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1.
We have investigated non‐Darcian flow to a vertical fracture represented as an extended well using a linearization procedure and a finite difference method in this study. Approximate analytical solutions have been obtained with and without the consideration of fracture storage based on the linearization procedure. A numerical solution for such a non‐Darcian flow case has also been obtained with a finite difference method. We have compared the numerical solution with the approximate analytical solutions obtained by the linearization method and the Boltzmann transform. The results indicate that the linearized solution agrees generally well with the numerical solution at late times, and underestimates the dimensionless drawdown at early times, no matter if the fracture storage is considered or not. When the fracture storage is excluded, the Boltzmann transform solution overestimates the dimensionless drawdown during the entire pumping period. The dimensionless drawdowns in the fracture with fracture storage for different values of dimensionless non‐Darcian hydraulic conductivity β approach the same asymptotic value at early times. A larger β value results in a smaller dimensionless drawdown in both the fracture and the aquifer when the fracture storage is included. The dimensionless drawdown is approximately proportional to the square root of the dimensionless time at late times.  相似文献   

2.
Analytical solutions of the momentum and energy equations for tidal flow are studied. Analytical solutions are well known for prismatic channels but are less well known for converging channels. As most estuaries have a planform with converging channels, the attention in this paper is fully focused on converging tidal channels. It will be shown that the tidal range along converging channels can be described by relatively simple expressions solving the energy and momentum equations (new approaches). The semi-analytical solution of the energy equation includes quadratic (nonlinear) bottom friction. The analytical solution of the continuity and momentum equations is only possible for linearized bottom friction. The linearized analytical solution is presented for sinusoidal tidal waves with and without reflection in strongly convergent (funnel type) channels. Using these approaches, simple and powerful tools (spreadsheet models) for tidal analysis of amplified and damped tidal wave propagation in converging estuaries have been developed. The analytical solutions are compared with the results of numerical solutions and with measured data of the Western Scheldt Estuary in the Netherlands, the Hooghly Estuary in India and the Delaware Estuary in the USA. The analytical solutions show surprisingly good agreement with measured tidal ranges in these large-scale tidal systems. Convergence is found to be dominant in long and deep-converging channels resulting in an amplified tidal range, whereas bottom friction is generally dominant in shallow converging channels resulting in a damped tidal range. Reflection in closed-end channels is important in the most landward 1/3 length of the total channel length. In strongly convergent channels with a single forward propagating tidal wave, there is a phase lead of the horizontal and vertical tide close to 90o, mimicking a standing wave system (apparent standing wave).  相似文献   

3.
The behaviour of river waves is described using a simplified dimensionless form of the momentum equation in conjunction with the continuity equation. Three dimensionless parameters were derived based on a quantitative linear analysis. These parameters, which depend on the Froude number of the steady uniform flow and the geometric characteristics of the river, permit quantification of the influence of inertia and pressure in the momentum equation. It was found that dynamic and diffusion waves occur mainly on gentle channel slopes and the transition between them is characterized by the Froude number. On the other hand, the kinematic wave has a wide range of applications. If the channel slope is greater than 1%, the kinematic wave is particularly suitable for describing the hydraulics of flow. Since slopes in natural channel networks are often greater than 1%, an analytical solution of the linearized kinematic wave equation with lateral inflow uniformly distributed along the channel is desirable and was therefore derived. The analytical solution was then implemented in a channel routing module of an existing simple rainfall–runoff model. The results obtained using the analytical solution compared well with those obtained from a non‐linear kinematic wave model. Copyright © 2000 John Wiley & Sons, Ltd.  相似文献   

4.
Analytical solution for drainage and recession from an unconfined aquifer   总被引：1，自引：0，他引：1  
Liang X  Zhang YK 《Ground water》2012,50(5):793-798
One-dimensional transient groundwater flow from a divide to a river in an unconfined aquifer described by the Boussinesq equation was studied. We derived the analytical solution for the water table recession and drainage change process described with a linearized Boussinesq equation with a physically based initial condition. A method for determining the average water table in the solutions was proposed. It is shown that the solution derived in the form of infinite series can be well approximated with the simplified solution which contains only the leading term of the original solution. The solution and their simplification can be easily evaluated and used by others to study the groundwater flow problems, such as drainage and base flow estimation, in an unconfined aquifer.  相似文献   

5.
In the present paper, an analytical expression of the Green’s function of linearized Saint-Venant equations (LSVEs) for shallow water waves is provided and applied to analyse the propagation of a perturbation superposed to a uniform flow. Independently of the kinematic character of the base flow, i.e., subcritical or supercritical uniform flow, the effects of a non-uniform vertical velocity profile and a non-constant resistance coefficient are accounted for. The use of the Darcy-Weisbach friction law allows a unified treatment of both laminar and turbulent conditions. The influence on the wave evolution of the wall roughness and the fluid viscosity are finally discussed, showing that in turbulent regime the assumption of constant friction coefficient may lead to an underestimation of both amplification and damping factors on the wave fronts, especially at low Reynolds numbers. This conclusion has to be accounted for, particularly in describing hyper-concentrated suspensions or other kinds of Newtonian mixtures, for which the high values of the kinematic viscosity may lead to relatively low Reynolds numbers.  相似文献   

6.
Reply     
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.  相似文献   

7.
In this article, we propose a new model, called LBLR for Linear Backwater Lag-and-Route, which approximates the Saint-Venant equations linearized around a non-uniform flow in a finite channel (with a downstream boundary condition). A classical frequency approach is used to build the distributed Saint-Venant transfer function providing the discharge at any point in the channel in the Laplace domain with respect to the upstream discharge. The moment matching method is used to match a second-order-with-delay model on the theoretical distributed Saint-Venant transfer function. Model parameters are then expressed analytically as functions of the pool characteristics. The proposed model efficiently accounts for the effects of downstream boundary condition on the channel dynamics.  相似文献   

8.
An analytical solution is presented for the slug tests conducted in a partially penetrating well in an unconfined aquifer affected from above by an unsaturated zone. The solution considers the effects of wellbore skin and oscillatory responses on underdamped slug tests. The flow in the saturated zone is described by a two‐dimensional, axially symmetric governing equation, and the flow in the unsaturated zone above the water table by a linearized one‐dimensional Richards' equation. The unsaturated medium properties are represented by the exponential constitutive relationships. A Laplace domain solution is derived using the Laplace and finite Fourier transform and the solution in the real‐time domain is evaluated using the numerical inverse Laplace transform method. The solution derived in this study is more general and reduces to the most commonly used solutions for slug tests in their specified conditions. It is found that the unsaturated flow has a significant impact on the slug test conducted in an unconfined aquifer. The impact of unsaturated flow on such a slug test is enhanced with a larger anisotropy ratio, a shorter well screen length, a shorter distance between the well screen and the water table, or a larger well screen radius. The impact of unsaturated flow on slug tests decreases as the degree of penetration (the length of well screen) increases. For a fixed well screen length, the impact of unsaturated flow on slug tests decreases as the distance between the centre of screen and the water table increases. A large dimensionless well screen radius (>0.01) leads to significant effects of unsaturated flow on slug tests. The unsaturated flow reduces the oscillatory responses to underdamped slug tests. The unsaturated zone has significant impact on slug test under high‐permeability wellbore skin.  相似文献   

9.
An analytical solution of a linearized Boussinesq equation is obtained to predict water table fluctuations as a result of time varying recharge from a strip basin for any number of recharge cycles. The analytical solution is obtained by using finite Fourier sine transform. Applications of the solution for the prediction of water table fluctuations and sensitivity analysis are demonstrated with the help of example problems. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

10.
This paper addresses the integral conservation of linear and angular momentum in the steady hydraulic jump in a linearly diverging channel.The flow is considered to be divided into a mainstream that conveys the total liquid discharge, and a roller where no average mass transport occurs. It is assumed that no macroscopic rheological relationship holds, so mass, momentum and angular momentum integral balances are independent relationships. Normal stresses are assumed to be hydrostatic on vertical, cylindrical surfaces. Viscous stresses are assumed to be negligible with respect to turbulent stresses. Assuming that the horizontal velocity distribution in the mainstream is uniform and that the horizontal momentum and angular momentum in the roller are negligible with respect to their mainstream counterparts, an analytical solution is obtained for the free surface profile of the flow. This solution is fundamental for finding the sequent depths and their positions. Consequently, it permits solving for the length of the jump, which is assumed to be equal to the length of the roller. Mainstream and roller thicknesses can also be derived from the present solution. This model may also be theoretically used to derive the average shear stresses exerted by the roller on the mainstream and the power losses per unit weight. This second relationship, which returns the well-known classical expression for total power loss in the jump, demonstrates that the strongly idealized mechanical model proposed here is internally consistent.  相似文献   

11.
Jiann‐Mou Chen 《水文研究》2008,22(26):5037-5047
Most methods developed to represent water flow phenomena in an unconfined aquifer with a fully penetrated pumping well are either numerical, such as the well‐known FEMWTER model, or experimental; analytical models of a partially penetrated pumping well are rare. This study employs the linearized Richards equation as the governing equation, with the aid of Fourier Integral Transformation, to obtain an analytical solution of the water content distribution in an unconfined aquifer with a partially penetrated pumping well. The results from this study could serve to substantiate in some sense results from numerical models. In addition, the theory developed herein can be modified to simulate a vacuum‐pressured pumping well since it is derived by considering, among others, the location and length of a well screen with fluxes. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

12.
Vedat Batu 《Ground water》2010,48(4):560-568
Using a steady-state mass conservative solute transport analytical solution that is based on the third-type (or flux-type or Cauchy) source condition, a method is developed to estimate the degradation parameters of solutes in groundwater. Then, the inadequacy of the methods based on the first-type source-based analytical solute transport solution is presented both theoretically and through an example. It is shown that the third-type source analytical solution exactly satisfies the mass balance constraint at the inlet location. It is also shown that the first-type source (or constant source concentration or Dirichlet) solution fails to satisfy the mass balance constraint at the inlet location and the degree of the failure depends on the value of the degradation as well as the flow and solute transport parameters. The error in the first-type source solution is determined with dimensionless parameters by comparing its results with the third-type source solution. Methods for estimating the degradation parameter values that are based on the first-type steady-state solute transport solution may significantly overestimate the degradation parameter values depending on the values of flow and solute transport parameters. It is recommended that the third-type source solution be used in estimating degradation parameters using measured concentrations instead of the first-type source solution.  相似文献   

13.
The adoption of the Diffusive Wave (DW) instead of the Full Dynamic (FD) model in the analysis of mud flood routing within the shallow-water framework may provide a significant reduction of the computational effort, and the knowledge of the conditions in which this approximation may be employed is therefore important. In this paper, the applicability of the DW approximation of a depth-integrated Herschel–Bulkley model is investigated through linear analysis. Assuming as the initial condition a steady hypocritical decelerated flow, induced by downstream backwater, the propagation characteristics of a small perturbation predicted by the DW and FD models are compared. The results show that the spatial variation on the initial profile may preclude the application of DW model with a prescribed accuracy. Whenever the method is applicable, the rising time of the mud flood must satisfy additional constraints, whose dependence on the flow depth, along with the Froude number and the rheological parameters, is deeply analyzed and discussed.  相似文献   

14.
《国际泥沙研究》2022,37(6):794-808
The current research deals with the dispersion of fine settling particles in a fluid flowing through an ice-covered channel under the laminar flow condition. An analytical solution of the two-dimensional convection–diffusion equation, based on the multi-scale homogenization technique, is obtained. To validate the current study, analytical results for the dispersion coefficient are compared with the available earlier research work. Moreover, the proposed analytical solutions for mean concentration distributions of the tracers are compared with the numerical results obtained from the finite difference technique. From the industrial and environmental points of view, the vertical concentration distribution provides a very significant information. The downstream evolution of the concentration distribution also is shown for typical time periods at different values of the settling velocity. The approach to the vertical uniformity shows that it is too slow a process in comparison to that of longitudinal normality. It was found that settling velocities of particles disturb the vertical uniformity and the centroid of the solute cloud rises due to the increase in settling velocity. Results illustrate that in the downstream direction, the vertical concentration distribution increases near the bed surface and it decreases in the proximity of the ice-covered surface of the channel with the increase of settling velocity, but the mean concentration of the solute increases. The current study may play an important role to understand the mechanism of the sedimentation process in a closed channel system.  相似文献   

15.
Addressing non-uniqueness in linearized multichannel surface wave inversion   总被引：1，自引：0，他引：1  
The multichannel analysis of the surface waves method is based on the inversion of observed Rayleigh-wave phase-velocity dispersion curves to estimate the shear-wave velocity profile of the site under investigation. This inverse problem is nonlinear and it is often solved using 'local' or linearized inversion strategies. Among linearized inversion algorithms, least-squares methods are widely used in research and prevailing in commercial software; the main drawback of this class of methods is their limited capability to explore the model parameter space. The possibility for the estimated solution to be trapped in local minima of the objective function strongly depends on the degree of nonuniqueness of the problem, which can be reduced by an adequate model parameterization and/or imposing constraints on the solution.
In this article, a linearized algorithm based on inequality constraints is introduced for the inversion of observed dispersion curves; this provides a flexible way to insert a priori information as well as physical constraints into the inversion process. As linearized inversion methods are strongly dependent on the choice of the initial model and on the accuracy of partial derivative calculations, these factors are carefully reviewed. Attention is also focused on the appraisal of the inverted solution, using resolution analysis and uncertainty estimation together with a posteriori effective-velocity modelling. Efficiency and stability of the proposed approach are demonstrated using both synthetic and real data; in the latter case, cross-hole S-wave velocity measurements are blind-compared with the results of the inversion process.  相似文献   

16.
A sudden acceleration of the Kuroshio jet appears off Cape Shionomisaki in the high-resolution (horizontal resolution of 1/36°) JCOPE 2 ocean reanalysis data. Using this dataset, we investigated the structure of the Kuroshio acceleration. The increase in the velocity of the current is accompanied by a downstream flow separation from the coast and an outcrop of cold temperature inshore. The acceleration of Kuroshio appears when it takes a near-shore path. Cape Shionomisaki amplifies the responses to the Kuroshio flow by creating the zonal velocity acceleration toward the downstream region when the Kuroshio flows closer to the coast. The Kuroshio acceleration coincided with the topographic ridge on the continental shelf near Cape Shionomisaki. This relation suggests that the dynamics of the acceleration is linked to the topographic feature. We proposed an explanation of the Kuroshio acceleration using a hydraulic control theory. An analytical solution was applied to the coastal topography around the Kii Peninsula. The solution captured some aspects of the Kuroshio acceleration.  相似文献   

17.
In interconnected microcracks, or in microcracks connected to spherical pores, the deformation associated with the passage of mechanical waves can induce fluid flow parallel to the crack walls, which is known as squirt flow. This phenomenon can also occur at larger scales in hydraulically interconnected mesoscopic cracks or fractures. The associated viscous friction causes the waves to experience attenuation and velocity dispersion. We present a simple hydromechanical numerical scheme, based on the interface-coupled Lamé–Navier and Navier–Stokes equations, to simulate squirt flow in the frequency domain. The linearized, quasi-static Navier–Stokes equations describe the laminar flow of a compressible viscous fluid in conduits embedded in a linear elastic solid background described by the quasi-static Lamé–Navier equations. Assuming that the heterogeneous model behaves effectively like a homogeneous viscoelastic medium at a larger spatial scale, the resulting attenuation and stiffness modulus dispersion are computed from spatial averages of the complex-valued, frequency-dependent stress and strain fields. An energy-based approach is implemented to calculate the local contributions to attenuation that, when integrated over the entire model, yield results that are identical to those based on the viscoelastic assumption. In addition to thus validating this assumption, the energy-based approach allows for analyses of the spatial dissipation patterns in squirt flow models. We perform simulations for a series of numerical models to illustrate the viability and versatility of the proposed method. For a 3D model consisting of a spherical crack embedded in a solid background, the characteristic frequency of the resulting P-wave attenuation agrees with that of a corresponding analytical solution, indicating that the dissipative viscous flow problem is appropriately handled in our numerical solution of the linearized, quasi-static Navier–Stokes equations. For 2D models containing either interconnected cracks or cracks connected to a circular pore, the results are compared with those based on Biot's poroelastic equations of consolidation, which are solved through an equivalent approach. Overall, our numerical simulations and the associated analyses demonstrate the suitability of the coupled Lamé–Navier and Navier–Stokes equations and of Biot's equations for quantifying attenuation and dispersion for a range of squirt flow scenarios. These analyses also allow for delineating numerical and physical limitations associated with each set of equations.  相似文献   

18.
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   

19.
ROGER MOUSSA 《水文研究》1996,10(9):1209-1227
The diffusive wave equation is generally used in flood routing in rivers. The two parameters of the equation, celerity and diffusivity, are usually taken as functions of the discharge. If these two parameters can be assumed to be constant without lateral inflow, the diffusive wave equation may have an analytical solution: the Hayami model. A general analytical method, based on ‘Hayami’s hypothesis, is developed here which resolves the diffusive wave flood routing equation with lateral inflow or outflow uniformly distributed over a channel reach. Flood routing parameters are then identified using observed inflow and outflow and the Hayami model used to simulate outflow. Two examples are discussed. Firstly, the prediction of the hydrograph at a downstream section on the basis of a knowledge of the hydrograph at an upstream section and the lateral inflow. The second example concerns lateral inflow identification between an upstream and a downstream section on the basis of a knowledge of hydrographs at the upstream and downstream sections. The new general Hayami model was applied to flood routing simulation and for lateral inflow identification of the River Allier in France. The major advantages of the method relate to computer simulation, real-time forecasting and control applications in examples where numerical instabilities, in the solution of the partial differential equations must be avoided.  相似文献   

20.
Summary The zonally asymmetric stationary component of the general circulation is studied for small Rossby number without the beta-plane approximation. The equations for this component are linearized about a mean flow. An analytic solution for the meridional wind is found when the zonal wind and static stability of the mean flow are independent of the vertical coordinate. The solution is used to compute the transports of angular momentum and heat. The angular momentum transports give rise to a net convergence of the order of Rossby number and are balanced by the zonal mean Coriolis torque. However, the heat transports vanish at this order of magnitude.  相似文献   

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