Existence for a time-dependent rainfall infiltration model with a blowing up diffusivity

A difficulty in the modelling of water infiltration into an unsaturated soil is due to the presence of a diffusion coefficient that blows up at the moisture saturation value. This is put in evidence in some well-known hydraulic models like those of Broadbridge and White and van Genuchten. In this paper, we obtain results concerning the existence, uniqueness and regularity properties of the solution of unsaturated water flow determined by a time-dependent rainfall, with a nonlinear flux boundary condition on the outflow boundary and a singular diffusion coefficient. Some considerations related to the possibility of saturation occurrence and the extension of the results to the model describing the infiltration into an nonhomogeneous stratified soil are finally made.     

Hydraulic hysteresis effects on the coupled flow–deformation processes in unsaturated soils: Numerical formulation and slope stability analysis

The hysteresis of water retention curve has a profound influence on the coupled hydro-mechanical behaviors in unsaturated soils, but numerical implementation with consideration of this property was rarely reported due to the difficulties in the integration of the coupled constitutive models. In this study, a numerical formulation is proposed for modeling the coupled flow–deformation processes with hydraulic hysteresis. A return mapping scheme is developed to integrate the water retention curve model with hydraulic hysteresis and the elasto-plastic model simultaneously within a time step, and the deformation-dependent nature of the water retention curve is considered rigorously by modifying the coefficient matrices in the discretized governing equations. The performance and efficiency of the proposed numerical formulation is validated by two existing laboratory tests and a computational example, demonstrating better performance and convergence of the proposed formulation. The proposed procedure is then applied for modeling the coupled flow–deformation processes in a soil slope under rain infiltration. The simulated results reveal the significant effects of hydraulic hysteresis on the coupled water–air two-phase flow and elasto-plastic deformation processes. The solid deformation and the evolution of the shear band would be remarkably overestimated, and the slope failure would be early predicted when neglecting hydraulic hysteresis.     

Uncertainty analysis of transport of water and pesticide in an unsaturated layered soil profile using fuzzy set theory

Incomplete information is notoriously common in planning soil and groundwater remediation. For making decisions groundwater flow and transport models are commonly used. However, uncertainty in prediction arises due to imprecise information on flow and transport parameters like saturated/unsaturated hydraulic conductivity, water retention curve parameters, precipitation and evapo-transpiration rates as well as factors governing the fate of pollutant in soil like dispersion, diffusion, degradation and chemical transformation. Different methods exist for quantifying uncertainty, e.g. first and second order Taylor’s Series and Monte-Carlo method. In this paper, a methodology based on fuzzy set theory is presented to express imprecision of input data, in terms of fuzzy number, to quantify the uncertainty in prediction. The application of the fuzzy set theory is demonstrated through pesticide (endosulfan) transport in an unsaturated layered soil profile. The governing partial differential equation along with fuzzy inputs, results in a non-linear optimization problem. The solution gives complete membership functions for flow (suction head) and pesticide concentration in soil column.     

非饱和土壤水流问题基于POD 方法的降阶有限体积元格式及外推算法实现

利用特征投影分解(proper orthogonal decomposition, 简记为POD) 方法对非饱和土壤水流问题的经典有限体积元格式做降阶处理, 建立一种具有足够高精度维数较低的降阶有限体积元格式, 并给出这种降阶有限体积元解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的. 进一步表明了基于POD 方法的降阶有限体积元格式对求解非饱和土壤水流问题数值解是可靠和有效的.     

Saturated-unsaturated flow in unconfined aquifers

An asymptotic theory is developed for the saturated-unsaturated flow in unconfined aquifers. It is found that in the first approximation the flow is governed by a nonlinear parabolic equation which reduces to the Boussinesq equation when the terms associated with the unsaturated zone are omitted. By assuming certain hydraulic properties of the porous medium it will be shown that the effects of the unsaturated zone are significant in most practical cases of interest.

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Zusammenfassung Eine asymptotische Theorie für die gesättigte-ungesättigte Strömung in porösen, wassertragenden, auf der Landfläche sich befindenden Schichten wird entwickelt. Wir fanden, daß in der ersten Annäherung die Strömung von einer nicht linearen parabolischen Gleichung bestimmt wird, die zur Gleichung von Boussinesq zurückgeführt werden kann, wenn die Glieder für die ungesättigte Zone ausgelassen werden. Wenn man dem porösen Material gewisse Eigenschaften zuschreibt, dann läßt sich zeigen, dass der Einfluß der ungesättigten Zone in den meisten in der Praxis auftretenden Fällen von Bedeutung sind.

     

Numerical method for the simultaneous determination of the hydraulic properties of unsaturated porous media

This paper presents a numerical algorithm for solving the inverse coefficient problem for nonlinear parabolic equations. This problem arises in simultaneous determination of the hydraulic properties of unsaturated porous media from a simple outflow experiment. The novel feature of the method is that it is not based on output least squares. In this method, the unknown functions are represented as polygons (continuous and piecewise linear functions) every new linear pieces that are determined in each time step by using information based only on previous time intervals. The results of some numerical experiments are displayed.     

An accuracy analysis of mass matrix lumping for three-dimensional,unsaturated flow,finite element,porous media computations using analytical solutions

Lumping of the mass matrix for transient problems is the state of the practice for finite element groundwater programs. This improves nonlinear convergence where there is unsaturated flow, but the discovery was made that the accuracy of the results is reduced significantly for a test problem. This paper compares results from a recently derived transient three-dimensional (3-D) analytical solution for a given unsaturated flow test problem with those obtained from a standard parallel Galerkin 3-D finite element groundwater program with the mass matrix lumping option turned both on and off. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

土壤水流与溶质耦合运移问题的混合元-迎风广义差分法研究

本文研究了一维非饱和土壤水流与溶质耦合运移问题的数学模型, 建立了求其数值解的守恒混合元-迎风广义差分格式. 对非线性土壤水分入渗方程, 采用守恒混合元法进行离散模拟, 同时得到了土壤含水量和水分通量; 而对对流-扩散形式的溶质运移方程, 利用迎风的广义差分法离散求解. 且分析了解的存在唯一性, 并讨论了误差估计. 最后给出数值算例, 模拟结果表明利用本文格式来求解非饱和土壤水流与溶质耦合运移问题是可靠的, 且该格式具有稳定性和可实用性.     

Mathematical Modeling to Simulate the Movement of Contaminants in Groundwater

The objectives of this paper are twofold. Firstly, we formulate a system of partial differential equations that models the contamination of groundwater due to migration of dissolved contaminants through unsaturated to saturated zone. A closed form solution using the singular perturbation techniques for the flow and solute transport equations in the unsaturated zone is obtained. Indeed, the solution can be used as a tool to verify the accuracy of numerical models of water flow and solute transport. The second part of this paper, deals with how the water level in a water reserve drops due to pumping water out of a well that is some distance away.     

Nonlinear Volume Flow Control of a Variable Displacement Vane Pump

Variable displacement vane pumps are used for mid-pressure applications in a variety of systems. Due to their potential for power loss reduction they are most prominently featured in automotive applications. They provide a hydraulic consumer with varying levels of volume flow, depending on the eccentricity of the cam ring of the pump. Eccentricity control is typically achieved by means of hydraulic actuation. In this contribution, a model of a hydraulic aggregate featuring a VDVP with a nonlinear control technique based on input-output-linearization for volume flow tracking requirements is presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

污染物在非饱和带内运移的流固耦合数学模型及其渐近解

污染物在非饱和带中运移过程是多组分多相渗流问题.在考虑气相的存在对水相影响的前提下,基于流固耦合力学理论,建立了污染物在非饱和带内运移的流固耦合数学模型.对该强非线性数学模型采用摄动法及积分变换法进行拟解析求解,得出了解析表达式.对非饱和带内的孔隙压力分布、孔隙水流速以及污染物的浓度在耦合与非耦合气相条件下的分布规律进行解析计算.对该渐近解与Faust模型的计算结果进行了对比分析,结果表明:该模型解与Faust解基本吻合,且气相作用以及介质的变形对溶质的输运过程产生较大的影响,从而验证了解析表达式的正确性和实用性.这为定量化预报预测污染物在非饱和带中迁移转化和实验室确定压力-饱和度-渗透率三者之间的关系提供了可靠的理论依据.     

二维非饱和土壤水分运动问题的半离散有限体积元模拟

本文利用基于重心对偶剖分的有限体积元法建立了二维非饱和土壤水分运动问题的数值逼近格式,讨论了离散有限体积元解的存在唯一性,并给出了最优误差估计的证明.最后给出数值算例,模拟结果表明,利用有限体积元格式来求解二维非饱和土壤水分运动问题是可靠的,且该格式具有稳定性和可实用性.     

Analytical solutions for seepage near material boundaries in dam cores: The Davison-Kalinin problems revisited

Steady Darcian seepage through a dam core and adjacent shells is analytically studied. By conformal mappings of the pentagon in the hodograph plane and triangle in the physical plane flow through a low-permeable dam core is analyzed. Mass-balance conjugation of flow in the core and downstream highly-permeable shell of the embankment is carried out by matching the seepage flow rates in the two zones assuming that all water is intercepted by a toe-drain. Seepage refraction is studied for a wedge-shaped domain where pressure and normal components of the Darcian velocities coincide on the interface between the core and shell. Mathematically, the problem of R-linear conjugation (the Riemann-Hilbert problem) is solved in an explicit form. As an illustration, flow to a semi-circular drain (filter) centered at the triple point (contact between the core, shell and impermeable base) is studied. A piece-wise constant hydraulic gradient in two adjacent angles making a two-layered wedge (the dam base at infinity) is examined. Essentially 2-D seepage in a domain bounded by an inlet constant head segment, an outlet seepage-face curve, a horizontal base and with a straight tilted interface between two zones (core and shell) is investigated. The flow net, isobars, and isotachs in the core and shell are reconstructed by computer algebra routines as functions of hydraulic conductivities of two media, the angle of tilt and the hydraulic head value at a specified point.     

Excess pore water pressure due to ground surface erosion

The Laplace transform is applied to solve the groundwater flow equation with a boundary that is initially fixed but that starts to move at a constant rate after some fixed time. This problem arises in the study of pore water pressures due to erosional unloading where the aquifer lies underneath an unsaturated zone. We derive an analytic solution and examine the predicted pressure profiles and boundary fluxes. We calculate the negative pore water pressure in the aquifer induced by the initial erosion of the unsaturated zone and subsequent erosion of the aquifer.     

Complex analytical solutions for flow in hydraulically fractured hydrocarbon reservoirs with and without natural fractures

Reservoir drainage towards producer wells in a hydraulically and naturally fractured reservoir is visualized by using an analytical streamline simulator that plots streamlines, time-of-flight contours and drainage contours based on complex potentials. A new analytical expression is derived to model the flow through natural fractures with enhanced hydraulic conductivity. Synthetic examples show that in an otherwise homogeneous reservoir even a small number of natural fractures may severely affect streamline patterns and distort the drainage contours. Multiple parallel natural fractures result in a drainage region that is narrower in the direction normal to the natural fractures while the drainage reach is larger in the natural fracture direction. Reservoirs with numerous natural fractures are shown to be characterized by more tortuous drainage patterns than reservoirs without natural fractures. Finally, the analytical flow model for naturally fractured reservoirs is applied to a natural analog of flow into hydraulic fractures. The tendency of the injected fluid to stay confined to the fracture network as opposed to matrix flow is entirely controlled by the hydraulic conductivity contrast between the fracture network and the matrix.     

Modeling of water surface profile in subterranean channel by differential quadrature method (DQM)

This study, investigates the hydraulic of flow in a subterranean channel headspring. The continuity and momentum equations of flow in porous media considering real conditions were used and the basic equation of flow in a subterranean channel was resulted. This equation is very similar to the spatially varied flow with increasing discharge. An equation, defining the hydraulic parameters of a subterranean channel section was adopted. Then differential quadrature method (DQM), was applied to the equation of flow in subterranean channel, consequently the water surface profile was resulted. To illustrate the rightness of model, the hydraulic parameters of flow in the Gavgard branch of the Joopar Goharriz Qanat were measured and the water surface profile was determined. This water surface profile was compared to the water surface profile computed by the model, which are in good agreement.     

Numerical solution of steady-state porous flow free boundary problems

Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.     

基于二次插值的非饱和土壤水流问题的特征差分方法及数值模拟

本文针对一类非饱和土壤水流问题,提出了基于二次插值的特征差分格式,得到了严谨的L2模误差估计.并作了数值试验,指明方法的有效性.     

Mixing Closures for Conservation Laws in Stratified Flows

A closure for shocks involving the mixing of the fluids in two-layer stratified flows is proposed. The closure maximizes the rate of mixing, treating the dynamical hydraulic equations and entropy conditions as constraints. This closure may also be viewed as yielding an upper bound on the mixing rate by internal shocks. It is shown that the maximal mixing rate is accomplished by a shock moving at the fastest allowable speed against the upstream flow. Depending on whether the active constraint limiting this speed is the Lax entropy condition or the positive dissipation of energy, we distinguish precisely between internal hydraulic jumps and bores. Maximizing entrainment is shown to be equivalent to maximizing a suitable entropy associated to mixing. By using the latter, one can describe the flow globally by an optimization procedure, without treating the shocks separately. A general mathematical framework is formulated that can be applied whenever an insufficient number of conservation laws is supplemented by a maximization principle.     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.