Unsaturated Flow Equation Research Articles

Modeling plant water deficit by a non-local root water uptake term in the unsaturated flow equation

In this paper we present a novel way to mathematically frame the concept of ecological memory of plant water stress in the context of root water uptake in unsaturated flow equations. Inspired by recent eco-hydrological papers, we model the water absorption by roots with a non-local sink term, accounting also for a memory effect. In order to model such a memory effect, an integral equation is defined; the main purpose of this work is to provide sufficient conditions on the functions at play for ensuring existence and uniqueness of its solution. Finally, tailored numerical methods are implemented, and numerical simulations are also provided.

Identification of a soil water flow equation using a two-step data-driven method

Governing equations of soil water flow are traditionally derived from the conservation of mass and the Buckingham-Darcy law. However, due to uncertainties in many aspects, such as physical processes and constitutive relationships, a thorough understanding of soil water flow in a theory-driven way often poses many challenges. With the fast improvement of data acquisition ability, data-driven approaches have gained considerable scientific interest in discovering the governing equations of physical problems. However, most existing studies focused on linear or weakly nonlinear problems with constant coefficients in their partial differential equations. This study attempted to propose a modified two-step data-driven method to identify the strongly nonlinear equation of unsaturated flow. The Richardson-Richards equation and coefficients were jointly identified by hybridizing sparse regression and iterative smoother. The automatic differentiation scheme was introduced to reduce the risk of performance collapse from data noise common in sparse regression methods. We demonstrated the ability, and the challenge as well, of identifying the soil water flow governing equations in a data-driven way with the aid of a series of synthetic cases. Our results showed that the modified method was capable of accurately identifying the governing equation of unsaturated flow, whose performance was almost independent of prior knowledge of the equation form and constitutive relationships. If the underlying physical processes are poorly understood, then a more relaxed equation form was recommended. The reduction in the spatiotemporal resolution of sampling may cause higher-order terms to be difficult to derive accurately or some terms that were not originally present to appear in the identified equation. Although not completely eliminated, the detrimental effect of data noise on the identification of the governing equations can be greatly mitigated by introducing the automatic differentiation scheme.

Reconstructing the Unsaturated Flow Equation From Sparse and Noisy Data: Leveraging the Synergy of Group Sparsity and Physics‐Informed Deep Learning

AbstractData‐driven scientific discovery methods have been developed and applied to discover governing equations from data, involving the attempt to discover the unsaturated flow equation in soils from data. However, an important but unresolved problem is how to reconstruct the unsaturated flow equation from highly noisy and scarce discrete data. In this study, we present a new deep‐learning framework: DeepGS (deep‐learning‐based group sparsity framework), that leverages the synergy of group sparsity and physics‐informed deep learning (PIDL) to reconstruct the latent governing equation for unsaturated flow. In particular, we design a strategy that decomposes the identification of the unsaturated flow equation into two tasks: the determination of the partial differential equation structure and the reconstruction of the nonlinear coefficients. The tasks can be seamlessly handled by group sparse regression and the PIDL approach. Through the training, it realizes the simultaneous reconstruction of soil moisture dynamics and unsaturated flow governing equation. A series of comprehensive numerical experiments are conducted to determine the optimal architecture and test its performance. The results show the efficacy and robustness of DeepGS, which significantly outperform previous methods. We also conclude that accurately reconstructing soil moisture dynamics and spatiotemporal derivatives from noisy and scarce data play a critical role in governing equation discovery. This study further demonstrates the potential of discovering the governing equation for unsaturated flow from data in more complex scenarios, where rich and accurate soil moisture observations are generally intractable to access.

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

Reconstructing the Unsaturated Flow Equation from Sparse Noisy Data: Leveraging the Synergy of Group Sparsity and Physics-Informed Deep Learning

On the Discretization of Richards Equation in Canadian Land Surface Models

ABSTRACT Both the Canadian Land Surface Scheme (CLASS) and the Soil, Vegetation, and Snow (SVS) land surface models employ a potentially problematic discretization of Richards equation for unsaturated vertical flow in the soil column. It is shown here that this discretization will always overestimate the vertical moisture gradient compared to a better-constructed first-order scheme, which under some circumstances could lead to erroneous moisture drawdown. The problem stems from an interpolation calculation on the irregularly spaced grids traditionally used by these models. While vanishing on uniform grids, this numerical error progressively worsens with increasing layer thickness differences. In this brief technical note a systematic method for developing first and higher order schemes on irregular, staggered grids is presented. To demonstrate the potential impact of the new first – order scheme, multi-year simulations of five FLUXNET sites are presented and discussed. A dramatic improvement in first layer soil moisture is found for two of the sites, which contributes to potentially significant differences in evapotranspiration. Higher order schemes are also possible but must be constructed carefully, in concert with a judicious choice of soil layer spacing in order to minimize discretization error. Given the extensive use of CLASS and SVS in Canadian environmental prediction systems, and the freedom with which users can specify soil layer thicknesses, it is recommended that modellers consider this issue carefully in their applications.

The difference between semi-continuum model and Richards’ equation for unsaturated porous media flow

Semi-continuum modelling of unsaturated porous media flow is based on representing the porous medium as a grid of non-infinitesimal blocks that retain the character of a porous medium. This approach is similar to the hybrid/multiscale modelling. Semi-continuum model is able to physically correctly describe diffusion-like flow, finger-like flow, and the transition between them. This article presents the limit of the semi-continuum model as the block size goes to zero. In the limiting process, the retention curve of each block scales with the block size and in the limit becomes a hysteresis operator of the Prandtl-type used in elasto-plasticity models. Mathematical analysis showed that the limit of the semi-continuum model is a hyperbolic-parabolic partial differential equation with a hysteresis operator of Prandl’s type. This limit differs from the standard Richards’ equation, which is a parabolic equation and is not able to describe finger-like flow.

DEM modelling of unsaturated seepage flows through porous media

A new hybrid discrete–continuum numerical approach that explores the key advantages of both discrete and continuum approaches is proposed to model unsaturated seepage flows through porous media. In contrast to existing approaches where a porous medium is often represented by a continuum medium or required a background mesh, the proposed approach explicitly exploits the discrete contact network formed by an assembly of discrete solid particles. Each solid particle is assumed to occupy an equivalent-continuum space, over which the governing equations for unsaturated seepage flow are derived. These governing equations are then discretised and solved on the discrete contact network through a new numerical procedure that links micro-diffusivity to the macro-one. Thanks to this concept, the proposed approach is capable of describing the nature of flow in unsaturated porous media at the microscale level. This unique feature also enables the proposed approach to naturally simulate the water flow through the heterogeneous porous media without any ad hoc treatments. In this paper, the mathematical concept of the proposed approach together with its implementation features and performances for a rigid porous media is presented and discussed. The focus is placed on its application to the discrete element method (DEM), although the proposed concept, in general, can be applied to any other methods possessing similar features.

Verification of tree induced suction with numerical model

This study discusses the validation between explorations of transpiration on the unsaturated soil zone with unsaturated flow equation of tree water uptake model. Numerical simulation was performed to the model of the slope in relation to moisture migration patterns in the unsaturated zone within the vicinity of mature tree. The results of numerical simulation conducted on typical slope geometry model with tree at the toe of the slope of induced transpiration were later presented in this study. The results of numerical simulation and field monitoring of suction against depth are in acceptable condition. It is important to note that, several of the differences between simulated numerical and field measurement are related to inconsistence effect of root density.

Evaluation of antecedent rainfall effects in the analysis of the probability of transient failure in unsaturated slopes

The occurrence of extreme events of precipitation is commonly responsible for inducing surface mass movements in slopes. Landslides are due to the soil-atmosphere interactions. The main inducer of failure in unsaturated slopes is the infiltration, which triggers the reduction of suction, favoring the decrease of stability. Thus, this paper presents the implementation of analytical models to evaluate the probability of transient failure of slopes under unsaturated conditions for a local scale when subjected to antecedent precipitation. In this sense, the model proposed involves an analytical solution to Richards equation for unsaturated flow under transient conditions, capable of analyzing the soil behavior in the infiltration process. Therefore, these analytical models and the geotechnical parameters taken from the case study of a landslide that occurred in Nova Friburgo (Brazil) allowed the simulation of the infiltration process using the precipitation data corresponding to the day of the landslide, as well as four months previous to the collapse. The calculation routine is capable of generating the decreasing performance of the transient factor of safety and the increasing probability of transient failure through a probabilistic approach, allowing easy predicting failure and management of threat of landslides.

Parameter estimation of soil hydraulic characteristics by inverse modeling of the analytical equation for unsaturated subsurface water flow

AbstractIn drip-irrigated systems, the understanding of the soil wetting pattern is essential in defining the area effectively irrigated, the spacing between the emitters and their installation depth, and the irrigation rate. Thus, this study aims to estimate soil hydraulic characteristics through inverse modeling of an analytical equation used in wetting bulb simulation based on soil moisture measurements obtained in the field. The parameters of the Gardner model, which describes the unsaturated soil hydraulic conductivity, and the van Genuchten model that describes the soil moisture retention curve, were estimated by inverse modeling techniques. The following options were considered: (A) estimating parameter β while considering the other parameters, Ko, θr, θs, α, n, and m, as known and obtained experimentally; (B) estimating parameters Ko and β while considering the experimental retention curve as known; (C) estimating parameters Ko, β, α, n, and m while considering the values of the θr and θs volumetric moisture as known; (D) estimating all the parameters of Gardner and van Genuchten models (Ko, θr, θs, α, n, and m). The results indicate that option D showed better concordance between the estimated and observed moisture values. Thus, the inverse modeling of the analytical equation is an important tool for irrigation design and management.

Estimation of the soil hydraulic properties from field data by solving an inverse problem

Estimating unsaturated soil hydraulic properties to predict water dynamics through a vertical soil profile under the effects of irrigation, drainage and evapotranspiration is imperative for managing soils in the arid regions. The aim of this work was to determine the soil water retention curve and the hydraulic permeability function of a bare soil profile in a Tunisian oasis threatened by salinization. The developed model combines a numerical inversion of the unsaturated flow equation with the BOBYQA optimisation algorithm. The direct model solved the Richards equation using a cell-centred finite difference model. Hydraulic properties were described by van Genuchten-Mualem models. Input data for the inverse problem are the infiltration flow, soil water contents and pressure heads measured during ponded infiltration and internal drainage tests. Numerical simulations of these two tests were performed considering a homogeneous single-layer soil profile but a better fitting between measured and simulated water contents was obtained when the soil profile was divided into five sub-layers. The hysteresis phenomenon was highlighted from the soil water retention and the relative permeability curves corresponding to the ponded infiltration and internal drainage tests.

OpenCAL system extension and application to the three-dimensional Richards equation for unsaturated flow

OpenCAL is a scientific software library specifically developed for the simulation of 2D and 3D complex dynamical systems on parallel computational devices. It is written in C/C++ and relies on OpenMP/OpenCL and MPI for parallel execution on multi-/many-core devices and clusters of computers, respectively. The library provides the Extended Cellular Automata paradigm as a high-level formalism for modeling complex systems on structured computational grids. As a consequence, it can be used as a parallel explicit solver of ordinary and partial differential equations through classical methods, including finite-difference and finite-volume. Here the latest version of the library is described, introducing the MPI infrastructure over the 3D OpenCL and 2D/3D OpenMP components. The implementation of a three-dimensional unsaturated flow model based on a direct discrete formulation of the Richards’ equation is also shown, corresponding to a finite-difference scheme. Computational performances have been assessed on both a scientific workstation equipped with a dual Intel Xeon socket and three Nvidia GPUs, and a 16 nodes cluster with a fast interconnection network. The OpenCAL embedded quantization optimization is also discussed and exploited to drastically reduce computing time.

Influence of time-dependent ground surface flux on aquifer recharge with a vadose zone injection well

Vadose zone well (VZW) injection is an effective method for aquifer recharge in semiarid and arid regions where groundwater table is deep or sufficiently permeable soils and/or sufficiently large land areas for surface infiltration are not available. Several analytical and numerical models have been developed for the injection well in the vadose zone. However, those models usually ignore the ground surface flux (GSF) generated by infiltration or evapotranspiration, which may cause errors and hinder the success of artificial recharge and groundwater management. This study takes GSF generated by these two processes into consideration with the purpose of understanding their influences on the hydraulic responses and recharge efficiency. An innovative mathematical model is established based on a three-dimensional (3D) governing equation for saturated flow and a linearized 3D Richards’ equation for unsaturated flow. The Laplace-Hankel transforms are applied to derive solutions for the hydraulic head increments and recharge rate. The solutions are tested by a finite-element numerical model developed using COMSOL Multiphysics. The obtained solutions are utilized to assess the influences of time-dependent GSF generated by infiltration or evapotranspiration. The magnitude of these influences depends on unsaturated and saturated aquifer properties and the value of GSF. The impacts of three important properties are investigated: the constitutive exponent of the unsaturated zone ω, the specific yield Sy, and the hydraulic conductivity anisotropy KD=KzKr. Smaller values of dimensionless constitutive exponent ωD and dimensionless specific yield SyD lead to larger changes of hydraulic head increments in both unsaturated zone (uD) and saturated zone (sD) when GSF is the same. A small value of KD can delay the changes of uD and sD, but the value of KD does not affect the final values of uD and sD. Larger values of GSF generated by surface infiltration lead to larger increases of uD and sD; larger absolute values of GSF generated by evapotranspiration lead to larger decreases of uD and sD. The proposed semi-analytical solutions can be used to improve the feasibility and efficiency of artificial recharge in semiarid and arid regions.

Infiltration behaviour of dike covers made of dredged material on the Rostock Research dike

The knowledge about the infiltration behaviour of a dike body is an important issue in dike design and flood risk assessment. To reduce the risk of stability failures during flood events, the infiltration rate of a dike should be kept low and the elapsing time to reach the saturated steady state should be long. A number of full-scale physical flood simulation experiments on the Rostock research dike, on which different processed dredged materials were applied in the dike body (cover layers in particular) showed that the infiltration time and rate is much higher as can be expected from comprehensive laboratory test data. To obtain information about the unsaturated/saturated hydraulic soil properties of the installed dredged materials in situ, an automatic multi-objective model calibration method for a minimizing problem is used to reduce the residuals between simulated and observed time series (hydraulic head pressure, suction pressure and rate budget). The measured suction pressure time series of these physical experiments lets us suppose, that it is not possible to simulate the suction pressure behaviour of the installed dredged material using RICHARD’s unsaturated flow equation with a proper water retention function and unsaturated flow conductivity. It seems that the hysteresis effects are negligibly small and the hydraulic behaviour is governed by the inter-aggregate pore structure. In the paper, different back-calculation methods to fit the soil hydraulic parameters to the full-scale measurements are discussed and compared a proposal for the evaluation of the data is provided.

Natural dam failure in slope failure mode triggered by seepage

Slope failure is one of the failure modes of natural dams. Turbid seepage flow is often observed on the downstream slope of a natural dam in the field and indicates the migration of fine particles in the dam. However, the coupling of internal erosion and seepage in natural dams during the impounding process is still unclear. This paper studies fine particles migration in a dam and the slope failure mode in the laboratory. Then, a coupling model was proposed with the fines migration equation, unsaturated seepage flow equation, and local stability analysis equation. In addition to the variation in the permeability coefficient due to fine particles migration, the strength parameters and density of the soil are also changed, which is considered in the model. Based on the conditions of the laboratory tests, simulations were conducted with the coupling model, and some parameters, such as pore pressure, porosity, principle stress, and factor of safety, were analyzed. Meanwhile, the influence of internal erosion and seepage on the stability of dams under different conditions is studied. Furthermore, the influence of some model parameters on the internal structure of the dam and its stability are discussed.

Modelling the Unsteady Flow of Water into a Partly Saturated Soil

Abstract Flows in unsaturated medium are frequent in the field of civil engineering and more particularly in geotechnics. The study undertaken here tries to solve the unsaturated transient flow equation in porous media using the finite element method. Numerical solution of a finite element discretization is used along with an implicit integration scheme of the time stepping. A functional one that makes it possible to find the distribution of the hydraulic load has been proposed and a calculation program has been developed. The results obtained by this program called TFAP (Transient Flow Analysis Program) are compared to other previous work in the subject. The authors show the importance of this study through two real examples. Liakopoulos conducted several experiments on the water drainage through a vertical column filled with Del Monte sand. One of these experiments was chosen to perform a simulation by the model. The results of the calculation are compared with the experimental data as well.