scholarly journals A Study on Solving the Seepage Flow Model of Three-Area Composite Reservoir

2014 ◽  
Vol 670-671 ◽  
pp. 599-603 ◽  
Author(s):  
Xiao Xu Dong ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Feng Jiu Zhang
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Flow Model ◽  
Constant Pressure ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Well Bore ◽  
Analysis Software

This paper studies the seepage flow mathematical model of three-area composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary), in which influences of well-bore storage and skin factor are not taken into consideration. On the basic of theory of similar structure of solution of boundary value problem of differential equation, this paper obtain the solution of the seepage flow model of three-area composite reservoir. The study is not only conducive to further analyze the inherent law of the solution and solve corresponding application problems, but also easy to compile corresponding analysis software.

A New Method and Applications of the Boundary Value Problem of Differential Equation

2014 ◽  
Vol 937 ◽  
pp. 695-699
Author(s):  
Hong E Li ◽  
Xiao Xu Dong ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Cong Yin Fan
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Kernel Function ◽  
Boundary Value ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Petroleum Engineering ◽  
Constructive Method ◽  
Linear Homogeneous Differential Equation ◽  
The Mathematical Model

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

scholarly journals On a nonlinear boundary value problem involving a small parameter

1962 ◽  
Vol 2 (4) ◽  
pp. 425-439 ◽  
Author(s):  
A. Erdéyi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Boundary Value Problem ◽  
Image Position

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

Similar Constructing Method for Solving the Model of the Plane Radial Flow of Dual Permeability Reservoir

2013 ◽  
Vol 419 ◽  
pp. 43-50 ◽  
Author(s):  
Cong Yin Fan ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Ming Hu ◽  
Hui Chun Li
Keyword(s):  
Boundary Value Problem ◽  
Radial Flow ◽  
Boundary Value ◽  
Outer Boundary ◽  
Structural Formula ◽  
Constructive Method ◽  
Well Bore ◽  
Bottom Hole ◽  
Constant Rate ◽  
Seepage Model

A seepage model of producing at a constant rate, based on the three outer boundary (infinite, closed, constant value) conditions and regardless of the well-bore storage and skin effects, is established for the problem of the plane radial flow of dual permeability reservoirs. Firstly, we get dimensionless model by introducing the dimensionless variables. Second, we obtain a boundary value problem of ordinary differential equation in the Laplace space by using the Laplace transformation. Finally, we prove that the solution to the boundary value problem has similar structural formula. Therefore a new method for solving such seepage model is obtainedSimilar Constructive Method (shortened as SCM). And then, according to the modified numerical inversion formula of Stehfest, we draw the curves of bottom-hole under the three kinds of outer boundary conditions by using MATLAB.

scholarly journals Survey of studies on degenerate boundary conditions and finite spectrum

2019 ◽  
Vol 14 (3) ◽  
pp. 184-201
Author(s):  
A.M. Akhtyamov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Complex Plane ◽  
Boundary Value ◽  
Finite Spectrum ◽  
Diffusion Operator ◽  
Odd Order ◽  
Entire Complex

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.

scholarly journals Mathematical model of heat transfer in roll caliber

2019 ◽  
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
Temperature Field ◽  
Temperature Distribution ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Rolling Mill ◽  
Thermal Process ◽  
Stationary Heat

A physical model of the thermal process in the roll caliber during the rolling of the tape on a two-roll rolling mill was constructed. A mathematical model of the temperature field of a rolling hollow roll of a rolling state of a cylindrical shape rotating about its axis with constant angular velocity is proposed. The mathematical model takes into account different conditions of heat exchange of the inner and outer surfaces of the roll with the belt and its surrounding environment. The temperature field of a hollow roll of a rolling mill is considered as an initial boundary-value problem for a homogeneous non-stationary heat equation with inhomogeneous, nonlinear boundary conditions, which also depend on the angle of rotation of the roll around its axis. The equation describes the temperature field of the rolls during uncontrolled heat transfer during rolling. It significantly depends on the time and number of revolutions around its axis. With a large number of revolutions of the roll around its axis, a quasi-stationary temperature distribution occurs. Therefore, the simplified problem of determining a quasistationary temperature field, which is associated with a thermal process that is time-independent, is considered further in the work. In this case, the temperature field is described using the boundary value problem in a ring for a homogeneous stationary heat equation with inhomogeneous boundary conditions and heat transfer conditions outside the ring, which lie from the angular coordinate. After the averaging operation, the solution of this problem is reduced to solving the equivalent integral equation of Hammerstein type with a kernel in the form of the Green's function. The Mathcad computer mathematical system builds the temperature distribution of the roll surface. An algorithm for solving a inhomogeneous problem was developed and the temperature distribution of the roll was constructed.

Equilibrium of a Radially Symmetric Inhomogeneous Plate Under Tension and Load

1997 ◽  
Vol 64 (1) ◽  
pp. 233-236
Author(s):  
A. R. Pineda ◽  
R. G. Root
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Normal Load ◽  
Physical Significance ◽  
Dirichlet Boundary Value Problem ◽  
Order Ordinary Differential Equation ◽  
Injection Molded

This paper studies a mathematical model for small deflections of a radially symmetric strongly inhomogeneous elastic plate under a normal load and tension in the center plane. The type of inhomogeneity studied occurs in injection molded plastics. The model consists of a Dirichlet boundary value problem for a linear fourth-order ordinary differential equation. A general expression is derived for physically reasonable solutions under relatively broad restrictions on the inhomogeneity and load. The solutions are analyzed and interpreted in terms of their physical significance.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

scholarly journals Weak Solutions for ap-Laplacian Antiperiodic Boundary Value Problem with Impulsive Effects

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Keyu Zhang ◽  
Jiafa Xu ◽  
Wei Dong
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Critical Point ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Impulsive Differential Equation ◽  
Point Theory ◽  
Existence Of Weak Solutions

By virtue of variational method and critical point theory, we will investigate the existence of weak solutions for ap-Laplacian impulsive differential equation with antiperiodic boundary conditions.

Ambrosetti–Prodi Periodic Problem Under Local Coercivity Conditions

2018 ◽  
Vol 18 (1) ◽  
pp. 169-182 ◽  
Author(s):  
Elisa Sovrano ◽  
Fabio Zanolin
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Problem ◽  
Neumann Boundary ◽  
Continuous Functions ◽  
Periodic Boundary Value Problem ◽  
Multiplicity Result ◽  
Coercivity Conditions

AbstractIn this paper we focus on the periodic boundary value problem associated with the Liénard differential equation{x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s}, wheresis a real parameter,fandgare continuous functions andgisT-periodic in the variablet. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions ong. Analogous results are also obtained for Neumann boundary conditions.

Similar Constructive Method of Solution for the Nonlinear Percolation Model in Composite Reservoir

2014 ◽  
Vol 670-671 ◽  
pp. 678-682
Author(s):  
Feng Jiu Zhang ◽  
Xi Tao Bao ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Xiao Xu Dong
Keyword(s):  
Constant Pressure ◽  
Outer Boundary ◽  
Percolation Model ◽  
Constructive Method ◽  
Well Test ◽  
Analysis Software ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Method Of Solution ◽  
Quadratic Gradient

This paper presents a percolation model for the composite reservoir, in which quadratic-gradient effect, well-bore storage, effective radius and three types of outer boundary conditions: constant pressure boundary, closed boundary and infinity boundary are considered. With Laplace transformation, the percolation model was linearized by the substitution of variables and obtained a boundary value problem of the composite modified zero-order Bessel equation. Using the Similar Constructive Method this method, we can gain the distributions of dimensionless reservoir pressure for the composite reservoirs in Laplace space. The similar structures of the solutions are convenient for analyzing the influence of reservoir parameters on pressure and providing significant convenience to the programming of well-test analysis software.

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