scholarly journals Similar Construction Method of Solution for Solving the Mathematical Model of Fractal Reservoir with Spherical Flow

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Cui-Cui Sheng ◽  
Jin-Zhou Zhao ◽  
Yong-Ming Li ◽  
Shun-Chu Li ◽  
Hu Jia
Keyword(s):  
Boundary Conditions ◽  
Boundary Value ◽  
Outer Boundary ◽  
Petroleum Engineering ◽  
Constructive Method ◽  
Porous Media Model ◽  
Method Of Solution ◽  
Complex Boundary ◽  
Linear Differential ◽  
Set Up

On the basis of similar structure of solution for a second-order linear differential equation's boundary value problem, and our analysis of the relationship between this similar structure and its kernel function, the differential equation, and the boundary conditions, we propose a new simple solution—similar constructive method of solution (SCMS)—and sum up its detailed steps. We set up a porous media model under three kinds of outer boundary conditions (infinite, constant pressure, and closed), in which the influences of fractal dimension, spherical flow, skin effect, and storage are taken into consideration. And then we use SCMS to solve it. The SCMS is a straightforward method for differential equation's boundary value problems with complex boundary conditions, especially for solving the reservoir models in petroleum engineering.

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.

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

2014 ◽  
Vol 700 ◽  
pp. 597-601
Author(s):  
Feng Jiu Zhang ◽  
Xu Xia Xiao ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Qiang Wang
Keyword(s):  
Flow Model ◽  
Boundary Value ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Analytic Solutions ◽  
Flow Equation ◽  
Constructive Method ◽  
Gradient Term ◽  
Transform Method ◽  
Method Of Solution

The nonlinear spherical seepage flow model has been established for the composite reservoir model. The nonlinear spherical seepage flow model considers the well produce at a constant rate, and the quadratic gradient term under three outer boundary conditions (closed, constant pressure and infinite). Firstly, through variable substitutions, the seepage flow equation is linearized; then the model is transformed into the boundary value problem of an ordinary differential equation by employing the Laplace transform method. It has been confirmed that the Laplace space analytic solutions of such boundary value problems has a formula under different external boundaries, using the Similar Constructive Method(it is a simple and effective new idea for solving this class seepage flow model, complicated calculus calculation is avoided). The prospect of this new method is promising for understanding and studying the inherent laws of fluids flow.

Adjoint Interior-Point Boundary Conditions for Linear Differential Operators

1977 ◽  
Vol 20 (4) ◽  
pp. 447-450 ◽  
Author(s):  
Robert Neff Bryan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Interior Point ◽  
Differential Operators ◽  
Boundary Value ◽  
Point Boundary ◽  
Linear Differential Operators ◽  
Linear Differential

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.

On Simplifying Complex Equations by Applying Block Elements

2020 ◽  
pp. 8-12
Author(s):  
V.A. Babeshko ◽  
O.V. Evdokimova ◽  
O.M. Babeshko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Block Element ◽  
Helmholtz Equations ◽  
Complex Boundary ◽  
Block Element Method ◽  
Element Method

There are several approaches aimed at simplifying complex partial differential equations or their systems involved in the formulation of boundary value problems by introducing simpler, but in a larger number of differential equations. Their solutions allow us to describe solutions to complex boundary value problems. However, to implement this approach, it is necessary to construct solutions of simplified boundary value problems for arbitrary boundary conditions in solvability spaces boundary value problem. In some cases, this can be done using the block element method. The block element method, which has a topological basis, reveals both global and local properties of solutions to boundary value problems for partial differential equations. At the same time, it can be used to study and solve more complex boundary value problems by applying relations that describe certain equations of the continuum by means of relatively simple equations, for example, Helmholtz. To do this, we need to construct solutions of the Helmholtz equations that satisfy boundary conditions that contain completely arbitrary values, rather than partial values, set at the boundary of functions. In relation to the Helmholtz equations, this is achieved using the block element method. Examples of constructing solutions to boundary value problems for Helmholtz equation for Dirichlet and Neumann problems and a comparative analysis of solutions are given in this article.

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 Study of the Boundary Value Problems for Nonlinear Wave Equations on Domains with a Complex Structure of the Boundary and Prehistory

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1888
Author(s):  
Vasyl Marynets ◽  
Kateryna Marynets ◽  
Oksana Kohutych
Keyword(s):  
Wave Equations ◽  
Boundary Value ◽  
Complex Structure ◽  
Nonlinear Partial Differential Equations ◽  
Hyperbolic Type ◽  
Nonlinear Wave Equations ◽  
Constructive Method ◽  
Boundary Constraints ◽  
Complex Boundary ◽  
The Given

We study a boundary value problem for nonlinear partial differential equations of the hyperbolic type on the plain in a domain with a complex boundary. To find the missing data for the given boundary constraints, we solve a supplementary nonlinear problem. For the approximation of solutions, one constructive method is built.

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.

scholarly journals Spectral theory of some non-selfadjoint linear differential operators

2013 ◽  
Vol 469 (2154) ◽  
pp. 20130019 ◽  
Author(s):  
B. Pelloni ◽  
D. A. Smith
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Differential Operators ◽  
Boundary Value ◽  
Bounded Interval ◽  
Linear Differential Operators ◽  
Linear Differential

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂ t q ±i Sq =0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

scholarly journals Numerical solution of a boundary value problem for a loaded parabolic equation with nonlocal boundary conditions

2020 ◽  
pp. 15-28
Author(s):  
В.М. Абдуллаев
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Test Problems ◽  
Method Of Solution ◽  
Numerical Method Of Solution

В работе с использованием метода прямых исследуется численное решение краевой задачи относительно нагруженного параболического уравнения с нелокальными краевыми условиями. Получены расчетные формулы и приводится алгоритм для решения задачи. Приведены результаты численного решения двух тестовых задач, иллюстрирующие эффективность предложенного подхода In the work, we propose a numerical method of solution to the boundary-value problem with respect to the loaded parabolic equation with nonlocal boundary conditions. We have obtained formulas and derived an algorithm for the solution of the problem. We provide the results of numerical solution to two test problems, which illustrates the efficiency of the approach proposed.

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