Keywords: composite reservoir, boundary value problem, nonlinear spherical seepage, similar structure of the solution, similar kernel function
American Journal of Applied Mathematics and Statistics, 2015 3 (1),
pp 1722.
DOI: 10.12691/ajams314
Received January 12, 2015; Revised January 28, 2015; Accepted February 02, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
In recent years, mathematician have perfected the seepage theory. However, the problem of nonlinear seepage theory and application research are still at the primary stage. Error caused by the quadratic gradient effect is very big, which should not be ignored. while the traditional model of reservoir was established which not taking the quadratic gradient into consideration. This can cause larger error as a model of linear seepage. Therefore, solving the nonlinear seepage problem has important theoretical significance and practical value.
Ref 1 established the homogeneous reservoir seepage model, obtained the similar structure of solution based on the Theory of Similar Structure of Solution. Ref 2 put forward the similar structure of the solution to radialspherical seepage problem considering quadratic gradient effect. Ref 3 discovered the similar constructive method of solutions of nonlinear seepage model for homogeneous reservoir. Ref 4 studied the similar constructive method for solving a nonlinearly spherical percolation model. Ref 5 discussed the similar constructive method of solution for the model of nonlinear spherical seepage in composite reservoir under the three kinds of outer boundary conditions (infinite, constant pressure and closed). Ref 6 studied how to solve the mathematical model of threearea composite reservoir.
Some of these researches studied the similar constructive method of solution for the model of nonlinear spherical seepage in composite reservoir, others discussed how to solve the mathematical model of threearea composite reservoir. Therefore, this paper establishes the nonlinear spherical seepage model of threearea composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary), not considering wellbore storage and skin factor. Based on the theory of similar structure of solution, the solution of the nonlinear spherical seepage model of threearea composite reservoir can be obtained.
2. Establishing the Nonlinear Spherical Seepage Model of ThreeArea Composite Reservoir
Here we study the model which not considering wellbore storage and skin factor. The main assumptions are as follows:
1) Singlephase micro compressible fluid obeys Darcy’s law;
2) Reservoir types is horizontal thickness, isotropic composite reservoir;
3) The well product is Isothermal flow instability;
4) At the interface of two seepage zone does not present the additional pressure drop;
5) Considering the effect of micro pressure gradient, neglecting the capillary singlephase horizontal flow without gravity effect;
6) Spherical seepage flow to a wall of the center reservoir.
The nonlinear spherical seepage equations of threearea composite reservoir which not considering wellbore storage and skin factor are as follows:
Inner area:
 (1) 
Middle area:
 (2) 
Outer area:
 (3) 
Initial condition:
 (4) 
Inner boundary condition:
 (5) 
Convergence condition:
 (6) 
Outer boundary condition:
1) When the outer boundary condition is infinite:
 (7) 
2) When the outer boundary condition is constant pressure:
 (8) 
3) When the outer boundary condition is closed:
 (9) 
3. Dimensionless the Seepage Mathematical Model
The dimensionless mathematical model of nonlinear spherical seepage of threearea composite reservoir is obtained:
 (10) 
Where dimensionless variables are introduced as follows:
4. Linearization of Nonlinear Seepage Mathematical Model
In order to make the seepage equation linearly, variable is replaced by the following:
 (11) 
 (12) 
 (13) 
5. Solution in Laplace Space and Similar Structure of the Solution
Take the Laplace transform on the dimensionless time , i.e.
 (14) 
Then the boundary value problem of the ODE in Laplace space with parameter (whereis Laplace space variable) is obtained:
 (15) 
6. Solving the Nonlinear Spherical Seepage Model of ThreeArea Composite Reservoir
On the basis of the Similar Constructive Method, the Laplace space analytic solutions of each area of the boundary value problem (15) are obtained, respectively:
 (16) 
 (17) 
 (18) 
Where are called the similar kernel function of outer area. Its expression is delimited as below:
1) When the outer boundary condition is infinite:
 (19) 
2) when the outer boundary condition is constant:
 (20) 
3) when the outer boundary condition is closed:
 (21) 
Where are called the similar kernel function of middle area. Its expression is delimited as below:
1) When the outer boundary condition is infinite:
 (22) 
2) when the outer boundary condition is constant:
 (23) 
3) when the outer boundary condition is closed:
 (24) 
Where are called the similar kernel function of inner area. Its expression is delimited as below:
1) when the outer boundary condition is infinite:
 (25) 
2) when the outer boundary condition is constant:
 (26) 
3) when the outer boundary condition is closed:
 (27) 
Where,
 (28) 
And , are modified Bessel functions of order ^{[7]}.
Then let , we can obtain the solution of dimensionless bottomhole pressure in Laplace space:
 (29) 
By using Laplace inversion formula, we can obtain the solution of dimensionless pressure in the real space as follows:
Similarity, we can obtain the solution of dimensionless bottomhole pressure in the real space as follows:
7. Conclusions
Base on the theory of similar structure of the solution of the boundary value problem of differential equation, this paper gets the unified expression of reservoir pressure and bottom hole pressure solution of Laplace space under three different outer boundary conditions, and kernel function is only different in the expression for the different outer boundary conditions.
Many the nonlinear spherical seepage models of threearea composite reservoir can be adapted to the boundary value problem of threearea composite modified Bessel equation. So this paper can easily get the solution of the nonlinear spherical seepage model of threearea composite reservoir, after a simple algebra calculation.
In general, we can obtain the solution in the real space by using Laplace inversion formula, which is convenient to compile corresponding test software.
Nomenclature
—Formation volume factor (dimension); —Permeability (); —Reservoir pressure (); —Wellbore pressure (); —Production rate or injection rate (); —Porosity (dimensionless); —The outer boundary radius (); —Wellbore radius (); —Viscosity (); —Initial reservoir pressure (); —Laplace space variable;—Time (); —Laplace domain; —Dimensionless; —Wellbore parameter; —total compressibility (); —fluid compressibility ();
Acknowledgment
The research was supported by the Nature Science Key Projects of the Sichuan Education Bureau of China(Grant No.12ZA164).
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