Theoretical analysis of aquifer response due to dewatering at Welland

1970 ◽  
Vol 7 (2) ◽  
pp. 205-216 ◽  
Author(s):  
E. O. Frind
Keyword(s):  
Differential Equation ◽  
Horizontal Plane ◽  
Linear Equations ◽  
Model Parameters ◽  
Two Dimensional ◽  
Construction Site ◽  
System Of Linear Equations ◽  
Flow Conditions ◽  
Observation Wells ◽  
And Storage

The investigation of the regional response of an aquifer due to depressurization at the Welland Canal construction site is discussed in this paper. The aquifer consists of the upper fractured zone of the bedrock, overlain by a poorly-permeable confining layer which permits recharge and discharge through leakage. The permeability in the aquifer varies throughout the area.As no exact solution exists, a mathematical model is developed. The model is two-dimensional in the horizontal plane and represents the non-homogeneous continuum of the aquifer by a finite number of nodes arranged in a grid. From the differential equation of flow, a system of linear equations is derived and solved by computer. Unsteady flow conditions are approximated by solving the system for discrete steps in time, from commencement of pumping up to equilibrium. Model parameters, consisting of the nodal values of transmissibility, leakance, and storage coefficient, are established by simulation of an actual period of pumping, for which the regional response is determined from readings at observation wells. After using one observed case, the model can be employed to solve any number of practical problems relating to the aquifer response within the area.

scholarly journals To the Solution of Geometric Inverse Heat Conduction Problems

2021 ◽  
Vol 24 (1) ◽  
pp. 6-12
Author(s):  
Yurii M. Matsevytyi ◽  
◽  
Valerii V. Hanchyn ◽  
Keyword(s):  
Heat Conduction ◽  
Iterative Process ◽  
Regularization Parameter ◽  
Outer Boundary ◽  
Linear Equations ◽  
The Body ◽  
Two Dimensional ◽  
Inverse Heat Conduction ◽  
System Of Linear Equations ◽  
Inverse Heat Conduction Problems

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

SPECTRAL ANALYSIS OF GRAVITY AND MAGNETIC ANOMALIES DUE TO TWO‐DIMENSIONAL STRUCTURES

Geophysics ◽  
1975 ◽  
Vol 40 (6) ◽  
pp. 993-1013 ◽  
Author(s):  
B. K. Bhattacharyya ◽  
Lei‐Kuang Leu
Keyword(s):  
Linear Equations ◽  
Magnetic Anomalies ◽  
Dimensional Structure ◽  
Approximation Technique ◽  
System Of Equations ◽  
Two Dimensional ◽  
System Of Linear Equations ◽  
Unknown Parameters ◽  
Low Frequencies ◽  
Gravity And Magnetic

The expressions for the spectra of both gravity and magnetic anomalies due to a two‐dimensional structure consist of (except for a factor) sums of exponentials. The exponents of these exponentials are functions of frequency and the locations of the corners of the polygonal cross‐section of the structure. Two computationally feasible methods for determining the exponents from a given spectrum are described in this paper; they are essentially based on the generation of a system of linear equations. The unknown coefficients in this system of equations are functions of the corner locations. The first method requires expansion of the exponentials in the expressions for the spectra in the form of a series and works reliably when the amplitudes of low frequencies are analyzed. The unknown parameters are determined fairly accurately with this method by suitable combinations of the spectra of the observed anomaly and its moments. The second method utilizes an exponential approximation technique for producing the system of linear equations. If only the spectrum of the anomaly is used, the system of equations becomes ill‐conditioned in most cases resulting in grossly inaccurate solutions. However, particular combinations of the spectra of the anomaly and its first and second order moments are found to improve significantly the behavior of the system of equations and thus the quality of results. It has also been found that the mean values of corner locations can be calculated fairly accurately by taking the ratios of the spectra of the anomaly and its moments. Once the corner locations are found, computation of the density contrast in the case of a gravity anomaly and the magnetization contrast for a magnetic anomaly is straightforward.

AUTOMATIC CONTOURING OF FAULTED SUBSURFACES

Geophysics ◽  
1976 ◽  
Vol 41 (6) ◽  
pp. 1377-1393 ◽  
Author(s):  
G. Bolondi ◽  
F. Rocca ◽  
S. Zanoletti
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Horizontal Plane ◽  
Elliptic Partial Differential Equation ◽  
Computer Time ◽  
Two Dimensional ◽  
The Third ◽  
Different Types ◽  
Data Points ◽  
Automatic Contouring

The problem of contouring a faulted surface known at randomly spaced points is analyzed and different types of solutions are proposed. The data may in fact be from a field which satisfies an elliptic partial differential equation; if the equation is harmonic, the surface corresponds to the displacement of a membrane properly raised from a horizontal plane in correspondence to the data points and cut along the faults. If the equation is biharmonic, the surface corresponds to the displacement of an elastic plate, properly riveted in correspondence to the data points and again cut along the faults. A third method analyzed, that corresponds to a family of interpolation methods, is that of two‐dimensional estimation. The technique used is that of modeling the autocovariance of the data as a function of the distance between the points only. The surface will depend upon the particular function chosen and it will tend to be peaked at the data points, if the function is peaked at the origin, and smoother if the autocovariance is smoother. When faults are present, the distance between two points is defined to be the length of the shortest linking path, not cutting a fault. In the latter case, it is shown that the set of functions eligible to be chosen as autocovariances is very limited. The first method has the useful property that maxima and minima of the surface are data points. The second method generates smoother surfaces that sometimes may overshoot. Both methods are implemented by iteratively smoothing the interpolated lattice (except in the neighborhood of data points), and therefore are rather expensive in terms of computer time. The third method is not iterative and is less expensive; since the surfaces that it generates are noisy, it may be used to supply a tentative solution to be refined with an iterated smoothing. These different techniques arc discussed in detail and some examples of their application are shown.

scholarly journals Generalized Refinement of Gauss-Seidel Method for Consistently Ordered 2-Cyclic Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Gashaye Dessalew ◽  
Tesfaye Kebede ◽  
Gurju Awgichew ◽  
Assaye Walelign
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Rate Of Convergence ◽  
Difference Method ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Seidel Method ◽  
Minimum Number

This paper presents generalized refinement of Gauss-Seidel method of solving system of linear equations by considering consistently ordered 2-cyclic matrices. Consistently ordered 2-cyclic matrices are obtained while finite difference method is applied to solve differential equation. Suitable theorems are introduced to verify the convergence of this proposed method. To observe the effectiveness of this method, few numerical examples are given. The study points out that, using the generalized refinement of Gauss-Seidel method, we obtain a solution of a problem with a minimum number of iteration and obtain a greater rate of convergence than other previous methods.

scholarly journals UNAMBIGUOUS SOLVABILITY OF A PARTICULAR CASE OF SYSTEMS OF INTEGRODIFFERENTIAL EQUATIONS WITH A PULSED KAEV DISTANCE CONTAINING A PARAMETER

2020 ◽  
Vol 72 (4) ◽  
pp. 78-84
Author(s):  
Kh.I. Usmanov ◽  
◽  
A.S. Zhappar ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Integral Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
System Of Linear Equations ◽  
New Variables

We consider a special case of systems of integro-differential equations with a momentum boundary condition containing a parameter when the derivative of the desired function is contained in the right side of the equation. By integrating in parts, an integro-differential equation with a pulsed boundary condition is reduced to a loaded integrodifferential equation with a pulsed boundary condition. it is given in the system of integral-differential equations with impulse boundary conditions parametrically loaded. Then, by entering new parameters, as well as passing to new variables based on these parameters, the problem is reduced to an equivalent problem. Switching to new variables makes it possible to get the initial conditions for the equation. Based on this, the solution of the problem is reduced to solving a special Cauchy problem and a system of linear equations. Using the fundamental matrix of the main part of the differential equation, an integral equation of the Volterra type is obtained. The method of sequential approximation determines the unique solution of the integral equation. Based on this, we find a solution to the special Cauchy problem and put it in the boundary conditions. On the basis of the obtained system of linear equations, necessary and sufficient conditions for an unambiguous solution of the initial problem are established.

scholarly journals Approximate Solution ofnth-Order Fuzzy Linear Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaobin Guo ◽  
Dequan Shang
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Fuzzy Numbers ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Linear Differential

The approximate solution ofnth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

A discussion on advanced methods of energy conversion- Magnetohydrodynamic power generation - Flow of current in a cross connected m.h.d. generator with four electrodes

1967 ◽  
Vol 261 (1123) ◽  
pp. 455-460 ◽  
Keyword(s):  
Current Flow ◽  
Linear Equations ◽  
Cathode Side ◽  
Two Dimensional ◽  
Anode Side ◽  
Flow Conditions ◽  
Fluid Properties ◽  
Simultaneous Linear Equations ◽  
Christoffel Transformation ◽  
Four Electrodes

The current flow patterns in an m.h.d. generator with four electrodes are derived. There are two electrodes each side of the duct; the upstream one on the cathode side is connected to the downstream one on the anode side and a load across the other pair of electrodes produces a rudimentary cross-connected generator. The fluid properties, flow conditions and magnetic field are supposed uniform across the duct and a two-dimensional analysis is made. A Schwarz-Christoffel transformation is employed to find potential and current distributions. Three simultaneous linear equations must be solved to give explicit equations for currents and potentials. The method is extendable to the case of more electrodes.

Controllable Synthesis of Ultrathin Layered Transition Metallic Hydroxide/Zeolitic Imidazolate Framework-67 Hybrid Nanosheets for High-Performance Supercapacitors

2021 ◽  
Author(s):  
Chunli Liu ◽  
Yang Bai ◽  
Ji Wang ◽  
Ziming Qiu ◽  
Huan Pang
Keyword(s):  
Charge Transfer ◽  
Energy Conversion ◽  
High Performance ◽  
2D Materials ◽  
Two Dimensional ◽  
Controllable Synthesis ◽  
Zeolitic Imidazolate Framework ◽  
Fast Charge ◽  
Energy Conversion And Storage ◽  
And Storage

Two-dimensional (2D) materials with structures having diverse features are promising for application in energy conversion and storage. A stronger layered orientation can guarantee fast charge transfer along the 2D planes...

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