This study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractal reservoir which can avoid the complicated mathematical deduction. The models are presented in three kinds of outer boundary conditions (infinite, constant pressure, and closed). The influence of wellbore storage effect, skin factor, and variable flow rate production is also involved in the inner boundary conditions. The analytical solutions are constructed in the Laplace space and presented in a pattern with one continued fraction—the similar structure of solution. The pattern can bring convenience to well test analysis programming. The mathematical beauty of fractal is that the infinite complexity is formed with relatively simple equations. So the relation of reservoir parameters (wellbore storage effect, the skin factor, fractal dimension, and conductivity index), the formation pressure, and the wellbore pressure can be learnt easily. Type curves of the wellbore pressure and pressure derivative are plotted and analyzed in real domain using the Stehfest numerical invention algorithm. The SCMS and type curves can interpret intuitively transient pressure response of fractal spherical flow reservoir. The results obtained in this study have both theoretical and practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS.
The mechanics of oil and gas seepage is a discipline which researches the law and state of fluid flow in porous media. The practical development of oil and gas reservoirs shows that reservoir distribution and its space structure are awfully complicated. In 1982, Mandelbrot and Blumen first proposed the fractal geometry theory which used self-similar to characterize the complexity of things [
Different versions of cylindrical flow model have been studied on the assumption that wells were completely opened and fully penetrated the productive formation. However, the assumption may seem too restrictive for practical application. Reservoirs have vertical permeability and the length of injection or extraction region was small compared to thick formations; spherical flow model can provide a good approximation in practice [
In the above proposed studies, the solution processes of their mathematical models are very complicated. Based on some study in a boundary value problem of the second-order linear homogeneous differential equation, Li and Liao recently introduced a new idea, similar construction method. The similar structure theory and its application were developed maturely [
This paper presents a mathematical model for the analysis of the pressure transient response of fluid spherical flow in fractal reservoir, which considers wellbore storage and skin effect in inner boundary conditions. According to a mathematical method, called SCMS, the expressions of the dimensionless formation pressure and the dimensionless wellbore pressure are constructed in the Laplace space, which avoid complicated calculations. Type curves of the wellbore pressure and pressure derivative responses are plotted and analyzed in real domain using the Stehfest numerical invention algorithm [
Nonpenetrating wells that occur in a thick formation can be treated as spherical systems, as shown in Figure
Schematic for spherical flow in a reservoir.
The dimensionless mathematical model of the reservoir is made up of four parts, such as continuity equation, initial condition, inner boundary condition, and outer boundary condition. Appendix
The continuity equation of fluid spherical flow in a fractal reservoir is
Initial condition:
Inner boundary condition:
Three kinds of outer boundary conditions are the following cases.
Infinite outer boundary condition:
Constant pressure outer boundary condition:
Closed outer boundary condition:
The dimensionless mathematical model (
The SCMS is based on a boundary value problem of a second-order linear differential equation and is firstly proposed by Sheng et al. [
Solve the linearly independent solutions of
Construct a binary function
Construct similar kernel functions
Construct the similar structure of solution with the similar kernel function equation
Equation
The analytical solution of the mathematical model of fluid flow in fractal reservoir was rewritten as continued fraction, like real numbers, the structure of continued fraction is very beautiful. It is very convenient to study the influence of both wellbore storage effect and skin factor using equation
Figure
The type curves of the fractal reservoir with spherical flow.
The fluid flow is principally affected by wellbore storage effect. In this flow period, both the dimensionless pressure change and its rate are aligned in a unite slope trend.
The period represents the transition to early time spherical flow (Stage
During this period, either the dimensionless pressure change curve or the derivative curves show a straight line with a slope of zero. The greater the radius of the outer boundary is, the longer the duration of the fluid flow period will be.
When the outer boundary is infinite, the fluid flow still holds the same response. When pressure is constant in the outer boundary, the dimensionless pressure curve nearly remains level, but its derivative curve decreases quickly. For a closed outer boundary condition, both the dimensionless pressure and derivative curves are cocked up and overlapped. And the values of their slope curve are about 1.
As Figure
Figure
Effect of
Figure
Effect of
The effect of the fractal dimension,
Effect of
Figure
Effect of
Based on the reasoning mentioned above, the following conclusions can be drawn. The mathematical models of fluid spherical flow in fractal reservoirs with three kinds of outer boundary conditions (infinite, constant pressure, and closed) were presented, which comprehensively took into consideration the effect of wellbore storage effect, the skin factor, fractal dimension, and conductivity index. The dimensionless formation pressure in the Laplace space was constructed by SCMS and was written as a continued fraction. The analytical solution presented in this paper has afforded theoretical basis to understand the effect of reservoir parameters and practical significance to make the well test analysis software. The dimensionless pressure and derivative type curves were plotted and their characteristics were analyzed. For three kinds of outer boundary conditions, the pressure transient responses make no difference when the fluid does not reach to the outer boundary condition. The time and the high of the hump of the dimensionless wellbore pressure derivative have connection with the coefficient of wellbore storage
In Acuna et al’.s research [
The fluid flow from reservoir into the wellbore follows Darcy’s law. The fluid flow in the reservoir is isothermal and single-phase with constant fluid viscosity. Based on the principle of mass and energy conservation, the mathematical equations of single fluid phase in fractal reservoir can be written in the form as
Initial condition:
Inner boundary conditions:
The three kinds of outer boundary condition are as follows.
The solution in real space can be obtained by the method of the Stehfest numerical invention algorithm [
Here
Formation volume factor, RB/STB
Coefficient of wellbore storage, m3/Pa
Total compressibility, MPa−1
Euclid dimension
Fractal dimension
Permeability, mD
Reservoir pressure, MPa
Production rate or injection rate, m3
Radial distance of the outer boundary, m
Radial distance in spherical coordinate, m
Skin factor
Time, h
Laplace transform variable.
Conductivity index
Viscosity, mPa·s
Porosity.
Dimensionless
Initial
Wellbore parameter.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous referee for his/her helpful suggestions and comments. The research is supported by the National Natural Science Foundation of China (no. 51274169), the National Basic Research Program of China (973 Program) (no. 2013CB228004), the New Century Excellent Talents in University (no. NCET-11-1062), and the Scientific Research Fund of Sichuan Provincial Education Department of China (no. 12ZA164).