Optimal placement of a wellbore seal impeding seepage from a tilted fracture

Steady, one-phase, 2D Darcian seepage in a homogeneous pay from one horizontal isobar to another with a tilted fracture intersecting the two isobars is studied by the methods of complex analysis. The shortcut through the fracture is foiled by an impermeable seal of a constant length which is placed at the fracture-isobar intersection with an objective to minimise either the total flow rate from the fracture or the travel time of marked particles moving along the seal. Conformal mappings of the slanted-face half-strip in the physical domain onto a half-strip in the complex potential domain give explicit expressions of the seepage rate and travel time through hypergeometric functions. The optimal seal position is found by computer algebra routines for any given distance between isobaric horizons, tilt angle, pressure drop between isobars and hydraulic conductivity of the rock.     

The design of the iso-velocity contour for the flow past the base of a dam with a confining bed

The underground contour of a submerged rectangular dam, the angles of which are rounded off along curves of constant magnitude of the seepage flow velocity is constructed in the case when the water-permeable base is underlain by a confining bed, consisting of one horizontal and two curved sections, which are the iso velocity lines of the seepage flow. The corresponding multiparameter mixed problem in the theory of analytical functions is solved using the Riemann-Schwartz symmetry principle and a semi-inverse version of the velocity hodograph method, first proposed by Polybarinova-Kochina and Kochina. The results of numerical calculations are presented and a hydrodynamic analysis of the effect of the basic physical parameters of the model on the shape and dimensions of the underground contour of the dam and of the horizontal and curved sections of the confining bed is given.     

Modelación del movimiento del agua en medios porosos no lineales mediante un esquema de diferencias finitas. Aplicación al sobrevertido en presas de escollera

Flow through coarse porous media, as rockfill, gravel or sand of big size, exhibit a non linear relation between hydraulic gradient and seepage velocity. This behavior is observed when water flows over the crest of a rockfill dam, seeps into the downstream shoulder and moves through the rockfill voids. A finite difference scheme is used here to solve this problem through numerical modeling. It is implemented in the code MNLEE.RDS for solving dam overtopping problems. In addition, a procedure is provided to solve the coupled problem of flow over and through the rockfill dam. For that purpose, the concept of «law of flow exchange»is introduced.     

Seepage beneath a concrete dam with a downstream filter

A study is presented on the seepage occurring through homogeneous soil beneath a concrete dam with a downstream filter and taking into account the existence of a lower horizontal impervious sublayer. The problem is hydrodynamically solved by using the Schwartz-Christoffel transformation and a seepage formula is presented. A study on a non-dimensional graphical representation of the seepage formula led to the prediction of the recommended lengths of downstream filters. In addition, the author presents a formula for evaluating the uplift pressure distribution on the base of the dam. Examination of the pressure distribution formula shows that two important boundary conditions are satisfied.     

Admixture stratification in the stagnation region of two streams

A self-similar 2D steady-state flow of two immisible viscous fluids with inertial particles is considered. It is assumed that two viscous streams, one of which contains the particles, collide forming a flat interface with a stagnation point. The general case is discussed, when the fluids have different viscosities and densities and the streams are directed at arbitrary angles. The far field corresponds to inviscid vortex flow near an “oblique” stagnation point. The limiting case of viscous dusty flow near a rigid plane is studied in detail within one-way and two-way coupling approximations. Thin zones of particle accumulation are detected. Threshold parameters corresponding to the change of flow regime are found. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady flow of a viscous liquid in a porous elliptic tube

Flow of a viscous-liquid in a porous tube of elliptic cross-section is studied using the generalized momentum equation. As a particular case, flow of the liquid in a tube of a circular cross-section is obtained. It is observed that the classical Darcian effect is realized only in a core very near to the axis of the tube while the non-Darcian phenomenon is felt predominantly near the boundary of the tube.     

The Dupuit Approximation for the Rectangular Dam Problem

A porous rectangular dam is above a horizontal impermeable base.There is a steady flow in which water seeps through the damfrom one reservoir (on the left)to a lower reservoir (on theright). Because of gravity, the water does not flow throughthe entire dam and the dam is dry near its upper-right corner.The interface separating the dry and wet regions of the damis a free boundary (in the hydrology literature called the phreaticsurface). Dupuit, in 1863, derived the formal approximationthat the phreatic surface was nearly a parabola. In recent years,mathematicians have obtained existence results. Here, relationsare established between the formal approximations and the solutionsobtained from the existence proofs.     

The cauchy problem for the radiative transfer equation with generalized conjugation conditions

The initial boundary problem for the nonstationary radiative transfer equation in a nonhomogeneous plane layer with generalized conjugation conditions on the material interface is studied. A generalized Monte Carlo algorithm is proposed for solving the problem, and numerical experiments are discussed.     

Seepage refraction in a semicircular lens located at the boundary of two porous massifs

The problem of calculating the two-dimensional seepage field in a structurally inhomogeneous three-component medium in the form of two infinitely porous massifs with a semicircular inclusion in their plane boundary of contact is considered. The distribution of the seepage rate, when two matching conditions along the lines of contact of unlike zones are strictly satisfied, is obtained in closed analytic form by methods of complex analysis. Limiting cases of the conduction of the components of the medium and cases of the degeneration of a three-component medium into a two-component medium are considered.     

Periodic behaviour for the evolutionary dam problem and related free boundary problems Evolutionary dam problem

The time dependent dam problem describing the seepage of a compressible or incompressible fluid in a porous dam is studied. We prove existence of solutions in a suitable weak sense, and uniqueness for rectangular dams. Existence or periodic solutions is established and questions ofstability and periodic behavior for large time are studied.     

Modelling the impact of dam failure scenarios on flood inundation using SPH

Flooding resulting from collapse of a dam is a highly destructive event. It is important to accurately predict the flow behaviour so that potential mitigation strategies can be investigated for disaster management planning. The meshless SPH method has previously been able to model this class of extreme flow events. In this paper, we extend the method to include modelling of dam wall fragments. Collisions between dam wall fragments, between fragments and terrain and full two-way coupling between fragments and the free surface water flow is included. This gives a method that can specifically investigate the impact of the dam wall failure scenario on the subsequent inundation. The historical St. Francis dam failure is used to demonstrate the impact of including the dam fragments. It also provides a means of quantitatively investigating their effect in terms of arrival time and water height at a downstream power station. The scenario with multiple independently timed failures of different parts of the wall (which closely matches the historical failure) gives excellent agreement with the observed data and gives the best match of all failure scenarios. Traditionally such modelling is performed by solving the two dimensional shallow water equations which is not able to capture the three dimensional nature of the flow in earlier stages of dam flooding. We specifically investigate the three dimensional nature of flow structures and formation of multiple downstream hydraulic jumps. These strongly influence water height and therefore control the extent of flooding of tributary valleys.     

Temperature Stresses in a Piecewise-Homogeneous Cylindrical Shell Caused by the Presence of a Longitudinal Crack

By using the method of generalized conjugation problems, we propose a numerical scheme for investigation of the redistribution of temperature stresses in a piecewise-homogeneous cylindrical shell caused by a longitudinal crack. This scheme is based on systems of integral equations (some of them are singular) to determine the unknown jumps of integral characteristics of the disturbed temperature field and displacements and their derivatives on the line of a crack and on the interface as well as the derivatives of these integral characteristics with respect to the longitudinal coordinate at the interface.     

Integral equations in the wave scattering problem at an irregular interface of domains

Modeling the scattering of electromagnetic waves at an interface of media with different characteristics, one encounters the conjugation problem. Using the method of boundary integral equations and the theory of generalized potentials, we prove the classical resolvability of this problem. The boundary is assumed to be irregular. This means that the plane is divided into two domains by a curve which coincides with a straight line, except for a finite part, producing the irregularity. We propose algorithms for the approximate solution of the conjugation problem based on the spline methods for the solution of integral equations. We theoretically substantiate the computational scheme, namely, we prove the convergence and estimate the convergence rate.     

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.     

A periodic system of electrically permeable cracks at the interface between two piezoelectric materials

We constructed a closed solution for a bimaterial plane consisting of two dissimilar piezoelectric half-planes with a periodic system of electrically permeable cracks at the interface between these materials. The presence of zones of smooth contact of the crack lips near their tips was taken into account. By representing the characteristics of electromechanical fields via piecewise analytic functions, we reduced the problem to a Dirichlet–Riemann periodic problem, which was solved exactly. As a result of numerical analysis of the derived solution, we studied the dependence of the relative length of the contact zones and stress intensity factors on the ratio between the crack length and period for different combinations of piezoelectric materials.     

A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle

We consider the steady two-dimensional flow under gravity ofwater from one reservoir (on the left) to a lower reservoir(on the right) through a porous rectangular isotropic homogeneousdam with impervious bottom. Because of gravity the water doesnot flow through the entire dam and the dam is dry near itsupper right corner. The interface separating the dry and wetregions of the dam is a free boundary. Recently, Friedman &Jensen (1977) have proved that the free boundary is convex.We give a different proof which uses only the maximum principleand its generalizations.     

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.     

Studies of an interface relaxation domain decomposition technique using finite elements on a parallel computer

Studies are presented for an interface relaxation domain decomposition technique using finite elements on an iPSC/2 D5 Hypercube Concurrent computer. The general type of problem to be solved is one governed by a partial differential equation. The application of the approach, however, will be extended to a free boundary value problem by appropriate modification of the numerical scheme. Using the domain decomposition technique, the computation domain is subdivided into several subdomains. In addition, on the interfaces between two adjacent subdomains are imposed a continuity condition on one side and an equilibrium condition on the other side. Successive overrelaxation iterative processes are then carried out in all subdomains with a relaxation process imposed on the interfaces. With this domain decomposition technique, the problem can be solved parallelly until convergence is reached both in the interiors and on the interfaces of all subdomains. Moreover, the formulation includes a simple domain decomposer that automatically divides a finite element mesh into a list of subdomains to guarantee load balancing. Furthermore, it is shown, through numerical experiments performed on an example problem of free surface seepage through a porous dam, how the values of the relaxation parameters, the choice of imposed boundary conditions, and the number of subdomains (i.e., the number of processors used) affect the solution convergence in this parallel computing environment. © 1993 John Wiley & Sons, Inc.     

A variational inequality formulation for unconfined seepage problems in porous media

In the existing variational inequality formulations for the unconfined seepage problem in porous media, the seepage point, namely the exit point of the free surface, is a singular point and how to locate the seepage point exactly has been an open issue. By generalizing Darcy’s law applied solely to the saturated zone in an earth dam to the entire dam including the no-flow zone, a new variational inequality formulation is presented. The new formulation imposes a boundary condition of Signorini’s type on the potential seepage boundary and the seepage point turns out to be such a point that makes both inequalities in Signorini’s complementary condition become equalities. Singularity of the seepage point is accordingly eliminated. A strategy is developed for overcoming the mesh-dependency in the finite element implementation.