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To investigate a model to simulate wheat dry matter accumulation, three wheat cultivars with different tillering abilities were grown at three densities each in a field experiment. Five simulation models with high correlation coefficients for relative dry matter accumulation were established by the method of normalized. Among these models, the Richards equation was the best in fitting and forecasting, i.e., y = 1.1435/(1 + e0.2776 − 4.6558x )1/0.1130, r = 0.9927. Correlation coefficient of grey comprehensive relationship degree between actual dry matter accumulation in occurrence time of maximum rate of dry matter accumulation and dry matter weight was highest, so a higher actual dry matter weight in occurrence time of maximum rate of dry matter accumulation played an important role in stabilizing and improving dry matter weight of wheat.

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.

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.

In this work, two-phase flow in porous media is simulated numerically with Discontinuous Galerkin methods. The Symmetrical Interior Penalty Galerkin method (SIPG), the Non-symmetrical Interior Penalty Galerkin method (NIPG) and the scheme from Oden, Babuska and Baumann (OBB) are considered. The terminology and the examples are taken from soil science. First, the Richards equation is solved, then a two-phase flow problem in the pressure-saturation formulation. The results of a final year project in computer science are presented and it is shown what a student can achieve with Dune-PDELab within six months.

The soil-root-stem water pathway is a major component of the subsurface hydrological system. We model water uptake and water transport through the soil and through the root system as coupled processes. The model is based on a differential equation approach. The coupled non-linear equation system is solved numerically. On the macroscale, fluid flow through the soil matrix within and outside of the rooted area is modelled using a two-dimensional, axial-symmetric Richards equation. Flow through the root system is described by a Darcy equation. Water uptake and transport through the root system are considered to be passive processes; osmotic effects are neglected. In order to obtain the source and sink distribution of the root system, local soil water potentials (near a single root) are computed in a one-dimensional approximation in a further step. The model is capable of simulating hydraulic lift, which is the transport of water from moist into drier soil layers through plant root systems. The calculated amount of passively shifted water is consistent with experimental data. We discuss under which conditions hydraulic lift could be an optimised strategy for the plant and how water uptake patterns are modified by the multi-scale (local) approach.

Field and theoretical efforts indicate that the presence of entrapped gas bubbles below the water table may influence the propagation dynamics of excess pore pressure caused by load changes. The process of pore pressure propagation depends on the volume of the occluded gas phase and on the relation between hydraulic conductivity of the dynamics of (hydraulic or mechanical) loading changes at the boundary. In the context of safety considerations the estimation of excess pressure dissipation dynamics may be essential. Based on Boyle-Mariotte and Henry principles predictions on water saturation versus pore pressure were established and the soil-water characteristic curve was extended from the unsaturated zone throughout the zone of air occlusion. Analogous to unsaturated zone concepts a pore pressure dependent water-capacity relationship was derived for the zone of the residual gas phase and set in 2D FE-model based on Richards equation. The extended model accounting for gas entrapment was evaluated against an explicit 2-phase model, laboratory experiments and field observations. The approach presented enables the inspection of the effect of gas entrapment on pressure propagation considering the transition from saturated to unsaturated state.

Groundwater flow in unsaturated soil is governed by Richards equation, a nonlinear convection-diffusion equation. The process is normally convection-dominated, and steep fronts are common in solution profiles. The problem is further complicated if the medium is heterogeneous, for example when there are two or more different soil layers. In this paper, the least squares finite element method is used to solve for flow through 5 layers with differing hydraulic properties. Solution-dependent coefficients are constructed from smooth fits of experimental data. The least squares finite element approach is developed, along with the method for building an optimized, nonuniform grid. Numerical results are presented for the 1D problem. Generalization to higher dimensions is also discussed.

In this work we simulate numerically the transport and biodegradation of an organic contaminant (BTEX) in the subsurface. A “real world” contaminated site is considered and realistic laboratory-derived or field-measured input parameters are used. The biodegradation of the dissolved contaminant plume is modelled by Monod type kinetics. For the computations we use an approximation scheme which is based on a higher order finite element method and the two step backward differentiation formula and was proposed and carefully analyzed by the author in a recent work [3]. It was successfully applied to test (benchmark) problems reported in the literature (cf. [10, 13]) as well as complex scenarios with an additional numerical computation of the flow field by solving the parabolic-elliptic degenerate Richards equation; cf. [3]. The higher order approximation scheme has shown to reduce significantly the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process and wrong prediction is avoided.

This paper deals with numerical algorithms for solving elliptic-parabolic problems. An example of such problem is given by the Richards equation for modeling the saturated-unsaturated water flow in porous media. We consider a linear model problem and investigate the convergence of two finite-volume schemes. The first one uses the implicit approximation in the whole domain, and the second scheme is constructed using the splitting method. Results of numerical experiments are also given.

The many simultaneously occurring processes in unsaturated-saturated heterogeneous soils and fractured rocks can cause field observations to become imprecise and incomplete. As a result, field observations can become inconsistent with deterministic and stochastic mathematical models used for predictions. The performance of a system that includes soil, rock, and a monitoring network can become uncertain because of vagueness or “fuzziness,” which is inherent in the system behavior, rather than being purely random. Fuzzy systems modeling, already widely used in such fields as engineering, physics, chemistry, and biology, is expected to become useful in simulating hydrogeologic system behavior. After presenting a hydrogeologic system as a fuzzy system, we derive a fuzzy-logic form of Darcy’s equation. Based on this equation, a fuzzy logic form of the parabolic-type partial differential equations is derived. The elliptic-type (Laplace equation) and the parabolic-type (Richards equation) partial differential equations were approximated using fuzzy variables and solved using the basic principles of fuzzy arithmetic. The results of fuzzy systems modeling are then compared with those obtained using deterministic models. The application of fuzzy ordinary and partial differential equations to various earth sciences problems, such as flow and transport in the subsurface, is an emerging area of research.

For the Richards equation, a nonlinear elliptic-parabolic partial differential equation modelling saturated-unsaturated flow in porous media, we present a hybrid mixed finite element discretization. The efficiency of the algorithm is improved by local time step and grid adaption. The adaption algorithms are based on rigorous error estimators, whose derivation is indicated. Examples elucidate the performance of the algorithm

We know that hydrological processes are highly complex. We know that the techniques available to assess the complexity of hydrological processes and parameters are inadequate, even for small scale field studies. The observability problem is particularly acute for subsurface flow processes where most measurement techniques can give only a very local indication of the hydrological response. In a series of recent papers I have highlighted some of the problems of subsurface flow modelling in the context of predicting hydrological responses at the hillslope, catchment and larger scales ; the interlinked problems of process, of heterogeneity and of scale (Beven 1987,1989a,b, 1991, 1993). There have also been a number of recent papers that have discussed these problems such as Grayson et al. (1992) but, as yet, very little in the way of progress towards resolving those problems. With some exceptions, hydrological modelling of subsurface processes continues to rely on a Darcian or Richards equation framework coupled to a concept of effective parameters with an a priori assumption that such a concept is valid regardless of the scale of application.