Stable pumping rates for horizontal wells in bank filtration systems

We consider common bank filtration systems and develop an explicit analytic solution representing steady, two-dimensional, groundwater flow to a horizontal well near a river in an unconfined aquifer. For the boundary-value problem investigated, we find that a unique solution exists for all negative well discharges. For positive discharges, a maximum value exists which corresponds to the formation of a cusp on the free surface. For positive discharges less than the maximum, the solution is not unique, consisting of two alternate configurations of the free surface. One solution includes a stable free surface on a single-valued physical plane, while the alternate solution includes a looped free surface lying on two sheets of a Riemann surface. Imposing a stability condition on the free surface results in a unique solution to the problem. We use the solution to investigate the behavior of the free surface under varying well discharges to identify stable pumping rates and predict well yield. In particular, we examine the well yield and the stability of the free surface when the head in the horizontal well is maintained at the top of the well screen. This condition is shown to produce a stable free surface for a wide range of well radii; the stability is independent of the hydraulic conductivity of the aquifer, the location of the well, or the presence of a skin resistance at the well.     

Determination of moment tensor and location of microseismic events under conditions of highly correlated noise based on the maximum likelihood method

We examine the problem of localization of a single microseismic event and determination of its seismic moment tensor in the presence of strongly correlated noise. This is a typical problem occurring in monitoring of microseismic events from a daylight surface during producing or surface monitoring of hydraulic fracturing. We propose a solution to this problem based on the method of maximum likelihood. We discuss mathematical aspects of the problem, some features and weak points of the proposed approach, estimate the required computing resources, and present the results of numerical experiments. We show that the proposed approach is much more resistant to correlated noises than diffraction stacking methods and time reverse modeling.     

Geoid Determination Through Ellipsoidal Stokes Boundary-Value Problem by Splitting Its Solution to the Low-Degree and the High-Degree Parts

The ellipsoidal Stokes boundary-value problem is used to compute the geoidal heights. The low degree part of the geoidal heights can be represented more accurately by Global Geopotential Models (GGM). So the disturbing potential is splitted into a low-degree reference potential and a higher-degree potential. To compute the low-degree part, the global geopotential model is used, and for the high-degree part, the solution of the ellipsoidal Stokes boundary-value problem in the form of the surface integral is used. We present an effective method to remove the singularity of the high-degree of the spherical and ellipsoidal Stokes functions around the computational point. Finally, the numerical results of solving the ellipsoidal Stokes boundary-value problem and the difference between the high-degree part of the solution of the ellipsoidal Stokes boundary-value problem and that of the spherical Stokes boundary-value problem is presented.     

Uncertainty of moment tensor determination from surface wave analysis for shallow earthquakes

We treat the problem of the non-uniqueness of the moment tensor inversion for shallow earthquakes from long period surface wave data. We give an existence condition for double couples radiating the same long period surface waves as the deviatoric moment tensor obtained by such inversion. We describe the family of such double couples and show that they may provide better estimates of double couple mechanisms than the traditional “best double-couple” solution. We demonstrate the results of application to four large shallow earthquakes with different source signatures.     

A complex variable technique to study notch-effects in geodynamics

Summary The techniques of Complex Variables are used to obtain a solution for a general notch problem having a load distributed over any portion of its free surface. This solution is very general in that it can be used for many different shapes of notches, including protrusions, by simply varying a few parameters. The same techniques are then used to solve the problem of a notched plate acted on by loads at infinity. This solution is of a very general nature. One particular solution holds for the problem studied in an earlier publication.     

On the geometrical form of volcanoes

Many volcanic edifices have a remarkably symmetric geometrical form. An example is Mount Fuji in Japan. We model this form assuming that the surface of the volcano is a surface of uniform hydraulic potential; that an erupting magma will follow the path of minimum resistance to the surface. In order to model the resistance to fluid flow we assume the volcanic edifice is a uniform porous medium. The vertical flow of magma is also resisted by the gravitational body force. If the volcano becomes too tall flank eruptions will widen it; if the volcano becomes too wide summit eruptions will increase its elevation. Using the Dupuit approximation for an unconfined aquifer it is shown that the percolation equation is applicable. As magma reaches the surface it is assumed to extend the solid, porous matrix. A similarity solution is obtained to this moving boundary problem. The solution predicts a uniform shape for all volcanoes. This shape is shown to be in excellent agreement with the geometrical form of Mount Fuji.     

First‐arrival traveltime tomography with modified total‐variation regularization

First‐arrival traveltime tomography is a robust tool for near‐surface velocity estimation. A common approach to stabilizing the ill‐posed inverse problem is to apply Tikhonov regularization to the inversion. However, the Tikhonov regularization method recovers smooth local structures while blurring the sharp features in the model solution. We present a first‐arrival traveltime tomography method with modified total‐variation regularization to preserve sharp velocity contrasts and improve the accuracy of velocity inversion. To solve the minimization problem of the new traveltime tomography method, we decouple the original optimization problem into the two following subproblems: a standard traveltime tomography problem with the traditional Tikhonov regularization and a L₂ total‐variation problem. We apply the conjugate gradient method and split‐Bregman iterative method to solve these two subproblems, respectively. Our synthetic examples show that the new method produces higher resolution models than the conventional traveltime tomography with Tikhonov regularization, and creates less artefacts than the total variation regularization method for the models with sharp interfaces. For the field data, pre‐stack time migration sections show that the modified total‐variation traveltime tomography produces a near‐surface velocity model, which makes statics corrections more accurate.     

Discussion on “Implications of surface wave data measurement uncertainty on seismic ground response analysis” by Jakka et al.

The discussion deals with the effect of shear wave velocity uncertainties on 1D seismic ground response analysis. In particular, the paper refers to uncertainties deriving from the solution of the inverse problem in surface wave methods. We address some issues related to the evaluation of “equivalent” profiles from surface wave data, the inversion strategy and the numerical simulation of seismic site response. The pitfalls in the analyses point out the need for more refined studies to draw general conclusions on the subject.     

ASPECTS OF CHARGE ACCUMULATION IN d. c. RESISTIVITY EXPERIMENTS1

When an electric current is introduced to the earth, it sets up a distribution of charges both on and beneath the earth's surface. These charges give rise to the anomalous potential measured in the d. c. resistivity experiment. We investigate different aspects of charge accumulation and its fundamental role in d. c. experiments. The basic equations and boundary conditions for the d. c. problem are first presented with emphasis on the terms involving accumulated charges which occur wherever there is a non-zero component of electric field parallel to the gradient of conductivity. In the case of a polarizable medium, the polarization charges are also present due to the applied electric field, yet they do not change the final field distribution. We investigate the precise role of the permittivity of the medium. The charge buildup alters the electric fields and causes the refraction of current lines; this results in the well-known phenomenon of current channelling. We demonstrate this by using charge density to derive the refraction formula at a boundary. An integral equation for charge density is presented for whole-space models where the upper half-space is treated as an in-homogeneity with zero conductivity. The integral equation provides a tool with which the charge accumulation can be examined quantitatively and employed in the practical forward modelling. With the aid of this equation, we investigate the effect of accumulated charges on the earth's surface and show the equivalence between the half-space and whole-space formulations of the problem. Two analytic examples are presented to illustrate the charge accumulation and its role in the d. c. problem. We investigate the relationship between the solution for the potential via the image method and via the charge density. We show that the essence of the image method solution to the potential problem is to derive a set of fictitious sources which produce the same potential as does the true charge distribution. It follows that the image method is viable only when the conductivity structure is such that the effect of the accumulated charge can be represented by a set of point images. As numerical examples, we evaluate quantitatively the charge density on the earth's surface that arises because of topography and the charge density on a buried conductive prism. By these means, we demonstrate the use of the boundary element technique and charge density in d. c. forward modelling problems.     

On the effective inversion by imposing <Emphasis Type="Italic">a priori</Emphasis> information for retrieval of land surface parameters

The anisotropy of the land surface can be best described by the bidirectional reflectance distribution function (BRDF). As the field of multiangular remote sensing advances, it is increasingly probable that BRDF models can be inverted to estimate the important biological or climatological parameters of the earth surface such as leaf area index and albedo. The state-of-the-art of BRDF is the use of the linear kernel-driven models, mathematically described as the linear combination of the isotropic kernel, volume scattering kernel and geometric optics kernel. The computational stability is characterized by the algebraic operator spectrum of the kernel-matrix and the observation errors. Therefore, the retrieval of the model coefficients is of great importance for computation of the land surface albedos. We first consider the smoothing solution method of the kernel-driven BRDF models for retrieval of land surface albedos. This is known as an ill-posed inverse problem. The ill-posedness arises from that the linear kernel driven BRDF model is usually underdetermined if there are too few looks or poor directional ranges, or the observations are highly dependent. For example, a single angular observation may lead to an under-determined system whose solution is infinite (the null space of the kernel operator contains nonzero vectors) or no solution (the rank of the coefficient matrix is not equal to the augmented matrix). Therefore, some smoothing or regularization technique should be applied to suppress the ill-posedness. So far, least squares error methods with a priori knowledge, QR decomposition method for inversion of the BRDF model and regularization theories for ill-posed inversion were developed. In this paper, we emphasize on imposing a priori information in different spaces. We first propose a general a priori imposed regularization model problem, and then address two forms of regularization scheme. The first one is a regularized singular value decomposition method, and then we propose a retrieval method in I ¹ space. We show that the proposed method is suitable for solving land surface parameter retrieval problem if the sampling data are poor. Numerical experiments are also given to show the efficiency of the proposed methods. Supported by National Natural Science Foundation of China (Grant Nos. 10501051, 10871191), and Key Project of Chinese National Programs for Fundamental Research and Development (Grant Nos. 2007CB714400, 2005CB422104)     

A Brovar-type solution of the fixed geodetic boundary-value problem

For more than 150 years gravity anomalies have been used for the determination of geoidal heights, height anomalies and the external gravity field. Due to the fact that precise ellipsoidal heights could not be observed directly, traditionally a free geodetic boundary-value problem (GBVP) had to be formulated which after linearisation is related to gravity anomalies. Since nowadays the three-dimensional positions of gravity points can be determined by global navigation satellite systems very precisely, the modern formulation of the GBVP can be based on gravity disturbances which are related to a fixed GBVP using the known topographical surface of the Earth as boundary surface. The paper discusses various approaches into the solution of the fixed GBVP which after linearization corresponds to an oblique-derivative boundary-value problem for the Laplace equation. Among the analytical solution approaches a Brovar-type solution is worked out in detail, showing many similarities with respect to the classical solution of the scalar free GBVP.     

Steady flow to a well near a stream with a leaky bed

We present an explicit analytic solution for steady, two-dimensional ground water flow to a well near a leaky streambed that penetrates the aquifer partially. Leakage from the stream is approximated as occurring along the centerline of the stream. The problem domain is infinite and pumping on one side of the stream induces flow on the other side. The solution includes the effects of uniform flow in the far field and a sloping hydraulic head in the stream. We use the solution to investigate the interaction between ground water and surface water in the stream, the effects of pumping on the opposite side of the stream, and the effects of the leaky streambed on the capture zone envelope of the well. We develop a relationship between parameters such that the pumping well will not capture water from the stream, or from the opposite side of the stream. When the discharge of the well is large enough to capture water from the stream, the shape of the capture zone envelope depends on flow conditions on the side of the stream opposite the well.     

Surface-driven flows in long rectangular enclosures

This paper is devoted to numerical and theoretical studies of the motions of viscous fluids driven by a constant stress acting in long rectangular enclosures. This problem was originally addressed to study the flow behaviour of molten metals driven by thermocapillarity in open boats during the directional solidification by the Bridgman technique and the study has now been extended to a closely related problem, i.e., the flows occuring in a water-basin under a stress generated by an external wind. We handled this problem by solving the 2D Navier-Stokes equations by an efficient finite-difference technique. Computations have been done for a rectangular basin with various aspect ratios (length/height), and for a large range of the surface Reynolds number. A notable feature of the solutions is the totally different end circulations. Following a previous study of Bye¹, we also emphasized this problem through the linear eigenequation resulting from (small) spatial disturbances of a Couette flow solution. We performed this analysis by using an efficient Tau-Chebyshev technique.     

An integrated method for resolving the seismic complex near-surface problem

We describe an integrated method for solving the complex near‐surface problem in land seismic imaging. This solution is based on an imaging approach and is obtained without deriving a complex near‐surface velocity model. We start by obtaining from the data the kinematics of the one‐way focusing operators (i.e. time‐reversed Green's functions) that describe propagation between the acquisition surface and a chosen datum reflector using the common‐focus‐point technology. The conventional statics solutions obtained from prior information about the near surface are integrated in the initial estimates of the focusing operators. The focusing operators are updated iteratively until the imaging principle of equal traveltime is fulfilled for each subsurface gridpoint of the datum reflector. Therefore, the seismic data is left intact without any application of time shifts, which makes this method an uncommitted statics solution. The focusing operators can be used directly for wave‐equation redatuming to the respective reflector or for prestack imaging if determined for multiple reflecting boundaries. The underlying velocity model is determined by tomographic inversion of the focusing operators while also integrating any hard prior information (e.g. well information). This velocity model can be used to perform prestack depth imaging or to calculate the depth of the new datum level. We demonstrate this approach on 2D seismic data acquired in Saudi Arabia in an area characterized by rugged topography and complex near‐surface geology.     

Numerical and approximate solutions for lithospheric thickening and thinning

Lithospheric thickness, surface heat flow and subsidence are calculated numerically as functions of time since the formation of oceanic lithosphere at a ridge crest. These numerical results confirm the validity of a previously derived approximate solution to the same problem. A numerical solution is also calculated for lithospheric thinning due to a sudden increase of heat flux into the base of the lithosphere. With this solution as a standard, an approximate solution is derived which can accurately predict thickness, heat flow, and uplift as functions of time. Lithospheric thinning and surface uplift begin immediately with the increase of heat flux at depth, but the increase in surface heat flow begins 20–40 m.y. later.     

一种适合沙漠地区的静校正方法

静校正问题是我国西部复杂地形和复杂近地表地区地震勘探迫切需要解决的一个关键问题,也是一个十分复杂的问题.而复杂近地表地区的静校正问题往往归结为近地表层速度的求解问题,地表越复杂,近地表速度的求解就越困难.塔中地区表层地质条件变化剧烈,潜水面以上覆盖着大小不一的疏松沙体,沙体高度从几米到近百米,地震波在地表的传播速度随沙丘厚度而变化,因此,研究该区表层速度变化规律,成为该区静校正的主要问题.本文在总结前人所研究的沙漠地区静校正方法的基础上,提出了一种新的静校正方法,通过多项式拟合来求取表面速度,实际应用见到了良好的效果.     

On one method of solving the direct gravimetric problem

Summary The general solution of the direct gravimetric problem for homogeneous and inhomogeneous bodies was derived in[1] in the form of Green-type integrals. In the present paper it is proved that this solution simplifies considerably, if the solution of the external Dirichlet problem, defined by conditions(2), is known for a closed surface bounding the attracting body.Dedicated to RNDr Jan Pícha, CSc., on his 60th Birthday     

Scattering and diffraction of plane P-waves in a 2-D elastic half-space Ⅱ:shallow arbitrary shaped canyon

Scattering and Diffraction of elastic in-plane P- and SV- waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH waves on the same elastic canyon that is semi-circular in shape on the half-space surface is the first such problem that was solved by analytic closed form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because, the in-plane P- and SV- scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattered and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape.     

Solving the complex near‐surface problem using 3D data‐driven near‐surface layer replacement

In many land seismic situations, the complex seismic wave propagation effects in the near‐surface area, due to its unconsolidated character, deteriorate the image quality. Although several methods have been proposed to address this problem, the negative impact of 3D complex near‐surface structures is still unsolved to a large extent. This paper presents a complete 3D data‐driven solution for the near‐surface problem based on 3D one‐way traveltime operators, which extends our previous attempts that were limited to a 2D situation. Our solution is composed of four steps: 1) seismic wave propagation from the surface to a suitable datum reflector is described by parametrized one‐way propagation operators, with all the parameters estimated by a new genetic algorithm, the self‐adjustable input genetic algorithm, in an automatic and purely data‐driven way; 2) surface‐consistent residual static corrections are estimated to accommodate the fast variations in the near‐surface area; 3) a replacement velocity model based on the traveltime operators in the good data area (without the near‐surface problem) is estimated; 4) data interpolation and surface layer replacement based on the estimated traveltime operators and the replacement velocity model are carried out in an interweaved manner in order to both remove the near‐surface imprints in the original data and keep the valuable geological information above the datum. Our method is demonstrated on a subset of a 3D field data set from the Middle East yielding encouraging results.     

Finite element analysis of three-dimensional groundwater flow problems

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.