Similarity solutions to the second boundary value problem of unsaturated flow through porous media

Similarity solutions to the second boundary value problem of unsaturated flow are studied in one-dimensional, semi-infinite porous media with the soil-water diffusivity proportional to some power of the water content. The existence and uniqueness of two types of similarity solutions to the problem are investigated and the properties of these solutions are presented. It is shown that these two types of similarity solutions exist and that they may not be unique for every parameter range studied. The use of the similarity solutions is discussed for the experimental determination of soil-water diffusivity.     

Analytical solutions of kinematic equations for erosion on a plane II. Rainfall of finite duration

This paper, part II of the series, derives analytical solutions to kinematic equations for erosion from a sloping plane subject to rainfall of finite duration. Both equilibrium and partial equilibrium cases are distinguished. Complete solutions of these cases have not appeared previously in hydrologic literature. Properties of these solutions are briefly discussed. Finally a procedure is suggested for estimating the parameters contained in these solutions.     

A post-processing technique and an a posteriori error estimate for the newmark method in dynamic analysis

In this paper, we present a post-processing technique and an a posteriori error estimate for the Newmark method in structural dynamic analysis. By post-processing the Newmark solutions, we derive a simple formulation for linearly varied third-order derivatives. By comparing the Newmark solutions with the exact solutions expanded in the Taylor series, we achieve the local post-processed solutions which are of fifth-order accuracy for displacements and fourth-order accuracy for velocities in one step. Based on the post-processing technique, a posteriori local error estimates for displacements, velocities and, thus, also the total energy norm error estimate are obtained. If the Newmark solutions are corrected at each step, the post-processed solutions are of third-order accuracy in the global sense, i.e. one-order improvement for the original Newmark solutions is achieved. We also discuss a method for estimating the global time integration error. We find that, when the total energy norm is used, the sum of the local error estimates will give a reasonable estimate for the global error. We present numerical studies on a SDOF and a 2-DOF example in order to demonstrate the performance of the proposed technique.     

Exact analytical solutions for two-dimensional advection-dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.     

Periodic forcing in composite aquifers

Observations of periodic components of measured heads have long been used to estimate aquifer diffusivities. The estimations are often made using well-known solutions of linear differential equations for the propagation of sinusoidal boundary fluctuations through homogeneous one-dimensional aquifers. Recent field data has indicated several instances where the homogeneous aquifer solutions give inconsistent estimates of aquifer diffusivity from measurements of tidal lag and attenuation. This paper presents new algebraic solutions for tidal propagation in spatially heterogeneous one-dimensional aquifers. By building on existing solutions for homogeneous aquifers, comprehensive solutions are presented for composite aquifers comprising of arbitrary (finite) numbers of contiguous homogeneous sub-aquifers and subject to sinusoidal linear boundary conditions. Both Cartesian and radial coordinate systems are considered. Properties of the solutions, including rapid phase shifting and attenuation effects, are discussed and their practical relevance noted. Consequent modal dispersive effects on tidal waveforms are also examined via tidal constituent analysis. It is demonstrated that, for multi-constituent tidal forcings, measured peak heights of head oscillations can seem to increase, and phase lags seem to decrease, with distance from the forcing boundary unless constituents are separated and considered in isolation.     

Numerical validation of analytical solutions and their use for equivalent-linear seismic analysis of circular tunnels

The first part of this paper presents an extensive validation of four analytical solutions for the seismic design of circular tunnels. The validation is performed with a quasi-static finite element (FE) model which conforms to the assumptions of the analytical solutions. Analyses are performed for a wide range of flexibility ratios, slippage conditions at soil–lining interface, assuming both drained and undrained behaviour. Based on the numerical predictions the relative merits of the considered analytical solutions are discussed and recommendations are given for their use in design. The second part of this paper explores the use of equivalent linear soil properties in analytical solutions as an approximate way of simulating nonlinearity. The results of equivalent linear site response analyses are used as an input for the analytical solutions. The comparison of the analytical predictions with nonlinear numerical analysis results is very satisfactory. The results of this study suggest that analytical solutions can be used for preliminary design using equivalent linear properties and the corresponding compatible strain as an approximate way of accounting for nonlinear soil response.     

Analytical and semi-analytical solutions of horizontal well capture times under no-flow and constant-head boundaries

The closed-form analytical solutions and semi-analytical solutions of capture times to horizontal wells are derived for different recovery scenarios. The capture time is the time a fluid particle takes to flow to the well. The first scenario is recovery from a confined aquifer in which the influence of regional groundwater flow upon the capture time is included. The second scenario is recovery from underneath a water reservoir in which the top boundary of the aquifer is constant-head. The third scenario is recovery from a low-permeability layer bounded above and below by much higher permeability media. Closed-form solutions are provided for the cases with: (1) a center or a bottom well for the first scenario; (2) a bottom well for the second scenario; and (3) a center well for the third scenario. Semi-analytical solutions are provided for general well locations for those scenarios. Solutions for both isotropic and anisotropic media are studied. These solutions can be used as quick references to calculate the capture times, and as benchmarks to validate numerical solutions. The limitations of the analytical solutions are analyzed. Our results show that the top and bottom no-flow boundaries of an aquifer constrain the vertical flow, but enhance the horizontal flow, resulting in elongated iso-capture time curves. When constant-head boundaries are presented, water can infiltrate vertically across those boundaries to replenish the aquifers, resulting in less elongated iso-capture time curves.     

山西数字遥测地震台网震源机制解速报方法及结果分析

使用振幅比法计算地震的震源机制解软件，要求条件低、计算简便易行，可满足地震台网在地震速报中同时提供震源机制解参数的需要，利用山西数字遥测地震台网考核运行期间的数字记录资料。进行了震源机制解的速报与分析，结果表明，虽然山西数字遥测地震台网运行的时间很短，能够计算震源机制解的地震有限，但是结果仍可显示出受华北构造应力场一定的控制作用，半数中小地震的震源机制解和华北强震的震源机制解相同。     

Effects of convective-diffusive vertical mixing on the conception of rectangular settling basins

Collection efficiency η of rectangular settling basins is studied in this note for two cases: (i) with no mixing and (ii) with perfect vertical mixing. At first, well known reference solutions corresponding to suspended solids with constant settling velocity are presented. The effects on deposition of suspended solids with varying settling velocities are then studied. Finally, original exact solutions for η are obtained for gamma distributed settling velocities. The reference solutions are perfectly compatible with these more general solutions.     

Discontinuous Steady‐State Analytical Solutions of the Boussinesq Equation and Their Numerical Representation by Modflow

Known analytical solutions of groundwater flow equations are routinely used for verification of computer codes. However, these analytical solutions (e.g., the Dupuit solution for the steady‐state unconfined unidirectional flow in a uniform aquifer with a flat bottom) represent smooth and continuous water table configurations, simulating which does not pose any significant problems for the numerical groundwater flow models, like MODFLOW. One of the most challenging numerical cases for MODFLOW arises from drying‐rewetting problems often associated with abrupt changes in the elevations of impervious base of a thin unconfined aquifer. Numerical solutions of groundwater flow equations cannot be rigorously verified for such cases due to the lack of corresponding exact analytical solutions. Analytical solutions of the steady‐state Boussinesq equation, associated with the discontinuous water table configurations over a stairway impervious base, are presented in this article. Conditions resulting in such configurations are analyzed and discussed. These solutions appear to be well suited for testing and verification of computer codes. Numerical solutions, obtained by the latest versions of MODFLOW (MODFLOW‐2005 and MODFLOW‐NWT), are compared with the presented discontinuous analytical solutions. It is shown that standard MODFLOW‐2005 code (as well as MODFLOW‐2000 and older versions) has significant convergence problems simulating such cases. The problems manifest themselves either in a total convergence failure or erroneous results. Alternatively, MODFLOW‐NWT, providing a good match to the presented discontinuous analytical solutions, appears to be a more reliable and appropriate code for simulating abrupt changes in water table elevations.     

Transverse response of underground cavities and pipes to incident SV waves

The transverse response of underground cylindrical cavities to incident SV waves is investigated. Analytical solutions are derived for unlined cavities embedded within an elastic half‐space using Fourier–Bessel series and a convex approximation of the half‐space free surface. The computed displacements at the half‐space free surface and the tangential stresses on the cavity are compared with the results of previous investigations. The analytical solutions are extended to formulate approximate solutions for assessing hoop stresses within cavity liners impinged by low‐frequency waves having wavelengths much longer than the cavity diameter. The approximate solutions are compared to existing numerical solutions, and used to evaluate the dynamic response of a flexible buried pipe shaken by the 1994 Northridge earthquake. The proposed approximate model for cavity liners is useful for the seismic analysis of underground pipes and small‐diameter tunnels. Copyright © 2001 John Wiley & Sons, Ltd.     

Centroid-moment tensor solutions for January–March, 1984

Centroid-moment tensor solutions are presented for 148 earthquakes that occurred during the first quarter of 1984. The solutions are obtained using corrections for aspherical Earth structure.     

Transient foundation solution of saturated soil to impulsive concentrated loading

The transient dynamic response of saturated soil under suddenly applied normal and horizontal concentrated loading is studied in this paper. The behavior of saturated soil is governed by Biot's consolidation theory. The general solutions for Biot equations of equilibrium are derived in terms of displacements and variations of fluid volume, using Laplace–Hankel integral transforms. The solutions in the time domain can be evaluated by numerical inverse Laplace–Hankel transforms. Selected numerical results for displacements, stresses, and pore pressures are presented. Comparisons with existing closed-form solutions for the elastic half-space are made to confirm the accuracy of the present solutions. The solutions can be used to study a variety of transient wave propagation problems and dynamical interactions between saturated soil and structures.     

Centroid-moment tensor solutions for April–June 1989

Centroid-moment tensor solutions are presented for 226 earthquakes that occurred during the second quarter of 1989. The solutions are obtained using corrections for aspherical Earth structure.     

Centroid-moment tensor solutions for October–December, 1984

Centroid-moment tensor solutions are presented for 175 earthquakes that occurred during the fourth quarter of 1984. The solutions are obtained using corrections for aspherical Earth structure.     

Centroid-moment tensor solutions for July–September, 1984

Centroid-moment tensor solutions are presented for 197 earthquakes that occurred during the third quarter of 1984. The solutions are obtained using corrections for aspherical Earth structure.     

Analytical solutions of kinematic equations for erosion on a plane I. Rainfall of indefinite duration

Kinematic equations have been utilized in the past for modeling erosion from a sloping plane subject to rainfall. This paper, part I in a series of two, presents analytical solutions of these equations under the assumption of constant effective rainfall of indefinite duration. These solutions contain a function which is related to a generalization of Dawson's integral. Some pertinent properties of this function are briefly discussed. The solutions obtained by previous investigators constitute a part of the solutions developed here. In part II the solutions are derived for the case when the duration of the effective rainfall is finite.     

Finite area beta-plane dipoles

Abstract

We study a class of vortex dipoles consisting of two patches of uniform potential vorticity in an otherwise quiescent flow on a β-plane. Steadily propagating solutions that are desingularised analogues of point vortex dipoles are found and compared with the point vortex solutions. Like the point vortex dipoles, both rapidly and slowly propagating solutions exist. Numerical simulations show that the slow solutions are unstable and break up under the influence of weak external perturbations. The fast solutions are more robust. The minimum dipole strength necessary for the existence of a steadily propagating solution is less than that found for point vortex dipoles.     

A tale of two beams: an elementary overview of Gaussian beams and Bessel beams

An overview of two types of beam solutions is presented, Gaussian beams and Bessel beams. Gaussian beams are examples of non-localized or diffracting beam solutions, and Bessel beams are example of localized, non-diffracting beam solutions. Gaussian beams stay bounded over a certain propagation range after which they diverge. Bessel beams are among a class of solutions to the wave equation that are ideally diffraction-free and do not diverge when they propagate. They can be described by plane waves with normal vectors along a cone with a fixed angle from the beam propagation direction. X-waves are an example of pulsed beams that propagate in an undistorted fashion. For realizable localized beam solutions, Bessel beams must ultimately be windowed by an aperture, and for a Gaussian tapered window function this results in Bessel-Gauss beams. Bessel-Gauss beams can also be realized by a combination of Gaussian beams propagating along a cone with a fixed opening angle. Depending on the beam parameters, Bessel-Gauss beams can be used to describe a range of beams solutions with Gaussian beams and Bessel beams as end-members. Both Gaussian beams, as well as limited diffraction beams, can be used as building blocks for the modeling and synthesis of other types of wave fields. In seismology and geophysics, limited diffraction beams have the potential of providing improved controllability of the beam solutions and a large depth of focus in the subsurface for seismic imaging.     

A Leaky Aquifer below Champlain Sea Clay: Closed‐Form Solutions for Natural Seepage

Closed‐form solutions are proposed for natural seepage in semiconfined (leaky) aquifers such as those existing below the massive Champlain Sea clay layers in the Saint‐Lawrence River Valley. The solutions are for an ideal horizontal leaky aquifer below an ideal aquitard that may have either a constant thickness and a constant hydraulic head at its surface, or a variable thickness and a variable hydraulic head at its surface. A few simplifying assumptions were needed to obtain the closed‐form solutions. These have been verified using a finite element method, which did not make any of the assumptions but gave an excellent agreement for hydraulic heads and groundwater velocities. For example, the difference between the two solutions was smaller than 1 mm for variations in the 5 to 8 m range for the hydraulic head in the semiconfined aquifer. Note that fitting the hydraulic head data of monitoring wells to the theoretical solutions gives only the ratio of the aquifer and aquitard hydraulic conductivities, a clear case of multiple solutions for an inverse problem. Consequently, field permeability tests in the aquitard and the aquifer, and pumping tests in the aquifer, are still needed to determine the hydraulic conductivity values.