A Finite Volume Characteristics Method for the One-Dimensional Simulation of Pollutant Transport in Groundwater A. Balaguer

A finite volume characteristics method for the one-dimensional simulation of pollutant transport in groundwater

A. Balaguer/ C. Conde/ A. Del Cerro,* J.A. Lopez/

V. Martinez" o Universitat Jaume I, Departament de Matematiques,

Campus Penyeta Roja, 12071 Castellon, Spain * Universitat Politecnica de Madrid, ETSI Minas, Dep. de

Matemdtica Aplicada y Metodos Informdticos, c/. Rios

ABSTRACT.

A new mass conservative method for solving the one- dimensional equation of pollutants transport in groundwater, based on the characteristics and finite volume method is derived and discussed. Test results demo st rate his accuracy and efficiency for transport problems that are dominated by advection.

1. INTRODUCTION.

Grounwater pollution due to human activities occurs in a number of ways such as seepage through landfills and waste deposits. The distance that can be covered within a certain time depends on the velocity of flow and the persist an ce of the pollutant. We assume that the flow field is known a priori. We also consider that the release of pollutant agents is performed by means of a Dirichlet boundary condition for the transport in a saturated zone and limit the further discussion to a saturated zone. In some field applications as well as in soil column experiments or when we average observed concentrations over the depth and the width of domain, the one- dimensional form of the transport equation is applicable. This equation, under conditions of uniform porosity, incompressible fluid and media and constant velocity flow, has the following form (see Bear [2])

+ ,«(*,*) = 0 (1) at ox Vze]o,L[, v*e]o,T[

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with appropriate initial and boundary conditions, being: v > 0 the constant pore average velocity, k > 0 the diffusion- dispersion coefficient, a > 0 the retention factor (we consider a linear adsorption isother- mal), q > 0 a first-order decay coefficient, and u(x,t) > 0, which is the only one unkonown function, the pollutant concentration at a spatial coordinate x and at time t.

Initial conditions are given by the concentration distribution "w(x,0)" at the starting time "t=0" of simulation. In this work we assume:

%(z,0) = 0 VzE]0,6[ (2)

Dirichlet boundary conditions are considered at x = 0 and x = L. These conditions specify prescribed concentrations on the boundary:

u(o,t) = MO vte[o,T] (3)

u(L,t) = 0 Vf€ [0,T] (4) being ho(t) a known function and where we have chosen the point L so far enought for its concentration is zero.

2. NUMERICAL METHODS.

We discretize space and time in intervals or cells, Ei i — 0, . . . , NX for space and Tn n = 0, . . . , NT — 1 for time. For the sake of simplicity we assume only cells of constant distances Ax and At respectively so:

n = 0, . . . , NT - I

Discretized equations are obtained by taking a pollutant mass balance over each cell of the space grid. The mass balance over the cell Ei requires that over a time interval ft™, £""*"*], inputs and outputs due to storage, advection, dispersion- diffusion and decay add up to zero. We choose a 0- scheme for time integration. Therefore we obtain fully implicit equations for 0 = 1, explicit equations for 9 — 0 and the central in time Crank- Nicholson scheme for 9 = 0.5. Thus, we obtain the following equations:

- 1 ^ = 0,..., 7VT - 1

Hydraulic Engineering Software 285 where denoting by z,-+i = (i + |) • Ax Vz = 0, . . . , (NX - 1):

and u™ denotes the approximate value of u at time f\ The integration over the extreme cells vanishes because of the Dirichlet boundary condition so their respective equations have been replaced by the equalities:

We have to approximate the integrals over each E^, the derivatives of concentration in the extremes of cells and the convective term:

as function of the discrete unknows at time t™ that are denoted by u? and are expected to be an approximation of the mean value of u on the cell "z" at time t™ Vz = 0, . . . , NX. In this work we use a centered finite difference scheme to approximate the partials of concentration in the extremes of cells:

l (6) <9z Az

In order to evaluate the integrals we assume that concentration is piece- wise constant within each cell. Therefore the used integration formula is:

4)'^ ^ _ ^ ^ /_

If we use an upwind approximation for the convective term:

i_ ' numerical solutions have artificial dispersion, even if we choose =

Ax and use an explicit scheme in time, more important for convection dominated problems. A more accurate scheme is obtained with a centered difference scheme in time and a linear interpolation for the concentration at the extremes of cells for the convective term,

(9)

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but though this scheme works very well in terms of accuracy and efficiency for diffusion dominated problems, we find that when convection dominates numerical solutions tend to show spurious oscillations that can only be avoided by suficiently fine discretization (see Balaguer et al. [1]). In the next sections we present a new numerical scheme based on a com- bination of characteristics and finite volume method to solve this problem.

3. FINITE VOLUME-CHARACTERISTICS METHOD.

In order to reduce the numerical diffusion we replace the above convective approximations (8) or (9) by the movement of tracer particles along the characteristic lines, each one with a time concentration assigned. At t = 0 we take NP particles in each interior cell and ^~ particles in & and ENX, because their length is equal to —-.

Each particle has an initial (at t=0) spatial coordinate that will be changing along time by his movement along the characteristic lines X(t, XQ, 2o) yielded by the next equation:

If we denote by Xp(t") the spatial coordinate of a particle "p" at f , its spatial coordinate at r+% is equal to X(r+\Ap(r),r) that we approach by solving (10) by means of an Euler scheme at time step [*",<"+*], so:

Thus we obtain the spatial coordinate X/f +*) at T+* of each particle "p" when %p(r) is known. Initially (t=0), if a particle "p" belongs to E, its spatial coordinate is:

= o where j £ {1, . . . , NP}, so each particle belongs to the interior of one cell.

In order to get numerical solutions with a small numerical diffusion each particle of E, can only be transported until E++I or until another point of Ei in a single time step. Therefore, we have to choose At such that the following condition is verified:

< 1 (ii) a-Ax - 2

Notice that f is the length of the smaller cell (Eo and ENX)-

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On the other hand, in each time step there are some particles "p*" that reach or surpass the spatial boundary x = L (i.e. Xp+(i"+^) > L). These particles are destroyed but then, we define a new particle "p" by each destroyed particle that has this spatial coordinate:

Because of (11) now all these new particles belong to EQ and hence, at f* there is not any cell that is clean of particles. Along each characteristic line X(t,xo,to) ~ X(t), it holds:

so the partial differential equation (1) is reduced to:

) = o we]o,T[ (12) that expresses the rate of concentration change in an infinitesimal volume that moves along the characteristic line X(t). By solving this equation we assign to each particle an initial concentration given by the boundary and initial conditions. After (3) and (4), and as we have assumed that:

%(z,0) = 0 VzE]0,f,[ being Cp(t") the concentration of the particle "p" at T with n > 0, then:

C = 0 z/pGE, ; = 1,...,WX

The temporal change of the particles concentration along the characteristics lines is calculated by means of the numerical resolution of:

a^ = k~ - qu(x.t) Vf€]0,r[ Vxe]0,L[ (13) dt ox* by a finite volume method based on (7) and (6) and a 0- scheme, with the same procedure described in the above section. If we denote by u* '""*"* the numerical solution of (13) at E, and at £"+*, based on the numerical solution of (1) u,-(f*) at f , and we calculate D?+* Vz = 1, . . . , NX — 1 as the average concentration of all particles which are at r+i into Ei, then:

= 0

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is the numerical solution of the original equation (1). u™+* i = 0,..., NX is hence the numerical solution that we want to calculate.

If we define:

A,.*//n-f-l\ _ L I t^i+l i __ _J 1ZJ ^*(^ ) - ^ (^ ^ Az

A^(^') = ^ ( ^±1^ "' " ^"' \ Az Az

then, «*'"** is given by the solution of the next linear system: a'Az-i/r^ = a-Az.%r + ^-Af.A^(r+^) + (l-6)).A^A^(r)

*,n+l _ ^ /^M+l\

= 0

In this way we may calculate the concentration of each particle at time fi+i. Each particle is assigned the change of concentration at the cell E,

2" = 0, . . . , WX where it is at T+\ Thus if Xp(r+i) E E, 2 = 1, . . . , we can calculate Q,(r+i) by solving the following equation:

a Az Cr+') = a Az Q,(r) + A^ A^(r+'

If £\Ui(t™^) is negative, care has to be taken so that a particle concentration does not become negative. Therefore at these cases, Aw,(r+i) is scaled up to obtain Cp(r+^) > 0 Vp such as Xp(r+i) G #.

4. NUMERICAL RESULTS.

We analyze the behavior of the numerical scheme presented in the above section in two test problems where advective transport dominates. We assume the next boundary Dirichlet condition at x = 0:

%(Q,f) = 1 Vf E[0,T]

We also suppose %(z,0) = 0 Vz €]0,i[ with I = 100 and %(I,,Z) = 0 VZ E [0,T].

In thefirs t test problem, pollutant transport through a one- dimensional column was simulated for a period of T=20 hours and the following coefficients are used:

o = l, u = 1, 6 = 0.03, 9 = 0

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Ogata & Banks [5] gives the analytical solution to this problem. In figure

(1) we compare the analytical with the numerical solution obtained if we choose differents Ax and At, with NP = 4. The same numerical solution is obtained for different 0-values. Therefore we can use a explicit scheme for solving (13). However, notice that the accuracy of the finite volume schemes described in section 2 depends on the 0-value chosen, so numerical diffusion increases when 0 grows towards 1. We can also see that if (11) is not verified then numerical diffusion arises.

In the second test problem, we use the following coefficients:

a= 1, v = 1, jfc = 0.03, q = 0.01

In this problem decay exists. The behavior of the numerical scheme is the same as in the above test problem (see figure (2)).

ACKNOWLEDGEMENTS

Angel Balaguer acknowledges the Conselleria de Cultura, Educacio i Ciencia of the Generalitat Valenciana for a fellowship. This work has been partially supported by the Fundacio Caixa Castello. The work of Carlos Conde has been partially supported by ENRESA under contract number 0700442.

References

[1] Balaguer A., Conde C., Del Cerro A., Lopez J.A., Martinez V. Simula- don numerica del problema de transporte unidimensional de sustancias radiactivas con tecnicas de volumenes finitos. Proceedings of the 1st

Simposium internacional de mo del ado en oceanograffa climatologfa y medio ambiente: aspectos matematicos y numericos, Malaga, Spain, 1994, (to appear).

[2] Bear J. Hydraulics of groundwater. Ed. McGraw-Hill, New York, 1979

[3] Healy R. W., Russell T.F. A finite-Volume Eulerian-lagrangian local-

ized adjoint method for solution of the advection-dispersion equation. Water resources Reseach, 1993, vol.29, no. 7, pp. 2399-2413.

[4] Kinzelbach W. Groundwater Modelling. An introduction with sample

programs in basic. Ed. Elsevier, Oxford, 1989

[5] Ogata A., Banks R.B. A solution of the differential equation of longi- tudinal dispersion in porous media. United States Geological Survey, Professional Paper Nr. 411-A., 1961.

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a = l i; = l, k = 0.03, 9 = 0, 9 = 0, 7VP - 4

0.8

exact — O 1

0 5 10 15 20 25 30 35 40 45 50 X figure (1) (*) eqn. (11) is verified, (**) eqn. (11) is not verified

a = 1, v = 1, k = 0.03, q = 0.01, 0 = 0, NP = 4

t=20 exact — §

0.2

. , V 0 5 10 15 20 25 30 35 40 45 50 X figure (2) (*) eqn. (11) is verified, (**) eqn. (11) is not verified