Diagrammatic solutions for hydraulic head moments in 1-D and 2-D bounded domains

We present a diagrammatic method for solving stochastic 1-D and 2-D steady-state flow equations in bounded domains. The diagrammatic method results in explicit solutions for the moments of the hydraulic head. This avoids certain numerical constraints encountered in realization-based methods. The diagrammatic technique also allows for the consideration of finite domains or large fluctuations, and is not restricted by distributional assumptions. The results of the method for 1-D and 2-D finite domains are compared with those obtained through a realization-based approach. Mean and variance of head are well reproduced for all log-conductivity variances inputted, including those larger than one. The diagrammatic results also compare favorably to hydraulic head moments derived by standard analytic methods requiring a linearized form of the flow equation.     

Diagrammatic solutions for hydraulic head moments in 1-D and 2-D bounded domains

We present a diagrammatic method for solving stochastic 1-D and 2-D steady-state flow equations in bounded domains. The diagrammatic method results in explicit solutions for the moments of the hydraulic head. This avoids certain numerical constraints encountered in realization-based methods. The diagrammatic technique also allows for the consideration of finite domains or large fluctuations, and is not restricted by distributional assumptions. The results of the method for 1-D and 2-D finite domains are compared with those obtained through a realization-based approach. Mean and variance of head are well reproduced for all log-conductivity variances inputted, including those larger than one. The diagrammatic results also compare favorably to hydraulic head moments derived by standard analytic methods requiring a linearized form of the flow equation.     

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhars work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.     

Explicit algebraic influence coefficients: two-dimensional aquifer model / Coefficients d'influence algébriques explicites: modèle d'aquifère bidimensionnel

Abstract

This work extends the algebraic expression of influence coefficients developed for one-dimensional aquifer models to a two-dimensional (2-D) case. First, the partial differential equation governing the flow in a 2-D semi-confined aquifer is discretized using a finite difference scheme. This results in a system of discrete equations presented in the form of water balance equations associated with a network of interconnected compartments centred on the grid nodes. The foregoing system is transformed into a series of uncoupled 1-D equations stated in terms of some generalized hydraulic head for which they are also solved. Second, the original hydraulic head is recovered from the generalized one via an appropriate linear transformation. Whence, the algebraic expression making the hydraulic head explicit versus sources and boundary conditions is derived. This discrete expression, mapped onto its continuous counterpart, helps to deduce an algebraic form of the inter-compartment influence coefficients. Finally, a comparison with the analytical Green function is carried out.     

可控源电磁三维频率域有限元模拟

本文采用电磁场的磁矢量位和电标量势,将Maxwell方程组化为位势的类似于Helmholtz型方程,并引入罚项及稳定化方法克服了电磁三维有限元算法中的伪解及数值不稳定性;采用人工边界把计算区域局域化,将均匀半空间中水平电偶极子源产生的位势值作为人工边界上的第一类边界条件以表示源的作用,减少了实际的计算区域.理论模型和复杂模型的计算结果均表明,可控源电磁三维有限元数值模拟给出了稳定、可靠的电磁场分布.     

Marine terraces

Summary The theory of the formation of marine terraces is studied. A model is set up which is believed to describe the physical processes involved. This model leads to a nonlinear partial differential equation which is integrated for some special cases by means of a digital computer. Asymptotic solutions of the differential equation are also found by direct analytical means. It is found that no terraces can develop if eustatic changes occur at a constant rate, contrary to some contentions that have been advanced in the literature.     

三维高阶深度偏移方程及其数值求解方法

从三维非均匀介质中的波动方程出发，利用拟微分算子理论，Ｐａｄｅ逼近方法及因式分解技巧，获得了非均匀介质的三维高阶深度偏移方程，相应地提出了逐次低阶方程方法、低阶方程组方法及分裂方法等３种求解方法．与二维情形不同，以上每一种方法在数值求解时均存在由测线坐标ｙ的出现而带来的困难．为了克服这一困难，我们提出了差分算子分解方法，避免了近年来人们竞相研究的ｘ－ｙ方向微分算子分裂带来的分裂误差，保持了应有的相容性，解决了这一令人烦恼的问题．     

三维高阶深度偏移方程及其数值求解方法

从三维非均匀介质中的波动方程出发,利用拟微分算子理论,Ｐａｄｅ逼近方法及因式分解技巧,获得了非均匀介质的三维高阶深度偏移方程,相应地提出了逐次低阶方程方法、低阶方程组方法及分裂方法等３种求解方法．与二维情形不同,以上每一种方法在数值求解时均存在由测线坐标ｙ的出现而带来的困难．为了克服这一困难,我们提出了差分算子分解方法,避免了近年来人们竞相研究的ｘ－ｙ方向微分算子分裂带来的分裂误差,保持了应有的相容性,解决了这一令人烦恼的问题．     

波动方程三维吸收边界分解与拟合深度偏移方法

在偏移问题中引入吸收边界条件,既可以消除由人工边界激发的虚假反射,从而提高剖面质量。又可以减少计算工作量．本文讨论了三维吸收边界条件方程,提出了求解具有吸收边界条件的三维波动方程偏移定解问题的分解与拟合方法。理论分析与合成记录及野外实际地震资料处理结果表明,本文方法为一有效的三维吸收边界深度偏移方法。     

波动方程三维吸收边界分解与拟合深度偏移方法

在偏移问题中引入吸收边界条件，既可以消除由人工边界激发的虚假反射，从而提高剖面质量。又可以减少计算工作量．本文讨论了三维吸收边界条件方程，提出了求解具有吸收边界条件的三维波动方程偏移定解问题的分解与拟合方法。理论分析与合成记录及野外实际地震资料处理结果表明，本文方法为一有效的三维吸收边界深度偏移方法。     

雅可比椭圆函数在大气和海洋动力学中的应用:二维非线性Rossby波研究

提出了扩展雅可比椭圆函数方法,来求得Petviashvili方程的精确解析解.Petviashvili方程被视为正压准地转位涡度方程的非地转扩展,应用该方法可以得到很多二维非线性Rossby波的周期波解,在取极限情况下,也可以得到二维Rossby孤立子解.     

Low-frequency dilatational wave propagation through fully-saturated poroelastic media

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.     

Mass-conservative finite volume methods on 2-D unstructured grids for the Richards’ equation

The solution to the 2-D time-dependent unsaturated flow equation is numerically approximated by a second-order accurate cell-centered finite-volume discretization on unstructured grids. The approximation method is based on a vertex-centered Least Squares linear reconstruction of the solution gradients at mesh edges.A Taylor series development in time of the water content dependent variable in a finite-difference framework guarantees that the proposed finite volume method is mass conservative. A Picard iterative scheme solves at each time step the resulting non-linear algebraic problem. The performance of the method is assessed on five different test cases and implementing four distinct soil constitutive relationships. The first test case deals with a column infiltration problem. It shows the capability of providing a mass-conservative behavior. The second test case verifies the numerical approximation by comparison with an analytical mixed saturated–unsaturated solution. In this case, the water drains from a fully saturated portion of a 1-D column. The third and fourth test cases illustrate the performance of the approximation scheme on sharp soil heterogeneities on 1-D and 2-D multi-layered infiltration problems. The 2-D case shows the passage of an abrupt infiltration front across a curved interface between two layers. Finally, the fifth test case compares the numerical results with an analytical solution that is developed for a 2-D heterogeneous soil with a source term representing plant roots. This last test case illustrates the formal second-order accuracy of the method in the numerical approximation of the pressure head.     

A numerical study of 1-D nonlinear P-wave propagation in solid

IntroductionBecauseoftheextensivedistributionofruptures,micro-cracksandcrystalfracturesintheearth,therelationshipsbetweenthestressandstrainarenolongerlinear,infact,theyarenonlinear.Inordertoinvestigateandusethenonlinearcharacteristicsofsolidmediumintheearth,weshouldconsidertheinfluenceofnonlinearresponseduringthecourseofseismicmodelingandinversion.Thisisoneoftheimportantstudyfieldsthathavebeenpaidgreatattentionsinthere-centyearsintheworld(Minster,etal,1991;ZHANG,TENG,1993).Thenonlinearchar…     

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Abstract

This paper develops a new analytical solution for the aquifer system, which comprises an unconfined aquifer on the top, a semi-confined aquifer at the bottom and an aquitard between them. This new solution is derived from the Boussinesq equation for the unconfined aquifer and one-dimensional leaky confined flow equation for the lower aquifer using the perturbation method, considering the water table over-height at the remote boundary. The head fluctuation predicted from this solution is generally greater than the one solved from the linearized Boussinesq equation when the ratio of the tidal amplitude to the thickness of unconfined aquifer is large. It is found that both submarine groundwater discharges from upper and lower aquifers increase with tidal amplitude–aquifer thickness ratio and may be underestimated if the discharge is calculated based on the average head fluctuation. The effects of the aquifer parameters and linearization of the Boussinesq equation on the normalized head fluctuation are also investigated.

Editor D. Koutsoyiannis; Associate editor J. Simunek

Citation Chuang, M.-H., Mahdi, A.-A. and Yeh, H.-D., 2012. A perturbation solution for head fluctuations in a coastal leaky aquifer system considering water table over-height. Hydrological Sciences Journal, 57 (1), 162–172.     

A generalized transformation approach for simulating steady-state variably-saturated subsurface flow

A numerical method is proposed to accurately and efficiently compute a direct steady-state solution of the nonlinear Richards equation. In the proposed method, the Kirchhoff integral transformation and a complementary transformation are applied to the governing equation in order to separate the nonlinear hyperbolic characteristic from the linear parabolic part. The separation allows the transformed governing equation to be applied to partially- to fully-saturated systems with arbitrary constitutive relations between primary (pressure head) and secondary variables (relative permeability). The transformed governing equation is then discretized with control volume finite difference/finite element approximations, followed by inverse transformation. The approach is compared to analytical and other numerical approaches for variably-saturated flow in 1-D and 3-D domains. The results clearly demonstrate that the approach is not only more computationally efficient but also more accurate than traditional numerical solutions. The approach is also applied to an example flow problem involving a regional-scale variably-saturated heterogeneous system, where the vadose zone is up to 1 km thick. The performance, stability, and effectiveness of the transform approach is exemplified for this complex heterogeneous example, which is typical of many problems encountered in the field. It is shown that computational performance can be enhanced by several orders of magnitude with the described integral transformation approach.     

Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations

In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators reduce the original multi-dimensional problem to the one-dimensional space, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the present work we discuss developments in the theoretical formulation of flow models with space-dependent coefficients in terms of space transformations. The formulation is based on stochastic Radon operator representations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tractable expressions of the space transformed flow equation. A Plancherel representation of the space transformation product of the head potential and the log-conductivity is also obtained. A test problem is first considered in detail and the solutions obtained by means of the proposed approach are compared with the exact solutions obtained by standard partial differential equation methods. Then, solutions of three-dimensional groundwater flow are derived starting from solutions of a one-dimensional model along various directions in space. A step-by-step numerical formulation of the approach to the flow problem is also discussed, which is useful for practical applications. Finally, the space transformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient.     

积分法三维电阻率成像

二维或是三维电阻率反演成像研究,最关键的环节是在反演系数矩阵即敏感矩阵(或雅可比矩阵)的求取上.本文从微分方程的积分解出发,推导了表达式极为简单的三维雅可比系数矩阵,构造了成像方程.根据反演方程系数的稀疏特征,采用改进的降维高斯赛德尔迭代法来求解该反演方程,并通过内外迭代的结合,对大型稀疏欠定方程能很快收敛,得到可靠的解答.合成数据模型结果及实际资料的成像结果表明积分法不但实现起来极容易,成像结果的精度也相当高.     

积分法三维电阻率成像

二维或是三维电阻率反演成像研究,最关键的环节是在反演系数矩阵即敏感矩阵(或雅可比矩阵)的求取上.本文从微分方程的积分解出发,推导了表达式极为简单的三维雅可比系数矩阵,构造了成像方程.根据反演方程系数的稀疏特征,采用改进的降维高斯赛德尔迭代法来求解该反演方程,并通过内外迭代的结合,对大型稀疏欠定方程能很快收敛,得到可靠的解答.合成数据模型结果及实际资料的成像结果表明积分法不但实现起来极容易,成像结果的精度也相当高.