). The results form an inomplete third order approximation, as indiated by the supersript () in that the algorithm exludes the third-moment z ĉ v v i.

6 36 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 Algorithm 3.: a) Solve (3.)-(3.4) for ĉ () b) Evaluate ˆQ () i and set ĉ old = ĉ (). using ĉ old instead of ĉ (). ) Solve (3.5)-(3.8) for ĉ () using ĉ () and ˆQ (). d) Compute ĉ new = ĉ () ĉ (). e) If ĉ new ĉ old f) Evaluate Ĉ () exeeds a predetermined tolerane, set ĉ old using ĉ new and ˆQ () instead of ĉ () and ˆQ (). = ĉ new and go bak to step b. Morales-Casique et al. [38] also examine a spae-loalized finite element solution (designated by the supersript L) with dispersive flux ˆQ L i (x,λ)= ˆD () (x,λ) (x,λ) L, whih beomes a time-onvolution integral in the time domain. 4 Comparison of FELT with another high-auray sheme We ompare below the Laplae transform finite element algorithm (FELT) used by Morales-Casique et al. [37, 38] with another high-auray approah, motivated by the following onsiderations. Solving the stohasti transport problem by numerial Monte Carlo simulation requires a omputational grid that is fine enough to resolve ω-sale random veloity flutuations. As ω-sale dispersivities are relatively small, eah Monte Carlo run tends to be dominated by advetion. Eah onditional Monte Carlo run produes a spae-time realization of ω-sale veloity, solute onentration and mass flux whih, upon averaging over numerous realizations, yields sample statistial moments of these quantities. An alternative method of solving stohasti transport problems is to ompute ensemble moments of these quantities diretly by solving a system of ensemble moment equations suh as those mentioned earlier. The orresponding zeroorder moment equations entail only a loal, ω-sale dispersion term that render them advetion-dominated in many ases of interest. Seond-order moment equations inlude a marodispersion term that is initially zero but grows with solute residene time toward values that may beome large in omparison to loal dispersion [56]. Hene seond-order moment equations are advetion dominated at early time while tending to be less so at later times. It follows that solving stohasti transport equations, by whatever method, is subjet to the usual numerial diffiulties assoiated with advetion-dominated problems. In partiular, standard upstream numerial shemes tend to exhibit artifiial smearing of sharp onentration fronts and entered shemes tend to generate spurious osillations that may manifest themselves as onentration overshoot, undershoot and lipping

7 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp of peaks. The first phenomenon results from trunation errors of insuffiiently high order, the seond from variable propagation speeds of short-wave omponents in a Fourier representation of the numerial solution []. Among modern omputational approahes designed to deal with some of these problems we ount the Eulerian-Lagrangian loalized adjoint method (ELLAM) [5, 9], the streamline method [, ], total-variation-diminishing (TVD) methods [35], essentially (ENO) and uniformly (UNO) nonosillatory methods [6, 7], and Laplae-transform finite elements (FELT) [4,46,53] in whih a finite element solution of Laplae-transformed transport equations is transformed bak into the time domain using numerial inversion. A summary of some of these and other approahes was published by Ewing and Wang [8]. Al-Lawatia et al. [] ompared Runge-Kutta harateristi ELLAM methods with Crank-Niholson finite element (Galerkin, Quadrati and Cubi Petrov- Galerkin), streamline diffusion, ontinuous and disontinuous Galerkin and two highauray finite volume shemes (monotone upstream-entered sheme for onservation laws MUSCL [5, 5] and MINMOD [35]) for one-dimensional advetion-dispersion. They found ELLAM to be more aurate and effiient than other shemes: whereas seond-order TVD shemes generated monotonous profiles with some numerial dispersion, all other shemes but ELLAM developed spurious osillations, all requiring finer spae-time grids and muh more omputer time than ELLAM. However, the authors noted that ELLAM might not work well in the ase of multidimensional problems with spatially varying veloity. Farthing and Miller [] onduted a thorough review and omparison of leading high-auray finite-volume shemes. Among 4 suh shemes (inluding MUSCL and MINMOD) they found two higher-order TVD shemes, ULTIMATE-QUICKEST [33] and the pieewise paraboli method PPM [6], to perform well on their test problems and to be generally more effiient than the best lower order TVD method (MUSCL) and the three uniformly nonosillatory (UNO) and essentially nonosillatory (ENO) shemes they had evaluated. In partiular ULTIMATE-QUICKEST seemed to offer the most attrative balane between auray and omputational effiieny. The sheme was reently inorporated in the popular groundwater transport ode MT3DMS [57]. Stohasti moment equations inlude temporal onvolution integrals whih FELT redues to simple produts in the Laplae-transformed domain. In addition, FELT provides time-ontinuous solutions that are free of restritions on the Courant number, mutually independent and so amenable to parallel omputation, and independent of solutions at previous time values. On the other hand FELT is limited to linear transport equations assoiated with time-independent veloity fields, may suffer from osillations (Gibbs phenomenon) near disontinuities due to trunation errors, and may be less effiient than time marhing shemes when a solution is required at many time values. Based on these onsiderations we deided to ompare FELT with ULTIMATE-QUICKEST identified by Farthing and Miller [] as one of the best performing higher-order TVD shemes. Our numerial examples inlude a omposite -D onentration profile in a uniform veloity field, -D transport due to an instantaneous point soure in suh a field, -D transport in a

8 38 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 randomly varying veloity field arising from random medium heterogeneity, and Monte Carlo simulated mean -D transport in mildly and strongly heterogeneous random media. 4. Bakground Total variation diminishing (TVD) finite volume shemes In one-dimension the advetion equation an be approximated by the finite differene onservation form n i = n i t x [F( i/ ) F( i / )], (4.) where i is sequential node number, n i is an approximation of the ell averaged onentration n i = xi/ x (x,t x i / n )dx at time t n (time step n) in a ell bounded by x i / and x i/, and F( i/ ) = tn t t n f( i/ )dt is time-averaged advetive solute flux f( i/ )=v i/ i/ at x i/. The total variation (TV) of n aross the grid is defined as [35] TV( n )= N n i n i, (4.) i= where N is the number of nodes. A numerial sheme is said to be total-variationdiminishing or TVD if TV( n ) TV( n ) for every n. Spurious osillations bring about an inrease in TV whih may ause (4.) to be violated. One way to suppress suh osillations is to enfore (4.) by adding numerial diffusion to a higher-order sheme near sharp fronts using flux- or slope-limiters [35]. Early TVD methods suh as the flux-orreted approah [4], Superbee limiter [43], MUSCL [5] and MINMOD [35] were seond-order aurate. Current shemes suh as that of Cox and Nishikawa [8], ULTIMATE-QUICKEST [33] and PPM [6] inlude modifiations to improve auray in smooth regions. We fous below on the ULTIMATE- QUICKEST sheme. ULTIMATE-QUICKEST sheme Consider a monotoni one-dimensional onentration profile n i n i n i n i adveted by a positive spatially varying veloity v >. The onservation expression (4.) now takes the form n i = n i t [ ] vi/ x i/ v i / i /, (4.3) where / is limited by n i i/ n i. Maintaining monotoniity requires n i n i n i.

9 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp By virtue of (4.3) and onsidering that v i / >, the left inequality transforms into v i/ i/ v i / i / ( n i n i ) x/ t, whih, for a worst-ase estimate of n i /, yields v i/ i/ v i / i x ( n t i n ) i. (4.4) Likewise, the right inequality in n i i/ n i transforms into v i / i / v i/ i/ ( n ) i n i x/ t. Viewing i/ as onentration at the left fae of a ell entered about i and adopting a worst-ase estimate for i3/ yields v i/ i/ v i3/ i x ( n t i n ) i, vi3/ >. (4.5) For onstant advetive veloity (4.5) is replaed by i/ n i in n i i/ n i. For loal extrema one sets i/ = n i. Inequalities (4.4) and (4.5) onstitute the universal limiter ULTIMATE [33]. In pratie, it needs to be applied only near sharp gradients where the onstraints ould be violated. Leonard [33] onsidered several high-order interpolation shemes to ompute the ell-fae values i/ among whih the simplest, yet suffiiently aurate, is the third order QUICKEST interpolator. The algorithm implemented in this work for a onstant positive veloity is [33]: Algorithm 4.: a) Compute DEL = n i n i and CURV = n i n i n i. b) If CURV.6 DEL use the unonstrained QUICKEST fae value QUICK i/ = ( n i n i where Cr = v t/ x is the Courant number. ) Cr ( n i n i ) If CURV DEL (loal extrema) set i/ = n i. d) Otherwise use the ULTIMATE limiter to set re f i/ = n i Cr ) ( Cr ) CURV, 6 [ n i n ] i. ( e) If DEL> (monotoni inreasing) limit i/ first by i/ = max ( ) i/ =min i/, re f i/,n i. ( f) If DEL< (monotoni dereasing) limit i/ first by i/ = min ( ) i/ =max i/, re f i/,n i. QUICK i/ QUICK i/ ),n i and then by ),n i and then by

10 4 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 The extension to two dimensions of the ULTIMATE-QUICKEST algorithm for advetive transport is detailed in Leonard and Niknafs [34]; we employ this algorithm in our two-dimensional simulations as implemented in the ode MT3DMS [57]. In applying the one- and two-dimensional algorithms to advetive-dispersive transport we adopt an alternating split operator approah [36, 45] as implemented by Farthing and Miller []: a) Solve a pure dispersion problem using a Crank-Niholson sheme over the first half of a time step [t,t t/] with (x,t) as initial onentration. b) Solve a pure advetion problem over the entire time step [t,t t/] with from (a) as initial onentration. ) Solve a pure dispersion problem using a Crank-Niholson sheme over the seond half of the time step [t t/,t t] with from (b) as initial onentration. FELT sheme Applying the Laplae transform to the advetion dispersion equation yields λĉ(x,λ) [v(x)ĉ(x,λ) D d ĉ(x,λ)] ĝ(x,λ) C (x)=, (4.6) where λ is a omplex parameter with Re(λ) >, ĉ and ĝ are Laplae transforms of and g, respetively, and C is initial onentration. In FELT (4.6) is solved by finite elements and the results are inverted numerially bak into the time domain. As the finite element equations entail omplex, generally nonsymmetri oeffiient matries, we presently solve them using the IMSL ( LUfatorization subroutine DLSACB with iterative refinement for improved auray. To obtain a solution in the time domain Sudiky [46] used the Crump inverse Laplae transform algorithm [9] due to its auray and reasonable performane near disontinuities. For a given point in spae, the orresponding inverse is given by { (x,t)= eλ t M ĉ(x,λ ) πk (i Re [ĉ(x,λ T k )e t)]} T, (4.7) k= where the inverse Laplae transform is disretized using a trapezoidal rule with mesh size π/t and the Laplae parameter is expressed as λ k =λ iπk/t, k=,,m. Although the optimal seletion of parameters is problem dependent, Crump reommended λ = µ ln(er)/t where T =.8t max, t max being the maximum time of interest, µ =, Er 8, Er being a measure of auray, and the inverse is omputed within the interval < t<t. To avoid instability at early time Gallo et al. [] reommend setting t>.t max. They also reommend limiting M 5 to minimize roundoff errors; however, we found that in the presene of sharp gradients a more aurate solution is obtained by setting M=3. Crump [9] employed a so-alled epsilon algorithm to evaluate (4.7). de Hoog et al. [4] improved the rate of onvergene by omputing a diagonal Padé approximation to the

11 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp series in (4.7), known as the quotient-differene (q-d) algorithm. They proposed an additional aelerated onvergene proedure that estimates the remainder of the ontinued fration. The q-d algorithm leads to a signifiant improvement in auray and is highly effiient when inversion is required for many time values. This is the algorithm we use below. To solve for ĉ(x,λ k ) we use standard finite elements with linear and bilinear interpolation funtions in -D and -D, respetively. FELT suffer from osillations (Gibbs phenomenon) arising from non-uniform onvergene of the series in (4.7) near disontinuities. This means that the number of terms in (4.7) needed for a given improvement in auray tends to beome arbitrarily large as one approahes a disontinuity (e.g. [3]). In general, ĉ(x,λ k ) is an osillatory funtion with period Φ that inreases with k and depends additionally on D, v, and λ k. Sudiky [46] showed that for fixed v and λ k, Φ in -D inreases with D; for fixed D and λ k, Φ dereases as v inreases. It is easy to show [46] that, under pure advetion in a semi-infinite -D domain with zero initial onentration and Dirihlet inlet ondition, the disretization interval needed for an aurate solution is x=vt max /n(m) where n is the number of nodes needed to resolve ĉ(x,λ k ) over a distane Φ and M is the last term in the series (4.7). Therefore, near onentration disontinuities in fixed grid, inreasing M in (4.7) leads to improved auray up to some limit beyond whih ĉ(x,λ k ) annot be aurately resolved; to further redue (though not neessarily eliminate) osillations, one must inrease M and refine the grid simultaneously. Yet when (v,t) is smooth, a oarse grid produes aurate results with a relatively small value of M [9, 46]. 4. Numerial omparison of ULTIMATE-QUICKEST and FELT -D omposite initial onentration profile in uniform veloity field To ompare ULTIMATE-QUICKEST with FELT we start by onsidering the -D dimensionless ADE t x Pe x =, Pe= vl >, (4.8) D on a finite interval x [,] subjet to Dirihlet boundary onditions (,t)=, (,t)= and a omposite initial onentration profile (x,) = if x [.,.5] (square), os ( πx ) if x [.5,.35] (squared osine), x ) / ( (x.5) if x [.45,.55] (semi-ellipse), (4.9).5 otherwise. Here x is normalized by the length L of the flow domain, = C/C is dimensionless onentration where C is the peak of the initial onentration profile, and t = vτ/l is

12 4 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 dimensionless time. We use a disretization interval x =.5 assoiated with a grid Pelet number Pe g = v x/d. For finite Pelet numbers Pe <, an analytial solution exists in the form [3] [ ] (x,t)=e Pe (x ) t G(x x,t)(x,)e Pe x dx.5 er f (A)e xpe er f (B), (4.) where G is a Green s funtion, A=(x t)/ 4t/Pe and B=(xt)/ 4t/Pe. For simpliity we adopt the semi-infinite Green s funtion (x x ) G(x x,t τ)= e Pe 4(t τ) e Pe (xx ) 4(t τ), (4.) 4π(t τ)/pe set τ = and evaluate the integral by Gaussian quadrature. To avoid dependene of the numerial solution on the downstream boundary we terminate it before any notable hange in onentration would take plae lose to it. Preliminary runs of FELT with (x,)= and (,t)= have suggested M = 3, µ= and Er = 8. We kept these parameters fixed for all simulations, inluding those assoiated with small Pe g values. We onsider two ases: a) Infinite Peg (pure advetion). Fig. (left) shows results obtained with ULTIMATE- QUICKEST (ULT) with Cr =.5 ( t=.5) and Cr =.5 ( t=.5) at t=. and.4; we note that even though optimal results would be obtained with Cr=, in variable veloity fields one often has to limit Cr to smaller values, hene these are of pratial interest. Fig. (right) shows orresponding results obtained with FELT using the q-d algorithm without (FELT ) and with (FELT ) improvement of onvergene based on estimates of the ontinued fration remainder [4]. FELT and produe osillatory solutions near disontinuities and appear to have omparable auray exept at t =. near x =.3 where FELT produes stronger osillations. Inreasing the value of M to 35 produes a less osillatory solution for FELT (not shown), similar to that of FELT in Fig. ; inreasing the value of M further redues auray in both FELT versions due to an enhanement of disretization and roundoff errors. Inreasing M to 6 while reduing x by /5 redues signifiantly the period of the osillations and the magnitude of the largest (overshoot) and smallest (undershoot) onentrations at t =. from.46 and -.38 (Fig. ) to. and -.3 (not shown), respetively. Fig. ompares the performane of ULT with Cr =.5 and FELT at t=. and.4. ULT generates monotonous results, free of spurious osillations, though it introdues some smearing of sharp fronts. FELT resolves better the profile of sharp fronts but suffers from spurious osillations. Sine these osillations depend on the number of terms in (4.7) and the spatial disretization, reduing them requires inreasing M and refining the grid to allow better resolution of the omplex (periodi) solution in Laplae spae. b) Pe g finite (advetion-diffusion). Fig. 3 ompares results obtained by the two methods for Pe g = (left) and Pe g = (right), respetively. Here FELT with and without im-

13 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp Conentration t=. Analytial FELT FELT Conentration t=. Analytial ULT Cr=.5 ULT Cr= x x Conentration t=.4 Conentration t= x x Figure : ULTIMATE-QUICKEST (left) and FELT (right) solutions for omposite initial onentration and infinite grid Pelet number. Conentration t=..5 Analytial ULT Cr=.5 FELT Conentration x t= x Figure : Results of ULT and FELT for omposite initial onentration and infinite grid Pelet number. provement of onvergene based on estimates of the ontinued fration remainder gave similar results, and we therefore present only those obtained with FELT. At Pe g = FELT still suffers from spurious osillations but to a lesser degree than under pure advetion. On the other hand, the FELT solution for Pe g = is free of suh osillations. FELT resolves the squared osine and semi-ellipti profiles more aurately than does ULT in both ases. The ULT solution is free of osillations at all times; however, it smears sharp fronts even at the relatively small grid Pelet value of Pe g =. Table ompares the solutions in terms of mean squared error (MSE) relative to the analytial solution. Smearing of sharp fronts has an adverse effet on the overall auray of ULT, leading to relatively high MSE values. FELT is onsistently more aurate exept at early time under pure advetion where it produes an osillatory solution. Table also ompares exeution times. Both shemes were run on a PC with AMD Athlon TM proessor. ULT with Cr=.5 required about four times less entral proessor (CPU) time than did FELT; however, with Cr=.5 ULT required almost an order of magnitude more

14 44 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 Conentration t= x.5.75 Analytial ULT Cr=.5 FELT Conentration t= x.5.75 Analytial ULT Cr=.5 FELT Conentration t= x x Conentration t=.4 Figure 3: Results of ULT and FELT for omposite initial onentration at Pe g = (left) and Pe g = (right). CPU time. Table also lists the effiieny index E f f = [(MSE)(CPUT)] for t =.4 where CPUT is CPU time: whereas the effiieny index of ULT with Cr =.5 is to 3 times higher than that of FELT, with Cr=.5 it is 3 to 7 times lower. Table : Results for omposite initial onentration. Method MSE ( 3) CPU time (mse) Eff ( 3) t=. t=.4 t=.4 Pe g = infinite ULT, Cr = ULT, Cr = FELT (3) a 3. FELT b.5 Pe g = ULT, Cr = ULT, Cr = FELT () a 9.3 FELT..75 b 8.8 Pe g = ULT, Cr = ULT, Cr = FELT () a 5. FELT.37.7 b 5. a The first number indiates total exeution time. The number in parenthesis indiates exeution time devoted to finite element solution in Laplae spae. b Computed together with FELT as orresponding omputational effort is low. FELT and yield virtually idential results for Pe g =5, 5, 5 at all t when t max =. The MSE of FELT inreases with Pe g. For a given Pe g it is relatively small in the time range.t max <t<.t max, the lower limit providing a onfirmation of similar findings by Gallo et al. []. This is expeted as at early time the solute has not undergone enough

15 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp spreading to smooth out sharp fronts at whih trunation errors have the greatest effet. At t>.t max the solution osillates at low values of x. -D initial delta pulse in uniform veloity field In our seond one-dimensional example we replae the Dirihlet by Neumann boundary onditions / x x=,x= = and the initial onentration by (x,)=δ(x.) where δ is the Dira delta funtion. In the ULT finite-volume method we interpret the onentration as a ell-averaged value (x,t)= x p (x,t)dx / x where p is point onentration and set the initial onentration aordingly to (x =.,) = / x. Results for Pe g = 5,, and t =.,.4 show (Fig. 4 does so for Pe g = ) that FELT performs better than ULT aross this range of grid Pelet numbers, beoming omparatively more aurate as Peg inreases. At Pe g = and ULT suffers from peak lipping and some numerial dispersion. On the other hand, FELT at Pe g = suffers from mild osillations at early time (t=.) whih disappear at late time (t=.4); this is due to the fat that spreading smoothes the profile, allowing a more aurate solution at late time. Table shows that FELT outperforms ULT in terms of the effiieny index E f f in all three ases. Table : Results for initial onentration pulse. Method MSE ( 3) CPU time (mse) Eff ( 3) t=. t=.4 t=.4 Pe g =5 ULT, Cr = ULT, Cr = FELT () a.6 FELT.36.6 b.6 Pe g = ULT, Cr = ULT, Cr = FELT () a 3.9 FELT b 3.9 Pe g = ULT, Cr = ULT, Cr = FELT () a.7 FELT b. a The first number indiates total exeution time. The number in parenthesis indiates exeution time devoted to finite element solution in Laplae spae. b Computed together with FELT as orresponding omputational effort is low. -D transport in single realization of random veloity field Next we onsider a retangular domain with disretization intervals x = x =. (Fig. 5). We start by generating a single unonditional realization of a random log hydrauli ondutivity field Y(x) =lnk(x) at element enters using the sequential Gaussian

16 46 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 Conentration 4 Pe g = t= x Analytial ULT Cr=.5 FELT Conentration 3 Pe g = t= x Figure 4: Results of ULT and FELT for initial delta onentration pulse x=.. Analytial ULT Cr=.5 FELT simulation ode SGSIM [6]; the point values are then asribed to the entire element. The generated field has zero mean, variane σy = 3. (made purposely large to generate large-amplitude spatial flutuations in Y) and isotropi exponential spatial orrelation C YY (r) = σy exp( r/i Y), where I Y =. is integral sale (made purposely small to generate rapidly flutuating variations in Y), r being separation distane between any two points in the domain. 4 3 x x Figure 5: Computational domain with uniform non-zero initial onentration zone (shaded). We test two solutions of the flow problem. First we use standard Galerkin finite elements with bilinear weight and interpolation funtions to ompute the steady state distribution of hydrauli heads at all grid nodes subjet to presribed heads of H=.8 at x = and H= at x =8 and no-flow aross x = and x =4. Then, using a uniform advetive porosity arbitrarily set equal to, we ompute veloity at the enter of eah element and asribe it to the entire element. The resulting veloity field is globally onservative but loally disontinuous and non-onservative whih, as will be illustrated, has an adverse affet on the transport solution (e.g. [3, 7]). The veloity field obtained by the standard Galerkin approah an be post-proessed to reover loal mass onservation [7, 47]. Alternatively we solve the flow problem by ell-entered finite differenes and ompute veloity at the faes of the elements; the resulting veloity field is loally ontinuous normal to element faes and loally and globally onservative. By onstrution ULT requires veloity to be defined at the faes of eah element. In the ase of FELT, we aommodate

17 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp this variable veloity field by using linear interpolation, v (x) v (x )=w L (x )v L w R (x )v R, where w L (x )=.5( x ), w R (x )=.5(x ) and v L and v R are the veloity values at the left and right fae, respetively; v is interpolated in a similar manner in the x diretion. Linear interpolation yields disontinuous tangential veloities leading to disontinuities in a veloity-dependent dispersion tensor and affeting mass onservation [3]; however, sine in our examples we take loal dispersion to be onstant, this does not affet our results. y Pr Proo y Figure 6: Conentration at t=4 for D=. ( x = x =.): (a) FELT solution with loally disontinuous veloity field, (b) ULT and () FELT solutions with loally onservative veloity field. Transport is omputed for three salar dispersion oeffiients D =.,., and., orresponding to maximum grid Pelet numbers of approximately 5, 5 and 5 in the x diretion and.7, 7 and 7 in the x diretion, respetively. To ompute onentration we set [v D ] n = at x =, x = 4 and at x =, / x = at x = 8, C (x)= at.6 x.4 and.6 x.4, and C (x)= at all other points. The omputational grid is the same as that used for flow (Fig. 5). Fig. 6 depits onentrations at t =4 obtained for D =. using (a) FELT with a loally disontinuous veloity field, and both (b) ULT and () FELT with a loally onservative veloity field. Disontinuity of veloity at element faes yields osillatory onentration values; these unphysial osillations are

18 48 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 Figure 7: Conentrations at t=4 obtained for D=. (maximum grid Pelet number of 5) using (upper left) ULT and (upper right) FELT with x = x =. and (lower left) ULT and (lower right) FELT with x = x =.5. The veloity field is loally onservative. 3E-5 Mean AD E-5 E-5 A B C Variane of AD -8 A B C - Mean AD..8.4 A B C Variane of AD - A B C Time Time Time Time Figure 8: Average and variane of absolute differenes (AD) between oarse- and fine-grid solutions for (left) D =. and (right) D =. versus time. A = oarse- versus fine-grid ULT solution; B = oarse- versus fine-grid FELT solution; C = ULT versus FELT solutions on fine grid.

19 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp absent when the veloity field meets loal mass onservation riteria (Fig. 6). Hene in the remainder of this setion we present only results obtained using a loally onservative veloity field; however, in a later setion we will revisit the loally non-onservative veloity field in the ontext of Monte Carlo simulations. Upon dereasing D from. to. both ULT and FELT yield aurate solutions when the veloity field is loally onservative (not shown). Fig. 7 depits ULT and FELT solutions based on a loally onservative veloity field for D =. (inreasing the maximum grid Pelet number to 5) for a oarse ( x = x =.) and a fine ( x = x =.5) grid. In this ase ULT results show a marked sensitivity to grid refinement (Figs. 7(a) and 7(b)) while the oarse-grid FELT solution is plagued by nonphysial negative values of onentration (Fig. 7()); these osillations are greatly redued when the grid is refined (Fig. 7(d)). FELT osillations our in the viinity of sharp fronts whih, in heterogeneous media at high grid Pelet numbers, tend to develop in areas of large veloity ontrast. Fig. 8 shows how the average and variane of the absolute differenes AD between oarse- and fine-grid solutions for D =. and. vary with time. Values of AD(x i,t) = oarse (x i,t) f ine (x i,t) are omputed at grid nodes in omparing FELT results on oarse and fine grids, at element enters in omparing ULT results on oarse and fine grids, and at element enters in omparing FELT (interpolated by standard finite elements) and ULT results on fine grids. Fig. 8 onfirms that, in the ase of D=., FELT is less sensitive (has smaller average and variane of AD values) to grid refinement than is ULT, suggesting that FELT is more aurate. By the same standard, ULT and FELT are equally aurate in the ase of D =. (not shown) but ULT is more aurate than FELT in the ase of D=. (Fig. 8). In all three ases the variane of AD diminishes rapidly at early time and muh more slowly at later time. Global Mass Balane.4 ULTD=. FELTD=. ULTD=. FELTD=. ULTD=.. FELTD= Time Figure 9: Global mass balane fration versus time for oarse grid ( x = x =.) ULT and FELT solutions orresponding to various D values. Global mass balane is expressed by the fration MB=[M(t)BF(t)]/M(t ) where M(t ) and M(t) stand for total mass in the domain at times t and t, respetively, and

20 5 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 BF(t) is umulative net solute flux through the boundaries (positive out, negative in). Fig. 9 shows how MB varies with time in the ase of oarse grid ( x = x =.) ULT and FELT solutions orresponding to the above three D values. Both methods exhibit exellent global mass balane before the plume reahes the boundary at approximately t=5; for t>5 the mass balane of ULT deteriorates slightly due to insuffiient auray when omputing the time integral representing umulative mass flux at the boundary; the latter is easier to ompute in Laplae spae. FELT, despite its osillatory nature, maintains exellent global mass balane even at high grid Pelet numbers. Table 3 ompares exeution times for the above oarse grid solutions. Though FELT required.5 more CPU-time than did ULT, we note that the effiieny of FELT ould be improved substantially by taking advantage of its amenability to parallel omputation (e.g. [54]), whih we have not done in this work. Table 3: Comparison of exeution times for oarse grid solutions ( x = x =.). Cases Stability riterion for ULT Exeution time in onsistent units in seonds ULT FELT D = D =...43 D = Monte Carlo analysis of mean -D transport in random veloity field We end our omparative analysis of ULT and FELT by onduting Monte Carlo (MC) simulations of transport for σy =.3, D =.5 (ase I) and σ Y = 3, D =. (ase II) on the retangular domain depited in Fig. 5. We adopt the previous proedure to generate numerous ( in ase I and 3 in ase II) unonditional realizations of Y(x), orresponding realizations of veloity and time-varying onentration fields. In omputing eah realization of the veloity field we employ both approahes desribed in the previous setion, namely loally disontinuous veloities from standard Galerkin interpolation and loally onservative veloities from a finite differene solution. Comparing these two approahes to obtain the veloity field is of interest beause previous stohasti analysis of flow and transport [4, 5, 37, 38, 54] have relied on the first approah. Fig. depits mean onentration at t=4 for ases I and II when the underlying veloity field is loally onservative; both methods yield similar mean onentrations with minor differenes. We thus proeed to ompare FELT results when the veloity field is not loally onservative. Fig. shows that osillations present when the veloity field is not loally onservative (Fig. 6(a)) are smoothed when one averages many realizations (dashed in Fig. ). For small values of σy the mean and variane of onentration obtained with loally nononservative veloities are reasonably aurate, though they deteriorate as σy inreases. Fig. shows the same to hold for mean solute flux. This onfirms the auray of results presented elsewhere [37, 38] for σy.3, summarized in the next setion.

21 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp x x x x Figure : Mean onentration at t=4 obtained using ULT (dashed) and FELT (solid) in (left) ase I (σy =.3, D=.5, Monte Carlo runs) and (right) ase II (σy = 3, D=., 3 Monte Carlo runs) <> 3 x x x 4 3 <> x 4 3 x x x x Figure : Mean (left) and variane (right) of onentration at t = 4 obtained using FELT when the veloity field is loally onservative (solid) and loally nononservative (dashed) in ase I (σy =.3, D =.5, Monte Carlo runs, top) and ase II (σy = 3, D=., 3 Monte Carlo runs, bottom). 3 <J > 3 <J > x x x x <J > <J > 3 3 x x x x Figure : Mean solute flux t=4 obtained using FELT when the veloity field is loally onservative (solid) and loally nononservative (dashed) in ase I (σy =.3, D=.5, Monte Carlo runs, top) and ase II (σy = 3, D=., 3 Monte Carlo runs, bottom).

22 5 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 Table 4: Comparison of exeution times for oarse grid solutions ( x = x =.). Case σy Exeution time (h:min:s) FELT ULT I ( MC runs).3 :3:45 :5 II (3 MC runs) 3. 4:3:5 7:47:6 Table 4 ompares exeution times for MC simulations using the two methods. In ase I ULT required roughly one fourth the CPU time needed for FELT to attain omparable auray; in ase II this differene was redued to about one half. The inrease in exeution time was aused by a need to redue ULT time step size in order to omply with the Courant-Friedrih-Lewy ondition when higher values of σy produed larger ontrasts in veloity. We onlude by observing that whereas the plume of mean onentrations is symmetri in ase I, it is non-symmetri in ase II (Fig. ). The lak of symmetry is due to our imposition of zero solute mass flux aross the upstream boundary, whih prevents spread (marodispersion) of the mean plume aross it. We emphasize that this spread does not represent physial dispersion but loss of information about the unknown true (random) onentration, as explained by Morales-Casique et al. [37, 38]. 5 Conditional nonloal and loalized FELT solution ompared with Monte Carlo simulations We review below seleted omputational results of those previously reported by Morales- Casique et al. [38]. Computational domain and random veloity field Morales-Casique et al. [38] onsider advetive-dispersive transport in a retangular domain of length 8I Y and width 4I Y, disretized into square elements of size x = x =.I Y (Fig. 3); all variables are expressed in arbitrary onsistent units. Though the moment equations are distribution-free, for purposes of Monte Carlo simulation they take Y to be multivariate Gaussian with mean zero, variane σy and isotropi exponential spatial orrelation funtion C YY (r)=σy exp( r/i y) where r is separation distane. The authors generate a single unonditional realization of Y at element enters using the sequential Gaussian simulation ode SGSIM [6] and asribing it to the entire element. They then measure Y exatly at 8 onditioning points shown by solid squares in Fig. 3, use SGSIM in the above manner to generate numerous realizations of Y onditioned on these measurements, and evaluate the orresponding first and seond sample moments of the generated onditional Y fields. To generate orresponding statistis of time-independent advetive veloity, Morales- Casique et al. [38] use onditional Monte Carlo simulations and the reursive finite ele-

23 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp Conditioning point (hydrauli ondutivity measurement) 3 x Figure 3: Computational domain and grid, onditioning points and loation of initial non-zero onentration (shaded). x ment moment algorithm of Guadagnini and Neuman [4, 5], both on the above grid. In the ase of Monte Carlo simulations, they use standard finite elements to solve a deterministi steady state flow problem for eah onditional realization of Y and ompute first and seond sample moments of the orresponding advetive veloity realizations. The resulting veloity in eah realization is onstant in eah element and thus not loally onservative; however, we are interested in moments of these realizations whih, as shown in the previous setion, are reasonably aurate for small values of σy. The moment algorithm uses first and seond onditional moments of Y to ompute diretly first and seond onditional veloity moments to seond order in σ Y. In both ases advetive porosity is set (for simpliity) equal to, head is set equal to H=.8 at x = and H= at x =8, and no-flow onditions are imposed aross x = and x =4. For Y =, σy =.3 and I Y =, 3 Monte Carlo simulations and the reursive algorithm yield omparable first and seond onditional veloity moments. The authors found that for σy =.3 the unonditional longitudinal veloity v follows losely a lognormal distribution. Unonditional transverse veloity deays more slowly than a Gaussian distribution at the tails, most likely due to the effet of boundaries. Conditioning brings the distributions of both veloity omponents loser to Gaussian. Transport omputations To ompute transport variables Morales-Casique et al. [38] set D d = D d I where I is the identity tensor, [v D d ] n= at x = and x =4, [v D d ] n= at x =, / x = at x =8, C (x)= at.6 x.4 and.6 x.4, and C (x)= at all other points. They present results in terms of a Pelet number Pe=u I Y /D d where u =. is the (theoretial) unonditional mean veloity parallel to x, and a grid Pelet number Pe g =u x /D d. All results, inluding Monte Carlo simulations, are obtained using FELT. For the reursive and iterative algorithms the authors use ensemble veloity moments obtained using the flow moment algorithm of Guadagnini and Neuman [4, 5]. They ondut Monte Carlo simulations of transport by solving the stohasti transport equations (.)-(.6), for eah onditional realization of veloity, using the above grid.

24 54 E. Morales-Casique and S. P. Neuman / Commun. Comput. Phys., 6 (9), pp. 3-6 t= t= t=3 Conentration.8.4 MC [] [] Conentration.8.4 Conentration x x x.4 t=4.4 t=5.4 t=6 Conentration. Conentration. Conentration x x x Figure 4: Conditional mean onentration profile at x =. and t=,, 3, 4, 5, 6 for σ Y =.3, I Y=, D d =., Pe= and Pe g = based on 3 Monte Carlo simulations MC, seond-order reursive [] and iterative [] nonloal moment equations. Fig. 4 depits onditional mean onentration profile at x =. and t=, 3, 4, 5, 6 for σ Y =.3, I Y =, D d =., Pe= and Pe g = based on 3 Monte Carlo simulations MC, seond-order reursive [] and iterative [] nonloal moment equations. There is a pronouned bimodal behavior of the reursive mean onentration. Neither the Monte Carlo nor the iterative mean onentration, whih are in lose agreement with eah other, exhibit suh bimodality. These results and D simulations (not shown) indiate that seond and third-order perturbation terms are osillatory, that this osillatory behavior inreases with time, and that many terms in a standard perturbation series may be needed for onvergene. This osillatory behavior is not aused by the mean veloity being onstant in eah element (and thus loally nononservative). This is evidened by (a) the smooth and nonosillatory nature of the zero-order onentration () (Fig. 9 in [38]) and (b) the frequeny of the osillations in the seond-order orretion () (Fig. 9 in [38]) being muh larger than that due to disontinuities in the veloity field (see Fig. 6(a)). The iterative solution provides more aurate results than does standard perturbation beause it approximates the dispersive flux more losely and ould be improved further by aounting for veloity moments of order higher than two. Fig. 5 shows onditional mean onentration profiles at x = and t=, 5 for the above parameters based on Monte Carlo simulations ompared with iterative nonloal and spae-loalized moment solutions. The spae-loalized solution is seen to be inferior, exhibiting spurious osillations (undershoot) at early time while underestimating