Modeling of the base of a hydraulic structure with constant flow velocity sections and a curvilinear confining layer

We construct the underground contour of an embedded rectangular dam, whose corners are rounded by curves of constant flow velocity. We consider the case of a water-permeable base underlain by a curvilinear confining layer with a horizontal part, whereas the remainder parts of the layer are characterized by a constant flow velocity. We obtain an analytical solution to the corresponding mixed problem of the theory of analytic functions, we present results of numerical computations and consider the limiting case studied earlier by P. Ya. Polubarinova-Kochina and I. N. Kochina.     

海水入侵数值模拟的特征块中心差分法

本文研究二维海水入侵数值模拟的有限差分法．对关于压力的流动方程本文采用块中心差分法,对关于含盐浓度方程的对流扩散方程采用基于完全二次矩形插值的特征差分法,运用先验估计的理论和技巧得到了最佳阶L~2误差估计的结果．     

Investigation of the squeezing out of the current in some flows in coastal water-bearing layers

Mathematical models of certain flows of fresh ground waters, in a semi-infinite pressurized water-bearing layer, to a salt water sea (basin, reservoir, pot hole, etc.), above the surface of which there is a layer of fresh water, are considered within the framework of the two-dimensional theory of steady seepage. To investigate them, mixed boundary-value problems in the theory of analytic functions are formulated and solved using Polubarinova-Kochina's method. On the basis of these models, algorithms are developed for calculating the squeezing out (that is, the process of the forcing out of the seeping fresh waters by the heavier salt waters, leading to deformation of the interface of the liquids) in cases when the ground water flows enter the sea from the side and from below. A detailed analysis of the structure and characteristic features of the processes, as well as of the effect of all the physical characteristics of the models on the nature and degree of the squeezing out of the fresh water, is carried out using the exact analytical relations obtained as well as numerical calculations. In the special case when there is no layer of fresh water above the surface of the sea, a comparison of the results of the calculation is given for both inflow schemes, and the nature of the dependences of the degree of squeezing out of the water from the initial position of contact of the liquids is discussed.     

Some mathematical models related to the Zhukovskii problem of the flow around a sheet pile

The seepage under a Zhukovskii sheet pile through a layer of soil underlain by a highly permeable pressurized horizon is considered. The left semi-infinite part of the roof of this horizon is simulated by an impermeable foundation. The flow when the velocity on the edges of the sheet pile is equal to infinity and, on the two water permeable parts of the boundary of the domain of motion, the flow rate takes extremal values, is investigated. The limiting cases, associated with the absence of both a backwater and an impermeable inclusion, are mentioned. The problem of seepage from a foundation pit formed by two Zhukovskii sheet piles is solved within the limits of a flow with a highly permeable pressurized stratum lying below. In the case when there is no infiltration onto the free surface, a solution of the well-known Vedernikov problem is obtained. A contact scheme, arising when there are no such indicated critical points, is considered; it is described outside the scope of the constraints imposed on the unknown conforming mapping parameters ensuring the realization of the basic mathematical model. Solutions are given for two schemes of motion in a semi-inverse formulation. The classical Zhukovskii problem is the limiting case of one of them. The special features of such models are mentioned. The Polubarinova-Kochina method is used to study all the above-mentioned flows. This method enables exact analytical representations of the elements of the motion to be obtained. The results of numerical calculations and an analysis of the effect of all the physical factors on the seepage characteristics are presented.     

Mixed finite element methods for computing groundwater velocities

Central to the understanding of problems in water quality and quantity for effective management of water resources is the development of accurate numerical models to stimulate groundwater flows and contaminant transfer. We discuss several important difficulties arising in modeling of subsurface flow and present promising numerical procedures for alleviating these problems. Furthermore, we describe mixed-finite element techniques for accurately approximating fluid velocities, and review computational results on a variety of hydrologic problems.     

Proof of Dupuit’s Assumption for the Free Boundary Problem in an Inhomogeneous Porous Medium

For free boundary problems describing steady groundwater flows, the asymptotic behavior of solutions is studied in the situation where the scale in one of the spatial directions is much less than that in the other directions. The convergence of solutions to a certain limit is proved. Properties of the limit solution agree with the assumption known as Dupuit’s assumption in engineering applications, which customarily serves as a basis for constructing approximate models of groundwater flows in thin aquifers.     

Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions

In a rectangular domain, we consider the two-dimensional Poisson equation with nonlocal boundary conditions in one of the directions. For this problem, we construct a difference scheme of fourth-order approximation, study its solvability, and justify an iteration method for solving the corresponding system of difference equations. We give a detailed study of the spectrum of the matrix representing this system. In particular, we obtain a criterion for the nondegeneracy of this matrix and conditions for its eigenvalues to be positive.     

A two scale cellular automaton for flow dynamics modeling (2CAFDYM)

In this paper we propose a Two scale Cellular Automaton for Flow DYnamics Modeling (2CAFDYM) in a lowland region. Cells are terrain meshes with a predefined size, arranged in a bi-dimensional hexagonal lattice. The state of the cell consists of two scales: groundwater and surface water, in order to combine flows over saturated soil (Dunne flow) and flows exceeding the infiltration capacity (Hortonian flow). This allows for survey flood events and water resources. Each cell has intrinsic terrain attributes: altitude, soil type and land use. The obtained slopes are considered towards all the neighboring cells such that water flows simultaneously in multiple directions during the same time step. This helps us characterize laminar and turbulent flows. The model is subjected to climatic constraints: rainfall and temperature. The flow dynamics are regulated by mass conservation laws on hydraulic balance sheets (received, evaporated, infiltrated and drained water). Using Java Object Oriented Programming we have designed decision-aided software for the real-time monitoring of flow processes in 2D or 3D scenes through 2CAFDYM. We give some simulations for a basin in northern Morocco covering 34.3 km², including some areas that are potentially vulnerable to flooding. Digital terrain models, geological maps and satellite images are used to extract input data.     

Initial-boundary value problem of three phase flow in porous media

We study the mathematical model of three phase compressible flows through porous media. Under the condition that the rock, water and oil are incompressible, and the compressibility of gas is small, we present a finite element scheme to the initial-boundary value problem of the nonlinear system of equations, then by the convergence of the scheme we prove that the problem admits a weak solution.     

Numerical simulations for a seawater intrusion problem in a free aquifer

We simulate a sharp interface model issuing from a seawater intrusion problem in a free aquifer. We model the evolution of the sea front and of the upper free surface of the aquifer. We use a P1 finite element method for the space discretization combined with a semi-implicit in time scheme.     

Isospectral Flows for the Inhomogeneous String Density Problem

We derive isospectral flows of the mass density in the string boundary value problem corresponding to general boundary conditions. In particular, we show that certain class of rational flows produces in a suitable limit all flows generated by polynomials in negative powers of the spectral parameter. We illustrate the theory with concrete examples of isospectral flows of discrete mass densities which we prove to be Hamiltonian and for which we provide explicit solutions of equations of motion in terms of Stieltjes continued fractions and Hankel determinants.     

On the Dirichlet problem for a mixed-type equation of the second kind with strong degeneration

By separation of variables, we construct particular solutions of an equation of mixed elliptic-hyperbolic type of the second kind with strong degeneration in a rectangular domain. We pose certain transmission conditions on the singular line. The constructed solutions satisfy the equation in the entire mixed domain and are used to study the uniqueness of the solution of the Dirichlet problem for this domain.     

Domain-decomposition approach to local grid refinement in finite element collocation

Advection-dominated flows occur widely in the transport of groundwater contaminants, the movements of fluids in enhanced oil recovery projects, and many other contexts. In numerical models of such flows, adaptive local grid refinement is a conceptually attractive approach for resolving the sharp fronts or layers that tend to characterize the solutions. However, this approach can be difficult to implement in practice. A domain decomposition method developed by Bramble, Ewing, Pasciak, and Schatz, known as the BEPS method, overcomes many of the difficulties. We demonstrate the applicability of BEPS ideas to finite element collocation on trial spaces of piecewise Hermite cubics. The resulting scheme allows one to refine selected parts of a spatial grid without destroying algebraic efficiencies associated with the original coarse grid. We apply the method to steady-state problems with boundary and interior layers and a time-dependent advection-diffusion problem.     

Simulation of seepage flows from channels

Plane steady-state seepage in a homogeneous isotropic ground from channels through a layer of soil with an underlying highly permeable pressurized water-bearing layer when the ground possesses capillarity and there is evaporation from the free surface is considered in a hydrodynamic formulation. To investigate it, a mixed multiparametric boundary-value problem of the theory of analytical functions is formulated, which is solved using the Polubarinova-Kochina method and conformal mapping of the regions of a special form, typical of problems of underground hydromechanics. On the basis of this model, an algorithm for calculating the capillary spreading of water and seepage flow is developed in situations when the ground capillarity is taken into account in the seepage of water from channels, as well as evaporation from the free surface of the ground waters, and also backwater of the underlying highly permeable stratum. Using the exact analytical relations obtained and numerical calculations, a hydrodynamic analysis of the structure and characteristic features of the simulated process, and also of the effect of all the physical parameters of the system on the seepage characteristics, is carried out. Limit and special cases, related to the absence of one or two of the three factors, characterizing the simulated process are considered: the ground capillarity, evaporation from the free surface, and also backwater of the underlying highly permeable water-bearing layer. The results of the calculations are compared with similar seepage characteristics with a similar scheme, but in which the flow region is underlaid by an impenetrable base.     

Two‐phase flows in karstic geometry

Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil, and gas), and cloud and fog (water vapor, water, and air). Multiphase flows also play an important role in many engineering and environmental science applications. In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits, and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been performed on multiphase flows in karstic geometry. In this paper, we present a family of phase–field (diffusive interface) models for two‐phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws. A uniquely solvable numerical scheme that preserves the associated energy law is presented as well. Copyright © 2013 John Wiley & Sons, Ltd.     

A family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices

In this paper, we continue to develop a general scheme to study a broad class of integrable systems naturally associated with the coboundary dynamical Lie algebroids. In particular, we present a factorization method for solving the Hamiltonian flows. We also present two important classes of new examples, a family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices. To illustrate our factorization theory, we show how to solve these Hamiltonian systems explicitly. © 2004 Wiley Periodicals, Inc.     

Hydrogeochemical modeling for groundwater management in arid and semiarid regions using MODFLOW and MT3DMS: A case study of the Jeffara of Medenine coastal aquifer,South-Eastern Tunisia

The study of water quality and the quantification of reserves and their variations according to natural and anthropogenic forcing is necessary to establish an adequate management plan for groundwater resources. For this purpose, a modeling approach is a useful tool that allows, after calibration phase and verification of simulation, and under different scenarios of forcing and operational changes, to estimate and control the groundwater quantity and quality. The main objective of this study is to collect all available data in a model that simulates the Jeffara of Medenine coastal aquifer system functioning. To achieve this goal, a conceptual model was constructed based on previous studies and hydrogeological investigations. The regional groundwater numerical flow model for the Jeffara aquifer was developed using MODFLOW working under steady-state and transient conditions. Groundwater elevations measured from the piezometric wells distributed throughout the study area in 1973 were selected as the target water levels for steady state (head) model calibration. A transient simulation was undertaken for the 42 years from 1973 to 2015. The historical transient model calibration was satisfactory, consistent with the continuous piezometric decline in response to the increase in groundwater abstraction. The developed numerical model was used to study the system's behavior over the next 35 years under various constraints. Two scenarios for potential groundwater extraction for the period 2015–2050 are presented. The predictive simulations show the effect of the increase of the exploitation on the piezometric levels. To study the phenomenon of salinization, which is one of the most severe and widespread groundwater contamination problems, especially in coastal regions, a solute transport model has been constructed by using MT3DMS software coupled with the groundwater flow model. The best calibration results are obtained when the connection with the overlying superficial aquifer is considered suggesting that groundwater contamination originates from this aquifer. Recommendations for water resource managers

• The results of this study show that Groundwater resources of Jeffara of Medenine coastal aquifer in Tunisia are under immense pressure from multiple stresses.
• The water resources manager must consider the impact of economic and demographic development in groundwater management to avoid the intrusion of saline water.
• The results obtained presented some reference information that can serve as a basis for water resources planning.
• The model runs to provide information that managers can use to regulate and adequately control the Jeffara of Medenine water resources.

     

A robust central scheme for the shallow water flows with an abrupt topography based on modified hydrostatic reconstructions

We study a second-order central scheme for the shallow water flows with a discontinuous bottom topography based on modified hydrostatic reconstructions (HRs). The first HR scheme was proposed in Audusse et al, which may be missing the effect of the large discontinuous bottom topography. We introduce a modified HR method to cope with this numerical difficulty. The new scheme is well-balanced for still water solutions and can guarantee the positivity of the water depth. Finally, several numerical results of classical problems of the shallow water equations confirmed these properties of the new scheme. Especially, the new scheme yields superior results for the shallow water downhill flow over a step.     

The existence of a periodic solution in a tide dynamics problem

We examine a quasilinear boundary-value problem describing the dynamics of tidal flow in the sea. For small values of ground friction coefficient and under the fulfillment of a certain condition on the function giving the depth of the sea, we prove the existence of a generalized periodic solution. We construct a difference scheme for the numerical solution of the problem being examined and we prove its stability.Translated from Problemy Matematicheskogo Analiza. No. 4: Integralnye i Differentsial'nye Operatory. Differentsial'nye Uraveniya, pp. 3–9, 1973.     

Unsteady mixed flows in non uniform closed water pipes: a Full Kinetic Approach

We recall the Pressurized and Free Surface model constructed for the modeling of unsteady mixed flows in closed water pipes where transition points between the free surface and pressurized flow are treated as a free boundary associated to a discontinuity of the gradient of pressure. Then we present a numerical kinetic scheme for the computations of unsteady mixed flows in closed water pipes. This kinetic method that we call FKA for “Full Kinetic Approach” is an easy and mathematically elegant way to deal with multiple transition points when the changes of state between free surface and pressurized flow occur. We use two approaches namely the “ghost waves approach” and the “Full Kinetic Approach” to treat these transition points. We show that this kinetic numerical scheme has the following properties: it is wet area conservative, under a CFL condition it preserves the wet area positive, it treats “naturally” the flooding zones and most of all it is very easy to implement it. Finally numerical experiments versus laboratory experiments are presented and the scheme produces results that are in a very good agreement. We also present a numerical comparison with analytic solutions for free surface flows in non uniform pipes: the numerical scheme has a very good behavior. A code to code comparison for pressurized flows is also conducted and leads to a very good agreement. We also perform a numerical experiment when flooding and drying flows may occur and finally make a numerical study of the order of the kinetic method.