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.     

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.     

The Zhukovskii problem of the flow around a sheet pile

The solution of the Zhukovskii problem of the flow around a sheet pile is given using the principles of two-dimensional steady-state seepage in the case when, accompanying the motion of the seeping water, there is a layer of saline ground waters at a certain depth under the sheet pile and this layer is located above an impermeable thickness of rock salt. The mixed boundary-value problem of the theory of analytic functions which arises is solved using Polubarinova-Kochina's method, which is based on the application of the analytical theory of linear differential equations and, also, the method, developed by us, of the conformal mappings of circular polygons in polar meshes, which are extremely typical for the velocity hodograph domains of such flows. While reflecting the specific details and individual properties of such flows, the solution constructed below turns out to be expressed in closed form in terms of elementary functions and, consequently, it is the simplest and most convenient solution. In addition, it is the most general solution for the class of problems being considered. The well known results Zhukovskii, Vedernikov and others are obtained from it as special and limiting çases A detailed hydrodynamic analysis and the specific features of the seepage process being considered, as well as the effects of all the physical parameters of the model on the pattern of the phenomenon, are presented using this solution and by numerical calculations.     

A computational procedure for planar seepage problems in unbounded domains

We consider the boundary-value problem modeling planar seepage of groundwater from a single source contour to a regular infinite system of perfect drainage ditches. A fast converging iterative method is developed for such problems, based on the P-transformation method and the method of accelerated successive over relaxation.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 86–92, 1987.     

Mathematical modeling of systematic drainage of irrigated soil over an impermeable layer

We solve the problem of plane steady-state seepage of groundwater in a homogeneous isotropic soil layer from a periodic system of irrigation canals under conditions of both infiltration and horizontal drainage. A detailed hydrodynamic analysis is performed of the structure and properties of seepage flow and the effect of physical parameters of the flow scheme.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 72–75, 1990.     

A conformal mapping approach for investigating the free-boundary seepage from symmetric channels

We present a new method to investigate the two-dimensional free-boundary groundwater seepage from symmetric soil channels into a homogeneous isotropic porous medium. We use Levi–Civitá’s function to construct an integral representation for the conformal mapping of the complex potential domain onto the physical flow domain. A genetic algorithm (GA) is used to calculate the coefficients of the Maclaurin series expansion of Levi–Civitá’s function. The coordinates of the points from the channel contour, calculated by means of the integral representation, must satisfy the analytic equation of the contour. We use this condition to define the objective function of the genetic algorithm. Levi–Civitá’s function is afterwards used to calculate the seepage loss, the free lines, the streamlines, the equipotential lines, the isobars and the velocity field. Some examples illustrate the method.     

Estimates for the inverse of tridiagonal matrices arising in boundary-value problems

By considering tridiagonal matrices as three-term recurrence relations with Dirichlet boundary conditions, one can formulate their inverses in terms of Green's functions. This analysis is applied to three-point difference schemes for 1-D problems, and five-point difference schemes for 2-D problems. We derive either an explicit inverse of the Jacobian or a sharp estimate for both uniform and nonuniform grids.     

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.     

Numerical inversion of heat exchange coefficient and source sink term for a thermal-fluid coupled model in porous medium

Abstract

In this paper, we focus on three inverse problems for a coupled model from temperature-seepage field in high-dimensional spaces. These inverse problems aim to determine an unknown heat transfer coefficient and a source sink term in seepage continuity equation with specified initial-boundary conditions and additional measurements. Some finite difference schemes of coupled equations are presented and analyzed.Three algorithms for these inverse problems are proposed. Some numerical experiments are provided to assert the accuracy and efficiency of proposed algorithms.     

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.     

Spline Techniques for the Numerical Solution of Singular Perturbation Problems

One-dimensional singularly-perturbed two-point boundary-value problems arising in various fields of science and engineering (for instance, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics, etc.) are treated. Either these problems exhibits boundary layer(s) at one or both ends of the underlying interval or they possess oscillatory behavior depending on the nature of the coefficient of the first derivative term. Some spline difference schemes are derived for these problems using splines in compression and splines in tension. Second-order uniform convergence is achieved for both kind of schemes. By making use of the continuity of the first-order derivative of the spline function, a tridiagonal system is obtained which can be solved efficiently by well-known algorithms. Numerical examples are given to illustrate the theory.     

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.     

Seepage optimization of the shape of an earth channel allowing for capillarity

The channel shape minimizing the seepage losses for a given cross-section area is determined under the conditions of headless steady plane seepage. The solution is constructed using the theory of inverse boundaryvalue problems in the form of dependences of geometrical and seepage characteristics of the sought channel on given parameters — the area and the height of capillary rise.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 70–74, 1987.     

Comparison theorems for dam foundation design from back pressure diagram

Comparison theorems are established for the determination of the subsurface contour of the dam foundation from the seepage back pressure p(x). The corresponding problems are considered in a general setting, which allows curvilinear boundary sections, nonhomogeneous soil, and multiply connected regions. The theorems analyze the sensitivity of the contour to changes in initial data, such as the back pressure diagram, aquifer geometry, race bottom shape, and distribution of seepage coefficients. Similarly to Polozhii's comparison theorems [8] for problems of seepage under head, the proposed theorems can be applied to obtain majorizing bounds for inverse boundary-value problems.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 76–82, 1988.     

Optimization of solutions of one class of problems in seepage theory

Optimal control of prevention of groundwater flooding is considered. Optimization is performed by construction and operating costs of the protective structures. The seepage problem is solved for homogeneous Isotropic regions protected by perfect vertical drain ditches. A computer program is developed to solve the problem. Numerical results are reported.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 84–89, 1986.     

Investigation of the effect of seepage or evaporation from a free surface by the method of circular polygons

It is proposed to use a technique developed for polygons in polar nets to integrate equations of the Fuchs class that arise when solving a wide range of problems of plane steady seepage flow using the Polubarinova-Kochina method, based on the use of the analytical theory of linear differential equations. It is shown that, for a large class of pentagons in domains where the flows,which are very characteristic of seepage problems when there is infiltration or evaporation from the free surface, have a complex velocity, the solution of the problem of determining the unknown parameters which appear in the conformal mapping can be completed. In this case, the mapping is carried out in closed form in terms of elementary functions and it is simple and convenient for subsequent application. The results obtained are used to solve the problem of seepage from a channel, taking account of the capillarity of the ground when there is evaporation from the free surface. The results of numerical calculations are presented and a hydrodynamic analysis of the effect of the basic physical parameters of the model on the dimensions of the saturation zone is given.     

Effect of soil capillarity on seepage from a channel with evaporation

The method of Polubarinova-Kochina is applied to solve the seepage problem for a channel with evaporation and capillary soil. The capillary spread of water and the seepage discharge from the channel are investigated as a function of the height of capillary rise and evaporation rate. Results of numerical experiments are presented. The limiting case of flow without evaporation, previously studied by Rizenkampf, is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 40–43, 1986.     

Efficient Time Integration of IMEX Type using Exponential Integrators for Compressible,Viscous Flow Simulation

We investigate the adaption of the recently developed exponential integrators called EPIRK in the so-called domain-based implicit-explicit (IMEX) setting of spatially discretized PDE's. The EPIRK schemes were shown to be efficient for sufficiently stiff problems and offer high precision and good stability properties like A- and L-stability in theory. In practice, however, we can show that these stability properties are dependent on the parameters of the interior approximation techniques. Here, we introduce the IMEX-EPIRK method, which consists of coupling an explicit Runge-Kutta scheme with an EPIRK scheme. We briefly analyze its linear stability, show its conservation property and set up a CFL condition. Though the method is convergent of only first order, it demonstrates the advantages of this novel type of schemes for stiff problems very well. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the underground contour of a submerged apron with a region of constant velocity when there is a salt backwater

Within the framework of two-dimensional seepage theory, the underground contour of a submerged apron with a region of constant velocity in the case where there is a layer of stagnant salt water under the apron is constructed. The solution of the corresponding boundary-value problem is found by Polubarinova-Kochina's method [1] using the results obtained in [2]. The results of numerical calculations are given and the influence of the fundamental defining parameters of the model on the shape and size of the underground contour of the apron is analysed. Mention is made of special and limiting cases: a scheme with a water-confining stratum [3], an unsubmerged apron [2] and flow around a tongue [4,5].     

Numerical solution by the quasi-reversibility method of unstable problems for the first-order evolution equation

Prior bounds are derived on the solution of the perturbed problem in different versions of the quasi-reversibility method used for approximate solution of unstable problems for first-order evolution equations. An example of such a problem is provided by the problem backward in time for the equation of heat conduction. Approximate solution of perturbed problems by difference methods is considered. The investigation of the difference schemes of the quasi-reversibility method relies on the general theory of p-stability of difference schemes. Specific features of solution of problems with non-self-adjoint operators are considered. Efficient difference schemes are constructed for multidimensional problems.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 93–124, 1993.