共查询到20条相似文献，搜索用时 62 毫秒
1.
《Advances in water resources》1998,21(7):581-590
Advances in computer power and in reservoir characterization allow simulation of pressure transients in complex reservoirs generated stochastically. Generally, interpretation of these transient gives useful information about the reservoir hydraulic properties: a major goal is to interpret these transients in the stochastic context. First we ensemble average the pressure over all the random permeability field realizations to derive an equation which drives the ensemble averaged pressure. We use the Fourier transform in space and the Laplace transform in time, in conjuction with a perturbation series expansion in successive powers of the permeability fluctuations to obtain an explicit solution. The Nth order term of this series involves the hydrodynamic interaction between N permeability heterogeneities and after averaging we obtain an expansion containing correlation functions of permeability fluctuations of increasing order.Next, Feynman graphs are introduced allowing a more attractive graphical interpretation of the perturbation series. Then series summation techniques are employed to reduce the graph number to be summed at each order of the fluctuation expansion. This in turn gives useful physical insights on the homogenization processes involved. In particular, it is shown that the sum of the so-called ‘one-particle irreducible graphs’ gives the kernel of a linear integro-differential equation obeyed by the ensemble average pressure. All the information about the heterogeneity structure is contained in this renormalized kernel, which is a limited range function.This equation on its own is the starting point of useful asymptotic results and approximations. In particular it is shown that interpretation of pumping tests yields the steady-state equivalent permeability after a sufficiently long time for an infinite reservoir, as expected. 相似文献
2.
A stochastic approach to nonlinear unconfined flow subject to multiple random fields 总被引：3，自引：3，他引：0
Liangsheng Shi Jinzhong Yang Dongxiao Zhang 《Stochastic Environmental Research and Risk Assessment (SERRA)》2009,23(6):823-835
In this study, the KLME approach, a moment-equation approach based on the Karhunen–Loeve decomposition developed by Zhang
and Lu (Comput Phys 194(2):773–794, 2004), is applied to unconfined flow with multiple random inputs. The log-transformed hydraulic conductivity F, the recharge R, the Dirichlet boundary condition H, and the Neumann boundary condition Q are assumed to be Gaussian random fields with known means and covariance functions. The F, R, H and Q are first decomposed into finite series in terms of Gaussian standard random variables by the Karhunen–Loeve expansion. The
hydraulic head h is then represented by a perturbation expansion, and each term in the perturbation expansion is written as the products of
unknown coefficients and Gaussian standard random variables obtained from the Karhunen–Loeve expansions. A series of deterministic
partial differential equations are derived from the stochastic partial differential equations. The resulting equations for
uncorrelated and perfectly correlated cases are developed. The equations can be solved sequentially from low to high order
by the finite element method. We examine the accuracy of the KLME approach for the groundwater flow subject to uncorrelated
or perfectly correlated random inputs and study the capability of the KLME method for predicting the head variance in the
presence of various spatially variable parameters. It is shown that the proposed numerical model gives accurate results at
a much smaller computational cost than the Monte Carlo simulation. 相似文献
3.
This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function. 相似文献
4.
G. Christakos C. T. Miller D. Oliver 《Stochastic Environmental Research and Risk Assessment (SERRA)》1993,7(1):14-32
This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function. 相似文献
5.
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
6.
G. Christakos C. T. Miller D. Oliver 《Stochastic Environmental Research and Risk Assessment (SERRA)》1993,7(3):213-239
As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations. 相似文献
7.
This paper gives the exact solution in terms of the Karhunen–Loève expansion to a fractional stochastic partial differential equation on the unit sphere \({\mathbb {S}}^{2} \subset {\mathbb {R}}^{3}\) with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen–Loève expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background are given to illustrate the theoretical results. 相似文献
8.
In any numerical solution of the DC resistivity experiment, care must be taken to deal with strong heterogeneity of electrical conductivity. In order to examine the importance of conductivity contrasts, we develop a scattering decomposition of the DC resistivity equation in the sparse differential domain as opposed to the traditional dense integral formulation of scattering‐type equations. We remove the singularity in the differential scattered series via separation of primary and secondary conductivity, thereby avoiding the need to address the singularity in a Green's function. The differential scattering series is observed to diverge for large conductivity contrasts and to converge for small contrasts. We derive a convergence criterion, in terms of matrix norms for the weak‐form finite‐volume equations, that accounts for both the magnitude and distribution of heterogeneity of electrical conductivity. We demonstrate the relationship between the differential scattering series and the Fréchet derivative of the electrical potential with respect to electrical conductivity, and we show how the development may be applied to the inverse problem. For linearization associated with the Fréchet derivative to be valid, the perturbation in electrical conductivity must be small as defined by the convergence of the scattered series. The differential scattering formulation also provides an efficient tool for gaining insight into charge accumulation across contrasts in electrical conductivity, and we present a derivation that equates accumulated surface charge density to the source of scattered potential. 相似文献
9.
《Advances in water resources》2005,28(10):1091-1101
Certain nonlinear diffusion equations of degenerate parabolic type display a finite speed of propagation of disturbances. This mathematical behavior can be used to describe a wide range of nonlinear phenomena such as the penetration distance of a thermal layer, the boundary of a reaction zone, or a wetting front in unsaturated soil moisture flow. However, there are two main difficulties in obtaining solutions to problems of this class. One is that the location of the interface is not known a priori and must be discovered during the analysis. The other is the fact that the differential equation is singular in the neighborhood of the interface. The solution technique developed and presented in this work overcomes these difficulties by extracting a local solution of the differential equation in the neighborhood of the diffusing front. One profound result is the discovery that the velocity of the front is entirely controlled by the first term of the spectral series expansion. Also, by capturing the critical behavior of the solution in the region of the singularity and incorporating the behavior as a dominant factor, the series expansion is provided a means for very rapid convergence. The versatility of the solution technique is demonstrated by solving various boundary value problems covering a broad range of interest and the solutions are tested against previously published results. 相似文献
10.
Tariq Alkhalifah 《Geophysical Prospecting》2011,59(4):627-634
The wavefield dependence on a virtual shift in the source location can provide information helpful in velocity estimation and interpolation. However, the second‐order partial differential equation (PDE) that relates changes in the wavefield form (or shape) to lateral perturbations in the source location depends explicitly on lateral derivatives of the velocity field. For velocity models that include lateral velocity discontinuities this is problematic as such derivatives in their classical definition do not exist. As a result, I derive perturbation partial differential wave equations that are independent of direct velocity derivatives and thus, provide possibilities for wavefield shape extrapolation in complex media. These PDEs have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The solutions of the perturbation equations provide the coefficients of a Taylor's series type expansion for the wavefield. The new formulas introduce changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The accuracy of the representation, as demonstrated on the Marmousi model, is generally good. 相似文献
11.
采用不排水条件下孔隙水压力发展模式,作为Terzagh i一维固结方程中考虑波浪循环作用所引起的孔隙水压力源项,对于成层海床建立了推广的一维动力固结方程,运用数理方程中的分离变量法与G reen函数求解了成层海床在波浪作用下残余孔隙水压力的发展规律,进而对成层海床的液化势进行了评判。对比计算与分析表明,海床表层土的渗透性及其厚度对于海床的整体抗液化性能具有显著的影响,低渗透性的表层导致海床孔隙水压力的显著积累,此时表层置换法是防治液化的有效途径。 相似文献
12.
The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients. Based on the asymptotic solutions of this system, a linear, quasilinear, and nonlinear stage of perturbation dynamics is identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. As a supplement, an asymptotic analysis of nonlinear dynamics of long-wave perturbations superimposed on a damped by small viscosity Kolmogorov flow (very large, but finite Reynolds numbers) is made. It is strictly shown that all velocity components of the perturbed flow remain bounded in this case. 相似文献
13.
Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations. 相似文献
14.
Kui Xiang Kjersti Solberg Eikrem Morten Jakobsen Geir Nvdal 《Geophysical Prospecting》2022,70(1):3-18
We have derived a convergent scattering series solution for the frequency-domain wave equation in acoustic media with variable density and velocity. The convergent scattering series solution is based on the homotopy analysis of a vectorial integral equation of the Lippmann–Schwinger type. By using the Green's function and partial integration, we have derived the vectorial integral equation of the Lippmann–Schwinger type that involves the pressure gradient field as well as the pressure field from the wave equation. The vectorial Lippmann–Schwinger equation can in principle be solved via matrix inversion, but the computational cost of matrix inversion scales like , where is the number of grid blocks. The computational cost can be significantly reduced if one solves the vectorial Lippmann–Schwinger equation iteratively. A simple iterative solution is the Born series, but it is only convergent when the scattering potential is sufficiently small. In this study, we have used the so-called homotopy analysis method to derive an iterative solution for the vectorial Lippmann–Schwinger equation which can be made convergent even in strongly scattering media. The computational cost of our convergent scattering series scales as . Our algorithm, which is based on the homotopy analysis method, involves a convergence control operator that we select using hierarchical matrices. We use a three-layer model and a resampled version of the SEG/EAGE salt model to show the performance of the developed convergent scattering series. 相似文献
15.
Hakan Sirin 《Stochastic Environmental Research and Risk Assessment (SERRA)》2006,20(6):381-390
Solute plume subjected to field scale hydraulic conductivity heterogeneity shows a large dispersion/macrodispersion, which is the manifestation of existing fields scale heterogeneity on the solute plume. On the other hand, due to the scarcity of hydraulic conductivity measurements at field scale, hydraulic conductivity heterogeneity can only be defined statistically, which makes the hydraulic conductivity a random variable/function. Random hydraulic conductivity as a parameter in flow equation makes the pore flow velocity also random and the ground water solute transport equation is a stochastic differential equation now. In this study, the ensemble average of stochastic ground water solute transport equation is taken by the cumulant expansion method in order to upscale the laboratory scale transport equation to field scale by assuming pore flow velocity is a non stationary, non divergence-free and unsteady random function of space and time. Besides the stochastic explanation of macrodispersion and the velocity correction term obtained by Kavvas and Karakas (J Hydrol 179:321–351, 1996) before a new velocity correction term, which is a function of mean pore flow velocity divergence, is obtained in this study due to strict second order cumulant expansion (without omitting any term after the expansion) performed. The significance of the new velocity correction term is investigated on a one dimensional transport problem driven by a density dependent flow field. 相似文献
16.
《Advances in water resources》2002,25(4):439-453
A three-dimensional, reactive numerical flow model is developed that couples chemical reactions with density-dependent mass transport and fluid flow. The model includes equilibrium reactions for the aqueous species, kinetic reactions between the solid and aqueous phases, and full coupling of porosity and permeability changes that result from precipitation and dissolution reactions in porous media. A one-step, global implicit approach is used to solve the coupled flow, transport and reaction equations with a fully implicit upstream-weighted control volume discretization. The Newton–Raphson method is applied to the discretized non-linear equations and a block ILU-preconditioned CGSTAB method is used to solve the resulting Jacobian matrix equations. This approach permits the solution of the complete set of governing equations for both concentration and pressure simultaneously affected by chemical and physical processes. A series of chemical transport simulations are conducted to investigate coupled processes of reactive chemical transport and density-dependent flow and their subsequent impact on the development of preferential flow paths in porous media. The coupled effects of the processes driving flow and the chemical reactions occurring during solute transport is studied using a carbonate system in fully saturated porous media. Results demonstrate that instability development is sensitive to the initial perturbation caused by density differences between the solute plume and the ambient groundwater. If the initial perturbation is large, then it acts as a “trigger” in the flow system that causes instabilities to develop in a planar reaction front. When permeability changes occur due to dissolution reactions occurring in the porous media, a reactive feedback loop is created by calcite dissolution and the mixed convective transport of the system. Although the feedback loop does not have a significant impact on plume shape, complex concentration distributions develop as a result of the instabilities generated in the flow system. 相似文献
17.
本文讨论用分裂算法解高阶偏移方程时所用的有限差分方程与原偏微分方程的相容性和算法的稳定性。根据Lax等价原理,这种相容性和稳定性可以保证数值方程收敛于原偏微分方程。我们证明了该分裂算法满足相容性和稳定性要求。这就不但以文献[1]中的实例说明了这种算法的实用性,而且从原理上论证了算法的正确性。 相似文献
18.
19.
Renormalization group analysis of permeability upscaling 总被引：1，自引：1，他引：0
D. T. Hristopulos G. Christakos 《Stochastic Environmental Research and Risk Assessment (SERRA)》1999,13(1-2):131-161
The heterogeneity of the subsurface permeability is considered as the most influential factor in determining groundwater flow and the transport of toxic contaminants. Numerical simulators cannot handle the large grids required to represent the small-scale variability of permeability, and thus explicit estimates of the large-scale behavior in terms of coarse-grained parameters are often required. Perturbation formulations of the effective permeability are based on simplifying assumptions that are valid only for certain probability distributions and weak heterogeneity. A generalized perturbation ansatz that involves higher orders has been proposed (Gelhar and Axness, 1983), but to our knowledge its validity has not been rigorously proved before in three dimensions. In this work we propose a general upscaling formulation valid for strong heterogeneity, general permeability distributions, and media with impermeable zones. We show that the effective permeability is determined by the self-energy series of the permeability fluctuations at zero frequency. Using the diagrammatic representation, we obtain a Dyson equation that involves only irreducible diagrams of the proper self-energy series. We develop a renormalization group (RG) analysis for isotropic lognormal media that proves the generalized perturbation ansatz to all orders. We show that the RG result accurately estimates laboratory permeability measurements in limestone (strong heterogeneity) and sandstone (weak heterogeneity). We also propose an explicit RG estimate for the preasymptotic effective permeability. We compare our results with an approach based on a leading order Green's function expansion (Paleologos et?al., 1996), which, however, requires intensive numerical computations. Finally, we investigate the relation between the RG expression and the algebraic means used in numerical upscaling. 相似文献
20.
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