The Laplace Transform Galerkin technique for efficient time-continuous solution of solute transport in double-porosity media

Abstract A new and efficient technique is presented for numerical solution of solute transport problems in multi-dimensional double-porosity media. The method is based on the Laplace Transform Galerkin (LTG) technique (Sudicky, 1989) and employs a convolution integral in the governing advection-dispersion equation to express the diffusive exchange of solute between the mobile and immobile regions. Because the Laplace transformation is applied to the advection-dispersion equation describing transport in the mobile region, the convolution integral, in addition to the temporal derivative, is effectively removed. After the transformed equation is solved using the Galerkin finite element method, nodal concentrations in the time-domain are obtained using a robust and efficient algorithm for inversion of the Laplace-domain nodal concentrations. The method permits the solution to be evaluated at any future point in time without any need for time steps and permits the use of grids having a relatively coarse spatial discretization while avoiding significant numerical dispersion. Further computational efficiency is achieved for large-grid problems by employing an ORTHOMIN-accelerated iterative solver (Vinsome, 1976) to solve the system of Laplace-domain finite element equations. Influence functions describing the mobile-immobile region solute transfer are presented for the case of slab and sphere geometries for the immobile-fluid zone and for the case of first-order exchange theory, along with expressions which unify the three approaches. The utility of the model is demonstrated by applying it to the problem of solute transport in a sandy-type aquifer having a random, spatially-correlated hydraulic conductivity field and comprised of slightly porous grains that admit intragranular diffusion but do not conduct fluid.

[1]  Emil O. Frind,et al.  Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures , 1982 .

[2]  P. K. W. Vinsome,et al.  Orthomin, an Iterative Method for Solving Sparse Sets of Simultaneous Linear Equations , 1976 .

[3]  Robert Bibby,et al.  Mass transport of solutes in dual‐porosity media , 1981 .

[4]  M. V. Genuchten,et al.  Mass transfer studies in sorbing porous media. I. Analytical solutions , 1976 .

[5]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[6]  A. Mantoglou,et al.  The Turning Bands Method for simulation of random fields using line generation by a spectral method , 1982 .

[7]  E. A. Sudicky,et al.  Contaminant transport in fractured porous media: Analytical solution for a single fracture , 1981 .

[8]  A. Cheng,et al.  An efficient laplace transform solution for multiaquifer systems , 1989 .

[9]  E. A. Sudicky,et al.  The Laplace Transform Galerkin Technique: A time‐continuous finite element theory and application to mass transport in groundwater , 1989 .

[10]  B. Davies,et al.  Numerical Inversion of the Laplace Transform: A Survey and Comparison of Methods , 1979 .

[11]  Ivars Neretnieks,et al.  Diffusion in the rock matrix: An important factor in radionuclide retardation? , 1980 .

[12]  Robert W. Gillham,et al.  Experimental Investigation of Solute Transport in Stratified Porous Media: 2. The Reactive Case , 1985 .

[13]  Kenny S. Crump,et al.  Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation , 1976, J. ACM.

[14]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[15]  J. W. Mercer,et al.  An efficient finite element technique for modeling transport in fractured porous media: 1. Single species transport , 1983 .

[16]  G. Pinder,et al.  Computational Methods in Subsurface Flow , 1983 .

[17]  Peter A. Forsyth,et al.  Incomplete Factorization Methods for Fully Implicit Simulation of Enhanced Oil Recovery , 1984 .

[18]  M. V. Genuchten,et al.  Models for simulating salt movement in aggregated field soils , 1986 .

[19]  J. Ross Macdonald,et al.  Accelerated Convergence, Divergence, Iteration, Extrapolation, and Curve Fitting , 1964 .

[20]  W. P. Ball,et al.  Characterization of a sandy aquifer material at the grain scale , 1990 .

[21]  K. H. Coats,et al.  Dead-End Pore Volume and Dispersion in Porous Media , 1964 .

[22]  A. Rasmuson.  Migration of Radionuclides in Fissured Rock: Analytical Solutions for the Case of Constant Source Strength , 1984 .

[23]  Johan Bouma,et al.  Soil morphology and preferential flow along macropores , 1981 .

[24]  W. G. Gray,et al.  Finite Element Simulation in Surface and Subsurface Hydrology , 1977 .

[25]  R. E. Jessup,et al.  EXPERIMENTAL AND THEORETICAL ASPECTS OF SOLUTE DIFFUSION IN SPHERICAL AND NONSPHERICAL AGGREGATES , 1982 .

[26]  E. Sudicky,et al.  An Advection‐Diffusion Concept for Solute Transport in Heterogeneous Unconsolidated Geological Deposits , 1984 .

[27]  Emil O. Frind,et al.  Comparative error analysis in finite element formulations of the advection-dispersion equation , 1985 .

[28]  A. Warrick,et al.  A Two-Phase Model for the Miscible Displacement of Reactive Solutes in Soils1 , 1974 .

[29]  I. Herrera.  Theory of Multiple Leaky Aquifers , 1970 .

[30]  E. Sudicky.  A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process , 1986 .