Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue3.1999:

Hybrid Laplace transform ®nite element method for solving the

convection±dispersion problem

Li Ren

a

, Renduo Zhang

b,*

a_{Department of Soil and Water Sciences, China Agricultural University, Beijing 100094, People's Republic of China} b_{Department of Renewable Resources, University of Wyoming, Laramie, Wyoming 82071-3354, USA}

Received 8 September 1998; accepted 18 March 1999

Abstract

It can be very time consuming to use the conventional numerical methods, such as the ®nite element method, to solve convection± dispersion equations, especially for solutions of large-scale, long-term solute transport in porous media. In addition, the conven-tional methods are subject to arti®cial di€usion and oscillation when used to solve convection-dominant solute transport problems. In this paper, a hybrid method of Laplace transform and ®nite element method is developed to solve one- and two-dimensional convection±dispersion equations. The method is semi-analytical in time through Laplace transform. Then the transformed partial di€erential equations are solved numerically in the Laplace domain using the ®nite element method. Finally the nodal concentration values are obtained through a numerical inversion of the ®nite element solution, using a highly accurate inversion algorithm. The proposed method eliminates time steps in the computation and allows using relatively large grid sizes, which increases computation eciency dramatically. Numerical results of several examples show that the hybrid method is of high eciency and accuracy, and capable of eliminating numerical di€usion and oscillation e€ectively. Ó _{1999 Elsevier Science Ltd. All rights reserved.}

Keywords:Hybrid method; Laplace transform; Finite element; Numerical diculties

1. Introduction

Convection±dispersion type equations are being widely used to model solute transport in soils and groundwater systems. However, owing to the particular combination of hyperbolic and parabolic terms, serious diculties, such as numerical di€usion and oscillation, are often encountered in obtaining accurate numerical solutions of the equations. Besides the numerical di-culties, it would be very time consuming to use the conventional numerical methods, such as the ®nite el-ement method, to solve convection±dispersion equa-tions, especially for solutions of large-scale, long-term solute transport in porous media. A variety of numer-ical schemes have been developed to deal with the dif-®culties and to improve numerical eciency. It has been shown that combination of integral transforms with numerical methods has advantages to overcome the numerical problems and enhance computation e-ciency ([13,11]).

Liggett and Liu [10] used the Laplace transform in conjunction with the boundary element method to solve the unsteady groundwater ¯ow equation. The method is semi-analytical and semi-numerical in nature. The La-place transform is ®rstly applied to the governing equation as well as initial and boundary conditions describing the physical problem. Then a numerical method is employed in the Laplace domain to solve the transformed partial di€erential equation. Finally the solution of the original problem is obtained by inverting the Laplace solutions numerically. In recent years, these kinds of new numerical methods has been developed and applied in the ®eld of heat transfer and solute transport in subsurface ¯ow ([1±4,13,14]). Chen and Chen [1±3] solved the transient heat conduction prob-lems by combining the Laplace transformation with the ®nite di€erence method or ®nite element method. Su-dicky and McLaren [14] applied the Laplace transform Galerkin technique with the numerical inversion algo-rithm of Crump [5] for large-scale simulation of mass transport in discretely fractured porous formations. Recognizing the de®ciency of the Crump [5] algorithm, Ren [11,12] developed a hybrid method of Laplace transform and ®nite element (HLTFEM), using the *_{Corresponding author. Tel.: +307 766 5032; fax: +307 766 6403;}

e-mail: renduo@uwyo.edu

algorithm by Honig and Hirdes [9] for numerical in-version of Laplace solutions. The new algorithm of Laplace transform inversion was developed based on the Fourier series approximation of Durbin [6]. The algorithm combines procedures to diminish discretiza-tion error, accelerate the convergence of Fourier pro-gression, and to select optimal parameters. Therefore, the discretization and truncation errors of the inversion algorithm by Honig and Hirdes [9] do not depend on how to choose the free parameters. On the other hand, the Crump [5] algorithm uses a di€erent method to speed up the convergence of the Fourier series ([7,6]). The method does not provide a consistent reduction of truncation errors. Its computation eciency heavily depends on the choice of the parameters, which is somewhat arbitrary. The main disadvantage of the Crump [5] algorithm is that the discretization and truncation errors depend on the choice of the free pa-rameters. For example, a choice of the parameters can result in an arbitrarily small discretization error; how-ever, at the same time the truncation error grows to in®nity and vice versa. Fortunately, the problem of the Crump [5] algorithm is resolved in the algorithm of Honig and Hirdes [9]. Because of high accuracy of the algorithm, the HLTFEM has been successfully used to solve solute transport problems in the subsurface ([11,12]).

In this paper, the HLTFEM is further developed and utilized to simulate one- and two-dimensional solute transport under uniform ¯ow conditions. Examples are analyzed to illustrate the numerical accuracy and e-ciency of the present method. The performance of the method is evaluated against results from analytical so-lutions and other numerical methods.

2. Theory

One-dimensional solute transport through a porous medium can be described by the following governing equation with speci®ed boundary conditions

c…0;t† ˆc0; …2a†

c…1;t† ˆ0 t>0 2b_† and the initial condition

cx;0_{† ˆ}0 06x6₁3_†

Herec…x;t†is the concentration_‰MLÿ3_{Š,Dthe dispersion}

coecient ‰L2Tÿ1Š, u the average pore velocity ‰LTÿ1Š, andc0 the initial concentration‰MTÿ3Š.

For solute transport in a two-dimensional porous medium under a steady water ¯ow condition, the gov-erning equation is coordinates‰LŠ;Vx andVy are the average pore velocities

LTÿ1

‰ Šin x andy directions, respectively, andDxx,Dxy,

Dyx, Dyy are the components of the hydrodynamic

dis-persion tensor for an anisotropic medium. The boun-dary and initial conditions for Eq. (4) are

c…x;y;t† ˆc1 …x;y† 2C

c…x;y;0_{† ˆ}c0 …x;y† 2X;

…5_†

whereCis the exterior boundary of the solution domain X;c1is the concentration speci®ed alongC, andc0is the

initial concentration distribution.

To remove the time derivatives from the governing equations, the method of Laplace transform is utilized. The Laplace transform of a real function f…t† and its inversion are de®ned as

Fs_{† ˆ}L_‰ft_{†Š ˆ}

wheresis the Laplace transform parameter andsˆv‡

iwwithv;w2R.

3. Formulation of ®nite element equations

3.1. One-dimensional solute transport problem

After taking the Laplace transform with respect to time, Eqs. (1), (2a) and (2b) become

o

In the ®nite element method, an approximate solution for~cis in the form of

Here u_i is the linear interpolation function, N the number of nodes in the grid,sthe Laplace transformed variable, and c~i…s† the concentration in s space at the

Applying the Galerkin procedure to Eq. (8) leads

wherePis the number of elements that are joined to the nodei, andLe _{is the length of elemente. Incorporating}

Eq. (11) into Eq. (12) and applying Green's theorem to reduce the order of the second derivative term yields a system of algebraic equations as follows

Eic~iÿ1‡Fic~i‡Gic~i‡1ˆHi iˆ1;2;. . .;n; …13†

in whichDxis the descretized size of the space.

3.2. Two-dimensional solute transport problem

After taking the Laplace transform, Eq. (4) becomes The transformed boundary condition is

~

c…x;y;s† ˆc1=s: …16†

The approximate solution of~cis de®ned by

~

the transformed concentration at node j, and N is the total number of nodes in the triangular element mesh.

Using the Galerkin procedure and applying Green's theorem to Eq. (15), we have

Z Z

Substituting Eq. (17) into Eq. (18), we obtain

Z Z

The elements of matrix‰AŠand vectorFare expressed as follows:

HereDis the area of the triangular elementewith nodes i;j, andknumbered in the counterclockwise order and

aiˆxjykÿxkyj; biˆyjÿyk; ciˆxkÿxj;

ajˆxkyiÿxiyk; bjˆykÿyi; cjˆxiÿxk;

ak ˆxiyjÿxjyi; bk ˆyiÿyj; ckˆxjÿxi:

…23_†

The coordinates of nodes i;j and k are designated as

…xi;yi†;…xj;yj†, and …xk;yk†, respectively. The element

matrices and vectors are formed sequentially and the contributions to the global matrix are summed. The matrix equation is completed with the boundary condi-tion of Eq. (16). Solving Eqs. (13) and (20), we can ob-tain the transformed concentration values at the nodes.

4. Inversion of the Laplace transformation

Letting Lÿ1 _{denote the inverse transformation, from}

Eq. (11) we have the one-dimensional solution as follows

c…x;t† ˆX

inversion algorithm requires knowledge of the value of the transformed variable for di€erent values of sˆsk;kˆ1;2;. . .; as shown below (Eqs. (30)±(34)).

Similarly, from Eq. (17) the two-dimensional solution is

c…x;y;t† ˆX

The inversion process must be perfomed numerically. The numerical inversion form of the Laplace transform can be written ([1±3]):

F…s† ˆ

Substituting Eq. (26) into Eq. (7) yields

ft_{† ˆ} 1 Combining Eqs. (26) and (27) leads

f…t† ˆ e therefore, the second integral is zero and the equation is simpli®ed as

Based on the method of Durbin [6] for the Fourier series expansion,f…t†on the interval_‰0;2T_Šcan be derived as

Since the Fourier series in Eq. (30) can only be summed to a ®nite number of terms, a truncation error is introduced in the form of

FT…M;v;t;T† ˆ

where M is the ®nite number of terms of the Fourier series expansion andT is the half period of the Fourier series approximating the inverse on the interval [0, 2T]. Then the approximate solution off…t†is ([9,1±3])

fM…t† ˆ

following two conditions, respectively, (1) the parameter vis optimal if the absolute values of discretization and truncation error are equal, and (2) the parameter v is optimal if the sum of the absolute values of discretiza-tion and truncadiscretiza-tion errors is minimal.

For the numerical Laplace inversion, the only input is the timet at which the concentration is required. Then ft_{† ˆ}Lÿ1

‰Fs_†Š is computed based on the Laplace transform F…s†. More speci®cally, the time values are given by ([9])

tk ˆT1‡ …TN ÿT1†k=…N‡1†; kˆ1;. . .;N: …35†

HereN is the number oft-values for whichf…t†is to be computed,T1 and TN are the lower and upper limits of

the interval in whichft_†is calculated. For instance, to obtain concentration results at t1ˆ160 and t2ˆ320

day (Fig. 3), we selectedT1ˆ0,TN ˆ480 andN ˆ2. In

the Laplace inversion (Eq. (33)), the chosen value t in the time domain is related to a corresponding particular value ofsin the Laplace domain by

sˆv‡ikp=T; kˆ1;. . .;M: …36_†

5. Numerical applications

5.1. Example 1

This example has become the standard test of a nu-merical scheme designed to solve the one-dimensional convection±dispersion equation. The initial and boun-dary conditions are given by

c…x;0_{† ˆ}0 tˆ0; c0;t_{† ˆ}c0 t>0;

c…1;t† ˆ0 t>0:

…37_†

The analytical solution to this problem is ([16])

cx;t_{† ˆ}c0

To discuss di€erent transport modes, a local Peclet number is used and de®ned by

Pe_ˆuDx=D: …39_†

A length xˆ30 m and the solution time tˆ15 d were chosen such that at all times between 0 and t, the boundary condition c…1;t† ˆ0 was satis®ed. Other parameters used include c0ˆ10 g/l, uˆ1 m/d, and

Dxˆ0:1 m. Di€erent dispersion coecients were used: the range of D values from 0.1 to 0.001 m2_/d,

corre-sponding to Pe_ˆ1 to 100. The HLTFEM solutions were compared with numerical solutions of the ®nite element method (FEM) and the analytical solution. As shown in Fig. 1(a)±(c), the numerical solutions of the HLTFEM and the FEM are very close to the analytic solution for Pe<2. For Pe P2, the FEM produced breakthrough

curves more di€used than the analytical solution and the numerical di€usion increased with Pe. In contrast, the HLTFEM solution did not produce any numerical dif-fusion and oscillation for any Pe.

5.2. Example 2

This example was used to test the accuracy and ro-bustness of the HLTFEM for simulating transport in a two-dimensional aquifer under one-dimensional con-stant groundwater velocity ®eld. The governing equa-tion and initial and boundary condiequa-tions are given by

o_c dinal dispersivity L†; V the average pore velocity, Dy

the e€ective molecular di€usivity in y direction, t the time, andLx andLy the total lengths of seepage ®eld in

xandy directions, respectively.

The parameters used in this example area_ˆ1:0 m, V _ˆ0:1 m/day,Dyˆ810ÿ5 m2/day, Lxˆ200 m and

Lyˆ2 m. The in¯ow boundary condition is expressed

by

Frind [8] solved the same problem using the alternation direction Galerkin element method (ADG) and the conventional Galerkin ®nite element technique (FEM). Therefore, we compared our numerical results with Frind solutions [7] as well as the analytical solution.

First, using Dxˆ2:0 m and Dyˆ0:2 m, we com-pared the HLTFEM results with those of the FEM

…Dtˆ20 days). Fig. 2(a) and (b) show the longitudinal and the transverse pro®les, respectively. The results in-dicate that the HLTFEM is of good accuracy and ca-pable of e€ectively overcoming the numerical diculties, whereas the FEM introduced numerical oscillation and numerical di€usion.

The second case was examined withDxˆ8:0 m and

very close to the analytical solutions, whereas the ADG produces both numerical oscillation and numerical dif-fusion.

5.3. Example 3

The third example is a two-dimensional semi-in®-nite convection±dispersion problem [15] with the governing equation, initial and boundary conditions as follows:

r …Drc† ˆo_c=o_{t‡ r …uc†}

c…x;y;0† ˆ0

c…0;y;t† ˆ1:0 26y63

c…0;y;t† ˆ0 06y<2;3<y65

o_c=o_{y…x;0;t† ˆ}o_c=o_{y…x;5;t† ˆ}0 x>0

o_c=o_{x…1;y;t† ˆ0 06y65}

…42_†

Simulations were conducted in a studying domain of 105, whose lower half is shown in Fig. 4 because of the symmetry of the problem at yˆ2:5. Two hundred triangular elements…N ˆ200†and 126 nodes were used to discretize the computational domain.

Following Taigbenu and Liggett [15], we examined three transport cases in the aquifer: dispersion dominant transport (Pe_ˆ0.05), dispersion and convection trans-port (Pe _ˆ1.0), and convection dominant transport (Pe

ˆ50). Other parameters used in the simulations for these cases are listed in Table 1.

The numerical results from the HLTFEM and the boundary element method (BEM) ([15]) were com-pared along with the analytical solutions. As an ex-ample, Fig. 5 presents concentration breakthrough curves along the axisyˆ2:5 for Peˆ50. The Courant number of the BEM was 0.4. The ®gure shows a

reasonable agreement between the HLTFEM results and the analytical solution. There are both profound numerical oscillation and di€usion in the BEM solu-tion. The HLTFEM solution exhibits no numerical di€usion and some numerical oscillation in the upwind nodes.

The HLTFEM is an inherently non-time-marching method. In other words, the method calculates nodal concentrations at a speci®c time using one time step. In contrast, the conventional FEM must compute nodal concentrations through numerous intermediate time steps until the required time is reached. In many cases, the time step for the FEM has to be very small to guarantee numerical convergence and accuracy. There-fore, the HLTFEM is a time-saving procedure to solve long-time chemical transport problems in large ®eld domains. Fig. 6 shows a typical relationship between the required CPU time as a function of the total simulation time for a ®xed node number. The CPU time of the HLFEM is mainly used for numerical inversion, therefore, more or less constant for any total simulation time. However, the CPU time of the FEM increases

Fig. 2. Concentration pro®les (Example 2,Dxˆ2 m andDyˆ0:2 m) calculated with the HLTFEM, the Galerkin FEM, and the analytical solution for (a) longitudinal (yˆ0) and (b) transverse (xˆ8) pro®les.

Fig. 3. Concentration curves (Example 2,Dxˆ8 m andDyˆ0:2 m) calculated with the HLTFEM, the alternation direction Galerkin ele-ment method (ADG), and the analytical solution for the longitudinal pro®le…yˆ0†.

approximately linearly or exponentially as the total simulation time increases.

6. Conclusions

A hybrid Laplace transform ®nite element method (HLTFEM) was developed for solving

convection±dis-persion problems. Several application examples have shown that the HLTFEM has great potential in mod-eling of solute transport in one- and two-dimensional ®elds under steady-state ¯ow conditions.

Compared with the conventional Galerkin ®nite ment technique, the alternation direction Galerkin ele-ment method and the boundary eleele-ment method, the HLTFEM shows many advantages. The HLTFEM has higher accuracy in simulating sharp solute fronts in convection-dominant problems and overcomes numeri-cal di€usion as well as oscillation e€ectively. The HLTFEM is an inherently non-time-marching method to calculate nodal concentrations at any speci®c time. Therefore, the method eliminates the disadvantage of the step by step computation in the time domain and provides high computation eciency as well as low computation cost. The HLTFEM is especially useful to predict long-time soil and groundwater contamination in large ®eld domains with stable and highly accurate solutions.

Acknowledgements

The authors express appreciation to Professors W. Zhang and Y. Zhang at Wuhan University of Hydraulic and Electric Engineering for their guidance and en-couragement.

References

[1] Chen HT, Chen CK. Hybrid Laplace transform/®nite di€erence method for transient heat conduction problems. Int J Numer Meth Eng 1988;26:1433±47.

[2] Chen HT, Chen CK. Hybrid Laplace transform/®nite-element method for two-dimensional heat conduction. J Thermophys 1988;2:31±6.

[3] Chen HT, Chen CK. Application of hybrid Laplace transform/ ®nite-di€erence method to transient heat conduction problems. Numer Heat Transfer 1988;14:343±56.

[4] Chen HT. Hybrid method for transient response of circular pins. Int J Numer Meth Eng 1990;29:303±14.

[5] Crump KS. Numerical inversion of Laplace transform using Fourier series approximation. J Assoc Comput Mach 1976;23:89± 96.

[6] Durbin F. Numerical inversion of Laplace transforms: ecient improvement to Dubner and Abate's method. Comput J 1973;17:371±76.

[7] Dubner H, Abate J. Numerical inversion of Laplace transforms by relating them to the ®nite Fourier cosine transform. J. Assoc Comput Mach 1968;15:115±23.

Fig. 5. Longitudinal concentration pro®les (Example 3,y_ˆ2:5 m, and Pe_ˆ50) calculated with the HLTFEM, the boundary element method (BEM), and the analytical solution.

Fig. 6. A typical relationship between the required CPU time as a function of the total simulation time for a ®xed node number. Table 1

Parameters of Example 3

Type of transport Pe u DxxˆDyy DxˆDy Solution time,t

Dispersion dominant 0.05 1 10 0.5 4.0

Dispersion±convection 1.0 1 0.5 0.5 4.0

[8] Frind EO. The principal direction technique: a new approach to groundwater contaminant transport modeling. In: Proceedings of the International Conference on Finite Element in Water Resources 1982;13:25±42.

[9] Honig G, Hirdes U. A method for the numerical inversion of Laplace transform. J Comput Appl Meth 1984;10:113±32. [10] Liggett JA, Liu P. The boundary integral equation method for

porous media ¯ow. Winchester: Allen and Unwin, 1983. [11] Ren L. Research on calculating method of solute transport

in subsurface water and on numerical simulation of moisture movement and heat transfer in unsaturated soils, Ph.D. Thesis, Univerity of Wuhan Hydraulic. Electr. Eng., China, 1994.

[12] Ren L. A hybrid Laplace transform ®nite element method for solute radial dispersion problem in subsurface ¯ow. J Hydrody-namics 1994;9:37±43.

[13] Sudicky EA. The Laplace transform Galerkin technique a time-continuous ®nite element theory and application to mass transport in groundwater. Water Resour Res 1989;25:1833±46.

[14] Sudicky EA, Mclaren RG. The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations. Water Resour Res 1992;28:499±514.

[15] Taigbenu A, Liggett JA. An integral solution for the di€usion-advection equation. Water Resour Res 1986;22:1237±46. [16] van Genuchten MT, Alves WJ. Analytical solutions of the