Unsaturated Flow Equation Research Articles

Finite Analytic Method for Solving the Unsaturated Flow Equation

The unsaturated zone plays an important role in groundwater recharge, discharge, and the ecological environment. Mathematical models are essential to the investigations of flow processes in the unsaturated zone. The governing equations of the mathematic models for unsaturated flow are nonlinear since their unsaturated hydraulic parameters of the equations depend on their solutions (i.e., pressure or moisture content). The nonlinearity can become very strong with flow in relatively dry soils, and the equation has hyperbolic characteristics. Therefore, the traditional numerical methods, such as the finite difference method and the finite element method, could lead to the numerical oscillation, dispersion, and the divergence of the solution if improper time and space steps were selected. In this study, a finite analytic method (FAM) to solve Richards' equation was developed. The basic idea of the FAM is the incorporation of the analytic solution in a small local element to formulate the algebraic representation of the partial differential equation of the unsaturated flow. Therefore, the finite analytic numerical method can effectively control the numerical oscillation and dispersion. The convergence and stability of finite analytic numerical scheme are then proven by a rigorous mathematical analysis. Numerical experiments are then used to show that FAM is highly accurate by comparing its results with those obtained using analytical solutions by a modified Picard finite difference method. In addition, we also demonstrate that FAM reproduces results of laboratory experiments. Therefore, it can be considered an appropriate simulation method for the unsaturated flow.

The foam drainage equation for unsaturated flow in porous media

[1] A class of capillary flows in unsaturated porous media is characterized by quasi steady viscous flow confined behind curved air-water interfaces and within liquid bodies held by capillary forces along crevices and grain contacts. The geometry of the connected capillary liquid network within the pore space resembles channels that form between adjacent bubbles in foam (Plateau borders) with solid grains representing gas bubbles in foam. For simplified channel geometry, we combine expressions for viscous flow with continuity considerations to describe the evolution of the channels cross-sectional area during gravity drainage. This formulation enables modeling of unsaturated flow without invoking the Richards equation and associated hydraulic functions. We adapt a formalism originally developed for foam “free drainage” (drainage under gravity) or “forced drainage” (infiltration front motion) to a class of unsaturated flows in porous media that require a few input parameters only (mean channel corner angle, air entry value, and porosity) for certain initial and boundary conditions. We demonstrate that the reduction in capillary channel cross section yields a consistent description of self-regulating internal fluxes toward attainment of the so-called “field capacity” in soil and provides an alternative method for interpretation of outflow experiments for prescribed pressure boundary conditions. Additionally, the geometrically explicit formulation provides a more intuitive picture of capillary flows across textural boundaries (changes in channel cross section and number of channels). The foam drainage methodology expands the range of tools available for analyses of unsaturated flow processes and offers more realistic links between liquid configuration and flow dynamics in unsaturated porous media.

Experimental Study and Numerical Modeling of Surface/Subsurface Flow at Field Scale

Infiltration which occurred in the process of runoff on farmland surface causes the increasing of soil moisture in unsaturated zone. It’s necessary to comprehend the dynamic process during farmland runoff process, analyze the coupling mechanism of water movement and solute transport among surface/subsurface zone. In this paper, a physically-based, spatially-distributed model was built for simulation of surface/subsurface flow and the interactions between two domains. The system is represented by the three-dimensional saturated–unsaturated flow equation for the subsurface, coupled with the diffusion wave equation for overland flow, Ground surface unevenness at the grid scale is incorporated via the concept of detention storage. The system of equations is discretized using a fully implicit procedure, with the Newton-Raphson method to handle non-linearity of the equations. In order to assess this modeling approach, the simulations compared with the experiment results. It shows that the model can simulate the process of farmland runoff production and soil water movement accurately. Finally, Statistics and mass balance analysis was also performed to examine the model accuracy and response to changes of specific input data.

A reduced-order finite volume element formulation based on POD method and implementation of its extrapolation algorithm for unsaturated soil water flow equation

A proper orthogonal decomposition(POD) method is used to treat a classical finite volume element(FVE) formulation for unsaturated soil water flow problem.A reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional unsaturated soil water flow problem is established and the error estimates between the reduced-order FVE solutions and the classical FVE solutions and the implementation of extrapolation algorithm for solving the reduced-order FVE formulation are provided.Some numerical examples illustrate that the results of numerical computation are consistent with theoretical conclusions.Moreover,it is shown that the reduced-order FVE formulation based on POD method is feasible and efficient for solving two-dimensional unsaturated soil water flow problem.

A new Iterative Alternating Direction Implicit (IADI) algorithm for multi-dimensional saturated–unsaturated flow

► This study has proposed a new IADI algorithm for Richards’ equation. ► The new scheme can be applied to two- and three-dimensional simulations. ► The new scheme has shown better performance than the original IADI scheme. Numerical simulation of saturated–unsaturated subsurface flows is widely used in many branches of science and engineering, and rapid developments in computer technology have enabled not only one-dimensional but also multi-dimensional simulations on a personal computer. However, a multi-dimensional subsurface flow simulation still incurs heavy load on computational resources, particularly for simulations in wide regions with long periods. An Iterative Alternating Direction Implicit (IADI) scheme has certain advantages in terms of computational cost and algorithmic simplicity. However, it is barely used at present because it occasionally incurs numerical instabilities and convergence difficulties. Another reason is that three-dimensional simulations cannot be performed by the original IADI scheme. This study has proposed an advanced IADI algorithm for solving the saturated–unsaturated flow equation; this advanced scheme is more numerically stable than the original IADI scheme and can be used for three-dimensional simulations. The performance of the proposed scheme was assessed through test simulations. In all the simulations, the new method was shown to be faster than the fully implicit scheme linearized by the modified Picard iteration method while still yielding very similar results.

확률적 정량모델을 이용한 토양에서의 바이러스 저감 평가

Abstract : The objective of this study was to analyze VIRULO model, a probabilistic quantitative model, which had been developedby US Environmental Protection Agency. The model could assess the viral attenuation capacity of soil as hydrogeologic barrier using Monte Carlo simulation. The governing equations used in the model were composed of unsaturated flow equations and viral transport equations. Among the model parameters, those related to water flow for 11 soil types were from UNDODA data, and those related to 5 virus species were from the literatures. The model compared the attenuation factor with threshold of attenuation to determine the probability of failure and presented the exceedances and Monte Carlo runs as output. The analysis indicated that among 11 USDA soil types, the viral attenuation capacity of loamy sand and sand were far lower than those of clay and silt soils. Also, there were differences in the attenuation in soil among 5 viruses with poliovirus showing the highest attenuation. The viral attenuation capacity of soil decreased sharply with increasing soil water content and increased nonlinearly with increasing soil barrier length. This study indicates that VIRULO model could be considered as a useful screening tool for viral risk assessment in subsurface environment.

An optimizing implicit difference scheme based on proper orthogonal decomposition for the two‐dimensional unsaturated soil water flow equation

SUMMARYAn optimizing reduced implicit difference scheme (IDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for the two‐dimensional unsaturated soil water flow equation is presented. An ensemble of snapshots is compiled from the transient solutions derived from the usual IDS for a two‐dimensional unsaturated flow equation. Then, optimal orthogonal bases are reconstructed by implementing SVD and POD techniques for the ensemble of snapshots. Combining POD with a Galerkin projection approach, a new lower dimensional and highly accurate IDS for the two‐dimensional unsaturated flow equation is obtained. Error estimates between the true solution, the usual IDS solution, and the reduced IDS solution based on POD basis are derived. Finally, it is shown by means of a numerical example using the technology of local refined grids that the computational load is greatly diminished by using the reduced IDS. Also, the error between the POD approximate solution and the usual IDS solution is proved to be consistent with the derived theoretical results. Thus, both feasibility and efficiency of the POD method are validated. Copyright © 2011 John Wiley & Sons, Ltd.

Infiltration in unsaturated soils – An analytical approach

An approximate analytical solution has been established for the well known Richards’ equation for unsaturated flow of transports in soils. Despite the importance of Richards’ equation in geotechnical and geoenvironmental applications, most solutions to the problem are generally based on numerical methods often having limitations in application. It is therefore advantageous to explore analytical solutions to the problem. The proposed solution is based on homotopy analysis method (HAM). The method is superior over traditional perturbation techniques in a sense that it is not dependent on the assumption of a small parameter as the initial step. Several simple cases have been explored to describe the application of HAM to Richards’ equation. The results obtained by HAM are also compared with existing solutions to demonstrate the accuracy of the proposed method.

Comparison of conductivity averaging methods for one-dimensional unsaturated flow in layered soils

One of the important factors influencing the accuracy of the numerical solution of 1D unsaturated flow equation (Richards’ equation) is the averaging method applied to compute hydraulic conductivity between two adjacent nodes of the computational grid. A number of averaging schemes have been proposed in the literature for homogeneous soil, including arithmetic, geometric, upstream and integrated means, as well as more sophisticated approaches, based on the local solution of steady state flow between the neighboring nodes (Darcian means). Another group of methods have been developed for the case when a material interface is present between the nodes. They range from simple arithmetic averaging to more complex schemes using the pressure- and flux-continuity conditions at the interface. In this paper we compare several averaging schemes for a number of steady and unsteady flow problems in layered soils. The first group of methods is applied in the framework of the vertex-centered approach to spatial discretization, where the nodes are placed at the material interfaces, while the second group is used with the cell-centered approach, where the material interfaces are located between computational nodes. The resulting numerical schemes are evaluated in terms of accuracy and computational time. It is shown that the averaging schemes based on Darcian mean principle [19] used in the framework of either vertex-centered or cell-centered approach compare favorably to other methods for a range of test cases.

Computing Internodal Conductivities in Numerical Modeling of two Dimensional Unsaturated Flow on Rectangular Grid

Abstract This paper compares numerical solutions of transient two-dimensional unsaturated flow equation by using different averaging schemes for internodal conductivities. Averaging methods such as arithmetic mean, geometric mean, upstream weighting, and integrated mean are taken into account, as well as a recent approach based on steady-state approximation. The latter method proved the most flexible, producing relatively accurate solutions for both downward and upward flow cases.

Mathematical Forms and Numerical Schemes for the Solution of Unsaturated Flow Equations

Prediction of water infiltration into the soil, fluid movement in the unsaturated soils, and groundwater recharge are important problems in different fields of science and engineering. Moreover, the transfer of the different pollutants (e.g., pesticides) from ground surface to groundwater occurs through the unsaturated zone. In this paper, a comprehensive evaluation of different finite difference schemes (e.g., fully implicit, Crank-Nicolson, and Runge-Kutta) is presented for the solution of head-based and mixed forms of the Richard’s equation. Two examples of water infiltration in very dry and relatively wet unsaturated soils are used for the evaluation of schemes. In addition, the effects of various approximations of moisture capacity function, convergence criteria, and time stepping methods on the performance of the schemes are investigated, and the results showed their significant influences on mass balance, number of iterations, and convergence condition of the numerical schemes. The results of numer...

Throughfall reduction in a cacao agroforest: tree water use and soil water budgeting

In a cacao agroforest with Gliricidia shade trees we studied the effects of experimentally reduced throughfall on tree sap flux densities and soil water budget. The objectives of our study were: (1) to identify the response of cacao and Gliricidia sap flux densities to reduced soil water availability, and (2) to analyze the changes in stand level soil water partitioning, measured as changes in transpiration, drainage and soil moisture storage under reduced throughfall. The study was conducted in Sulawesi, Indonesia at a site where ambient precipitation was 2841 mm yr −1. The research site was subdivided into three control plots and three plastic roof plots in which net precipitation was reduced by 71% through the use of a sub-canopy roof for 13 months, to create rainfall conditions similar to an extended El Niño event. The two species differed in their sap flux response to declining soil water content. Daily Gliricidia sap flux declined at higher soil water contents than cacao, while Gliricidia sap flux declined in response to all levels of vapor pressure deficit (VPD) and cacao only responded to high levels of VPD. Average monthly sap flux densities of cacao and Gliricidia in the roof plots decreased linearly with decreasing soil water content reaching a maximum reduction of 21% (cacao) and 29% ( Gliricidia) as compared to control plots. Cacao withdrew water mainly from the topsoil where small amounts of throughfall entering gaps in the roof frequently rewetted the soil. Therefore the competition between the two species for water resources during long periods with only little precipitation was low or in favor of cacao. Average daily stand transpiration was 1.3 mm in roof plots and 1.5 mm in control plots. Measured soil water contents were simulated satisfactorily by the application of a one-dimensional soil water model based on Richards unsaturated flow equation. The model suggested that drainage amounted to 1554 mm yr −1 in control plots. Drainage in roof plots was greatly reduced, totaling 299 mm yr −1. Soil moisture storage in the control underwent minor variations while it was heavily reduced in roof plots. Drainage in roof plots was a large driver of changes in soil moisture storage especially in deeper soil layers while precipitation and root water uptake were the main influencing factors for the topsoil. Deep root water uptake seemed to play a minor role in this experiment. Despite the strong reduction in net precipitation, declines in stand transpiration were moderate, suggesting that the stand did not react sensitively to the reduced throughfall scenario.

Steady state groundwater seepage in sloping unconfined aquifers

This paper reviews analytical solutions for steady state seepage of groundwater in an ideal sloping unconfined aquifer, recharged by an effective infiltration, when the unsaturated seepage is neglected. Available solutions are examined and compared. Their robustness is assessed with a finite element code, which solves the complete equations for saturated and unsaturated flow, thus making none of the simplifying assumptions of the theoretical solutions. An example is provided for a sloping aquifer on the north shore of the Saint-Lawrence River.

Review and Classification of Modeling Approaches of Soil Hydrology Processes

To use soil hydrology processes (SHP) models, which have increasingly extended during the last years, comprehensive knowledge about these models and their modeling approaches seems to be necessary. The modeling approaches can be categorized as either classical or non-classical. Classical approaches mainly model the SHP through solving the general unsaturated flow (Richards) equation, numerically or analytically. Due to some shortcomings of classical approaches, a trend toward the application of non-classical models has been initiated in recent years. Artificial neural networks and fuzzy logic systems are two main categories of non-classical approaches. In this study, existing modeling approaches of SHP are examined and compared, with an emphasis on recent trends. Also, modeling approaches of soil hydraulic functions are reviewed briefly as a main part of SHP models. Finally, different perspectives on classifications for SHP models are presented.

Approximation of internodal conductivities in numerical simulation of one‐dimensional infiltration, drainage, and capillary rise in unsaturated soils

When the unsaturated flow equation is solved with the finite difference method, the water flux between two adjacent nodes of the grid is approximated with a discrete form of Darcy's law. It requires the average value of the hydraulic conductivity in the considered grid block. Since the hydraulic conductivity in unsaturated soil is a highly nonlinear function of the water potential, the nodal conductivities can vary by multiple orders of magnitude, which makes the choice of the appropriate averaging procedure a nontrivial task. In this paper we present a new method to calculate the internodal conductivity for an arbitrary type of the conductivity function and arbitrary large grid size. It is based on the analysis of approximate profiles of the water potential head for steady flow between nodes. Numerical experiments show that the method is reasonably accurate for a wide range of soil types, for both steady and unsteady flow simulations.

Nonlinear Multigrid Methods for Numerical Solution of the Unsaturated Flow Equation in Two Space Dimensions

The need of accurate and efficient numerical schemes to solve Richards’ equation is well recognized. This study is carried out to examine the numerical performances of the nonlinear multigrid method for numerical solving of the two-dimensional Richards’ equation modeling water flow in variably saturated porous media. The numerical approach is based on an implicit, second-order accurate time discretization combined with a vertex centered finite volume method for spatial discretization. The test problems simulate infiltration of water in 2D saturated–unsaturated soils with hydraulic properties described by van Genuchten–Mualem models. The numerical results obtained are compared with those provided by the modified Picard–preconditioned conjugated gradient (Krylov subspace) approach.

Modeling Unsaturated Flow in Absorbent Swelling Porous Media: Part 1. Theory

The flow and deformation processes in swelling porous media are modeled for absorbent hygiene products (e.g., diapers, wipes, papers etc.). The first part of the article derives the fundamental equations for the hysteretic unsaturated flow, liquid absorption, and large deformation. The final set of model equations consists of balance equations of mobile and absorbed (immobile) liquid combined with a series of constitutive relationships. The resulting equation system is strongly nonlinear and requires advanced numerical strategies for solving. The second part of the article focuses on numerical solution and presents simulation results for 2D and 3D applications.

An adaptive multiscale finite element method for unsaturated flow problems in heterogeneous porous media

In this paper we present an adaptive multiscale finite element method for solving the unsaturated water flow problems in heterogeneous porous media spanning over many scales. The main purpose is to design a numerical method which is capable of adaptively capturing the large-scale behavior of the solution on a coarse-scale mesh without resolving all the small-scale details at each time step. This is accomplished by constructing the multiscale base functions that are adapted to the time change of the unsaturated hydraulic conductivity field. The key idea of our method is to use a criterion based on the temporal variation of the hydraulic conductivity field to determine when and where to update our multiscale base functions. As a consequence, these base functions are able to dynamically account for the spatio-temporal variability in the equation coefficients. We described the principle for constructing such a method in detail and gave an algorithm for implementing it. Numerical experiments were carried out for the unsaturated water flow equation with randomly generated lognormal hydraulic parameters to demonstrate the efficiency and accuracy of the proposed method. The results show that throughout the adaptive simulation, only a very small fraction of the multiscale base functions needs to be recomputed, and the level of accuracy of the adaptive method is higher than that of the multiscale finite element technique in which the base functions are not updated with the time change of the hydraulic conductivity.

Infiltration and deep flow over sloping surfaces: Comparison of numerical and experimental results

Summary Infiltration and deep flow on sloping surfaces were studied by combining controlled laboratory experiments with mathematical models. Experimental variables included soil types, rainfall intensities, and surface slopes. Preliminary analyses of the data indicated that increased slope has a positive influence on surface flow, and that this influence increases relatively with decreasing rainfall rate. However, current theories could not adequately explain the observed behavior of deep and surface flows for varying slopes. Three mathematical models of varying complexity were employed to supplement the experimental results. These models were (i) the 2-D Hydrus numerical model, (ii) numerical solution of 1-D saturated–unsaturated flow equation on sloping surfaces, and (iii) a simplified 1-D sharp-front model for sloping surfaces. For the latter two models, a surface flow component based on the kinematic wave approximation for shallow flows was externally coupled to the subsurface flows to route water over the soil surface. For each soil, one experiment at the lowest slope and rainfall rate was utilized for estimation of model parameters that could not be measured independently, while all the other results were used for model corroboration. The Hydrus model indicated that a 1-D analysis would be adequate as the water front moves essentially parallel to the slope. To account for the influence of slope and soil-type on experimental results, an effective saturated conductivity was proposed. Model results were found to be in reasonable agreement with observations of surface flow, deep flow, and water contents in the soil profile with the use of the proposed effective saturated conductivity. Limitations on the applicability of the sharp-front model in this context were discussed.

Measurements of water content distribution in vertical subsurface flow constructed wetlands using a capacitance probe: benefits and limitations

Water content profoundly affects the efficiency of vertical flow constructed wetlands. Detailed vertical watercontent profiles could help to improve our understanding of the behaviour of these systems. In this study we testeda capacitance probe to measure water content in a vertical flow constructed wetland pilot plant. Field measurementswere compared with data recorded in a purpose-built laboratory apparatus. Our results show that the on-siteassessment of water content profiles should help to identify the real layout of the vertical systems, and this in turnshould prevent or at least highlight possible running failures. Theoretical hydraulic retention time, oxygen gasavailability and the calibration of van Genuchten’s equation for unsaturated flow were easily assessed thanks to thewater content measurements. Furthermore, water content measurements performed in dynamic conditions enabledus to define the actual specific flow for the various phases of the filling-emptying processes and to check the influentponding for a uniform distribution of wastewater over the bed. Water content measurements are quick, cheap andsafe, providing a useful tool in many application areas. However, calibrating the probe for the media typically usedin vertical flow bed is not simple. In addition seriously misleading measurements can arise when steep variations inwater content occur in the system.