共查询到20条相似文献，搜索用时 46 毫秒
1.
A. C. LIAKOPOULOS Dr. Eng. 《水文科学杂志》2013,58(1):69-110
ABSTRACT The one-dimensional transient downward entry of water in unsaturated soils is investigated theoretically. The mathematical equation describing the infiltration process is derived by combining Darcy's dynamic equation of motion with the continuity and thermodynamic state equations adjusted for the unsaturated flow conditions. The resulting equation together with the corresponding initial and boundary conditions constitues a mathematical initial boundary value problem requiring the solution of a nonlinear partial differential equation of the parabolic type. The volumetric water content is taken as the dependent variable and the time and the position along the vertical direction are taken as the independent variables. The governing equation is of such nature that a solution exists for t > 0 and is uniquely determined if two relationships are defined, together with the specified state of the system, at the initial time t = 0 and at the two boundaries. The two required relations are those of pressure versus permeability and pressure versus volumetric water content. Since the partial differential equation has strong non-linear terms, a discrete solution is obtained by approximating the derivatives with finite-differences at discrete mesh points in the solution domain and integrated for the corresponding initial and boundary conditions. The use of an implicit difference scheme is employed in order to generate a system of simultaneous non-linear equations that has to be solved for each time increment. For n mesh points the two boundary conditions provide two equations and the repetition of the recurrence formula provides n—2 equations, the total being n equations for each time increment. The solution of the system is obtained by matrix inversion and particularly with a back-substitution technique. The FORTRAN statements used for obtaining the solution with an electronic digital computer (IBM 704) are presented together with the input data. Analysis of the errors involved in the numerical solution is made and the stability and convergence of the solution of the approximate difference equation to that of the differential equation is investigated. The method applied is that of making a Fourier series expansion of a whole line of errors and then following the progress of the general term of the series expansion and also the behavior of each constituent harmonic. The errors (forming a continuous function of points in an abstract Banach space) are represented by vectors with the Fourier coefficients constituting a second Banach space. The amplification factor of the difference equation is shown to be always less than unity which guarantees the stability of the employed implicit recurrence scheme. Experiments conducted on a vertical column packed uniformly with very fine sand, show a satisfactory agreement between the theoretically and experimentally obtained values. Many experimental results are shown in an attempt to explain the infiltration phenomenon with emphasis on the shape and movement of the wet front, and the effects of the degree of compaction, initial water content and deaired water on the infiltration rate. 相似文献
2.
Aleksey S. Telyakovskiy Gastão A. Braga Satoko Kurita Jeff Mortensen 《Advances in water resources》2010
For certain initial and boundary conditions the Boussinesq equation, a nonlinear partial differential equation describing the flow of water in unconfined aquifers, can be reduced to a boundary value problem for a nonlinear ordinary differential equation. Using Song et al.'s (2007) [7] approach, we show that for zero head initial condition and power-law flux boundary condition at the inlet boundary, the solution in the form of power series can be obtained with Barenblatt's (1990) [2] rescaling procedure applied to the power series solution obtained in Song et al. (2007) [7] for the power-law head boundary condition. Polynomial approximations can then be obtained by taking terms from the power series. Although for a small number of terms the newly obtained approximations may be worse than polynomial approximations obtained by other techniques, any desired accuracy can be achieved by taking more terms from the power series. 相似文献
3.
To calculate the dynamic-stiffness matrix at the structure–medium interface of an unbounded medium for the range of frequencies of interest, the consistent infinitesimal finite-element cell method based on finite elements is developed. The derivation makes use of similarity and finite-element assemblage, yielding a non-linear first-order ordinary differential equation in frequency. The asymptotic expansion for high frequency yields the boundary condition satisfying the radiation condition. In an application only the structure–medium interface is discretized resulting in a reduction of the spatial dimension by one. The boundary condition on the free surface is satisfied automatically. The consistent infinitesimal finite-element cell method is exact in the radial direction and converges to the exact solution in the finite-element sense in the circumferential directions. Excellent accuracy results. 相似文献
4.
5.
V. V. Spichak 《Izvestiya Physics of the Solid Earth》2006,42(3):193-200
A general approach to the construction of differential boundary conditions for vector fields satisfying the Helmholtz equation is proposed on the basis of the field expansion in multipole series and the application of annihilating operators to them. The resulting differential constraints can be used as boundary conditions in solving external boundary value problems. Examples of their application to the solution of forward geoelectric problems in three-dimensionally inhomogeneous media are examined. Their use at a finite distance from the source of an anomaly is shown to yield more accurate results than those obtained under the assumption that the anomalous field at this distance vanishes. Another effect of their application is a substantial decrease in the dimensions of the modeling domain and therefore in the time required to solve the forward problem. The “safe” distance for using the Dirichlet-type boundary conditions is estimated. 相似文献
6.
D. G. Zeitoun C. Braester 《Stochastic Environmental Research and Risk Assessment (SERRA)》1991,5(3):207-226
A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well. 相似文献
7.
Mehl S 《Ground water》2006,44(4):583-594
This study examines the use of Picard and Newton iteration to solve the nonlinear, saturated ground water flow equation. Here, a simple three-node problem is used to demonstrate the convergence difficulties that can arise when solving the nonlinear, saturated ground water flow equation in both homogeneous and heterogeneous systems with and without nonlinear boundary conditions. For these cases, the characteristic types of convergence patterns are examined. Viewing these convergence patterns as orbits of an attractor in a dynamical system provides further insight. It is shown that the nonlinearity that arises from nonlinear head-dependent boundary conditions can cause more convergence difficulties than the nonlinearity that arises from flow in an unconfined aquifer. Furthermore, the effects of damping on both convergence and convergence rate are investigated. It is shown that no single strategy is effective for all problems and how understanding pitfalls and merits of several methods can be helpful in overcoming convergence difficulties. Results show that Picard iterations can be a simple and effective method for the solution of nonlinear, saturated ground water flow problems. 相似文献
8.
Developing robust and efficient numerical solution methods for Richards' equation (RE) continues to be a challenge for certain problems. We consider such a problem here: infiltration into unsaturated porous media initially at static conditions for uniform and non-uniform pore size media. For ponded boundary conditions, a sharp infiltration front results, which propagates through the media. We evaluate the resultant solution method for robustness and efficiency using combinations of variable transformation and adaptive time-stepping methods. Transformation methods introduce a change of variable that results in a smoother solution, which is more amenable to efficient numerical solution. We use adaptive time-stepping methods to adjust the time-step size, and in some cases the order of the solution method, to meet a constraint on nonlinear solution convergence properties or a solution error criterion. Results for three test problems showed that adaptive time-stepping methods provided robust solutions; in most cases transforming the dependent variable led to more efficient solutions than untransformed approaches, especially as the pore-size uniformity increased; and the higher-order adaptive time integration method was robust and the most efficient method evaluated. 相似文献
9.
Based on Fourier-Bessel series expansion of wave functions,an analytical solution to 2-D scattering ofincident plane SV waves by circular cylindrical canyons with variable depthto-width ratios is deduced in this paper. Unlike other analytical solutions,this paper uses the asymptotic behavior of the cylindrical function to directly define the undetermined coefficients of scattered waves,thus,avoiding solving linear equation systems and the related numerical computation problems under high-frequency incident waves,thereby broadening the applicable frequency range of analytical solutions. Through comparison with existing analytical solutions,the correctness of this solution is demonstrated. Finally, the incident plane SV wave scattering effect under circular cylindrical canyons in wider frequency bands is explored. 相似文献
10.
B. I. Birger 《Izvestiya Physics of the Solid Earth》2011,47(6):541-554
Stationary solutions including wave solutions with constant amplitudes are found for nonlinear equations of thermal convection
in a layer with nonlinear rheology. The solution is based on the Fourier expansion of unknown velocities and temperatures
with only the first and first two terms retained in the velocity and temperature series, respectively. This method, which
can be regarded as the Lorenz method, yields the Lorenz equations that fairly well describe the thermal convection in a layer
with Newtonian rheology if the Rayleigh number is not very large. The obtained generalization of the Lorenz equations to the
case of an integral (having a memory) nonlinear rheology implies that only the first term is retained in the Fourier series
for the stress components, i.e., the nonlinear rheological equation is harmonically linearized. However, in the Fourier series
of temperature, it is essential to keep the second term: this term, which is independent of the horizontal coordinate, models
the thermal boundary layer that characterizes the developed convection. We constructed the bifurcation curves that describe
the stationary convection in the nonlinear integral medium simulating the rheology of the mantle, and analyzed the stability
of stationary convective flows. The Lorenz method is applied to study small-scale thermal convection in the lithosphere of
the Earth. 相似文献
11.
Horizontal gravity filtration of groundwater in soil is considered. Under Boussinesq approximation, the problem is reduced to a one-dimensional nonlinear parabolic equation in phreatic water level. The problem of linearizing the original equation is discussed. The comparison of gravity-filtration problem solutions in the nonlinear and linearized formulations shows considerable discrepancies to exist between the solutions, especially, for boundary problems with mixed boundary conditions, when the value of the function is not fixed on the right boundary. An analytical solution is obtained for steady-state flow from a water body into the soil with subsequent leakage into underlying beds. Two regimes are shown to exist: one with an infinite exponential tail, and another in the form of a finite groundwater mound. A new approach is proposed to the linearization problem—quasilinearization with the use of the Burgers equation. 相似文献
12.
Xingzhong Shi Richard T. McNider M. P. Singh David E. England Mark J. Friedman William M. Lapenta William B. Norris 《Pure and Applied Geophysics》2005,162(10):1811-1829
Previous studies of the stable atmospheric boundary layer using techniques of nonlinear dynamical systems (MCNIDER et al., 1995) have shown that the equations support multiple solutions in certain parameter spaces. When geostrophic speed is used as a bifurcation parameter, two stable equilibria are found—a warm solution corresponding to the high-wind regime where the surface layer of the atmosphere stays coupled to the outer layer, and a cold solution corresponding to the low-wind, decoupled case. Between the stable equilibria is an unstable region where multiple solutions exist. The bifurcation diagram is a classic S shape with the foldback region showing the multiple solutions. These studies were carried out using a simple two-layer model of the atmosphere with a fairly complete surface energy budget. This allowed the dynamical analysis to be carried out on a coupled set of four ordinary differential equations. The present paper extends this work by examining additional bifurcation parameters and, more importantly, analyzing a set of partial differential equations with full vertical dependence. Simple mathematical representations of classical problems in dynamical analysis often exhibit interesting behavior, such as multiple solutions, that is not retained in the behavior of more complete representations. In the present case the S-shaped bifurcation diagram remains with only slight variations from the two-layer model. For the parameter space in the foldback region, the evolution of the boundary layer may be dramatically affected by the initial conditions at sunset. An eigenvalue analysis carried out to determine whether the system might support pure limit-cycle behavior showed that purely complex eigenvalues are not found. Thus, any cyclic behavior is likely to be transient. 相似文献
13.
P. Avtar 《Pure and Applied Geophysics》1967,66(1):48-68
Summary The frequency equation is derived for the propagation of Love waves in a two layered crust overlying an inhomogeneous half-space. While exact solutions of the differential equation involved are obtained in some particular cases, both an asymptotic series and the WKBJ approximations are discussed for the solution in the general case. The asymptotic series solution and the WKBJ solution are compared with each other and with the asymptotic expansion of the exact solution. There is a good agreement between various results. It is shown that some of the existing results can be derived as particular cases of the general results obtained in this investigation. 相似文献
14.
A convection-diffusion equation arises from the conservation equations in miscible and immiscible flooding, thermal recovery, and water movement through desiccated soil. When the convection term dominates the diffusion term, the equations are very difficult to solve numerically. Owing to the hyperbolic character assumed for dominating convection, inaccurate, oscillating solutions result. A new solution technique minimizes the oscillations. The differential equation is transformed into a moving coordinate system which eliminates the convection term but makes the boundary location change in time. We illustrate the new method on two one-dimensional problems: the linear convection-diffusion equation and a non-linear diffusion type equation governing water movement through desiccated soil. Transforming the linear convection diffusion equation into a moving coordinate system gives a diffusion equation with time dependent boundary conditions. We apply orthogonal collocation on finite elements with a Crank-Nicholson time discretization. Comparisons are made to schemes using fixed coordinate systems. The equation describing movement of water in dry soil is a highly non-linear diffusion-type equation with coefficients varying over six orders of magnitude. We solve the equation in a coordinate system moving with a time-dependent velocity, which is determined by the location of the largest gradient of the solution. The finite difference technique with a variable grid size is applied, and a modified Crank-Nicholson technique is used for the temporal discretization. Comparisons are made to an exact solution obtained by similarity transformation, and with an ordinary finite difference scheme on a fixed coordinate system. 相似文献
15.
This paper presents a method for constructing polynomial-based approximate solutions to the Boussinesq equation with cylindrical symmetry. This equation models water injection at a single well in an unconfined aquifer; as a sample problem we examine recharge of an initially empty aquifer. For certain injection regimes it is possible to introduce similarity variables, reducing the original problem to a boundary-value problem for an ordinary differential equation. The approximate solutions introduced here incorporate both a singular part to model the behavior near the well and a polynomial part to model the behavior in the far field. Although the nonlinearity of the problem prevents decoupling of the singular and polynomial parts, the paper presents an approach for calculating the solution based on its spatial moments. This approach yields closed-form expressions for the position of the wetting front and for the form of the phreatic surface. Comparison with a highly accurate numerical solution verifies the accuracy of the newly derived approximate solutions. 相似文献
16.
《Advances in water resources》2001,24(5):531-550
The objective of this paper is to demonstrate the formulation of a numerical model for mass transport based on the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. To this end, the classical chemical transport equation is derived as the zeroth moment of the BGK Boltzmann differential equation. The relationship between the mass transport equation and the BGK Boltzmann equation allows an alternative approach to numerical modeling of mass transport, wherein mass fluxes are formulated indirectly from the zeroth moment of a difference model for the BGK Boltzmann equation rather than directly from the transport equation. In particular, a second-order numerical solution for the transport equation based on the discrete BGK Boltzmann equation is developed. The numerical discretization of the first-order BGK Boltzmann differential equation is straightforward and leads to diffusion effects being accounted for algebraically rather than through a second-order Fickian term. The resultant model satisfies the entropy condition, thus preventing the emergence of non-physically realizable solutions including oscillations in the vicinity of the front. Integration of the BGK Boltzmann difference equation into the particle velocity space provides the mass fluxes from the control volume and thus the difference equation for mass concentration. The difference model is a local approximation and thus may be easily included in a parallel model or in accounting for complex geometry. Numerical tests for a range of advection–diffusion transport problems, including one- and two-dimensional pure advection transport and advection–diffusion transport show the accuracy of the proposed model in comparison to analytical solutions and solutions obtained by other schemes. 相似文献
17.
《Advances in water resources》2005,28(10):1133-1141
We study the motion of wetting fronts for vertical infiltration problems as modeled by Richards’ equation. Parlange and others have shown that wetting fronts in infiltration flows can be described by traveling wave solutions. If the soil layer is not initially dry, but has an initial distribution of water content then the motion of the wetting front will change due to the interaction of the infiltrating flow with the pre-existing soil conditions. Using traveling wave profiles, we construct simple approximate solutions of initial-boundary value problems for Richards’ equation that accurately describe the position and moisture distribution of the wetting front. We show that the influences of surface boundary conditions and initial conditions produce shifts to the position of the wetting front. The shifts can be calculated by examining the cumulative infiltration, and are validated numerically for several problems for Richards’ equation and the linear advection–diffusion equation. 相似文献
18.
In this paper we derive some similarity solutions of a nonlinear equation associated with a free boundary problem arising
in the shallow-water approximation in glaciology. In addition we present a classical potential symmetry analysis of this second-order
nonlinear degenerate parabolic equation related to non-Newtonian ice sheet dynamics in the isothermal case. After obtaining
a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any
unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions
is described. This allows us to obtain several qualitative properties on the free moving boundary in the presence of an accumulation-ablation
function with realistic physical properties. 相似文献
19.
This series of articles present general applications of functional-analytic theory to the solution of the partial differential equation describing solid transport in aquifers, when either the evolution of the system, the sources, the parameters and/or the boundary conditions are prescribed as stochastic processes in time or in space. This procedure does not require the restricting assumptions placed upon the current particular solutions on which today's stochastic transport theory is based, such as small randomness assumptions (perturbation techniques), Montecarlo simulations, restriction to small spatial stochasticity in the hydraulic conductivity, use of spectral analysis techniques, restriction to asymptotic steady state conditions, and restriction to variance of the concentration as the only model output among others. Functional analysis provides a rigorous tool in which the concentration stochastic properties can be predicted in a natural way based upon the known stochastic properties of the sources, the parameters and/or the boundary conditions. Thus the theory satisfies a more general modeling need by providing, if desired, a systematic global information on the sample functions, the mean, the variance, correlation functions or higher-order moments based on similar information of any size, anywhere, of the input functions. Part I of this series of articles presents the main relevant results of functional-analytic theory and individual cases of applications to the solution of distributed sources problems, with time as well as spatial stochasticity, and the solution subject to stochastic boundary conditions. It was found that the stochastically-forced equation may be a promising model for a variety of random source problems. When the differential equation is perturbed by a time and space stochastic process, the output is also a time and space stochastic process, in contrast with most of the existing solutions which ignore the temporal component. Stochastic boundary conditions seems to quickly dissipate as time increases. 相似文献
20.
The strong coupling of applied stress and pore fluid pressure, known as poroelasticity, is relevant to a number of applied problems arising in hydrogeology and reservoir engineering. The standard theory of poroelastic behavior in a homogeneous, isotropic, elastic porous medium saturated by a viscous, compressible fluid is due to Biot, who derived a pair of coupled partial differential equations that accurately predict the existence of two independent dilatational (compressional) wave motions, corresponding to in-phase and out-of-phase displacements of the solid and fluid phases, respectively. The Biot equations can be decoupled exactly after Fourier transformation to the frequency domain, but the resulting pair of Helmholtz equations cannot be converted to partial differential equations in the time domain and, therefore, closed-form analytical solutions of these equations in space and time variables cannot be obtained. In this paper we show that the decoupled Helmholtz equations can in fact be transformed to two independent partial differential equations in the time domain if the wave excitation frequency is very small as compared to a critical frequency equal to the kinematic viscosity of the pore fluid divided by the permeability of the porous medium. The partial differential equations found are a propagating wave equation and a dissipative wave equation, for which closed-form solutions are known under a variety of initial and boundary conditions. Numerical calculations indicate that the magnitude of the critical frequency for representative sedimentary materials containing either water or a nonaqueous phase liquid is in the kHz–MHz range, which is generally above the seismic band of frequencies. Therefore, the two partial differential equations obtained should be accurate for modeling elastic wave phenomena in fluid-saturated porous media under typical low-frequency conditions applicable to hydrogeological problems. 相似文献