ELSEVIER

Advances in Warer Resources, Vol. 19, No. 4, 241-253, 1996 pp. Copyright 0 1996 Elwier Science Limited Printed in Great Britain. All rights reserved

PII: SO309-1708(96)00003-6 0309-l + 708/96/S 15.00 0.00

Comparison of solution approaches for the two-domain model of nonequilibrium

transport in porous media

Claudio Gallo 81 Claudio Paniconi Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna, Cagliari, Italy

Giuseppe Gambolati Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universith di Padova, Padua, Italy

(Received 20 June 1995; accepted 18 December 1995)

T:he two-domain concept is widely used in modelling transport in heterogeneous porous media and transport of rate-limited sorbing contaminants. When a first- order kinetic relationship is used to represent the transfer of mass between domains, the model can be expressed as a modified advection-dispersion equation describing general transport coupled to a first-order ordinary differential equation accounting for mass transfer. Different approaches can be used to solve the resulting system, including: simultaneously solving the coupled transport and kinetic equations; discretising and algebraically solving the mass transfer equation and substituting it into the transport equation; solving the mass transfer equation analytically and substituting the integral solution into the transport equation to obtain a single integro-differential equation; and solving the system in Laplace space. and back-transforming the solution into the time domain. These four approaches - coupled, algebraic substitution, integro-differential, and finite element Laplace transform (FELT) - are evaluated on the basis of their general features and on their performance in two test cases. The results indicate that the algebraic substitution approach is robust and, on scalar computers, very efficient. The FELT approach is easily parallelised and achieves good speed-up on supercomputers, but the method is restricted to time-invariant velocity and saturation fields, and is only useful for obtaining the solution at or not too far freom the maximum simulation time. The integro-differential method is as efficient as. but less robust than the algebraic substitution approach, requiring a small time step size when the mass transfer coefficient is very large. Finally, the coupled approach is robust and flexible, but requires the solution of a system of equations twice as large as the other methods. On balance, the algebraic substitution and, to a lesser extent, the integro-differential methods appear to be the most attractive approaches on scalar machines while FELT, when applicable, is an appealing alternative for coarse-grained multiprocessors. Copyright 0 1996 Elsevier Science Limited

1 INTRODUCTION

It is now recognised that there are many situations where the classical advection-dispersion equation cannot adequately describe the transport of contami- nants in porous media. Two of the most important factors responsible for ‘:nonideal’ transport are physical and chemical nonequilibrium phenomena related to heterogeneity and rate-limited sorption, respectively.435

A contaminant moving through a heterogeneous porous medium and a solute undergoing rate-limited sorption can exhibit similar behaviour (asymmetrical break- through profiles, tailing, apparent enhanced dispersion), and indeed the ‘dual-porosity’ model for physical nonequilibrium can be shown to be mathematically equivalent to the chemical ‘two-site’ modeL2’ This class of models has also been referred to in the literature as two-region, bicontinuum, first-order mass transfer,

241

242 C. Gallo et al.

mobile-immobile, dual-porosity, and two-domain, and in this paper we will use the latter term. Examples of the use of the two-domain concept to model transport in fractured, macroporous, aggregated, or stratified media can be found in Sudicky,22 Brusseau3 Gerke and van Genuchten13 and Li et al I9 while examples of the two- domain model applied to nonequilibrium sorption include Cameron and Klute,6 Goltz and 0xley,14 Thorbjarnar- son and Mackay23 and Haggerty and Gorelick.i6

Mathematically, the two-domain model consists of a coupled system of partial differential equations, with the precise form of the equations depending on the processes and level of detail being considered. We will use the mobile-immobile representation of van Genuchten and Wierenga,25 which results in a modified advection- dispersion equation coupled to a first-order mass transfer equation. This representation is widely used, and lends itself to a range of straightforward, efficient, or otherwise attractive numerical solution approaches.

The most obvious approach is to discretise and solve the coupled system of transport and kinetic equations directly. More efficient procedures can be obtained by decoupling the system, and three methods of achieving this will be described.

In the first decoupling method, the mass transfer equation is discretised in time and algebraically manipu- lated to obtain an expression for the immobile region concentrations at the current time level. This expression is substituted into the time-weighted transport equation, which is then spatially discretised and numerically solved for the mobile region concentrations at the current time level. The current immobile region concen- trations are then determined explicitly from the alge- braic expression.‘118 In the second approach, the mass transfer equation is solved analytically and its solution is substituted into the transport equation to obtain a single integro-differential equation that can be discretised and solved numerically for the mobile region concentra- tions.’ In the third decoupling method, a Laplace trans- formation is applied to the integro-differential equation, eliminating the time variable and exploiting the simple form of the convolution integral in Laplace space. The transformed equation is discretised and solved numeri- cally, and an efficient inversion algorithm is applied to back-transform the solution into the time domain.22

In this paper the coupled and the three decoupled solution approaches are presented and compared. We consider the general features of each approach and, based on these features and on results from two test cases, we assess their reliability, efficiency, and limitations.

2 THE TRANSPORT MODEL AND FOUR SOLUTION APPROACHES

In this paper the conceptualization of van Genuchten and Wierenga25 is used for the first-order kinetics

representation of the two-domain model. This model describes nonequilibrium contaminant transport in a variably saturated, aggregated porous medium where the saturated pore space is subdivided into a mobile water region and an immobile water region. Fluid flow and advective and dispersive solute transport occur in the mobile region only, and the exchange of solute between the mobile and immobile regions is controlled by a diffusive mechanism. The model is further enhanced by introducing linear equilibrium sorption and a first-order transformation reaction (biodegrada- tion or radioactive decay) in both the mobile and immobile regions. Under these assumptions, the general equations describing the two-domain model are”

+ A( Tm Cm + Tim Cim) + q(Cm - C*) -f

T, dcim - = (Y(Cm - Cim) - XTimCim ‘In dt

(14

(lb) where Xi is the ith Cartesian coordinate, t is time, c, and q,,, are the concentrations of the dissolved constituent in the mobile and immobile water regions respectively, X is the linear decay constant, (Y is the mass transfer coefficient for the diffusion process between the mobile and immobile water regions, q represents distributed source/sink terms (volumetric flow rate per unit volume), C* is the concentration of the solute injected or withdrawn with the fluid source or sink, and f is the distributed mass rate of the solute per unit volume. T,,, and Tim are modified retardation coefficients defined as T, = TV, SW, + ys( 1 - n, - nim)J’kd, and Tim = ni, + Ys(l - nm - nim)( 1 - F)kd,, where n, is the porosity of the mobile region (ratio of the volume of voids in the mobile region to the total volume), ni, is porosity of the immobile region, SW, is the water saturation in the mobile region, bus is the density of the solid grains, F is fraction of sorption sites in direct contact with the mobile water, and kd, and kdi, are the distribution coefficients in the linear Freundlich isotherms describing the instantaneous sorption in the mobile and immobile regions, respectively. The dispersion coefficient in eqn (la) is written as Dij = &TIWlSij + (QL - aT)viVj/lUl+ n, Swm DOTS,, where ‘YL and or are the longitudinal and transverse dispersivities, respectively, Vi is the Darcy velocity, I v I = e for a two-dimensional medium, S, is the Kronecker delta, Do is the molecular diffusion coefficient, and r is the tortuosity. Parameter (Y provides an indication of how close to equilibrium the system is. As (Y + 00, the mass exchange between the mobile and the immobile regions becomes instantaneous, and the transport model reduces to the classical advection- dispersion equation with a total porosity n = n, + ni,,” As a + 0, mass exchange becomes so slow that the model again reduces to the advection-dispersion equation, in this case with a total porosity IZ = n,.

Solution approaches for the two-domain transport model 243

2.1 Coupled approach

Using (1 b), we can write (la) as

+ XT,c, + q(c,, - c*) -f (2)

In the coupled approach, both eqns (2) and (lb) are integrated by finite elements in space and finite differences in time, using the Galerkin formulation and a weighted time stlepping scheme. Denoting by c, and q,,, the vectors containing the unknown mobile and immobile concentrations at each of the N nodes of the finite element mesh, the following algebraic system is obtained”

v&4+B+E+F):)1:+“1 +&Gk+vl c;+‘-yRCik,f’ 1

-!- GkfY1 -- vll (A + B + l? + F)k+YI I c;

+ v, 1 Rqk, - I.*~~+“~ Pa)

-&G* + u2R* cik,” - v2Rc~' I [ 1 G* - v;,~R* ct + v~~Rc;

= Atk 1 Pb)

where vI and ~2 (0 < pl, y 5 1) are weighting factors, VI] = 1 - 1/l, u22 = 1 - u2, k indicates time level, A, B, and G are the stiffness, aidvection, and capacity matrices, respectively, & F;, R, and R* are capacity-type matrices arising from the c, and qrn terms on the right-hand side of eqns (2) and (1 b) and from the advective component of Cauchy boundary (conditions, G’ is the capacity matrix for eqn (lb), and r* contains source/sink terms, Neumann boundary conditions, and the total solute flux across the Cauchy boundary.

The simultaneous solution of eqns (3a) and (3b) to obtain the 2N mobile iand immobile nodal concentra- tions can be costly in terms of storage and CPU, and in the following three solution approaches, an algebraic, integration, or transformation procedure is applied to reduce the size of the system to be solved, or to avoid having to solve it at each time step. This will result in a gain in efficiency, but ii should be noted that the three decoupled approaches may not be as generally applic- able as the coupled method to extensions, for instance to nonlinear equilibrium isotherms, of the two-domain model (1). Two of the decoupled methods will also be shown to have numerical1 limitations not associated with the coupled approach.

The time truncation error EC associated with the coupled scheme can be easily shown to be

where T represents T,,, or Tim and c represents c, or Cim. The order of accuracy of the coupled approach is therefore one for v # l/2 and two for v = l/2 (Crank- Nicolson scheme).

2.2 Algebraic substitution approach

Following Leismann et a1.,18 we first apply a weighted time difference approximation to the coupled eqns (2) and (1 b). For (1 b) we obtain”

cFL+’ = Timqi + aAtk[yci" + vZ~C~] Irn

v2Atk(a +XTi:,,)+ Tim k Atk(a + XTim)V22Cim

- y Atk(, + ATim) + Tim (5)

which is substituted for 8’ in the discretized form of (2). Integrating the equai:n in space by the Galerkin method gives a decoupled system in the N unknown

k+l. mobile region concentrations c, .

zq(A + B + i? + i)k+vl + & ek+“l - u2E* c;+’ 1 v&~+B+~+~)~+“’ -vZ2E* 1

x c- + E**qk. _ y*ik+y (6) where E * and E ** are capacity-type matrices involving the terms Ya2Atk/6 and a[Ti, + At:(a + ATim) x p - zq)/S], respectively, with S = y At (a + ATim) +

IIn. The two-domain model has been decoupled, and the

system to be solved numerically has been reduced to size N. Once ck” is known, cik,” can be computed explicitly at each node from the algebraic eqn (5).

The time truncation error EA of the algebraic substitution approach is the same as that of the coupled scheme, given in eqn (4). Hence the algebraic substitu- tion approach also becomes second order (in time) when v = l/2, and is expected to yield results with the same numerical accuracy as the coupled approach.

2.3 Integrodifferential approach

Integrating eqn (lb) analytically, assuming qrn = 0 at t = 0, and substituting the result into (la) leads to an integro-differential equation for the mobile region concentration’

= T, 2 + (a + XT, + q)c, - (qc* +f)

f - &p ,-(B+W f .I

,@+4 7 cm(~)d~

0 (7)

244 C. Gallo et al.

where /3 = a/Tim. The above equation is integrated in space by the Galerkin approach. The time derivative is approximated by a weighted (0 5 v < 1) time stepping scheme. The convolution integral is of a special form and can be efficiently executed, applying either the mean value theorem or the trapezoidal rule at each time step, and cumulated without saving the past history of cm(t).9 The integration error is consistent with the error of the time marching scheme used to discretise the term dc,/dt. In the test cases for this work we used the mean value theorem, in which case the following term is added to the left-hand side coefficient matrix v(A +,B_+ fi+P)k+” + Gk+“/Atk: Mi = - v@[l - e-(8+x)A’ ]S/ (2p + 2X), where 3 is the basic capacity matrix for transport problems, formed from the elemental integrals of the basis function products.

It can be shown that the additional terms introduced by the integro-differential formulation, in particular when the mean value theorem is used to execute the convolution integral, increase the importance of the diagonal coefficients of the left-hand side system matrix compared to the standard advection-dispersion equa- tion9 thereby accelerating convergence of conjugate gradient-like solvers. In other words, the integro- differential approach is particularly suited to enhance the performance of iterative solvers.i2

To compute the time truncation error when the convolution integral is executed by the mean value theorem, consider the integral in eqn (7). During the discrete time interval At this is calculated by the formula

s t+Ar eoTcm(T) dr M cm

t+*’ + c- e@(eo*t _ 1)

2 * P (8)

t

It can be shown that the error Ek associated with eqn (8) is

At time t the cumulative error EI in the execution of the convolution integral is

Hence eqn (10) represents a second order scheme, the same as the coupled and algebraic substitution approaches with v = l/2. It is interesting to note the difference between EC (for the coupled and algebraic substitution schemes) and EI. As ,L? grows large (i.e. when (Y + 00) the coefficient before At2 in eqn (10) also grows and tends to infinity if p + 00, while EC remains unaffected by the behaviour of p. This accounts for the inability of the integro-differential approach to repro- duce the limiting case of the two-domain model when

the mass transfer coefficient goes to infinity, i.e. when eqns (la) and (lb) reduce to the classical advection- dispersion equation with cm = cim. When (Y is very large, the time step At must be kept sufficiently small to reduce the magnitude of EI, which grows as cr does. As a rule of thumb, we have found that &At should not exceed 0.02.9

Although the integro-differential method decouples the system, producing a comparable storage and CPU saving to the algebraic substitution approach, it is not as robust as the latter scheme, as apparent from the truncation error analysis. However, for realistic (Y values that are characteristic of nonequilibrium transport, the integro-differential method should be superior to the coupled approach, and should compare favourably with the algebraic substitution method.

2.4 Finite element Laplace transform (FELT) approach

An interesting alternative to the previous time marching schemes is the FELT approach, first applied to the two- domain model of solute transport.22 The procedure consists of three successive steps: (1) application of the Laplace transform to (7) and (lb); (2) finite element discretization and solution of the transformed equation to obtain the mobile and immobile concentrations in Laplace space; (3) antitransformation (inversion) of the solution in the Laplace domain to obtain the solute concentrations in the time domain.

The Laplace transformation of a function f( t ), t 2 0, is defined as17

WWI =_?(P) = 1: f(t) ewpt dt (11)

where p is the Laplace transform parameter. Two useful results of the Laplace transformation are

Nfldtl = ~7 -f(O)

and

Applying (1 l), (12) and (13) to eqn (7) yields

a2z = LTm(P + x) + alcm - cm(p +$ + cr

+ q(Cm - i?*) -f- TmCm(Xi, t = 0)

The Laplace transformation of eqn (1 b) is

l?im = acrn

Tim(P + x) + a

The boundary conditions for the transport model are also easily transformed into the Laplace domain, or p-space.22 Equation (14) together with the transformed

(12)

(13)

(14)

(15)

Solution approaches for the two-domain transport model 245

boundary conditions represents a partial differential equation in p-space witlh the time variable eliminated. A solution of this equation at each of the complex p values indicated in (1:7) (see below) is needed in the antitransformation step. Applying a Galerkin finite element spatial discretisation to (14), and denoting by &a and ?,,t the real and imaginary parts of the vector of unknown transformed concentrations E,, we obtain

where GR = A + B + & + S,, S, and Si are the real and imaginary parts of a capacity-type matrix arising from the terms involving p in eqn (14), and bR and bt are the real and imaginary parts of the right-hand side vector containing boundary conditions and source/sink terms.

For the antitransformation step, we can express the Laplace transform para.meter p as7l22

p =pk =po +ikr/T, k=0,1,2,... (17) where i = a, T = 0.8t,,,, t,,, is the simulation period, and p. = -In (_E)/1.6t,,, with E in the range 10e4 to 10e6. The epsilon inversion algorithm of Grump,’ as refined by de Hoog et aL8 is used for the antitransformation. Denoting by cm,k,j the transformed mobile region concernration at node j for pk, the concentration at time t is given by

(18)

where Re and Im denote the real and imaginary parts of the i?,,,k,j values. Equation (18) represents a truncated Fourier series, and theoretically the truncation error goes to zero as M goes to infinity. In practice, M is chosen in the range 5 to 25 since discretization and machine roundoff errors become dominant over the truncation error for much larger values of M.8

Because of the nature of the Laplace transform, computing an accurate solution at increasingly small time values for a given tmax requires, theoretically, an increasing number of inversion terms. However, due to the limitations of Laplace inversion algorithms for very large M, it has been noted that the FELT approach should not be used to compute solutions for t < tm,,.2’ In our simulations we have observed that, as a rule of thumb, t > 0.1 t,,, shou.ld be used to avoid oscillations and significant inversion errors. This limits the applic- ability of the FELT approach to cases where the time history of a contaminant’s behavior is not of interest, but only its plume at a limited number of time values not too far from the simulation period. To obtain a solution at small t, tmx needs to be reduced. To obtain the solution at a large number of time values incurs the cost of repeated application of the inversion algorithm.

Other limitations of the FELT technique are that time-varying velocity and saturation fields are not handled, and nonlinear phenomena (such as nonlinear sorption isotherms) can be treated only in special cases, depending on the functional form of the nonlinearity. To handle time-varying velocities, for example, would require either explicit knowledge of the functions Vi = vi (xi, t ), which are in general not available in analytical form, or numerical integration of the term Jo00 e-rrzti (ac/axi) dt, which is quite costly to obtain.

One of the main advantages of the FELT approach over time marching schemes is that a solution at time t (not too far from t,,) can be computed directly without requiring the solution at previous times. Another advantage is that solving a small number of independent systems (eqn (16) for each pk, k = 0, 1, . . . ,2M) sufkes to compute all nodal concentrations at a given time t. This makes FELT very attractive for implementation on parallel computers.

3 NUMERICAL TESTS

Two test cases will be used to evaluate the accuracy, memory requirements, and computational efficiency of the coupled, algebraic substitution, integro-differential, and FELT approaches, examining the effects of grid Peclet number and heterogeneity in saturation and velocity fields, and the performance of the linear solver. We will also make some comments about the parallelisation of the FELT approach.

A comparison of numerical approaches based on test case results is not a substitute for rigorous mathematical analysis, although in many cases evaluations based on well-designed test cases serve to bring out physical insights into numerical behaviour which may escape or be too complex for theoretical analysis.

The sparse nonsymmetric linear systems resulting from discretisation are solved using the biconjugate gradient stabilised algorithm (Bi-CGSTAB) with incom- plete Crout (LU) decomposition as a preconditioner,24 and with the product of (X7)-’ and the right-hand side vector as solution estimate at the start of each new time step. All numerical simulations are run on an IBM RS6000/560 workstation, with the exception of the parallelised FELT simulations, which are run on a 32- node IBM SPl. For both test cases, DO, F, kd,, kdi,, and X are set to zero.

3.1 Test case 1: transport in a twodimensional soil

We consider nonequilibrium transport in a 100 x 1OOm region with a homogeneous velocity and saturation field (vX = 0*05m/day, w, = 0, Swm = O-8). A pulse condition is specified at the inflow boundary x = 0, with c, = 1 kg/m3 for 0 I t 5 400days and c, = 0 for

246 C. Gallo et al.

Table 1. Parameter values for test cases 1 and 2

Parameter Test Case 1 Test Case 2

Zone 1 Zone 2

At, d [l.O, lOO.O] 1.0 1.0 a~, m [0~025,10~0] 0.005 0.002 aT, m [0~025,10~0] 0.002 0.001 n, 0.2 0.2 0.2 ni, 0.2 0.1 0.3 a, d-’ 5.0e-3 1 .Oe-05 50e-05

t > 400days. On the remaining boundaries a zero- flux Neumann condition is prescribed, and the initial condition is c,,, = ci, = 0 throughout the region. Additional parameter values for this test case are given in Table 1.

To compare the accuracy of the four solution approaches, surrogate exact solutions S are computed using the coupled method, a fine grid discretisation of Ax = AZ = 1 m, and a small step size At = 1 day. The maximum Peclet value for the surrogate exact solutions is 10, and the solutions were carefully checked to ensure that no oscillations were present. The numerical solu- tions for the four approaches are computed using a discretisation of Ax = AZ = IOm, or N = 121 nodes, and the simulations are run to t,,, = 6000days. For each of the three time marching approaches, simulations are run using fixed time step sizes of 1, 3, 10, 30, and 100 days and a grid Peclet number Pe over the range [ 1 , 1001

1.0

O.Q-

0.8-

Pe = Ax/a,

XY. :* IL-.- :oo

(Pe = AX/CQ, with a longitudinal dispersivity range of [O. 1 , lO*O] m). For the FELT approach, simulations are run using M = 7,9, 15,20, and 25, and the same Peclet range as for the time marching schemes. The break- through curves for the Pe = 1, 2, 10, and 100 cases are shown in Fig. 1.

The maximum deviation or error

t = At, 2At,. . . , tmax

is calculated for each run, and is plotted in Fig. 2 as a function of Pe and At for the time marching schemes and as a function of Pe and M for the FELT method. The error (already normalised since max, i(x = 30, z = 30, t) = 1 for this test case) was computed at a point far enough inside the domain to be relatively insensitive to boundary effects. Moreover, for all simulations, the concentration wavefronts passed through this point.

For the FELT approach a ‘At’ of 10 days is used to calculate S, beginning at t = 600. That is, for each run the antitransformation eqn (18) is evaluated at t = 600, 610,620,. . . (6000. When 6 was also calculated using t < 600, the maximum FELT deviations behaved erratically and were up to 50 times larger than the time marching errors, thus supporting the rule of thumb t > O.lt,,, mentioned in the previous section. This adaptation of the 6 measure, which is best suited for

1.0

0.9

0.8

0.7

0.6

0.5 2

0 500 1000 1500 2000 o:J 500 1000 1500 2000

time [d] 5 time [d]

Fig. 1. Breakthrough curves for test case 1 at different values of Peclet number Pe for the four solution approaches: (a) time marching schemes with At = 10 days (identical results for all three schemes); (b) FELT approach with 2M = 50.

Solution approaches for the two-domain transport model 247

10.0~. a) At WI -1 . . . 3 -*- 10 - - 30 -*a* 100

‘o l.O- .*._..._...-...-...- . ..-...-. ..-...-. .._..._.I

***_...C...-*.*- 0.1-U _ _ ------______________----

4 O.Ol- I

1 10 100 Pe = AZ/a,

lO.O- b, avl

- 14 . . . . . . . . 18 __ __ 30 ---40 -..- 50

~ 1.0~.

I

10 PO = Ax/a,

Fig. 2. Accuracy of the solution for test case 1 expressed as the maximum deviation 6 vs Peclet number Pe for the four solution approaches: (a) time marching schemes at various values of At (identical results for all three schemes); (b) FELT approach at

various values of M.

time marching schemes, provides a consistent basis for comparing all four sohrtion approaches, even though in practice the maximum error for the FELT method was observed to occur close to the smallest time 0.1 tmax, making it not strictly necessary to compute the error at much later times.

The three time marching schemes show exactly the same accuracy behaviour, which is plotted in Fig. 2a. As expected, the accuracy decreases with increasing time step size. On the other hand, 6 increases only slightly with increasing Peclet number. The accuracy of the FELT approach is comparable to the small-At time marching runs, and the influence of M is negligible except for M = 7. For M = 7 the maximum deviation occurs closer to t = 600 than for the other M values, in accordance with the e,arlier remark that a larger M is required to compute a:n accurate solution at small time values.

In the above series of simulations we also observed the performance of the Bi-CGSTAB solver. The convergence behaviour of the three time marching schemes is identical, and is shown in Fig. 3a. The grid Peclet number has very little effect on the linear solver, with only a very slight increase in the number of iterations to convergence for highly advection domi- nated problems. The size of the time step also has very

little effect, and for all the time marching runs the linear solver converges in fewer than 10 iterations. For the FELT approach, Fig. 3b shows the number of iterations to convergence for each of the k = 0, 1, . . . ,2M linear systems to be solved in p-space, with 2M = 80 in this case. Each curve plotted is for a different Peclet number, ranging from 1 to 400. In contrast to the time marching schemes, the convergence of the linear solver is greatly affected by the Peclet number, becoming erratic, and exceeding 10,000 iterations for some k values, when the problem is highly advection dominated. We also observe from Fig. 3b that the erratic behaviour of the linear solver at high Peclet numbers occurs for the first 20 or 30 k values, with convergence becoming faster and more regular as k increases. Although this suggests that there is little extra linear solver cost incurred in using very large values of M, hardly any benefit is to be expected from doing so since, as mentioned previously, the inversion algorithm is prone to significant roundoff errors for very large M.

An indication of the computational efficiency of the four solution approaches is shown in Fig. 4, where for a fixed At, Pe, and A4 (10 days, 2, and 15 respectively) and t,,, = 3000days, each method is run for a series of grid discretisations, from N = 121 to N = 40,401 (Ax = AZ = 0.5 m). For these values of At,

248 C. Gallo et al.

0 1 I I I I I I I I I

1 2 4 7 10 20 40 70 100 200 400 Pe = Ax/a,

.

Pe = Ax/aL -_ 1 -..... 2 ??*lo -a- 50 --- 100 . . . . . . 200

- 400

0 10 20 30 40 k

50 60 70 80

Fig. 3. Test case 1 linear solver performance for the four solution approaches. Number of iterations to convergence is plotted: (a) as a function of Pe for the time marching schemes at various values of At (identical results for all three schemes); (b) as a function of k (k=O,l,..., 2M) for the FELT approach at various values of Pe. In (a) the number of iterations is the average over all time steps.

Pe, and A4 the accuracy of the four solution approaches is comparable, as shown in Fig. 2, so under these conditions we should obtain objective comparisons of computational efficiency. Pe = 2 is also selected in order to avoid erratic behaviour of the linear solver in FELT (see Fig. 3b). For these runs we use tmax = 30OOdays, and 10 antitransformations for the FELT approach. From Fig. 4 we can conclude that the algebraic sub- stitution method is slightly more efficient than the integro-differential method, while each of these is about

10’ 3 R I

/’

/ .’ ;;:.* - -e- Coupled

4’ _:P

1

;;fl:iT .*@- Alg. Sub. --I+- lnt. Diff. -Q-. FELT (2M=30)

Fig. 4. Test case 1 CPU time vs grid discretisation for the four solution approaches.

2.5 times faster than the coupled method. The curve for the FELT approach falls somewhere in between the algebraic substitution and coupled methods, and we can conclude that, for a reasonably small number of antitransformations, FELT is about 60% slower than the algebraic substitution method.

The memory requirements for the previous set of runs (with FELT simulations for A4 = 25 added) are given in Table 2. The algebraic substitution method is the most efficient in terms of storage, requiring slightly less memory than the integro-differential method. For problem sizes larger than about 1000, the coupled and FELT approaches require more than twice as much memory as the other two approaches. The memory requirement for the FELT approach increases with the number of p values used in the computation of the solution, being about 13% greater for A4 = 25 than for A4 = 15. FELT can be made more memory-efficient by storing the Laplace concentration values in a file and post-processing them to yield the time-domain concen- trations, at the expense of the added I/O costs associated with dumping and retrieving large amounts of dam from disk.

The breakdown of the total CPU cost is given in Table 3 for the runs using the finest mesh. For all methods the cost of solving the linear system of

Solution approaches for the two-domain transport model

Table 2. Memory requirement (MBytes) for test case 1

249

Problem sire Coupled Algebraic substitution Integro-differential FELT (2M = 30) FELT (2M = 50)

121 0.46 0.43 044 0.45 060 961 2.5 1.3 1.4 2.4 2.7 3721 9.3 4.5 4.6 9.1 10.3 10201 25.3 11.7 12.0 25.0 28.4 22 801 56.6 25.7 26.1 55.6 62.8 40401 100.3 45.0 46.0 93.0 110.0

Table 3. CPU breakdown (%) for test case 1

Code module Coupled Algebraic substitution Integro-differential FELT (2M = 30) FELT (2M = 50) Data input and initialisation 1.25 1.69 1.55 1.56 0.96 System assembly 13.32 26.72 29.10 1.11 1.61 Bight-hand side assembllr 0.93 1.11 6.37 0.03 0.06 Linear solver 83.16 64.05 58.77 76.2 70.12 Antitransformation 20.8 27.12 Other 1.34 6.43 4.21 0.3 0.13

equations is dominant, consuming the largest fraction of CPU in the case of .the coupled approach (more than 80%). The cost of assembling the system matrices is most significant for the algebraic substitution and integro-differential approaches (25-30%), whereas for the FELT approach an. equal fraction of CPU (25-30%) is instead used for thte 10 antitransformations of the concentrations. We note that as A4 increases, the ratio of antitransformation CPU to linear solver CPU increases. This is due to the fact that the conjugate gradient-like Bi-CGSTAB solver is superlinearly convergent, while the epsilon algorithm for the back-transformation is quadratic and thus becomes quite inefficient for large M.sY1’ System assembty is done at every time step for the time marching schemes, so the CPU costs are repre- sentative of more general problems with non-steady saturation and velocity fields.

3.2 Test case 2: transport in a two-dimensional heterogeneous aquifer

This test case is adapted from Gureghian,” and involves transport in a heterogeneous ditch-drained aquifer with incident steady rainfall and trickle infiltration of a reactive solute that undergoes nonequilibrium sorption. Figure 5 shows the vertical cross section of the left half of the aquifer, where the boundary x = 100 cm (line BC), located midway between two equally spaced ditches, corresponds to a line of symmetry. The soil surface is subjected to steady rainfall, with additional trickle infiltration on the retention pond represented by line ED, and the steady state saturation and velocity distributions obtained by solving the corresponding flow problem” are used as input for the transport simula- tions. The trickle infiltration on ED carries solute into

Unsaturated zone

fi B

Satumted zone zone 1

/I/////// I,, , , , , , , , , , , , , , , , , , , , ,

! -

I

5 2:

Fig. 5. Sketch of the problem configuration for test case 2.

250 C. Gallo et al.

the aquifer for a period of 15 days, which is represented by the Dirichlet boundary condition c, = 1 for t < 15, c m = 0 for t > 15. On the remaining boundaries a zero- flux Neumann condition is prescribed, and the initial condition is c, = Cim = 0 throughout. The heterogeneity of the aquifer stems from the 32 x 1Ocm slab, indicated in Fig. 5 as zone 2, that has different hydrogeologic characteristics from the rest of the aquifer (zone 1). The parameter values for the two zones are given in Table 1, along with other parameter values for this test case. The aquifer is discretised into N = 15 275 nodes and 29 952 elements, using a variable Ax and a fixed AZ.

Figure 6 shows the vertical concentration profiles along x = 60 cm at t = 3 days (a) and t = 25 days (b) for 25 and 120-day simulations. In these profiles we verify the expected difficulties that the FELT approach has in computing accurate solutions for t < t,,,. While the three time marching schemes produce smooth and consistent concentration profiles, the FELT solution at t = 3 is oscillatory for the M = 15, t,,, = 120 run. Refining the solution with M = 25 successfully reduces the oscillations, as does lowering the value of zmax to 25. For the larger time value, the M = 15 run does not produce oscillations. To obtain an accurate and efficient FELT solution at small time values such as t = 3, it is more practical to reduce t,,, than to keep tmax at 120 while increasing M, due to limitations in the inversion algorithm and the increase in CPU cost for large M.

In Fig. 6, for both t = 3 and t = 25, the FELT and time marching solutions are not identical, regardless of

1.0

0.8

0.6

0”

0.4

i

0.2

0.0

I

t max and the level of refinement used. As a possible explanation for this discrepancy, we note that in the Laplace inversion algorithm (eqn (18)), 2T (T = 0.8t,,,) is the period of the Fourier series approxi- mating the inverse function on the interval [0,2T].*’ Therefore, although the concentration profiles do not reach zone 2 by time 3 days and time 25 days, at the values of tmax used in the FELT simulations, 25 and 120 days, the concentration profiles most certainly cross zone 2. Thus, while the concentration profiles shown for the time marching schemes represent effectively the response to a ‘homogeneous’ domain, the FELT approach, on the other hand, is affected by the heterogeneity.

In Fig. 7 the performance of the Bi-CGSTAB solver for the FELT approach is again examined. As in Fig. 3b, the number of iterations to convergence is plotted for eachofthek=O,l,... ,2M linear systems to be solved in p-space, with 2M = 30 in this case. The runs shown are for the homogeneous case, with the parameter values for zone 2 equal to those given in Table 1 for zone 1. Each curve plotted is for a different t,,,, and we see that the simulation period has a pronounced effect on convergence of the linear solver, with a lo-fold increase in number of iterations between the lowest and highest t max for some of the k values. We also observe a trend that suggests that the k value at which the number of iterations starts increasing rapidly will be dependent on t max 3 with lower k (and hence M) values required for small tmax in order to avoid erratic behaviour of the linear solver.

-*- Coupled - - - Algebraic substitution -....- Integro-differential - FELT 2M=30, t,,, = 120 d -- -- FELT 2M=50, t,,, = 120 d . ?? FELT 2M=lO, t,,, =25 d

0 1 2 v 4 5 60 5 10 15 20

2 b4 = [cm1

,l.O

,o.a

,0.6

Fig. 6. Test case 2 mobile region concentration as a function of depth z at (a) x = 60 cm, t = 3 days and (b) x = 60 cm, t = 25 days.

Solution approaches for the two-domain transport model 251

10'

E .C! 10'

____20 _.- 50 % - 1000 b ;ri z lo2

iii $ 10'

loo 0 5 10 15 20 25 30

k

Fig. 7. Test case 2 linear solver performance for the FELT approach. Number of iterations to convergence is plotted as a function of k (k = 0, 1, . . . ,2M) at various values of t,,,.

3.3 Parallel performance of the FELT approach

Implicit time marching procedures are intrinsically serial due to the dependence ‘of the solution at time level k + 1 to the solution at level k, and the parallelisation of such methods is thus not a straightforward task.2 As indicated in Table 3, solving the sparse linear systems of equations that result from numerical discretisation is the most expensive task for all four solution approaches considered in this paper. As this is the case for many other numerical differential equation codes, there has been much work in recent years on parallel algorithms, both direct and iterative, for solving large sparse linear systems.2 Since a parallelised linear solver should incur approximately the same advantage to the three time marching schemes as to the FELT approach, we will not consider this aspect he:re. On the other hand, the FELT approach has a unique, and immediately parallelisable, property: the independence of the 2M + 1 linear systems to be solved in p-space. Another feature of the FELT method which can be easily exploited on a multi- processor is that the antitransformation step can be executed concurrently for each time t at which a solution is desired, or for each nodei of the spatial discretisation.

Parallelisation of this feature will not have as big an impact as concurrently solving the 2M + 1 systems, since, as shown in Table 3, the linear solver takes about 3 times more CPU than the inversion step.

Figure 8 shows the performance of the FELT method on the IBM SPl parallel computer, where the imple- mentation includes concurrent solution of 31 (M = 15) linear systems and concurrent antitransformation at 10 time values, Both homogeneous and heterogeneous runs are shown (for the homogeneous runs the parameter values for zone 2 are equal to those given in Table 1 for zone l), with not much difference in performance for the 2 cases. In Fig. 8, the speed-up Sr is the ratio of the CPU time to run the FELT code in serial mode (on one processor) to the CPU to run the code in parallel on P processors. Runs using P = 1, 2, 4, 6, 8, and 10 processors are shown. The parallel efficiency is defined as rip = Sp/P. We achieve a respectable speed-up and efficiency, reaching Sr = 5.1 (np just over 50%) with 10 processors. We would expect even better speed-up and efficiency for problems larger than the N = 15 275 case used here, since overhead and communication costs would diminish with respect to the total computational cost.

40.1 I I I I I I I I ) 0

1 2 3 4 5 6 7 8 9 10 number of processors (P)

Fig. 8. Test case 2 performance of the FELT approach on a parallel computer for homogeneous (solid line) and heterogeneous (dashed line) cases.

252 C. Gallo et al.

Although straightforward to implement and quite efficient in terms of CPU, the parallelisation of the FELT approach described here has three potential drawbacks. One is that solving a linear system of size N on each of P processors requires much more distributed memory than does, for instance, decom- posing a large system into P smaller parts. A second problem is that the concurrencies in the FELT algo- rithm (in p-space for the linear solver step and in either time or node space for the antitransformation step) are not consistent with each other, entailing large communi- cation or data exchange costs. Finally, parallelisation of the linear solver step is appealing for coarse-grained architectures, given the practical limit on M (~25) described earlier, but would not be efficient on fine- grained supercomputers, where the number of processors is on the order of hundreds or thousands.

4 CONCLUSIONS

We have examined four solution approaches for the two-domain model of physical and chemical non- equilibrium transport phenomena. The coupled approach is inefficient but was useful in providing the base case solutions needed to compare the performance of the decoupled methods. Since the coupled approach involves a straightforward numerical discretisation of the two-domain system of equations, any extensions to the model can be easily incorporated, such as nonlinear equilibrium sorption isotherms, multiple contaminants, multiple domains, and more complex chemical and transformation reactions. For these more comprehen- sive models, other solution strategies can be used to avoid the inefficiencies of the coupled approach, in particular methods based on sequential iteration.26

The two time marching decoupled methods are comparable in efficiency, but the integro-differential approach requires small time step sizes for problems with very rapid mass transfer between the mobile and immobile regions (i.e. large (Y values). On the other hand, the integro-differential approach has been shown to enhance the performance of iterative solvers for the sparse nonsymmetric linear systems resulting from discretisation of the transport equation, and could be a cost-effective method for realistic mass exchange rates characteristic of nonequilibrium transport, and for problems with At restrictions dictated by the Courant number criterion, for instance.

The finite element Laplace transform method requires more than twice as much memory as the other two decoupled methods, but runs very efficiently on coarse- grained multiprocessors. This method is limited to problems where the solution is required at only a small number of time values not too far from the maximum time tmax and does not allow time-varying velocity and saturation fields. Moreover, the FELT

method does not appear to be any less sensitive to grid Peclet number than time marching schemes, producing erratic and show linear solver convergence at high Peclet numbers. Increasing tmax also caused a deterioration in the performance of the iterative linear solver.

ACKNOWLEDGMENTS

This work has been supported by the Italian CNR (Gruppo Nazionale per la Difesa dalle Catastrofi Idrogeologiche, linea di Ricerca n. 4), by Fondi Ministeriali 40%, and by the Sardinia Regional Author- ities. We wish to thank the anonymous reviewers for their helpful comments.

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