The Similar Structure Method for Solving the Model of Fractal Dual-Porosity Reservoir

Abstract

This paper proposes a similar structure method (SSM) to solve the boundary value problem of the extended modified Bessel equation. The method could efficiently solve a second-order linear homogeneous differential equation’s boundary value problem and obtain its solutions’ similar structure. A mathematics model is set up on the dual-porosity media, in which the influence of fractal dimension, spherical flow, wellbore storage, and skin factor is taken into cosideration. Researches in the model found that it was a special type of the extended modified Bessel equation in Laplace space. Then, the formation pressure and wellbore pressure under three types of outer boundaries (infinite, constant pressure, and closed) are obtained via SSM in Laplace space. Combining SSM with the Stehfest algorithm, we propose the similar structure method algorithm (SSMA) which can be used to calculate wellbore pressure and pressure derivative of reservoir seepage models clearly. Type curves of fractal dual-porosity spherical flow are plotted by SSMA. The presented algorithm promotes the development of well test analysis software.

1. Introduction

Petroleum engineering and percolation mechanics have scale invariances in non-Euclidean heterogeneous porous media. References [1, 2] built mathematical models of reservoirs and described porous media by homogeneous or quasi homogeneous distribution. The reservoirs distribution and pore structure are complexities. Reference [3] demonstrated that the fractal characteristics of porous media affected fluid flow by studying many geological properties. Reference [4] discussed the geometric properties and spatial correlation structure of fractal distributions and reviewed methods for measuring the fractal character of field data and synthesizing fields with a similar correlation structure. References [5, 6] proposed a seepage model of Newtonian fluid in fractal reservoir and analyzed its well test curve. References [7, 8] showed that fractal reservoir seepage model can be explained much more effectively than traditional one and obtained a conclusion very consistent with well test curve. References [9, 10] discussed that the fractal theory was powerful in analysis of the fluid flow properties with complex and microscopic stochastic microstructures in porous media.

Model of flow through dual-porosity formations was proposed by [11]. Barenblatt et al. considered the medium of the blocks and that of the fractures as overlapping continua over entire flow domain. Darcy’s law and conservation of mass equation are written separately for the fluid in each medium. A dual-porosity media has fluid exchange between the two constitutive media. Many natural reservoirs fit with this scheme, so the fluid flow model of dual-porosity is particularly popular in mining engineering and petroleum engineering. The previous studies of fractal dual-porosity seepage flow model are mostly considered on radial flow. Reference [12] demonstrated that spherical flow analysis can model the well test interpretation and provide data insight for reservoir and production study more suitably.

Reference [13] first presented the concept of solutions’ similar structure for a second-order linear homogeneous differential equation and the partial differential equation which can be transformed into ordinary differential equation system via variable substitution or integral transform. Expression of their solutions can be simplified as a unified continued fraction only with different kernel functions. The solutions’ similar structure theory of differential equation has been widely applied in engineering problems, in particular, solving of reservoir mathematical models. References [14, 15] built mathematical models for fluid flow in porous media and got their solutions’ similar structure. However, most of research got their similar solutions and kernel functions via a complicated process of solutions. So, we sum up the detailed construction steps of SSM for the extended modified Bessel equation’s value problem. It is shown that the solutions of the extended modified Bessel equation can be simply constructed by the coefficients of boundary without too much calculating. A spherical flow of fractal model is designed in dual-porosity reservoir. The similar structure of formation pressure and wellbore pressure in Laplace space is conveniently calculated by SSM. In Section 3, combining SSM with Stehfest [16] numerical inversion we compile a similar structure method algorithm (SSMA). Type curves of the fractal dual-porosity spherical flow model are plotted with SSMA in real time domain. In Section 4, how the fractal dimension and wellbore storage affect formation pressure and its derivative of dual-porosity reservoir are discussed on infinite boundary. In a word, the SSM is a straightforward method for solving the differential equation’s boundary value problems with complex boundary condition, especially in petroleum engineering seepage models. SSMA has the significance meaning for making well test analysis.

2. The SSM for Solving the Model of the Fractal Dual-Porosity Reservoir

2.1. The Similar Structure Method (SSM)

To solve reservoir model, we first introduce SSM for the extended modified Bessel equation’s boundary value problem, which frequently occurred in reservoir engineering: 𝑥²𝑦^′′+𝐴𝑥𝑦^′+𝐵−𝐶𝑥^𝑞(),()𝑦𝑦=0𝑥>0𝐸𝑦+1+𝐸𝐹^′_𝑥=𝑎=𝐷,𝐺𝑦+𝐻𝑦^′_𝑥=𝑏=0, (1)(2)(3) where 𝐴, 𝐵, 𝐶, 𝐷, 𝐹, 𝐺, 𝐻, 𝑎, and 𝑏 are all known as real numbers and satisfy (1−𝐴)²>4𝐵,𝐶>0,𝐷≠0,𝐺²+𝐻²≠0,0<𝑎<𝑏,𝑞≠0. (4)(5)

The SSM steps are presented as follows.

Step 1. Find two linearly independent solutions of the extended modified Bessel equation (1) (see Appendix A for details). They are 𝑥^𝛼𝐼_𝑣𝑘𝑥^𝛽,𝑥^𝛼𝐾_𝑣𝑘𝑥^𝛽, (6) where 𝛼=(1−𝐴)/2, 𝛽=𝑞/2, 𝑘=√2𝐶/𝑞, and 𝑣=(1−𝐴)²−4𝐵/𝑞.

Step 2. A binary function 𝜑(𝑥,𝜉) is defined by the two linearly independent solutions 𝑥^𝛼𝐼_𝑣(𝑘𝑥^𝛽), 𝑥^𝛼𝐾_𝑣(𝑘𝑥^𝛽): 𝜑(𝑥,𝜉)=𝜑_0.0(𝑥,𝜉)=(𝑥𝜉)^𝛼𝐾_𝑣𝑘𝑥^𝛽𝐼_𝑣𝑘𝜉^𝛽−𝐼_𝑣𝑘𝑥^𝛽𝐾_𝑣𝑘𝜉^𝛽. (7)
If we define Ψ_𝑚⋅𝑛𝑥,𝑦,𝑡=𝐾_𝑚(𝑥𝑡)𝐼_𝑛+()𝑦𝑡−1^{𝑚−𝑛+1}𝐼_𝑚(𝑥𝑡)𝐾_𝑛𝑦𝑡, (8) where 𝑚, 𝑛 are real numbers, the binary function 𝜑(𝑥,𝜉) can be rewritten as 𝜑(𝑥,𝜉)=(𝑥𝜉)^𝛼Ψ_𝑣⋅𝑣𝑥^𝛽,𝜉^𝛽,𝑘. (9) Then, calculate the partial derivative of 𝜑(𝑥,𝜉) for 𝑥, 𝜉, respectively: 𝜑_1.0()=𝜕𝑥,𝜉𝜑𝜕𝑥^0.0(𝑥,𝜉)=𝑥^𝛼−1𝜉^𝛼𝛼+𝛽𝑣Ψ_𝑣⋅𝑣𝑥^𝛽,𝜉^𝛽,𝑘−𝑘𝛽𝑥^𝛽Ψ_{𝑣+1⋅𝑣}𝑥^𝛽,𝜉^𝛽,𝜑,𝑘_0.1()=𝜕𝑥,𝜉𝜑𝜕𝜉^0.0(𝑥,𝜉)=𝑥^𝛼𝜉^𝛼−1𝛼+𝛽𝑣Ψ_𝑣⋅𝑣𝑥^𝛽,𝜉^𝛽,𝑘+𝑘𝛽𝜉^𝛽Ψ_{𝑣⋅𝑣+1}𝑥^𝛽,𝜉^𝛽,𝜑,𝑘_1.1()=𝜕𝑥,𝜉𝜑𝜕𝑥^0.1()=𝜕𝑥,𝜉𝜑𝜕𝜉^1.0()=()𝑥,𝜉𝑥𝜉^𝛼−1𝛼+𝛽𝑣²Ψ_𝑣⋅𝑣𝑥^𝛽,𝜉^𝛽𝑥,𝑘−𝑘𝛽𝛼+𝛽𝑣^𝛽Ψ_{𝑣+1⋅𝑣}𝑥^𝛽,𝜉^𝛽𝜉,𝑘+𝑘𝛽𝛼+𝛽𝑣^𝛽Ψ_{𝑣⋅𝑣+1}𝑥^𝛽,𝜉^𝛽,𝑘−𝑘²𝛽²(𝑥𝜉)^𝛽Ψ_{𝑣+1⋅𝑣+1}𝑥^𝛽,𝜉^𝛽,𝑘. (10)(11)(12)

Step 3. A similar kernel function is constructed by using the coefficients of the right boundary (3) and Step 2 definition: Φ(𝑥)=𝐺𝜑_0.0(𝑥,𝑏)+𝐻𝜑_0.1(𝑥,𝑏)𝐺𝜑_1.0(𝑎,𝑏)+𝐻𝜑_1.1().𝑎,𝑏 (13) One can easily get Φ(𝑎).

Step 4. The similar structure of solutions is designed by the coefficients of the left boundary (2) and 1𝑦=𝐷⋅(((𝑎⋅1𝐸+1/𝐹+Φ)))(𝑎)(𝑥).𝐹+Φ⋅Φ (14)
Consequently, (15) is the similar solutions of the extended modified Bessel equation’s boundary value problem (1)–(3).

2.2. The SSM for Solving the Model of the Fractal Dual-Porosity Reservoir

The dimensionless mathematical model of a fractal dual-porosity reservoir is presented in Appendix B. In order to solve the model, we assume that the fractal dimension of fissure and matrix block are identical; that is, 𝑑_𝑓1=𝑑_𝑓2. Ideally, all the fluid can be seeped into fissure media from matrix block. We give the hypothesis that fractal exponential in matrix block approximates at zero (𝜃=0). Considering spherical symmetric flow, we can write the model in the Laplace space as follows.

Fundamental differential equations: 𝑑²𝑃_1𝐷𝑑𝑟²_𝐷+𝛽₀𝑟_𝐷𝑑𝑃_1𝐷𝑑𝑟_𝐷+𝜆𝑟^𝜃1_𝐷𝑒^2𝑠𝑃_2𝐷−𝑃_1𝐷()=0,1−𝜔𝑃_2𝐷+𝜆𝐶_𝐷𝑃_2𝐷−𝑃_1𝐷=0. (15)(16)

Initial boundary conditions: 𝑃_1𝐷𝑟_𝐷,0=𝑃_2𝐷𝑟_𝐷,0=0. (17)

Interboundary conditions: −𝑧𝑃_𝑤𝐷𝑟_𝐷+𝑑,𝑧𝑃_1𝐷𝑟_𝐷,𝑧𝑑𝑟_{𝐷𝑟𝐷=1}=−𝑞_𝐷(𝑧). (18)

The three types of outer boundary conditions: 𝑃_𝑖𝐷(∞,𝑧)=0,𝑃_𝑖𝐷𝑅_𝐷𝑑,𝑧=0or𝑃_𝑖𝐷𝑑𝑟_𝐷𝑅_𝐷,𝑧=0(𝑖=1,2). (19)

They denote that the outer boundaries of circular reservoir are infinite, constant pressure, and closed, respectively. All the notations are explained in the nomenclature section.

To simplify (17) yields 𝑃_2𝐷=𝜆𝐶_𝐷(1−𝜔)𝑧+𝜆𝐶_𝐷𝑃_1𝐷. (20)

𝑃_2𝐷 can be expressed by 𝑃_1𝐷 linearly. Next in the paper, we only consider 𝑃_1𝐷. Taking (21) into (16) yields 𝑑²𝑃_1𝐷𝑑𝑟²_𝐷+𝛽₀𝑟_𝐷𝑑𝑃_1𝐷𝑑𝑟_𝐷−𝑧^∗𝑟^𝜃1𝑃_1𝐷=0, (21) where 𝑧^∗(𝑧)=(𝑧/𝐶_𝐷𝑒^2𝑠)[(𝑧𝜔(1−𝜔)+𝜆𝐶_𝐷)/(𝑧(1−𝜔)+𝜆𝐶_𝐷)].

Equation (23) is a special form of the extended modified Bessel equation, so we can use SSM to construct the solutions’ similar structure of the dimensionless mathematical mode of fractal dual-porosity reservoir flow (16)–(20).

Equations (19), (20), and (23) comparing with (1)–(3) result in 𝑦=𝑃_1𝐷,𝑥=𝑟_𝐷,𝐴=𝛽₀,𝐵=0,𝐶=𝑧^∗𝑟^𝜃,𝐸=𝑧,𝐹=0,𝐷=−𝑞_𝐷(𝑧),𝑎=1. (22)𝐺=1, 𝐻=0, 𝑏→∞ or 𝐺=1, 𝐻=0, 𝑏=𝑅_𝐷 or 𝐺=0, 𝐻=1, 𝑏=𝑅_𝐷 representing three types of outer boundaries, respectively. So we can construct the similar structure of formation pressure in Laplace space as in the following expression: 𝑃_1𝐷𝑟_𝐷,𝑧=𝑞_𝐷(𝑧)1((1𝑧+1/Φ1,𝑧))Φ()Φ𝑟1,𝑧^𝐷,𝑧. (23)Φ(𝑟_𝐷,𝑧) is the similar kernel function.

For 𝑃_1𝐷(∞,𝑧)=0, using the properties of modified Bessel functions lim_𝑥→∞𝐼_𝑣(𝑥)=∞, lim_𝑥→∞𝐾_𝑣(𝑥)=0 with (14) obtaining the similar kernel function as follows: Φ𝑟_𝐷,𝑧=𝑟^(1−𝛽0_𝐷^)/2𝐾_𝑣2√𝑧^∗𝜃₁𝑟+2^{(𝜃1𝐷/2)+1}×√𝑧^∗𝐾_𝑣+12√𝑧^∗𝜃₁−+21−𝛽₀𝐾_𝑣2√𝑧^∗𝜃₁+2⁻¹. (24)

For 𝑃_1𝐷(𝑅_𝐷,𝑧)=0, the similar kernel function is Φ𝑟_𝐷,𝑧=𝑟^(1−𝛽0_𝐷^)/2×Ψ_𝑣⋅𝑣𝑟^(𝜃1_𝐷^/2)+1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗𝜃₁×+21−𝛽₀Ψ_𝑣⋅𝑣1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗𝜃₁−√+2𝑧^∗Ψ_{𝑣+1⋅𝑣}1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗𝜃₁+2⁻¹. (25)

For 𝑑𝑃_1𝐷/𝑑𝑟_𝐷|_{𝑟𝐷→𝑅𝐷}=0, the similar kernel function isΦ𝑟_𝐷,𝑧=𝑟^(1−𝛽0_𝐷^)/2×1−𝛽₀Ψ_𝑣⋅𝑣𝑟^(𝜃1_𝐷^/2)+1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+2+√𝑧^∗Ψ_{𝑣⋅𝑣+1}𝑟^(𝜃1_𝐷^/2)+1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+21−𝛽₀Ψ_𝑣⋅𝑣1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+2+√𝑧^∗Ψ_{𝑣⋅𝑣+1}1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁×+21−𝛽₀−√𝑧^∗(1−𝛽₀)Ψ_{𝑣+1⋅𝑣}1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+2+𝑅^(𝜃1_𝐷^/2)+1Ψ_{𝑣+1⋅𝑣+1}1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+2(1−𝛽₀)Ψ_𝑣⋅𝑣1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+2+√𝑧^∗𝑅^(𝜃1_𝐷^/2)+1Ψ_{𝑣⋅𝑣+1}1,𝑅^(𝜃1_𝐷^/2)+1,2√𝑧^∗/𝜃₁+2⁻¹. (26)

From formation pressure equation (24), the solution’s expression of wellbore pressure can be unified as follows: 𝑃_𝑤𝐷𝑟_𝐷,𝑧=𝑞_𝐷(𝑧)1((.𝑧+1/Φ1,𝑧)) (27)

Substituting (25), (26), and (27) into (24) and (A.3), respectively, we can get similar structure of formation pressure and the wellbore pressure of the fractal dual-porosity reservoir with spherical flow under three types of outer boundaries (infinite, constant pressure, and closed) in the Laplace.

3. The Similar Structure Method Algorithm (SSMA)

This section presents the work of fluid mechanics in porous medium to establish reservoir seepage mathematical models according to the given formation, testing well, and fluid physical parameters. The formation pressure changing law with time of test well is obtained by solving these models. The well test analysis is on quantitative interpretation of the measurable pressure change combined with production data. Based on this test, we can obtain the parameters which cannot be directly measured. The parameters are important to calculate reservoir reserves and make scheme for the oil/gas field exploration, including pollution situation, improvement degree, wellbore damage, wellbore effect, flow coefficient and permeability of the test formation, and reservoir boundary.

For improvement of the modern well test analysis method, this paper programmed SSMA via SSM and Stehfest numerical inversion to calculate wellbore pressure and its derivative of reservoir seepage models. The algorithm is shown in Figure 1.

Figure 1 

The similar structure method algorithm for solving the reservoir seepage models.

4. The Pressure Transient Characteristics and Flow Regimes

In this section, the dimensionless formation pressure and derivative responses for fractal dual-porosity reservoir spherical flow are calculated by SSMA proposed above. The well test analysis cures of wellbore pressure and pressure derivative are drew out with 𝑇_𝐷/𝐶_𝐷 as abscissa and 𝑃_𝑤𝐷, 𝑃^′_𝑤𝐷⋅𝑇_𝐷/𝐶_𝐷 as ordinate. We analyze the effects of relevant parameters on the behavior.

Figure 2 shows the dimensionless formation pressure and derivative responses for fractal dual-porosity reservoir affected by fractal dimension 𝑑_𝑓 in three types of outer boundaries. In early time wellbore storage period (stage 1) is observed in both pressure change and derivative aligned in unite slope trend which follow by a hump representing the transition flow (stage 2) to radial flow period (stage 3) impacted by the fractal dimension. After the radial flow, the fissure produces independently with the flow. This flow regime is followed by a characteristic dip in the pressure derivative, which is caused by cross flow (stage 4) in matrix. Finally, we attain the last-time compound radial flow (stage 5) characterized by different outer boundaries condition.

Figure 2 

Well testing curves in fractal dual-porosity reservoir with three types of outer boundaries (𝜔=0.01, 𝜆=0.00001, 𝜃=0.5, 𝑑𝑓=1.3, 𝐶𝐷𝑒2𝑠=3, and 𝑅=500m).

It should be noted that all the flow regimes described in the previous paragraph exist for all dual-porosity reservoirs. Depending on system properties, particularly wellbore storage parameter, some of the flow regimes above may be changed. In this section we present some results to reflect the effects of reservoir.

Figure 3 presents the dimensionless variables of wellbore storage and skin factor 𝐶_𝐷𝑒^2𝑠 on fractal dual-porosity reservoir with spherical flow. It can be seen that with all other parameters kept constant, the curve parameters 𝐶_𝐷𝑒^2𝑠 has obviously impacted on period of radial flow and cross flow. As the value of 𝐶_𝐷𝑒^2𝑠 increases, the duration time of radial flow in fissure decreases. When the faster cross flow takes place for the matrix to fissure, the earlier reflection on dimensionless curve appears.

Figure 3 

Effects of the curve parameters on pressure responses of a test well in fractal dual-porosity reservoir (𝜔=0.01, 𝜆=0.00001, 𝜃=0.5, 𝑑𝑓=1.3, and 𝑅=500m).

Figure 4 shows the effect of fractal dimension 𝑑_𝑓 on fractal dual-porosity reservoir with spherical flow. The properties of the reservoir are identical in this case. As shown in Figure 4 increasing the number of fractal dimension significantly influences the magnitude of the whole regime pressure and derivatives. That is because of the fact that increasing fractal dimension will improve the permeability around wellbore.

Figure 4 

Effect of fractal demission on pressure responses of a test well in fractal dual-porosity reservoir (𝜔=0.01, 𝜆=0.00001, 𝜃=0.5, 𝐶𝐷𝑒2𝑠=3, and 𝑅=500m).

Figure 5 shows the effect of fractal exponential 𝜃 on fractal dual-porosity reservoir with spherical flow. As show in Figure 4, increasing the value of fractal exponent significantly influences the magnitudes of whole regime pressures and derivatives. Fractal exponent depicts the connectivity of fractal object. With the fractal exponent increasing, the pressure curve is on the upward trend. It shows that the more tortuous fissure system is, the worse connectivity is.

Figure 5 

Effect of fractal exponent on the pressure responses of a test well in fractal dual-porosity reservoir (𝜔=0.01, 𝜆=0.00001, 𝑑𝑓=1.3, 𝐶𝐷𝑒2𝑠=3, and 𝑅=500m).

Figure 6 shows the effect of interporosity 𝜆 on fractal dual-porosity with spherical flow. Figure 6 indicates that the reflection of cross flow develops at later times for larger value of 𝜆. Also, duration of the radial flow in fissure depends on the value of 𝜆. This is understood as 𝜆 is directly proportional to cross flow time of oil (gas) in matrix block. Faster of the cross flow takes palace, the reflection on the dimension curve appears earlier.

Figure 6 

Effect of interporosity flow coefficient on the pressure responses of a test well in fractal dual-porosity (𝜔=0.01, 𝜃=0.5, 𝑑𝑓=1.3, 𝐶𝐷𝑒2𝑠=3, and 𝑅=500m).

Figure 7 shows the effect of storativity ratio 𝜔 on fractal dual-porosity with spherical flow. The storativity significantly affects on the radial flow and period of cross flow. The smaller the storativity is, the deeper and wider the dip in the derivative becomes. The fissure system flowing (stage 2) sustains short time, which will be probability covered by period of the wellbore storage.

Figure 7 

Effect of storativity ratio on the pressure responses of a test well in fractal dual-porosity reservoir (𝜆=0.00001, 𝜃=0.5, 𝑑𝑓=1.3, 𝐶𝐷𝑒2𝑠=3, and 𝑅=500m).

5. Conclusion

The SSM completed the theory of solutions’ similar structure. When we deal with the boundary value problem, a general second-order linear differential equation, the formation of the similar structure is decided by the coefficients of the left (inner) boundary condition. The kern functions are achieved by the coefficients of the right (outer) boundary condition. So the SSM is proposed to solve such complicated problem in reality.

Many existing reservoir seepage fluid models can be adapted to the boundary value problem of the extended modified Bessel equation. In this paper, we propose SSM which is a convenient, effective, creative method to solve the problem of fluid flow in porous media since it can avoid complex procedures. The proposed SMMA optimizes the programming process of well test analysis software and also simplifies the processing of Laplace inversion.

When 𝑑_𝑓1=𝑑_𝑓2 and 𝜃=0  (𝑣=0), it is easy to get a dual-porosity reservoir mathematic model. With the solutions’ similar structure on the model of fractal dual-porosity reservoir, it is easy to observe how the wellbore storage, skin factor, and fractal dimension affect the dimensionless formation pressure and the wellbore pressure. The model as well as the corresponding type curves are quite general that they are useful in predicting the production performance or analyzing the production data from this type of well-reservoir systems.

Appendices

A. Derivation of Two Linearly Independent Solutions of the General Extended Modified Bessel Equation

In this section, we demonstrate why (1) has two linearly independent solutions 𝐼_𝑣𝑘𝑥^𝛽,𝐾_𝑣𝑘𝑥^𝛽. (A.1)

Substituting the bellow variables into (1) 𝑢=𝑥^−𝛼𝑦,ℎ=𝑘𝑥^𝛽, (A.2) where 𝛼=(1−𝐴)/2, 𝛽=𝑞/2, and 𝑘=2√𝐶/𝑞, we can obtain the following standard Bessel equation: ℎ²𝑑²𝑢𝑑ℎ²+ℎ𝑑𝑢−ℎ𝑑ℎ²+𝑣²𝑢=0, (A.3) where 𝑣=(1−𝐴)²−4𝐵/𝑞.

The general solution of (1) is 𝑢=𝐶₁𝐼_𝑣(ℎ)+𝐶₂𝐾_𝑣(ℎ), (A.4) where 𝐶₁ and 𝐶₂ are arbitrary constants. Substituting (A.3) into (A.5), the general solution of (1) is 𝑦=𝑥^𝛼𝐶₁𝐼_𝑣(ℎ)+𝐶₂𝐾_𝑣(ℎ). (A.5)

We know that 𝐼_𝑣(𝑘𝑥^𝛽), 𝐾_𝑣(𝑘𝑥^𝛽) are two linearly independent solutions for the extended modified Bessel equation (1).

B. Mathematical Model of Fractal Dual-Porosity Reservoir

The fissure media and matrix block in dual-porosity media are homogeneously distributed. Due to taking small size in reservoir volume, the fissure media has low porosity with high permeability. Whereas, the porosity of the matrix block is high with low permeability because of sedimentation and diagenesis. We can ignore the permeability of matrix block for simplified model.

To formulate the derivation of the mathematical mode for fractal dual-porosity reservoir with spherical flow, the basic assumptions used for fluid flow in porous media are as follows:(i)flat pitch dual-porosity reservoir, uniform thickness, and isotropic;(ii)single-phase micro compressible fluid, spherical flow which has one production or injection well;(iii)darcy’s law applies and isothermal curve;(iv)neglecting the capillary single-phase horizontal flow without gravity effect;(v)fracture network becomes deformed under the pressure with fractal feature and the variety of fractal dimension can be ignored.

The permeability and porosity distributions of a fractal reservoir have been defined in various forms in different studies. Here, we consider skin factor’s influence by 𝑟_𝑤𝑒=𝑟_𝑤𝑒^−𝑠 and embed the fractal dimension 𝑑_𝑓 of fractal network into the Euclid space with the dimension 𝑑=2. We make use of definitions similar to those proposed by [7]: 𝜙_𝑖=𝜙_𝑤𝑖𝑟𝑟_𝑤𝑒^{𝑑𝑓𝑖−𝑑}(),𝑘𝑖=1,2_𝑖=𝑘_𝑤𝑖𝑟𝑟_𝑤𝑒^{𝑑𝑓𝑖−𝜃𝑖−𝑑}(𝑖=1,2). (B.1)(B.2)

Darcy’s law says that 𝑣_𝑖𝑘=−^𝑖𝜇Δ𝑃_𝑖(𝑖=1,2). (B.3)

Due to the conservation law, the continuity equation of the flow in fissure media is described by the following partial differential equation: 𝜕𝜙₁𝜌𝜕𝑡+∇𝜌𝑣₁−𝑞_𝜆=0. (B.4)

The flow in matrix block is described by the following partial differential equation: 𝜕𝜙₂𝜌𝜕𝑡+𝑞_𝜆=0. (B.5)

In (B.5) and (B.6), the second item in the right-hand side represents the cross flow intensity between fissure media and matrix block and is defined as follows: 𝑞_𝜆=𝛼𝑘₂𝜇𝑝₂−𝑝₁. (B.6)

Taking (B.2)–(B.4) and (B.7) into (B.5) yields 𝜕²𝑝₁𝜕𝑟²+𝛽₀𝑟𝜕𝑝₁𝑘𝜕𝑟+𝛼^𝑤2𝑘_𝑤1𝑟𝑟_𝑤𝑒^{(𝑑𝑓2−𝑑𝑓1)−(𝜃2−𝜃1)}𝑝₂−𝑝₁=𝜇𝜙_𝑤1𝐶_𝑡1𝑘_𝑤1𝑟𝑟_𝑤𝑒^𝜃1𝜕𝑝₁.𝜕𝑡 (B.7)

Taking (B.2)–(B.4) and (B.7) into (B.6) yields 𝜙_𝑤1𝐶_𝑡2𝑘_𝑤1𝑟𝑟_𝑤𝑒^𝜃2𝜕𝑝₂+𝜕𝑡𝛼𝑘_𝑤2𝜇𝑝₂−𝑝₁=0, (B.8) where 𝛽₀=𝑑_𝑓1−𝜃₁−𝑑+2, 𝐶_𝑡𝑖=𝐶₀+𝐶_𝑖  (𝑖=1,2) and 𝐶₀=1𝜌𝜕𝜌𝜕𝑝₂,𝐶_𝑖=1𝜙_𝑖𝜕𝜙_𝑖𝜕𝑝_𝑖. (B.9)

Initial condition is as follows: 𝑝₁(𝑟,0)=𝑝₂(𝑟,0)=𝑝₀. (B.10)

Inner boundary condition is as follows: 𝑝_𝑤=𝑝₁𝑟_𝑤𝑒,𝑟,𝑡₂𝜕𝑝₁𝜕𝑟_{𝑟=𝑟𝑤𝑒}=𝜇𝛿𝜋𝑘_𝑤1𝑞(𝑡)+𝐶𝑑𝑝_𝑤.𝑑𝑡 (B.11) Three types of outer boundary condition are as follows.

Case 1. Infinite pressure outer boundary condition is 𝑝₁(∞,𝑡)=𝑝₀. (B.12)

Case 2. Constant pressure outer boundary condition is 𝑝₁(𝑅,𝑡)=𝑝₀. (B.13)

Case 3. Closed outer boundary condition is 𝜕𝑝₁𝜕𝑟_𝑟→𝑅=𝑝₀. (B.14)

In terms of convenience of calculation, we define the dimensionless variables as follows: 𝑃_𝑖𝐷=𝛿𝜋𝑘_𝑤1𝑟_𝑤𝑒3.685×10⁻³𝑞_𝑒𝜇𝑝₀−𝑝_𝑖(),𝑟𝑖=1,2_𝐷=𝑟𝑟_𝑤𝑒=𝑟𝑟_𝑤𝑒^𝑠,𝑅_𝐷=𝑅𝑟_𝑤𝑒=𝑅𝑟_𝑤𝑒^𝑠,𝑞_𝐷𝑡_𝐷=𝑞(𝑡)𝑞_𝑒,𝑡_𝐷=3.6𝑘_𝑤1𝑡𝜙_𝑤1𝐶_𝑡1+𝜙_𝑤2𝐶_𝑡2𝑟²_𝑤𝜇,𝐶_𝐷=𝑘_𝑤1𝑒^𝑠𝛿𝜋𝜙_𝑤1𝐶_𝑡1+𝜙_𝑤2𝐶_𝑡2,𝜆=𝛼𝑟²_𝑤𝑘_𝑤2𝑘_𝑤1,𝜙𝜔=^𝑤1𝐶_𝑡1𝜙_𝑤1𝐶_𝑡1+𝜙_𝑤2𝐶_𝑡2,𝑇_𝐷=𝑡_𝐷𝐶_𝐷,𝛽_𝑖=𝑑_𝑓𝑖−𝜃_𝑖(𝑖=1,2),𝛿=2hemisphere,𝛿=4sphere. (B.15)

All the parameters are explained in the nomenclature. The dimensionless mathematical model for fractal dual-porosity reservoir model can be obtained as follows: 𝜕𝑃_1𝐷𝜕𝑟²_𝐷+𝛽₀𝑟_𝐷𝜕𝑃_1𝐷𝜕𝑟_𝐷+𝜆𝑟^{𝛽𝑖−𝜃𝑖}_𝐷𝑒^2𝑠𝑃_2𝐷−𝑃_1𝐷=𝜔𝑟_𝜃1𝐶_𝐷𝑒^2𝑠𝜕𝑃_1𝐷𝜕𝑇_𝐷,(1−𝜔)𝑟^𝜃1_𝐷𝜕𝑃_2𝐷𝜕𝑇_𝐷+𝜆𝐶_𝐷𝑃_2𝐷−𝑃_1𝐷𝑃=0,_1𝐷𝑟_𝐷,0=𝑃_2𝐷𝑟_𝐷𝑃,0=0,_𝑤𝐷=𝑃_1𝐷1,𝑇_𝐷,𝑟²_𝐷𝜕𝑃_𝑤𝐷𝜕𝑟_{𝐷𝑟𝐷=1}=−𝑞_𝐷𝑇_𝐷+𝑑𝑃_𝑤𝐷𝑑𝑇_𝐷,𝑃_{𝑖𝐷𝑟𝐷→∞}=0or𝑃_𝑖𝐷𝑅_𝐷,𝑇_𝐷=0or𝜕𝑃_𝑖𝐷𝜕𝑅_{𝐷𝑟→𝑅𝐷}=0. (B.16)

Taking the Laplace transform of 𝑃_1𝐷 with respect to 𝑇_𝐷, we get the boundary value problem of the ordinary differential equation with parameter 𝑧 (where 𝑧 denotes Laplace space variable); that is, 𝑑²𝑃_1𝐷𝑑𝑟²_𝐷+𝛽₀𝑟_𝐷𝑑𝑃_1𝐷𝑑𝑟_𝐷+𝜆𝑟^{𝛽𝑖−𝜃𝑖}_𝐷𝑒^2𝑠𝑃_2𝐷−𝑃_1𝐷=𝜔𝑟_𝜃1𝐶_𝐷𝑒^2𝑠𝑃_1𝐷,(1−𝜔)𝑟^𝜃1_𝐷𝑃_2𝐷+𝜆𝐶_𝐷𝑃_2𝐷−𝑃_1𝐷=0,𝑃_1𝐷𝑟_𝐷,0=𝑃_2𝐷𝑟_𝐷,0=0,𝑃_𝑤𝐷=𝑃_1𝐷1,𝑇_𝐷,𝑟²_𝐷𝑑𝑃_𝑤𝐷𝑑𝑟_{𝐷𝑟𝐷=1}=−𝑞_𝐷(𝑧)+𝑧𝑃_𝑤𝐷,𝑃_{𝑖𝐷𝑟𝐷→∞}=0or𝑃_{𝑖𝐷𝑟𝐷=𝑅𝐷}𝑑=0or𝑃_𝑖𝐷𝑑𝑟_{𝐷𝑟→𝑅𝐷}=0. (B.17)

Nomenclature

𝐶:	Wellbore storage coefficient, m3/Pa
𝐶0:	Fluid compressibility, Pa−1
𝐶𝑡𝑖:	Compressibility, Pa−1
𝑑:	Euclid dimension
𝑑𝑓𝑖:	Fractal dimension
𝑘𝑖:	Permeability, m2
𝑘𝑤𝑖:	Initial permeability, m2
𝑝𝑖:	Reservoir pressure, Pa
𝑝𝑤:	Wellbore pressure, Pa
𝑝0:	Initial reservoir pressure, Pa
𝑟:	Radial distance in spherical coordinate, m
𝑅:	Radial distance of outer boundary radius, m
𝑠:	Skin factor, dimensionless
𝑡:	Time, m
𝑣𝑖:	Spherical flow velocity, m/s
𝜌:	Fluid density, kg/m3
𝛼:	Form factor
𝜃𝑖:	Fractal exponent, dimensionless
𝜇:	Viscosity of fluid, Pa·s
𝜙𝑖:	Porosity, fraction
𝜙𝑤𝑖:	Initial porosity, fraction
𝜔:	Storativity ratio, dimensionless
𝜆:	Interporosity flow coefficient, dimensionless
𝐼𝑣(⋅):	Modified Bessel function of first-order 𝑣
𝐾𝑣(⋅):	Modified Bessel function of second-order 𝑣
𝜑(⋅):	A binary function
Φ(⋅):	Similar kernel function
𝑧:	Laplace space variable.

Superscript

−−:	Laplace domain.

Subscript

𝑖=1:	Fissure media
𝑖=2:	Matrix block
𝐷:	Dimensionless
𝑜:	Initial condition
𝑤:	Wellbore parameter
𝑤𝑒:	Well effective.

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.

Acknowledgments

This study is supported by National Natural Science Foundation for Petroleum and Chemical Industry Unite Fund Support Key Projects (Grant no. U1262209), the National Natural Science Fund (Grant no. 51274172), and the Scientific Research Fund of Sichuan Education Provincial Department of China (Grant no. 12ZA164).