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1.
A Laplace-transform analytic element method (LT-AEM) is described for the solution of transient flow problems in porous media. Following Laplace transformation of the original flow problem, the analytic element method (AEM) is used to solve the resultant time-independent modified Helmholtz equation, and the solution is inverted numerically back into the time domain. The solution is entirely general, retaining the mathematical elegance and computational efficiency of the AEM while being amenable to parallel computation. It is especially well suited for problems in which a solution is required at a limited number of points in space–time, and for problems involving materials with sharply contrasting hydraulic properties. We illustrate the LT-AEM on transient flow through a uniform confined aquifer with a circular inclusion of contrasting hydraulic conductivity and specific storage. Our results compare well with published analytical solutions in the special case of radial flow. 相似文献
2.
An analytic element approach is presented for the modeling of steady groundwater flow through multi-aquifer systems with piecewise constant aquifer and leaky layer properties. Different properties may be specified for domains bounded by closed polygons, referred to as polygonal inhomogeneities. The boundary of these inhomogeneities is modeled with two types of high-order line elements. First, a string of single-aquifer line-doublets is used; these elements cut through all aquifers and are valid both inside and outside the inhomogeneity. Second, two strings of multi-aquifer line-sinks are used, one string that is valid inside the inhomogeneity and one string that is valid outside; the comprehensive extraction of these line-sinks is zero at any point along the string. The proposed approach results in a comprehensive flow field of which the component normal to the boundary of the inhomogeneity is continuous across the boundary at any point. Within each individual aquifer, continuity of head and the component of flow normal to the boundary are met approximately across the boundary; the accuracy increases when the order of the line elements is increased and/or when shorter line elements are used. The proposed analytic element approach produces results that are virtually identical to the exact solution for a cylindrical inhomogeneity, and a high-resolution MODFLOW2000 model of two rectangular inhomogeneities with a shared boundary. The practical application of the approach is demonstrated through the solution of a problem with an irregularly shaped inhomogeneity with rivers crossing the inhomogeneity boundary. 相似文献
3.
In the analytic element method, strings of line-sinks may be used to model streams and strings of line-doublets may be used to model impermeable walls or boundaries of inhomogeneities. The resulting solutions are analytic, but the boundary conditions are met approximately. Equations for line elements may be derived in two ways: through integration of point elements (the integral solution) and through application of separation of variables in elliptical coordinates (the elliptical solution). Using both approaches, two sets of line elements are presented for four flow problems: line-sinks and line-doublets in (un)confined flow, and line-sinks and line-doublets in semi-confined flow. Elliptical line elements have the advantage that they do not need a far-field expansion for accurate evaluation far away from the element. The derivation of elliptical line elements is new and applicable to both (un)confined flow and semi-confined flow; only the resulting expressions for elliptical line elements for semi-confined flow have not been found in the current groundwater literature. Existing solutions for elliptical line elements for (un)confined flow were intended for the modeling of isolated features. Four examples are presented, one for each flow problem, to demonstrate that strings of elliptical line elements may be used to obtain accurate solutions; elliptical line-doublets for semi-confined flow can only be strung together in combination with two integral line-doublets. For a zigzag canal in (un)confined flow, a string of elliptical line-sinks performed better than a string of integral line-sinks of the same order when the smallest angle between two adjacent segments is less than 130°. Elliptical line-doublets performed better than integral line-doublets for a square inhomogeneity in a uniform, confined flow field; the difference was smaller for an octagonal inhomogeneity. For semi-confined flow, the difference between the integral and elliptical line-sinks was small when modeling a zigzag canal. 相似文献
4.
Strack OD 《Ground water》2006,44(1):91-98
We deal in this paper with an ongoing development of the analytic element method. We present in outline new analytic line elements that are suitable to model general flow fields, i.e., flow fields that possess a continuously varying areal inflow or outflow. These elements are constructed specifically to model the leakage through leaky layers that separate aquifers in leaky systems and to model transient effects. The leakage or release from storage underneath linear features is modeled precisely by the new elements; the singularity in leakage is matched exactly by the approximate solution. Applications are given for a problem involving leakage and for a case of transient flow. We note that the analytic elements can be used also to reproduce the effect of continuously varying aquifer properties, e.g., the hydraulic conductivity or the elevation of the base of the aquifer. In the latter case, the elements would reproduce the rotation of the flow field caused by the variation in properties, rather than the divergence as for the case of leakage. 相似文献
5.
A transient Green function due to suddenly applied line loads in an isotropic and homogeneous half-space is reported in this paper. The derivation of the half-space Green function in the Laplace and the Fourier transform spaces is first reviewed. Following an explicit inversion of the Fourier transform, the inverse Laplace transform is implemented along the contour integral on the p-complex plane in an integral form. The half-space Green function consists of full-space Green functions and a singularity-free complementary term. It can be easily incorporated into current transient boundary elements using the transient full-space Green function. Combined with finite elements, the half-space Green function can be used in a hybrid procedure to solve transient half-space problems without discretization of the free surface. Numerical results are presented to illustrate transient wave propagation in a half-space. 相似文献
6.
《Advances in water resources》2005,28(1):33-42
A numerical solution that is significantly more general than other semi-analytical solutions is presented for governing equations describing advective–dispersive transport with multirate mass transfer between mobile and immobile domains. The new solution approach is general in the sense that it does not impose any restrictive assumption on the spatial or temporal variability of advective and dispersive processes in the mobile domain. A single integro-differential equation (IDE) is developed for the concentration in the mobile domain by separating the concentration in the immobile domain from the set of two partial differential equations. The solution to the IDE requires the evaluation of a temporal integral of the concentration in the mobile domain, which is a function of the Laplace transform of the distribution of the mass transfer rate coefficient. The Laplace transform is not limited to flow fields with known constant velocities. The solutions for one- and two-dimensional examples obtained using the new approach agree with those obtained by existing semi-analytical and numerical approaches. 相似文献
7.
An analytic approach is presented for the simulation of variations in the groundwater level due to temporal variations of recharge in surficial aquifers. Such variations, called groundwater dynamics, are computed through convolution of the response function due to an impulse of recharge with a measured time series of recharge. It is proposed to approximate the impulse response function with an exponential function of time which has two parameters that are functions of space only. These parameters are computed by setting the zeroth and first temporal moments of the approximate impulse response function equal to the corresponding moments of the true impulse response function. The zeroth and first moments are modeled with the analytic element method. The zeroth moment may be modeled with existing analytic elements, while new analytic elements are derived for the modeling of the first moment. Moment matching may be applied in the same fashion with other approximate impulse response functions. It is shown that the proposed approach gives accurate results for a circular island through comparison with an exact solution; both a step recharge function and a measured series of 10 years of recharge were used. The presented approach is specifically useful for modeling groundwater dynamics in aquifers with shallow groundwater tables as is demonstrated in a practical application. The analytic element method is a gridless method that allows for the precise placement of ditches and streams that regulate groundwater levels in such aquifers; heads may be computed analytically at any point and at any time. The presented approach may be extended to simulate the effect of other transient stresses (such as fluctuating surface water levels or pumping rates), and to simulate transient effects in multi-aquifer systems. 相似文献
8.
Fitts CR 《Ground water》2006,44(1):99-101
Although most current applications of the analytic element method are formulated for isotropic hydraulic conductivity, anisotropic domains can be modeled with analytic elements using the well-known coordinate transformation where one coordinate axis is scaled by the square root of the anisotropy ratio. If the standard analytic solution for steady radial flow to a well is used with this coordinate transformation, the resulting solution correctly models the far field but it does not meet the constant head boundary condition at the well radius. This could be a significant shortcoming if you are interested in the flow field close to the well or want to estimate the head at the pumping well. A new solution for two-dimensional steady flow to a well in an anisotropic domain is presented. This solution satisfies the governing equations exactly and meets the constant head boundary condition at the well radius exactly. It was derived using a conformal mapping. 相似文献
9.
Bakker M 《Ground water》2006,44(1):81-85
An analytic element approach is presented for the modeling of multiaquifer domains embedded in a single-aquifer model. The inside of each domain may consist of an arbitrary number of aquifers separated by leaky layers. The analytic element solution is obtained through a combination of existing single-aquifer and multiaquifer analytic elements and allows for the analytic computation of head and leakage at any point in the aquifer. Along the boundary of an embedded multiaquifer domain, the normal flux is continuous everywhere; continuity of head across the boundary is met exactly at collocations points and approximately, but very accurately, in between. The analytic element solution compares well with an existing exact solution. A hypothetical example with a river intersecting two embedded domains illustrates the practical application of the proposed approach. 相似文献
10.
The Analytic Element Method (AEM) provides a convenient tool for groundwater flow analysis in unbounded continuous domains. The AEM is based on the superposition of analytic functions, known as elements, useful at both regional and local scales. In this study, analytic elements for strip aquifers are presented. Such aquifers occur in riverine or coastal deposits and in outcrop zones of confined aquifers. Local flow field is modelled indirectly, using a reference plane related to the aquifer domain through the Schwarz‐Christoffel transform. The regional flow is obtained as a solution of the one‐dimensional flow equation. The proposed methodology was tested by modelling two hypothetical situations, which were compared to exact solutions. It is shown that regional boundaries can be reproduced exactly while local fields are adequately reproduced with analytic elements. The developed elements are applied to simulate a real flow field in northeastern Brazil showing good agreement with measured water levels. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
11.
A new analytic solution approach is presented for the modeling of steady flow to pumping wells near rivers in strip aquifers; all boundaries of the river and strip aquifer may be curved. The river penetrates the aquifer only partially and has a leaky stream bed. The water level in the river may vary spatially. Flow in the aquifer below the river is semi-confined while flow in the aquifer adjacent to the river is confined or unconfined and may be subject to areal recharge. Analytic solutions are obtained through superposition of analytic elements and Fourier series. Boundary conditions are specified at collocation points along the boundaries. The number of collocation points is larger than the number of coefficients in the Fourier series and a solution is obtained in the least squares sense. The solution is analytic while boundary conditions are met approximately. Very accurate solutions are obtained when enough terms are used in the series. Several examples are presented for domains with straight and curved boundaries, including a well pumping near a meandering river with a varying water level. The area of the river bottom where water infiltrates into the aquifer is delineated and the fraction of river water in the well water is computed for several cases. 相似文献
12.
Zaadnoordijk WJ 《Ground water》2006,44(1):106-110
Analytic elements are well suited for the design of building pit dewatering. Wells and drains can be modeled accurately by analytic elements, both nearby to determine the pumping level and at some distance to verify the targeted drawdown at the building site and to estimate the consequences in the vicinity. The ability to shift locations of wells or drains easily makes the design process very flexible. The temporary pumping has transient effects, for which transient analytic elements may be used. This is illustrated using the free, open-source, object-oriented analytic element simulator Tim(SL) for the design of a building pit dewatering near a canal. Steady calculations are complemented with transient calculations. Finally, the bandwidths of the results are estimated using linear variance analysis. 相似文献
13.
In-Mo Lee 《Soil Dynamics and Earthquake Engineering》1996,15(3):151-159
A closed-form transient solution to blasting loading is presented. The blasting loading is modelled as a finite sheet dilatational source, rather than a finite line, so that the dimensions of the explosives are taken into account in two directions, i.e. one in the horizontal direction and the other in the vertical direction. The solution is obtained by using Laplace transform, with respect to the time, and Fourier transform with respect to the coordinates. Inverse Laplace transform is implemented analytically. The final solution is expressed in double integral form. The solution can be used to determine groundmotion in studying blasting impacts on underground or aboveground structures. 相似文献
14.
We obtained an exact solution in terms of the discharge potential for a constant-strength line-sink that satisfies the modified Helmholtz equation for groundwater flow, for example for semi-confined flow and transient flow. The solution is obtained by integrating the potential for a point sink (well) along a straight line element. The potential for the point-sink is the modified Bessel function of the second kind and zero order K0. Since K0 cannot be integrated directly (in closed form) along a line-element, earlier solutions for a line-sink have been obtained by integrating polynomial approximations to K0. These approximations, however, are only valid up to a certain distance from the well and consequently impose a limit on the length of the line-sink. In this paper we integrate an exact series representation for K0 that is valid at any distance from the well, thus allowing integration along line-elements of any length, at least in theory. Numerical difficulties arise when evaluating our expressions at large distances from the line-sink, but these are shown to be of little consequence in practice. We made use of Wirtinger calculus to facilitate integration and also to allow us to arrive at exact expressions for the integrated flux over a poly-line and the total leakage over a domain. These properties are essential when using the solution in the context of the Analytic Element Method (AEM). We demonstrate our solution for the case of semi-confined flow (with leakage) and for the case of transient flow in the context of the Laplace Transform Analytic Element Method (LT-AEM). 相似文献
15.
The solution describing the wellbore flow rate in a constant‐head test integrated with an optimization approach is commonly used to analyze observed wellbore flow‐rate data for estimating the hydrogeological parameters of low‐permeability aquifers. To our knowledge, the wellbore flow‐rate solution for the constant‐head test in a two‐zone finite‐extent confined aquifer has never been reported so far in the literature. This article is first to develop a mathematical model for describing the head distribution in the two‐zone aquifer. The Laplace domain solutions for the head distributions and wellbore flow rate in a two‐zone finite confined aquifer are derived using the Laplace transform, and their corresponding time domain solutions are then obtained using the Bromwich integral method and residue theorem. These new solutions are expressed in terms of an infinite series with Bessel functions and not straightforward to calculate numerically. A large‐time solution for the wellbore flow rate is therefore developed by employing the relationship of small Laplace variable versus large time variable and L'Hospital's rule. The result shows that the large‐time solution is identical to the steady‐state solution obtained after applying the Tauberian theorem into the Laplace domain solution. This large‐time solution can reduce to the Thiem equation in the case of no skin. Finally, the newly developed solution is used to investigate the effects of outer boundary distance and conductivity ratio on the wellbore flow rate. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
16.
Dionyssia-Pinelopi N. Kontoni Dimitri E. Beskos 《Soil Dynamics and Earthquake Engineering》1987,6(4):227-238
In this paper the applicability of an approximate Boundary Element Method to uniform half-plane elastodynamic problems is investigated. This method employs the concept of images to construct approximate fundamental solutions for the half-plane and does not require any half-plane surface discretization. The method is formulated in the frequency domain for the case of harmonic disturbances or the Laplace transform domain for the case of transient disturbances. In the latter case a numerical inversion of the transformed solution is necessary to obtain the time domain response. The proposed method can be used as an alternative to boundary element methods that either utilize the infinite plane fundamental solution and thus require a half-plane surface discretization, or employ the exact half-plane fundamental solution, which even though leads to no surface discretization, is of a very lengthy and complicated form. Two characteristics numerical examples are used to illustrate the proposed method and study its advantages and disadvantages. 相似文献
17.
A Fourier transform approach is applied to the transient analysis of dynamic soil–structure interaction under SH-motion. The governing equations are formulated in the frequency domain using a Finite Element–Boundary Element (FE–BE) coupling method. After solving the transformed problem, the transient solution is obtained using the discrete inverse Fourier transform with a fast Fourier transform algorithm. Two examples are presented in order to show the numerical performance of the proposed technique. 相似文献
18.
P.W. France 《Journal of Hydrology》1974,21(4):381-398
A numerical method is presented for analysing either steady state or transient three-dimensional groundwater flow problems. The governing equation is formulated in terms of the finite element process using the Galerkin approach, and cubic isoparametric elements are used to simulate the flow domain as these permit accurate modelling of curved boundaries. Particular attention is paid to the time dependent movement of the phreatic surface where an iterative technique based on the replacement of the original transient problem by a discrete number of steady state problems is used to effect a solution. Furthermore, in tracing the movement of the surface use is made of the element formulation theory in order to compute the normal to the boundary.The validity of the technique is first established by analysing a radially symmetrical problem for which an alternative analytical solution is available. Finally, a general three-dimensional flow system is studied for which there is no known analytical solution. It is shown that relatively few elements are required to yield practical solutions. 相似文献
19.
Charles R. Fitts Joshua Godwin Kathleen Feiner Charles McLane Seth Mullendore 《Ground water》2015,53(3):432-439
This paper presents the analytic element modeling approach implemented in the software AnAqSim for simulating steady groundwater flow with a sharp fresh‐salt interface in multilayer (three‐dimensional) aquifer systems. Compared with numerical methods for variable‐density interface modeling, this approach allows quick model construction and can yield useful guidance about the three‐dimensional configuration of an interface even at a large scale. The approach employs subdomains and multiple layers as outlined by Fitts (2010) with the addition of discharge potentials for shallow interface flow (Strack 1989). The following simplifying assumptions are made: steady flow, a sharp interface between fresh‐ and salt water, static salt water, and no resistance to vertical flow and hydrostatic heads within each fresh water layer. A key component of this approach is a transition to a thin fixed minimum fresh water thickness mode when the fresh water thickness approaches zero. This allows the solution to converge and determine the steady interface position without a long transient simulation. The approach is checked against the widely used numerical codes SEAWAT and SWI/MODFLOW and a hypothetical application of the method to a coastal wellfield is presented. 相似文献
20.
《Advances in water resources》2002,25(3):279-291
An analytical approach is presented for solving problems of steady, two-dimensional groundwater flow with inhomogeneity boundaries. A common approach for such problems is to separate the problem domain into two homogeneous domains, search for solutions in each domain, and then attempt to match conditions, either exactly or approximately, along the inhomogeneity boundary. Here, we use classical solutions to problems with inhomogeneity boundaries with simple geometries, and map conformally the entire domain onto a new one. In this way, existing solutions are used to solve problems with more complex, and more practical, boundary geometries. The approach is general, but subject to some restrictions on the mapping functions that may be used.Using this approach, we develop explicit analytical solutions for two problems of practical interest. The first problem addresses aquifer interaction across a gap in an impermeable separating layer; flow regimes are defined and the interaction is quantified. The second solution represents flow in the vertical plane to a partially clogged stream bed that is partially penetrating the aquifer; the stream bed is modeled as a thin layer of low-permeability silt. Flow regimes for groundwater surface–water interaction are quantified analytically. 相似文献