Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

Analytical and Numerical Solutions... 261
Using meliorated higher-order operator-splitting methods, we can improve our methods
for the solution of the full equations.
The methods are verified by benchmark problems and the numerical results are compared
with the analytical solutions. General initial conditions are verified by polynomial
initial conditions in an axisymmetric two-dimensional application. A real-life problem is
presented as a simulation of a waste-disposal with realistic parameters and underlying layers
in the porous media. The calculations are presented within the figures and convergence
results. Some applications containing these methods are computed with the underlying
programme tool R 3 T , and the main concepts are presented.
The paper is outlined as follows. We introduce our mathematical model of a contaminant
transport in flowing groundwater in Section 2.. The decomposition methods are presented
in Section 3.. In Section 4. we introduce the finite volume methods to be used as basic
discretization methods to develop higher-order methods based on the analytical solutions.
The analytical solutions with general initial conditions, such as embedded mass-conserved
equations for the finite volume methods, are explained in Section 5.. Our numerical results
with benchmark problems and realistic waste-disposals are described in Section 6.. Finally
we discuss our future works in Section 7. with respect to our research area.
2. Mathematical Model
The motivation for the study presented below comes from a computational simulation of
radionuclide contaminants transported in flowing groundwater [10], [13].
The mathematical equations are given by:
∂ t R α (c α ) + ∇ · (vc α − D∇c α ) + λ αβ R α (c α ) = ∑
λ γα R γ (c γ ) , (1)
γ∈γ(α)
R α (c α ) = (φ + (1 − φ)ρK(c α )) c α , (2)
with α = 1, . . .,m . (3)
The unknowns c α = c α (x, t) are considered in Ω×(0, T) ⊂ IR d ×IR, the space-dimension
is given by d. φ is the porosity, ρ is the density. The retardation R α (c α ) is given as a
linear or nonlinear function. For the linear retardation we have the Henry-Isotherm with
K(c α ) = Kd α, where Kα d is the element-specific K d-parameter. We simplify the model
and assume that for each element one isotope confers our complex model [18]. For the
nonlinear retardation-factor we have the Freundlich-Isotherm with K(c α ) = K nl (c α ) p−1 ,
where K nl is the specific sorption-constant and p is the exponent of the isotherm. Another
nonlinear retardation-factor is the Langmuir-Isotherm with K(c α ) =
κ b
1+b c α
, where b is the
specific sorption-constant and κ is the specific sorption-capacity. The other parameters are
λ αβ , the decaying rates from α to β, where γ(α) are the predecessor elements of element
α. D is the Scheidegger diffusion-dispersion tensor and v is the velocity.
The main aim of this paper is to present new methods, which are based on exact solutions
for simpler one-dimensional equations. For this purpose we derive analytical solutions
for the simpler one-dimensional convection-reaction equation and also for the nonlinear reaction
equations. The analytical solutions are used for the explicit time-discretization and
spatial-discretization with finite volume methods for d-dimensions.