Alternative split-operator approach for solving ... - Science Direct

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SO309-1708(96)00002-4

Advances in Water Resources. Vol. 19, No. 5, pp 26-275, 1996 Copyright 0 1996 Else&r Science Limited Printed in Great Britain. All rights reserved 0309-1708/96/$15.00+0.00

Alternative split-operator approach for solving chemical reaction/groundwater transport models D. A. Barry, K. Bajracharya Department of Environmental Engineering, University of Western Australia, Nedlandr, Western Australia 6907

C. T. Miller Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, North Carolina, 27599-7400 USA

(Received 2 January

1995; accepted 26 December 1995)

Various schemes are available to solve coupled transport/reaction mathematical models, one of the most efficient and easy to apply being the two-step splitoperator method in which the transport and reaction steps are performed separately. Operator splitting, however, does not solve exactly the fully coupled numerical model derived from the governing partial differential and algebiaic equations describing the transport and reaction processes. An error, proportional to At (the time step used in the numerical solution) is introduced. Thus, small time steps must be used to ensure that accurate solutions ‘result. An alternative scheme is presented, which iterates to the exact solution of the fully coupled numerical model. The new scheme enables accurate solutions to be calculated more efficiently than the two-step method, while maintaining separation of the transport and reaction steps in the calculations. As in the two-step method, the reaction calculations are performed node-wise throughout the computation grid. However, because the scheme relies on LU factorisation of the coefficient matrix in the transport equation solution, the reaction calculations must be performed in sequence, the sequence order being determined by the ordering of the nodes in the grid. Also, because LU factorisation is used, the scheme is limited to solute transport problems for which LU factorisation is a practical solution method. Copyright 0 1996 Elsevier Science Limited Key worak: numerical algorithm, Nicolson scheme. nonequilibrium transport, fixed-point &ration.

LU factorisation, iterative solution, Crankreactions, equilibrium reactions, multispecies

NOTATION A AL AR

b

bK B c

CF”

exact solution at the i* node, MLP3 ilum numerical solution at the i* node, MLP3 ci concentration at the inlet boundary, MLF3 co iti chemical species concentration, MLP3 ic

representing discretisation of the spatial operators, T-’ I - AAt/ I + AAt/ vector representing discretisation of boundary conditions, MLe3 T-’ average of b over two time steps, MLm3 constant in eqn (29) liquid phase solute concentration, MLm3

matrix

C

vector of solute concentrations numerical solution, MLm3

c*

approximation to c, MLP3 intermediate step in the calculation

C*

cii G G 261

calculated in the

of c, MLm3 immobile region concentration, MLm3 cation exchange capacity, MLe3 number of surface sites

262 D DU E f jfi f j

sX f

‘h i

I j k ‘k, jkb K K Kis 1

L jL m

;c Nx P R Rj S

S

jS

t UA ‘24 ju;

U Uu V X jx

Y Z zi

D. A. Barry et al.

diffusion/dispersion coefficient, L* T-’ matrix whose only non-zero elements are the diagonal elements of U fitting parameter in the sorption isotherm delined by eqn (18), LT-’ sorption isotherm, MLm3 function expressing ju in terms of the individual component concentrations, MLm3 function expressing ‘s in terms of the individual component concentrations, MLW3 j=l , . . . , Nx, mass exchange rate between solution and solid phases, MT-’ Lm3 counter identifying nodes in the discretised transport domain identity matrix counter time level in the numerical solution forward reaction rate, T-’ backward reaction rate, T-’ decay rate, T-’ equilibrium constant for the i* chemical species equilibrium constant for the i* surface adsorption reaction length of flow domain, L unit lower triangular matrix (elements Lij) advection-dispersion operator, T-’ number of bands above and below the diagonal of AL number of nodes in numerical solution number of chemical species number of chemical components counter retardation factor j = 1,2, constants in eqn (29) solid phase concentration, converted to liquid phase concentration units, MLe3 vector of sorbed solute concentrations calculated in the numerical solution, MLm3 ith sorbed phase chemical species concentration (liquid phase units), MLm3 time, T approximation to u, MLe3 total aqueous phase concentration of the j* component, MLm3 transport step result for the jth component at the it’ node, MLe3 upper triangular matrix (elements Uij) U with the diagonal elements replaced by zeroes fluid velocity, LT-’ position, L j* chemical component, ML-3 L-‘f, ML-3 L-‘s, ML-3 i=l , . . . , n, elements of the vector z, MLm3

Greek cx reaction rate, T-’

(oh am

i: a”r Ax ;. 0: &j V

rate coefficient defined below eqn (33), TV1 rate coefficient delined below eqn (33), T-’ rate coefficient in eqn (32), T-’ dimensionless parameter defined below eqn (34) dimensionless parameter defined below eqn (34) temporal discretisation in the numerical solution, T spatial discretisation in the numerical solution, L stopping criterion moisture content of the immobile region moisture content of the mobile region stoichiometric coefficient, i=l ,***, N,, j= 1,***, Nx fitting parameter used in the sorption isotherm deftned by eqn (18)

Other I. 1 Euclidean norm of.

1 INTRODUCTION

The numerical dilhculties related to solving coupling solute transport and reaction problems have been discussed at length previously.23Y33~41~42 A convenient technique that is relatively easy to apply is the splitoperator method, a in which the transport and reaction portions are split. This class of methods was reviewed by Miller and Rabideau.26 An advantage of the splitoperator method is that existing codes for nonreactive solute transport can be used directly. As the chemical reactions are treated separately, they can also be solved independently for either equilibrium or kinetic reactions. For example, in the case of equilibrium reactions, existing geochemical computer packages are often utilised.3 The convenience of operator splitting does, however, come at a price. Consider that the governing partial differential equations describing solute transport and reaction are approximated using some domain-based method, usually tlnite differences or mrite elements. The split-operator method amounts to replacing the finitedifference or finite-element scheme by another approximation. It was shown” for the simple case of linear retardation that additional numerical dispersion is induced by the standard operator-splitting method, the additional numerical dispersion being a constant proportional to At, the time step in the numerical solution. This additional numerical dispersion can be removed in this simple case by adjusting the dispersion coefficient used in the transport equation. In multispecies transport/reaction problems such a convenient correction is not available as the numerical dispersion depends on the solute concentration.7 In this paper a new approach to operator splitting is presented. The new approach solves exactly the numerical model produced by discretisation of the governing partial differential and algebraic equations modelling the

263

Reaction~transpott solutions using operator splitting transport and reaction processes. Previous approaches that also produce such exact solutions are summarized by Yeh and Tripathi.41 These approaches may be categorised as the Direct Substitution Approach (DSA, chemical equilibrium equations are substituted directly into the transport equations) and the Sequential Iteration Approach (SIA). Of these, Yeh and Tripath41 consider SIA as being the best solution method for general application. Essentially,, SIA amounts to treating all reaction terms as source/sinks in the transport equation. The source/sinks, in turn, are described by chemical reaction equations. When solving the transport equation, the source/sink terms are treated as known. The results from the transport model solution are then used in the chemical reaction equations. Iteration is performed between the two sets of equations until convergence is reached. The scheme thus solves the numerical model using tied-point iteration.12 Like DSA, it has the disadvantage that the transport model must be written such that it accounts for the sink/ sourcetirms, i.e. existing codes are subject to substantial rewriting before they can be used. The approach presented below avoids this. Unlike both DSA and SIA which both, in effect, include the chemistry in the transport step of the so:lution, the new approach leaves the transport solution to proceed unchanged. Rather, the chemical reaction calculations are modified so that the original model is solved exactly. That is, the transport is included in the chemistry step of the solution. Several examples are included to demonstrate the new approach.

2 THEORY We begin our exposition of the new algorithm by considering the single-species, one-dimensional reaction/ transport model. The single species case is convenient as it allows the algorithm to be explained in a straightforward manner. The approach taken using this model will subsequently be extended to include multiple species reaction and transport. The governing, single-species advection-dispersion transport model is2

We shall use also the third-type entrance condition14

w,

vco = ~(0, t) - D ax,

40, t) = co,

t > (3

(2)

t > 0.

The soil profile is assumed to be solute free initially, i.e. c(x, 0) = 0,

x > 0.

(4

The solute concentration is bounded for large x,~ giving the second-type boundary condition

ac = 0, 2% x-+00

t>O

(5)

In most numerical methods, eqn (5) is replaced by a condition at some finite distance, I, from the entrance surface, e.g.

ac

ax=, 0

x=l,t>O

An alternative condition to eqn (6), based on removal of solute immediately after it reaches the exit plane of the flow system, is8 c(2, t) = 0,

t>o

(7)

Suppose a finite difference or finite element numerical scheme is applied to the spatial derivatives on the rightside of eqn (1). The result of this operation, where the left-hand side of eqn (1) is kept in mass-conservative form, is the system of equations’ dc ds z+z=Ac+b

(8)

Equation (8) applies to a finite-difference discretisation. If the finite-element method was used, a mass matrix would appear on the left-hand side. However, eqn (8) still applies if it is assumed the mass matrix is incorporated into the right-hand side (in the definitions of A and b). If there are n nodes in the discretised spatial domain, then c is the n x 1 vector of concentrations at those nodes, s is the corresponding n x 1 vector of sorbed solute concentrations, A is a sparse (often banded) n x n matrix representing the discretisation of the spatial operators in eqn (l), and b is an n x 1 vector resulting from the application of the boundary conditions. DilTerencing of the time derivatives in eqn (8) is now performed. Various approachescanbeused,perhapsthemostcommonO(At’) scheme being Crank-Nicolson differencing,27 the scheme applied here. Equation (8) then becomes A&

where c is the solute concentration in the liquid phase, s is the concentration in the solid phase (converted to liquid phase concentration units), and sources and sinks are ignored. The time derivative of s in eqn (1) describes the sorption reaction, or interphase mass transfer, that is undergone by the solute. Equation (1) is to be solved with a first-type boundary condition at the inlet

t)

+ 1) + s(k + 1) = s(k) + ARC(k) + bK

(9)

where c(k) is the numerical solution at the kth time step, AL = I - AAt/2, AR = I + AAt/ and bK = [b(k + 1) + b(k)]At/2. The standard two-step solution of eqn (9) proceeds as follows. Step 1. Transport c*(k + 1) = Ai1 [A,c(k) + bK]

(10)

D. A. Barry et al.

264

where the asterisk indicates the intermediate approximation to c(k + 1) [sometimes denoted by other authors as c(k + l/2)]. Equation (10) is the formal solution of the transport step. The inverse of AL is used in eqn (10) merely as a notation aid since the method is independent of the solution method used to find c*(k + 1). In the method developed here, the transport step is unchanged, although it is specialised in that eqn (10) is solved using LU factorisation of AL. Step 2. Reaction c(k+l)+s(k+l)=c*(k+l)+s(k)

(11)

which simply expresses mass balance of solute at each node in the solution domain. Note that s(k + 1) is determined using a separate chemical reaction equation. The reaction equation and eqn (11) can be solved simultaneously, or s(k + 1) can be replaced to give an expression (in general nonlinear) to be solved for c(k + 1). Step 1 can be carried out with any solute transport code. The key advantage of Step 2 is that it can be carried out node-wise throughout the solution grid. Equation (9), which typically consists of a set of nonlinear algebraic equations to be solved simultaneously for all nodes in the solution domain, has therefore been reduced to the solution of n linear algebraic equations (the transport step), and n independent (possibly) nonlinear algebraic equations (the reaction step). These latter equations, because they are independent, can be solved in parallel. The solution of the fully coupled problem, eqn (9), necessitates a far greater computational overhead. If N, solutes were reacting in the porous medium, then there would be N, transport equations requiring, after discretisation, at least N, x n simultaneous nonlinear algebraic equations at each time step. Clearly, the standard two-step method, although it involves an additional error, is much more efficient than either DSA or SIA. Because of its convenience and efficiency, we shall use the two-step method as a basis for comparison in the examples presented subsequently. Note that eqn (11) is written somewhat formally, and is clearly a discretised version of the differential equation: aqat = -(as/at). The purpose of the second step is to solve this equation, i.e. compute c(k + 1) and s(k + 1). Equation (11) may be solved directly to achieve this end, or ordinary differential equation solvers could be used instead.26 In this paper, except where analytical results are available, results are all based on solving eqn (11) directly. Equations (10) and (11) are an approximation to the numerical scheme given in eqn (9). The formal, exact solution to eqn (9) can be obtained by solving c(k+ 1) +AL’s(k+

1) = c*(k+ 1) +AL’s(k)

(12)

Equation (12) maintains the separation of the transport and reaction steps, but does not separate the reaction

portion of the solution into independent nodal computations. Our approach is to solve eqn (12) exactly, while maintaining the advantage of performing the reaction step node-by-node. Note that moving the s(k + 1) term of eqn (12) to the right-hand side, then solving by lixedpoint iteration, would constitute an S&type algorithm. A comparison of eqns (11) and (12) shows immediately the truncation error induced by the standard two-step approach. Indeed, subtracting eqn (12) from eqn (11) shows that the standard approach has an O(At) error. Numerical solutions commonly use Crank-Nicolson temporal differencing, which has a truncation error of O(At2). A practical method of solving eqn (10) is to perform an LU factorisation of AL. If the coefficients on the right-hand side of eqn (1) are independent of time, then the LU factorisation is performed only once. In addition, if AL is banded, then the LU factorisation can take advantage of this fact, both in terms of computation and storage (Golub and Ortega18 give a thorough discussion). We assume that this step has been carried out, and that the factors L and U are available. To be more specific, we take L to be unit lower triangular, i.e. the diagonal elements of L are all unity. Further, assume that U = Uu + Do, where the nonzero portions of Uo and Du are the upper triangle and diagonal portions of U, respectively. Equation (12) is equivalent to Dt,c(k+l)+z(k+l)=z(k)+Uc*(k+l) - U”C(k + 1)

(13)

where Lz = s. Equation (13) with an approximation for c(k + 1) on the right-hand side, can be solved node-wise throughout the solution grid. To see this, first note that when c(k + 1) is replaced by an approximation, the right-hand side of eqn (13) is fully known. Then, observe that the computation of z can proceed sequentially from node 1 since L is lower triangular. After the solution for node 1 is calculated, an equation with the concentration at node 2 as the only unknown results. A similar statement applies as the solution steps through the solution domain. The derivation leading to eqn (13) is the basis of the new algorithm and used below. It is introduced by way of an example. 2.1 Single species equlllbrlnm reaction For instantaneous reactions, the sorbed solute concentration at each node is a function of the liquid phase concentration, i.e. s =f(c) (14) where f is a sorption isotherm. The substitution of eqn (14) into eqn (12) gives a system of algebraic equations. If f is nonlinear, then eqn (12) is a system of n nonlinear algebraic equations. Clearly, it is much more efficient to solve n individual nonlinear algebraic equations than a nonlinear system of n coupled equations. As mentioned

Reaction/transport solutions using operator splitting

already, eqn (12) will be manipulated so that it reduces to n independent equations. As an example of the computations needed to solve eqn (13) for a single time step, we write the explicit equations that am to be implemented in the ‘reaction’ portion of a computer program. We assume first that A is a banded matrix, with ~tlbands above and below the main diagonal. This assumption can be removed without difhculty. Let i=l , . . . , n indicate the nodes in the numerical solution. The approximation to c(k + 1) used on the right-side of eqn (13) is denoted by c*(k + 1). Matrix elements are given by the matrix name with appropriate subscripts. Iterative Improvement Algorithm: Single Species Equilibhm Reaction (Algorithm 1) Regin [c*(k + 1) as well as L and U are known from the transport equation solution] cbooeKe

= si(o)

-

C j=max(l,i-m)

Lijzj(o)

EndIf Guess initial c RepeatuntiIIc(k+ 1) -c*(k+ l)l/lc(k+ l)[ < E(test at end of loop) Replace c*(k + 1) by current estimate of c(k + 1) Cahlate for i from 1 to n: Uiici(k+

l)+si(k+

1)

min(n,i+m) =

z,(k) +

‘C

Uv[cJ(k + 1) - $(k

+ 1)]

j=i+l i-l +

Uiic;(k+ I)+

L,zj(k + 1)

C j=max(l,i-m)

[i.e. find ci(k + 1) and si(k + l)] and

(15)

i-l

zf(k + 1) = si(k + 1) -

C

Lijrj(k + 1)

(16)

max(l,i-m)

End Repeat Replace k by k + 1 End This algorithm solves eqn (13) using fixed-point iteration. To begin the algorithm, an approximation, c*(k + l), to c(k + 1) is needed. Each equation in the algorithm is solved in turn. In some of the examples presented here, eqn (1 l), the standard solution in the two-step scheme, was used to generate c*(k + 1). Otherwise, the results of the transport step, in this case c*(k + l), were used. 2.1 .l Examples: Single species transport and sorption 2.1.1.1 Linear retardation. A simple example is that

265

of linear retardation, i.e. in eqn (14) we have f(c) = (R - 1)~. For a tit-type boundary condition and a semi-infinite domain, the exact solution is given by39

(17) This case is considered here because its solution is exact, and because it demonstrates very simply the operation of Algorithm 1. Algorithm 1 was tested by first using a standard Crank-Nicolson m&e-different solution2’ to solve the transport equation (1) with s = 0. Note that the CrankNicolson solution is used in later examples also. This solution provided c*(k + 1) as shown in eqn (10). For a linear isotherm, the second step in the two-step algorithm becomes fully explicit. Figure 1 shows some typical results. The solid line is the exact solution, eqn (17), calculated for R = 1 at t = 5 (for any convenient set of units). This solution is unchanged for any other combination of t and R values such that t/R = 5. scheme results (pluses) show the The Crank-Nicolson (negligible) error inherent in this scheme for the indicated discretisation. The standard two-step scheme [eqns (10) and (ll)] results (dashes) were computed using 1000 time steps. Additional numerical dispersion,” proportional to At, is evident. The circles were derived using 50 time steps utilising Algorithm 1, i.e. 20 times less than that used for the standard two-step method. Clearly, the Algorithm 1 results are virtually identical to the tracer (R = 1) results produced by Crank-Nicolson scheme. For each node, two passes of the iterative improvement loop in Algorithm 1 were needed to satisfy the convergence condition (E = 10m4). Observe that the Courant number (vat/Ax) used in the transport part of the standard two-step scheme is 1 while the corresponding value in the new method is 20. The latter value would cause oscillations in the results if applied in the standard method.30 However, the results in Fig. la do not show oscillations, indicating that the new method, because it is a solution of eqn (1 l), acts to stabilise the large Courant number used in the transport equation solution. In this and subsequent examples the transport step of Algorithm 1 uses a coarser time step than the two-step method. As the transport step was performed using a Crank-Nicolson scheme, the temporal discretisation error is 0(At2). When comparing the results of the present scheme and the two-step method it should be borne in mind that the increased transport step size introduces an additional error. That is, even though Algorithm 1 iterates to the exact solution of eqn (12) the results may not be particularly accurate if too coarse a time step is selected. The O(At) error inherent in the standard two-step method is not reduced to O(At2) by alternating the transport and reaction steps, i.e. within a time step the

D. A. Barry et al.

0.2 ---.

order of the calculations change to: transport over At/2, reaction, transport over At/2. For single species radioactive decay, the alternating split-operator method has been found to produce accurate [i.e. O(At2)] results.26~37143 The alternating split-operator method is usually traced to Strang,36 and is sometimes referred to as Strang splitting. It is often assumed that Strang splitting improves the accuracy of the two-step method for any type of reaction.17 Strang’s analysis, however, was confined to spatial operators, not the temporal operators that are relevant in the present case. One can show, for instance, that an O(At) error results when the alternating split-operator method is applied to the case of linear retardation.6 A rough, but representative, way to compare the computational load needed by the two schemes (the standard two-step method and the new method) is to keep track of the number of times the reaction portion of each scheme is calculated. This count will constitute a reasonable comparison if the computational load of the chemical calculations outweighs the transport calculations. In this case, of course, the calculations themselves are trivial. However, for complex, multispecies problems, the chemical reaction calculations could be significant. If the computer time needed for the transport calculations in each time step is of the same magnitude as that needed for the reactions, then the transport part should be considered as well. For the standard two-step scheme, there is one reaction step per time step, whereas for Algorithm 1 many reaction steps may be needed until convergence is reached. In Fig. lb the accuracy of the results as a function of the number of reaction steps is explored systematically. By varying At, the number or reaction steps varies (reducing At increases the number of reaction steps). Figure lb shows that the rate of improvement of the solution is much better for Algorithm 1 than the two-step scheme. Both schemes eventually reach a plateau, where further decreases in error are ineligible. The two-step scheme plateau is two orders of magnitude above that of Algorithm 1. Similar calculations carried out for the next examples show that Algorithm 1 is consistently more efficient than the standard method.

+ Crank-Nicolson Standard P-Step

0.0 2

0

6

4

8

Position, x

(b)

,

10-l U

Standard e-Step Scheme New Scheme

H

b L 8 !! 9 a

t

1o-2

2.1.1.2 Nonlinear sorption. As a second example we take the exact solution for a nonlinear sorption isotherm presented by Barry and Sander,7 eqn (23) UC s(c)

10-s

I

1

I

ld

.

.

. . . . . .I

103

.

. . . . . . .I

lo4

.

.

.

.**...I

lo5

. . . .

+

c =

E(l

-Y)+Wc

(18)

lo6

Number of Reaction Steps Fig. 1. (a) Results for the linear retardation example. The solid line is the exact solution, eqn (17), to eqn (1) for R = 20, t = 100, D = 0.1 and v = 1. The plus symbols represent the solution for the standard Crank-Nicolson numerical solution for this problem solved using t = 5, R = 1, Ax = 0.1 and At = 0.1 (50 time steps). The dashes are the solution calculated

using the standard two-step procedure, eqns (10) and (11) with R = 20, Ax = 0.1 and At = O-1(1OOOtimesteps). Thecirclesare the results of applying Algorithm 1 with R = 20, E = 10m4, Ax = 0.1 and At = 2 (50 time steps). (b) Average error, defined as c;‘J( l/n) Cbl (c? - cfum)‘, as a function of the number of reaction steps for the problem considered in Fig. la. The effect of reducing At is to increase the number of reaction steps.

267

Reaction/transportsolutions using operator splitting

0.0

0.2

0.4 0.6 Concentration, c

0.8

1.0

Fig. 2. The nonlinear sorption isotherm defined by eqn (IS),

plotted for E = u = O-5and v = 1. where E (>O) and v (9 and ~1) are fitting parameters and both s and c are taken as being normalised by the inlet concentration, co. In Fig. 2, eqn (18) is plotted using the parameters E = v = 0.5. By contrast to most isotherms which increase monotonically with increasing liquid phase solute concentration, this isotherm is clearly unusual in that it displays a peak around c = 05. Because relatively less sorption takes place for higher concentrations, this isotherm will act to sharpen the solute front as it moves through the porous medium. It is for this reason that it was selected. The steep front will be balanced by diffusion such that a travelling wave front will result.38 In Fig. 3 we plot, for t = 5, the exact solution for the third-type entrance condition, eqn (3),

2.8

3.2 Po!3UO”,~

4.0

3. The exact solution (line) to eqn (1) for the isotherm given in Fig. 2 calculated for t = 5, v = 1 and D = 0.1. The

Fig.

results of the standard two-step method are given by the dashes (Ax = 0.1, At = O-0125, 400 time steps). The circles were calculated using Algorithm 1 (CT = 1O4, Ax = O-1, At = O-125, 40 time steps).

given by Barry and Sander’ for a soil profile containing no solute initially, using the same D and v as used in deriving the results presented in Fig. 1. A total of 400 time steps were used to generate the results for the standard two-step algorithm (dashes) calculated using, again the Crank-Nicolson solution of eqn (1) with s = 0 for the transport step. The solution of the reaction step, eqn (1 l), reduces to a simple quadratic for the isotherm eqn (18). The results of the standard method and Algorithm 1 (circles, 40 time steps) are nearly identical, although a much coarser time step was used for the latter scheme. In Algorithm 1, the stopping criterion, e, was set at 10e4, which was satisfied in 2 iterations at nearly all nodes, with a few taking 3 iterations. Although an analytical solution is available for this case, the scheme used here is applicable to any isotherm. The only difference is that not all isotherms will permit an analytical solution to eqn (15). Then, a numerical solution, such as Newton-Raphson iterations, or the bisection method would be used. 2.1.1.3 Radioactive &cay. The third example considered is that of transport with a reaction term proportional to the solute concentration, i.e. we set ds/dt = Kc in eqn (l), where K is the decay strength of the sink. This function is incorporated directly into eqn (11) and Algorithm 1 using the standard CrankNicolson discretisation s(k+ 1) = s(k) + y

[c(k + 1) + c(k)]

(19)

Valocchi and Mahnstead37 have presented an analysis of the standard two-step method for this case solved with a third-type condition in place of the fist-type condition (2), the condition used here. They show that the standard approach removes excessive mass from the soil pore fluid, with the mass balance error increasing in magnitude with KAt. The exact analytical solution for radioactive decay in a semi-infinite domain is readily available,” and will be used to check the standard twostep method and Algorithm 1. In Fig. 4 we plot the exact solution and the results of the two numerical schemes. Note that the standard two-step scheme used the exact analytical solution of the reaction model (&/at = -Kc) to calculate the second step of the scheme, i.e. C *k+l exp (-KAt) was used. This removes any discretisation error from within the second step. If an analytical expression was not used, then additional intermediate steps may be needed to resolve the second step accuratelyF6 That is, if K is large then the step size, 6t, used in the reaction part of the scheme should be such that KSt cc 1 (say KSt = 0.1) to ensure that minimal additional error is induced in solving for the reaction. This restriction does not apply for the scheme presented here. The line in Fig. 4 represents the exact solution. The results from the standard two-step scheme (pluses, 50 time steps), were computed such that KAt = 0.1, in

268

D. A. Barry et al.

0.8

-Exact + Two-Step Scheme Algorithm 1

contained in Algorithm 1 can be extended to deal with these processes as well. The extension of the algorithm to multiple species is conceptually straightforward, and does not rely on a spec5c formulations of the chemical reaction equations. For example, the approach taken by Zysset et a1.44l4’could be utilised. First, a set of Nx chemical components41 is chosen. All chemical complexes are formed as reactions between components. Mass action laws quantify species concentrations formed in complexation, i.e.

where *c is the p* chemical species (measured in moles per unit volume of fluid), N, is the number of complexes, KP is its equilibrium constant, ‘X is the j* chemical component, and Xpj is a stoichiometric coe5cient. Define ju as the total soluble concentration of the j* component, then .

'U =

NC

.

‘X +

c

pCApj,

j=

l,...,Nx

(21)

p=l

0.5

Position, x Fig. 4. Comparison of solutions for the radioactive decay case. The line is the exact solution for K = 5,t= 1, D = O-1 and v = 1. The pluses are the results of the standard two-step scheme using AX = O-015 and At = 0.02. The dashes were calculated using Algorithm 1 for E = 2 x 10m4,AX = 0.015 and At = 0.2.

which case reasonable accurate results are to be expected, although it is clear that too much solute has been removed from the s~stem.~ The dashes were calculated using Algorithm 1 (5 time steps) with KAt = 1. The results from the previous time step were used as the initial guess for CAin Algorithm 1. Clearly, the results are better than the standard two-step scheme, even though a much larger time step was used in the calculations. In the present example, Algorithm 1 required 8 iterations per time step, giving a total of 40 reaction steps. The standard scheme required 25% more reaction steps. 2.2 Multi~e

equilibrium reactions

There are a number of approaches to dealing with equilibrium reactions among multiple species. Here we adopt the equilibrium chemical modeling approach taken by Kirkner and Reeves,*l which we describe briefly below. For the sake of simplicity, we consider chemical reactions that take place between dilute solutions, such that the activity coefficients can be taken as unity for all chemical species. Further, below we ignore precipitation and dissolution reactions. These simplifications will be appropriate for some aquifer contamination events. In practice, however, precipitation and dissolution can occur also, and the method

The combination of eqns (20) and (21) gives ju as a function of the N, chemical components in the system. For brevity, this function is written as ju =‘?(‘X,.

j = 1,. . . , Nx

. . , NxX),

(22)

The same approach is used to model sorption. Sorption reactions, including surface complexation and ion exchange, are modelled as another nonlinear function, i.e. ps =pfSX(‘X,.

. . ,Nxx),

p=

l,...,Nx

(23)

This function can take various forms, e.g. the multicomponent Langmuir isotherm*l ps =

KpsCTpX

p=l

,-**> Nx

1 +&Q j=l

where KPs is the equilibrium constant for the p* surface adsorption reaction and C-r is the total number of surface sites available. Note that the sorbed phase concentrations in eqn (24) need to be converted to liquid phase concentration units for simulation in a solute transport model. Equations (22) and (23) give the total liquid phase concentration and the sorbed concentration of each chemical component. There are Nx transport equations for these components*’ j=

l,...,Nx

where ‘L is the generalisation of the advectiondispersion operator on the right-hand side of eqn (1) to include higher dimensions (providing that LU factorisation is still an efficient solution method),

Reaction/transport solutions using operator splitting variable advective velocities and solute specific dispersion coefficients. Let us consider the standard two-step solution of the Nx equations (25). As usual the solution at time step k is assumed known. First the reaction term is dropped and the transport of the chemical components is performed as in eqn (10). Then, the reaction step, eqn (ll), is solved. Each node in the numerical solution grid is considered separately. At the i* node, the chemical concentrations are determined by solving the Nx nonlinear algebraic equations: ‘ui(k + 1) + ‘si(k + 1) = ‘u;(k + 1) + ‘q(k),

j=

l,...,Nx

(26)

The left-hand sides of eqns (26) are replaced using eqns (22) and (23), while the right-hand sides are known. Thus, eqn (26) consists of Nx equations to find Nx unknowns, the latter quantities being the chemical components, jX, j = 1,. . . ,I&. When the component concentrations are found, eqns (22) and (23) are used to calculate the Nx component concentrations in the liquid and sorbed phases, allowing the numerical solution to proceed to the next time step. Proceeding from the standard two-step method to the modification of Algorithm 1 to allow for multispecies reactions is a simple matter. The transport step is carried out in precisely the same manner. Then, for each component, reaction equations of the form t.1) + 1) + ‘z(k ??

D&k

= jz(k) + Uju*(k + 1) - U,‘u(k + l), j=l

,***, Nx

exchange capacity has its exchange sites saturated with a given, isotopically labelled, cation, e.g. Ca, and a low concentration of Ca is present in the interstitial fluid. A high concentration influent solution, isotopically unlabelled Ca, is added to the soil profile. The influent solution, acting like snow-plow, rapidly removes the labelled ions. In the case of a soil column experiment, the removed solute appears in the effluent as a high concentration pulse around one pore volume. The reverse situation of a high initial concentrations being displaced by a relatively low concentration intluent solution is called the precursor effect.9Y35 Both phenomena are quantified by the model presented below. The labelled and unlabelled Ca ions are ‘u and ‘u, respectively. The governing transport model is given by eqns (25). For a soil column experiment under steady flow conditions the advection-dispersion transport operator, L, is given by the right-hand side of eqn (1). The exchange reactions, with sorbed phase concentrations ‘s and 2s, are modelled using”

is= s‘UC

1U+2U’

are to be solved. If the ,‘L operators in eqn (25) are not identical, then the discretisations of those operators will vary, such variation to be reflected in eqn (13). Algorithm 1 then provides the step-by-step solution to eqn (13), with minor notation changes. In the examples given above, a single equation was solved at each node. With Nx components, there will be Nx (in general nonlinear) equations to be solved at each node, i.e. eqn (15) applies to each chemical component. Observe that for each of these equations the right-hand side is always known, as it is for eqn (26). This means that methods used in numerical schemes using the standard two-step approach can be applied to Algorithm 1 with only minor modifications. We demonstrate the application of these ideas with an example of an ion-exchange transport/reaction problem. We consider the case of two-species ion exchange, i.e. we take Nx = 2 in eqn (25). For the ionic species we take a specific chemical for which different isotopes are available. A special case of this type of system, the snowplow effect, was analysed both experimentally and numerically by Starr and ParlangeM and Barry et aZ.,‘O respectively. A soil profile with a relatively low cation

j=

1,2

where C, is the cation exchange capacity of the soil, expressed in liquid phase concentration units. Because there are two species, the reaction step of the two-step method, as well as the corresponding equation in Algorithm 1, eqn (15), involve the solution of two equations at each node. These equations are all similar in form, viz. ‘ui(k + 1)B +

(27)

269

j=

1,2;i=

‘q(k + l)Cs ‘ui(k + 1) +2 ui(k + 1) = ”

(29)

1,***, n

where B, R1 and R2 are known quantities that change according to which equation in Algorithm 1 is being solved. The solution to eqn (29) is ju=%(l

-A),

j=

1,2

(30)

Thus, in this case no numerical error is introduced in the reaction portion of the calculations. We show some results derived using the new splitoperator algorithm. The main purpose of this example is to compare the number of calculations needed for both methods to give similar results. One of the experiments reported by Starr and Parlange demonstrating the snow-plow effect is simulated (Fig. 5, pluses), using parameter values reported by them. In the experiment, a 5-cm long soil column tYled with sandy loam was saturated with isotopically labelled (using 45Ca) Ca by flushing with O*OlN CaC12 solution. The influent was then abruptly changed to a 2N unlabelled solution. The value of C, for the soil is 0~065meq/ml.” Simulation results are shown in Fig. 5 (third-type entrance condition assumed). The thick line represents ‘exact’

270

D. A. Barry et al. Bather than consider the model given by eqns (1) and (31), the two-region (mobile-immobile) model will be examined. The two-region model, first developed by Coats Smith,13 is widely used in practice.15J6~31~32 Suppose c in eqn (1) is the concentration in the mobile region and ci, is the concentration in the immobile region (for an alternative interpretation of these different regions see Li et &24p). Then the two-region model consists of eqn (l), where s = 0bc&&, and (32)

0.8

Equation (32) generalises the linear two-region model, which is recovered iff(c) = c. Equation (9) resulted from a Crank-Nicolson discretisation of eqn (8). The same approach applied to eqn (32) gives15~31*32

1.2

Pore Volumes Fig. 5. Simulation of the snow-plow experiment reported by

Starr and Parlange.” Experimental data are represented by the plus signs. Algorithm 1 results are given by the dashes (E = 5 x 10m4, Ax = 0.1 cm, At = 1.25 x 10-2hr, D = 0.375 cm2/hr and v = 6*5cm/hr). Thick line: ‘exact’ numerical results.‘o Thin line: results from the two-step method using Ax=O~lcmandAt=5~10-~hr. numerical results calculated by Barry et al.” Note that the column P&et number for the laboratory experiment is 100, and so the downstream boundary condition will affect negligibly the breakthrough curve.” The numerical results in Fig. 5 were computed on a domain of length 1Ocm (breakthrough curves in Fig. 5 are the concentrations at x = 5 cm), and so represent the case of a semi-infinite column. Clearly, the results do not predict the experimental data very well. This lack of agreement is possibly due to the simplicity of model used (a slightly more complicated model is considered below). Almost coinciding with the thick line are two other curves. The thin line represents the results from the twostep method, while the dashes are the results of the new scheme. The results of the transport calculations in each time step were used as the initial guess needed to initiate Algorithm 1. Both sets of results are essentially identical. However, the standard two-step results needed 250 sets of reaction calculations, whereas Algorithm 2 needed 150 sets. 2.3 Single species nonequilibrium reaction A common model for time-dependent reactions is a linear exchange between the sorbed phase and the solution phase, i.e. g = a[f(c)

-s]

(31)

where (Yis the transfer rate coefficient andf(c) expresses the sorption reaction. As a -+ 00, (31) gives s =f(c), an equilibrium reaction. Tracer transport is recovered for (Y= 0.

9

[f(k+ 1) +f(k)]+

s(k + 1) = 1+2

(1 -+)s(k)

ohAt (33)

where f(k + 1) -f[c(k+ l)], (Y, = (Y*/& and cui, = a*/(&. Now, s(k + 1) in eqn (9) can be eliminated, giving ALc(k + 1) + &f(k + 1) = 2&s(k)

- &f(k)

+ ARC(k) + bK

(34)

where ,& = ((a,At)/2)/(1+ (ahAt/2)) and /3b = ((abAt)/2)/(1+ (ahAt)/2)). Note that eqns (33) and (34) apply also for the model in eqn (31). In both equations a, and oh are replaced by (Y. Equation (34) is now solved in two steps. As before, the first step consists of pure transport. This step involves the final two terms on the right-hand side of eqn (34), and gives the intermediate result, c*(k + 1). For the reaction step we proceed as for the single-species equilibrium reaction case. The coefficient matrix, AL, is again factor&d as AL =LU,withU=U,,+h.This allows eqn (34) to be rewritten as Duc(k

+ 1) + P,Y(k

+Uc*(k+

+ 1) = Win&>

1) -Uuc(k+

1)

- &y(k)

(35)

where Ly = f. Equation (35) is solved in a manner analogous to that used to solve eqn (13). To summa&e, the c(k + 1) term in the right-hand side of eqn (3.5) is initially approximated using the either the standard twostep approach or the results of the transport step, in which case a new solution can be found for c(k + 1). Then the solution to eqn (35) is found by fixed-point iteration. The iteration proceeds until convergence is reached. The detailed steps involved are very similar to those listed in Algorithm 1, with some minor variations to account ‘for the calculation of the vector y. For convenience, the mod&d algorithm is still referred to as Algorithm 1.

271

Reaction/transport solutions using operator splitting As an example we take f(c) = c in eqn (32) so that comparison with an exact analytical solution can be performed. Note that the computations in Algorithm 1 become fully explicit in this case. As a test problem we used the exact solution for eqns (1) and (32) solved for a third-type boundary condition in an initially solute free, semi-infinite domain. The solution is given, for instance, by De Smedt and Wierenha16 and implemented by The test problem results Parker and van Genuchten. were computed using the parameters listed in the caption of Fig. 6. The solid line is the exact solution, the dashes the standard two-step method and pluses the results of Algorithm 1. The latter results were computed using E = 10m4, which entailed two iterations for each node in the solution domain. In this example, the iteration scheme start& with the results of the standard scheme, giving a total of 3 iterations for each node. Recall that a reasonable comparison is to allow the standard two-step method the same number of iterations at each node by reducing the time step by a factor of 3, i.e. the dashes in Fig. 6 were calculated using 3 times as many time steps as used in Algorithm 1. Even so, the results of Algorithm 1 are essentially exact, while the standard method displays the usual numerical dispersion induced by operator splitting. Comparisons for different parameter values have been performed with similar results to those shown in Fig. 6.

‘.“T+--x t

I

2.4 Multispeci~ nonequilibrium reactions The nonequilibrium transport and reaction problem can be formulated in various ways. For the purpose of applying the new algorithm we shall refer to the model used by Kirkner et al*’ An assumption made by these authors is that only ion-exchange reactions are considered. For each component in solution the transport equation (25) is applicable. The transfer rate between the liquid and solid phases is described in a general way by g

=

jh(‘u,. . . ,Nxu,ls,.. . /“s)

where the functions jh are specified by the chemical assumptions in the model. Equation (36) is discretised using the Crank-Nicolson approach

js(k + 1) = js(k) + $ [jh(k + 1) + jh(k)], (37) j=l

,***9 Nx

where ‘h(k) means the arguments of eqn (36) are evaluated the k* time level. Equation (37) applies at each node in the solution grid. If’h is such that is(k + 1) can be isolated in eqn (37), then an explicit expression for that variable results. In that case, the solution proceeds in a manner similar to that outlined in Algorithm 1. The effect of eliminating js(k + 1) is that the reaction calculations at each node then involves the solution of Nx simultaneous equations. The more general case where js(k + 1) cannot be isolated is considered below. At each node, 2Nx simultaneous equations must be solved at each node for ‘s(k + 1) and ‘c(k + 1). Discretisation of the chemical component transport equations yields equations identical to eqn (12) U’u(k + 1) + L%(k

+ 1)

= Uju’(k + 1) + L%(k) j=l

0

2

4

0

10

,

(38)

,***, Nx

The combination of eqns (37) and (12) yields a system of 2Nx equations at each node in the solution domain that can be solved sequentially. The details of this solution are contained in Algorithm 2.

Position, t

Fig. 6. Comparison of Algorithm 1 and the standard two-step method for the mobile-immobile region model. The solid line is the exact solution for the two-region model at t = 5 calculated using the parameters: D =0-l,v=2, 0, =t9,=0*25 and (Y*= 1. The other two curves are numerical solutions given by the two-step method (dashes) and the results of Algorithm 1 (pluses). Both numerical solutions were computed using a standard Crank-Nicolson solution of the transport equation on adomainoflength lOdiscretisedintosegmentsoflengthO~l.The time step varied in each case: At = 0.08333 (total of 60 steps) for the two-step method and At = 0.25 (total of 20 steps) for Algorithm 1.

Iterative Improvement Algoritbmz Nonequilibrium Single Species Reaction (Algorithm 2) , Nx as well as L and U are IBegin[ju*(k+l),j=l,... known from the transport equation solution] CbW!X?~o

(42)

*u(x, 0) = 0,

x>o

(43)

+ l),

p=max(l,i-m)

j=l

defines ‘h in eqns (36) and (37), as well as in Algorithm 2. The rate equations (41) allow the js(k + 1) terms to be isolated after Crank-Nicolson discretisation of the temporal derivative. These terms can then be eliminated in eqn (39), giving two equations in two unknowns, ‘u and *u, to be solved in each case. Because these equations (including those solved in the standard twostep method) all have the same form, their analytical solution need only be calculated once. The solution is rather messy and so is not presented here, although it is easy to derive. The first example of Algorithm 2 is shown in Fig. 7. A third-type entrance boundary condition was applied for each solute. For ‘u, there was zero concentration in the influent solution, whereas for *u, the influent concentration was set at 2. The following initial conditions were used

Other parameters are listed in the figure caption. An ‘exact’ solution was calculated using a Crank-Nicolson finite-difference scheme, the scheme used being a generalisation of that described by Barry et al.” Gridindependent results are shown in the figure. The standard two-step method results were calculated also. Using a total of 100 time steps, the standard method gives excellent results for species 2, but poor results for species 1. Algorithm 2 gives excellent results for both

,,n

0.4

0.0 0

j=

1,2

a

4

10

POSitiO"a

(41)

where jf(‘u, *u) is given by the right-hand side of eqn (28), and ‘kf and jkb are the forward and backward reaction rates, respectively. Note that at equilibrium, the left-hand side of eqn (41) is zero in which case eqn (28) applies, with the additional factor ‘kffkb mul$ply$g the right-hand side. Thus, the product Cs’kf/‘kb represents the cation exchange capacity for this chemical species in this soil. The right-hand side of eqn (41)

2

7. Two-species kinetic reactions example. Parameters used are t = 5, D = 0.1, v = 1, ‘kf = 5, ‘kb = 4, ‘kf = O-04, and ‘kf = 0.05. The thick lines represent the ‘exact’ (meaning grid-independent) solution computed using the CrankNicolson solution of the coupled transport/reaction problem. The thin lines were computed using the standard two-step scheme using AX = O-1in the transport equation solution and At = 0.05 (100 steps). The dashes represent the results of Algorithm 2 using the same Ax with At = O-125 (40 steps) and e=2x lo+. Fig.

Reaction/transport solutions using operator splitting species, using only 40 time steps. The results of the transport equation solution were used as the initial guesses in the reaction calculations. For most nodes, two iterations of the Repeat section of Algorithm 2 were needed at each step (82 passes in total). That is, as far as the reactions are concerned, roughly 25% more calculations were needed for the two-step method, although Algorithm 2 still gives superior results. For the final example, we return to the snow-plow experiment of Starr and Parlange.M Bajracharya and Barry’ showed that the data could be fitted better using a kinetic model, instead of the equilibrium model used in the example of Section 2.2. In Fig. 8 the snow-plow data are plotted again. The results are calculated using the same parameters as before except that D is set to O-15cm*/h and C, = 0.06 meq/ml. The forward and backward reaction rates are all set to 17.5 h-l. The results of the Crank-Nicolson numerical solution (thick line) are an improvement over the equilibrium model results in Fig. 5. The thin line was derived using the standard method while the dashes represent the results of Algorithm 2. In Algorithm 2, the results of the transport step were used as an initial estimate to start the algorithm. A total of 229 reaction solutions were calculated, compared with 250 passes in the standard method. Clearly, Algorithm 2 has produced more accurate results. Reducing At further to, say, lo-‘h, gives results that are essentially identical to the CrankNicolson scheme for either method.

25

1

0.4

1.2

1.6

Pore &mes

8. Breakthrough curves from the kinetic reaction model compared with the snow-plow data of Starr and Parlange.” The thick line was computed using the ‘exact’Crank-Nicolson

Fig.

solution. The two-step method results are given by the thin line using Ax = O-1cm in the transport equation solution and At = 5 x 10e3 h (250 time steps). The dashes represent the results of Algorithm 2, calculated with e = 2 x 1O-4 and At = 7.8125 x lo-‘h (160 time steps).

273

3 DISCUSSION AND CONCLUSIONS The new iteration algorithm relies on an appropriate starting guess. Two obvious choices were tried for the examples used here, i.e. the solution from the standard two-step scheme or the solution at the end of transport step. The difference in the rate of convergence using either case is not significant. Similarly, the P&clet number and spatial discretisation had little or no effect on the convergence rate. The new algorithm has been tested on a variety of problems. It has been compared in each example with the commonly used two-step procedure, the latter procedure being taken as a benchmark in terms of efficiency. Within each time step, this latter method allows the transport portion of the problem to be solved independently of the chemical reactions. Further, the reaction calculations are solved only at each node. The standard two-step procedure is both easy to implement and efhcient to compute. The split-operator algorithm presented here was developed so that both these essential features are maintained as much as possible. The present algorithm converges to the exact solution of the discretised transport/reaction mode, whereas the two-step scheme does not. Note that the reaction equations have exactly the same form in both methods. Thus, the computational load necessary to compute one pass of the reaction calculations at any node is of the same magnitude. The new algorithm is not, however, a universal panacea for all transport/reaction problems. First, the scheme relies on having available the LU factorisation of the coefficient matrix used in the solution of the transport equation. Very large problems are usually solved using iterative methods for solving the linear algebraic systems of equations.4 The mesh size where one method will be more efficient than another depends on various factors. For instance, for steady flow problems, the LU factorisation need be computed only once while for transient flow it will be computed at each time step. Storage considerations may enter as well, depending on the coefficient matrix in the transport equation solution. Another feature of the two-step method is that the reaction calculations at a given node are independent of the same calculations elsewhere in the solution mesh. For the present algorithm, the reaction calculations are also independent so long as they are carried out in sequence, i.e. node-wise parallelism of the reaction calculations is not possible. It might be possible to circumvent the sequential nature of the method using techniques similar to the cyclic reduction methods applied to banded linear algebraic systems of equations,18 although we have not investigated this as yet. Based on the examples presented here, we conclude that the new algorithm offers efficiency and accuracy advantages over the two-step approach, although for

D. A. Barry et al. large problems LU factorisation solution method.

will not be an attractive

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