Simulations of Unsaturated Flow in Multiply Zoned Media by Green Element Models

Flows in variably saturated media that exhibit second-type heterogeneity, in which abrupt changes of medium parameters occur, are simulated by the Green element method (GEM). Such media are usually encountered where soil formations have arisen by different geological or geomorphological processes spread over different time scales. Two challenges are posed when simulating flows in multiply zoned unsaturated media: one is the highly nonlinear nature of the flow within each zone, and the other is dealing with sharp contrast in medium parameters at the interfaces of different zones. Both challenges are accommodated in this paper using a flux-based Green element formulation to simulate the flow and incorporating the Picard and Newton–Raphson (N-R) algorithms to simplify the nonlinear discrete equations. Calculations are carried out on three numerical examples of infiltration into unsaturated soils in two spatial dimensions. The convergence rate of the N-R algorithm is superior to the Picard algorithm only for the first example, while none of the algorithms has a clear advantage for the other two examples. The N-R algorithm suffers from repeated calculation of derivatives of the medium parameters with respect to the pressure head, thereby compromising the accuracy of the solution and increasing computational cost.     

Laboratory Investigation of Variable-Density Flow and Solute Transport in Unsaturated–Saturated Porous Media

In many groundwater systems, fluid density and viscosity may vary in space and time as a function of changes in concentration and temperature of the fluid. When dense groundwater plumes interact with less dense ambient groundwater, these density variations can significantly affect flow and transport processes. Under certain conditions, gravitational instabilities in the form of lobe-shaped fingers can occur. This process is significant because it can lead to more rapid and spatially extensive solute transport. This paper presents new experiments carried out in a sand filled glass flow container under both fully saturated and variably-saturated conditions and focuses upon the processes that occur at the capillary fringe and below the water table, as affected by a dense contaminant plumes migration through the unsaturated zone. Source fluids stained with Rhodamine-WT were introduced at the upper boundary of the tank at a range of low and high densities. In addition to the fluid density gradients and porous medium permeability that determine the onset conditions for instabilities in fully saturated experiments, volumetric water content appears critical in the variably-saturated laboratory runs. Plume behaviour at the water table appears dependent upon the density of the fluid that accumulates there. For neutral and low density fluids, plumes accumulate at the water table and then spread laterally above it and the water table forms a barrier to further vertical flow as pore water velocities reduce with increasing water content. For medium and high density fluids, vertical movement continues as instabilities form at the capillary fringe and fingers begin to grow at the water table boundary and move downwards into the saturated zone. In these cases, lateral spreading of the plume is small. Despite these more qualitative observations, the exact nature of the relevant stability criteria for the onset and growth of instabilities in variably-saturated porous media presently remain unclear. All experimental results suggest, however, that the unsaturated zone and position of the water table must be considered in contaminant studies in order to predict the migration pathways, rates and ultimate fate of dense contaminant plumes. It is possible that the results of experiments presented in this paper could form a useful basis for the testing of variable-density (and variably-saturated) groundwater flow and solute transport numerical codes because they offer controlled physical laboratory analogs for comparison. They also provide a strong basis for the development of more rigorous mathematical formulations that are likely to be either developed or tested using numerical flow and solute transport simulators.     

Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media

Abstract. The Picard and modified Picard iteration schemes are often used to numerically solve the nonlinear Richards equation governing water flow in variably saturated porous media. While these methods are easy to implement, they are only linearly convergent. Another approach to solve the Richards equation is to use Newton's iterative method. This method, also known as Newton–Raphson iteration, is quadratically convergent and requires the computation of first derivatives. We implemented Newton's scheme into the mixed form of the Richards equation. As compared to the modified Picard scheme, Newton's scheme requires two additional matrices when the mixed form of the Richards equation is used and requires three additional matrices, when the pressure head-based form is used. The modified Picard scheme may actually be viewed as a simplified Newton scheme.Two examples are used to investigate the numerical performance of different forms of the 1D vertical Richards equation and the different iterative solution schemes. In the first example, we simulate infiltration in a homogeneous dry porous medium by solving both, the h based and mixed forms of Richards equation using the modified Picard and Newton schemes. Results shows that, very small time steps are required to obtain an accurate mass balance. These small times steps make the Newton method less attractive.In a second test problem, we simulate variable inflows and outflows in a heterogeneous dry porous medium by solving the mixed form of the Richards equation, using the modified Picard and Newton schemes. Analytical computation of the Jacobian required less CPU time than its computation by perturbation. A combination of the modified Picard and Newton scheme was found to be more efficient than the modified Picard or Newton scheme.     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

3D simulation of radionuclide transport in porous media

We present an efficient and easily implementable finite volume method simulating radionuclide transport through highly heterogeneous grounds in three space dimensions. The numerical concentration of the transported chemicals are proved to remain nonnegative and stable. Then, we run a realistic test case in which some radioactive iodine I¹²⁹ particles are released from a leak in an underground nuclear waste disposal site. The question of whether the radionuclide invades the underground and reach the ground surface is investigated. Because of the 3D nature of the problem, a particular emphasis is made on the control of CPU time. Copyright © 2009 John Wiley & Sons, Ltd.     

Pore-Level Modeling of Gas and Condensate Flow in Two- and Three-Dimensional Pore Networks: Pore Size Distribution Effects on the Relative Permeability of Gas and Condensate

We present a mechanistic model of retrograde condensation processes in two- and three-dimensional capillary tube networks under gravitational forces. Condensate filling-emptying cycles in pore segments and gas connection–isolation cycles are included. With the pore-level distribution of gas and condensate in hand, we determine their corresponding relative permeabilities. Details of pore space and displacement are subsumed in pore conductances. Solving for the pressure field in each phase, we find a single effective conductance for each phase as a function of condensate saturation. Along with the effective conductance for the saturated network, the relative permeability for each phase is calculated. Our model porous media are two- and three-dimensional regular networks of pore segments with distributed size and square cross-section. With a Monte Carlo sampling we find the optimum network size to avoid size effects and then we investigate the effect of network dimensionality and pore size distribution on the relative permeabilities of gas and condensate.