Adam Szymkiewicz

Two-scale modeling of unsaturated water flow in a double-porosity medium under axisymmetric conditions

In this paper the development and experimental validation of a numerical model of two-dimensional unsaturated flow in a double-porosity medium is presented. The model is based on the coupled formulation for flow in macro- and micropores obtained by homogenization. It was applied to simulate the axisymmetrical tension disk infiltration experiments that were carried out in a double-porosity medium. The physical model was a three-dimensional periodic structure, composed of porous spheres made of sintered clay and embedded in Hostun fine sand HN38. The hydraulic parameters of both porous materials were determined by inverse analysis of independent infiltration experiments performed on sand and sintered clay. The effective parameters of the double-porosity medium were calculated from the solution of the local boundary value problem, obtained from the homogenization procedure. The cumulative infiltration curve and the global dimensions of the humidified zone obtained from the numerical solution are in good agre...

Simulations of freshwater lens recharge and salt/freshwater interfaces using the HYDRUS and SWI2 packages for MODFLOW

Abstract The paper presents an evaluation of the combined use of the HYDRUS and SWI2 packages for MODFLOW as a potential tool for modeling recharge in coastal aquifers subject to saltwater intrusion. The HYDRUS package for MODFLOW solves numerically the one-dimensional form of the Richards equation describing water flow in variablysaturated media. The code computes groundwater recharge to or capillary rise from the groundwater table while considering weather, vegetation, and soil hydraulic property data. The SWI2 package represents in a simplified way variable-density flow associated with saltwater intrusion in coastal aquifers. Combining these two packages within the MODFLOW framework provides a more accurate description of vadose zone processes in subsurface systems with shallow aquifers, which strongly depend upon infiltration. The two packages were applied to a two-dimensional problem of recharge of a freshwater lens in a sandy peninsula, which is a typical geomorphologic form along the Baltic and the North Sea coasts, among other places. Results highlighted the sensitivity of calculated recharge rates to the temporal resolution of weather data. Using daily values of precipitation and potential evapotranspiration produced average recharge rates more than 20% larger than those obtained with weekly or monthly averaged weather data, leading to different trends in the evolution of freshwater-saltwater interfaces. Root water uptake significantly influenced both the recharge rate and the position of the freshwater-saltwater interface. The results were less sensitive to changes in soil hydraulic parameters, which in our study were found to affect average yearly recharge rates by up to 13%.

Influence of heterogeneous air entry pressure on large scale unsaturated flow in porous media

The paper presents numerical simulations of water infiltration in unsaturated porous media containing coarse-textured inclusions embedded in fine-textured background material. The calculations are performed using the two-phase model for water and air flow and a simplified model known as the Richards equation. It is shown that the Richards equation cannot correctly describe flow in the presence of heterogeneities. However, its performance can be improved by introducing appropriately defined effective capillary and permeability functions, representing largescale behaviour of the heterogeneous medium.

Mathematical Models of Flow in Porous Media

In this chapter a general model for the two-phase fluid flow in porous media is presented, together with its simplified form, known as the Richards equation, which is applicable (under specific assumptions) to describe water flow in the vadose zone. In each case the governing equations are formulated at the Darcy scale, using the capillary pressure–saturation relationship and an empirical extension of the Darcy equation for the multiphase flow. Initial and boundary conditions for the governing equations are also discussed, together with conditions applicable at material interfaces.

Experimental and Numerical Analysis of Air Trapping in a Porous Medium with Coarse Textured Inclusions

The paper presents a 2D upward infiltration experiment performed on a model porous medium consisting of fine sand background with two inclusions made of coarser sands. The purpose of the experiment was to investigate the effects of structural air trapping, which occurs during infiltration as a result of heterogeneous material structure. The experiment shows that a significant amount of air becomes trapped in each of the inclusions. Numerical simulations were carried out using the two-phase water-air flow model and the Richards equation. The experimental results can be reproduced with good accuracy only using a two-phase flow model, which accounts for both structural and pore-scale trapping. On the other hand, the Richards equation was not able to represent the structural trapping caused by material heterogeneity.

Examples of numerical simulations of two- dimensional unsaturated flow with VS2DI code using different interblock conductivity averaging schemes

Abstract Flow in unsaturated porous media is commonly described by the Richards equation. This equation is strongly nonlinear due to interrelationships between water pressure head (negative in unsaturated conditions), water content and hydraulic conductivity. The accuracy of numerical solution of the Richards equation often depends on the method used to estimate average hydraulic conductivity between neighbouring nodes or cells of the numerical grid. The present paper discusses application of the computer simulation code VS2DI to three test problems concerning infiltration into an initially dry medium, using various methods for inter-cell conductivity calculation (arithmetic mean, geometric mean and upstream weighting). It is shown that the influence of the averaging method can be very large for coarse grid, but that it diminishes as cell size decreases. Overall, the arithmetic average produced the most reliable results for coarse grids. Moreover, the difference between results obtained with various methods is a convenient indicator of the adequacy of grid refinement.

Flow in Binary Media with Heterogeneous Hydraulic Diffusivity

Unsaturated flow in binary medium with inclusions dispersed in background material is considered. It is assumed that the Richards equation is a valid model for flow in both porous materials, the flow is locally dominated by the capillary forces and there is no air entry pressure in the capillary functions. The inclusions and background differ in the permeability and capillary diffusivity functions. Upscaled models obtained using the method of asymptotic homogenization for various ratios of the characteristic hydraulic diffusivities in the two porous materials are discussed. It is shown that the inclusion-background diffusivity ratio has important influence on the field-scale behaviour of the system. A generalized model suitable for the whole range of the diffusivity ratio values is described. Numerical results obtained with this model are compared to the Darcy-scale simulations with explicit representation of material heterogeneity and to experimental results for two-dimensional axi-symmetric infiltration.

Numerical Solution of Flow Equations

Numerical solution of nonlinear partial differential equations, which describe two-phase and unsaturated flow, includes several stages: spatial discretization, temporal discretization, linearization of the discrete equations and solution of linear algebraic equations. The basic spatial discretization techniques, i.e. finite difference, finite element and finite volume approaches are described, and the similarities and differences between them are emphasized. The problems related to the spatial averaging of nonlinear parameters appearing in the governing equations are outlined. A brief overview of the time discretization techniques is provided, with the focus on the fully implicit first order scheme, which is used in the subsequent simulations. Two basic methods for the solution of the systems of nonlinear algebraic equations, i.e. the Newton and Picard schemes are presented and the relationship between them is discussed. The choice of time step size and the solution of linear algebraic systems are also briefly covered.

Flow in Binary Media with Heterogeneous Air-Entry Pressure

In this chapter, the influence of heterogeneity in the air-entry pressure on the field-scale flow is examined. This issue is particularly important for media containing disconnected coarse-textured inclusions with low entry pressure embedded in a continuous fine-textured background with high entry pressure. During capillary-driven imbibition, the background material becomes water-saturated at a higher value of the capillary pressure than inclusions. The air phase loses its continuity and becomes trapped in inclusions. During drainage of fully water-saturated medium the inclusions can be drained only when the air entry pressure of the background material is exceeded. At the Darcy scale these two effects can be captured by the two-phase model, but not by the Richards equation, which does not account for air flow. However, at the field scale it is possible to represent these phenomena using either the two-phase formulation or the Richards model, on condition that the field-scale capillary and permeability functions are appropriately modified.

Computation of Inter-Nodal Permeabilities for Richards Equation

An important part of all finite difference and many finite volume discretization schemes developed for multiphase flow equations is the approximation of the average permeability value between two neighbouring nodes. Various averaging techniques are presented in this chapter, with particular focus on the case of one-dimensional unsaturated flow in a homogeneous medium, for which accurate inter-nodal permeability estimations based on steady flow analysis are available. It is shown that the relation between capillary and gravity forces at the scale of a single grid cell has key importance for the choice of the averaging scheme. An averaging method developed by the author for one-dimensional flow is presented in detail, and its extensions to heterogeneous materials and multidimensional problems are discussed. Implications for two-phase flow modelling are also considered.