Transient flow model and pressure dynamic features of tree-shaped fractal reservoirs*

TAN Xiao-hua (谭晓华), LI Xiao-ping (李晓平)

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University,

Chengdu 610500, China, E-mail:xiaohua-tan@163.com

Transient flow model and pressure dynamic features of tree-shaped fractal reservoirs*

TAN Xiao-hua (谭晓华), LI Xiao-ping (李晓平)

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University,

Chengdu 610500, China, E-mail:xiaohua-tan@163.com

(Received February 13, 2013, Revised September 2, 2013)

A transient flow model of tree-shaped fractal reservoirs is built by embedding a fracture network simulated by a tree-shaped fractal network into a matrix system. The model can be solved using the Laplace conversion method. The dimensionless bottom hole pressure can be obtained using the Stehfest numerical inversion method. The bi-logarithmic type curves for the tree-shaped fractal reservoirs are thus obtained. The pressure transient responses under different fractal factors are discussed. The factors with a primary effect on the inter-porosity flow regime include the initial branch numberN , the length ratioα, and the branch angleθ. The diameter ratioβhas a significant effect on the fracture radial flow, the inter-porosity and the total system radial flow regimes. The total branch levelMof the network mainly influences the total system radial flow regime. The model presented in this paper provides a new methodology for analyzing and predicting the pressure dynamic characteristics of naturally fractured reservoirs.

tree-shaped fractal, transient flow, pressure dynamic characteristic, naturally fractured reservoir, type curve

Introduction

The flow behavior of fluid in fractured reservoirs was widely studied by using a fractal network to simulate the non-uniform distribution of fractures in such reservoirs. Using the fractal reservoir model introduced by Kristanto et al.[1]identified a new type curves for handling multiple well test problems, particularly for analyzing interference tests. They calculated reservoir characteristics such as the permeability, the thickness, the fluid capacitance coefficient, and the fractal dimension.

By assuming that the network of fractures is fractal, Flamenco-Lopez and Camacho-Velazquez[2,3]derived an approximate analytical solution for dual-porosity systems in the Laplace space. They also investigated the transient pressure behavior of naturally fractured reservoirs in the context of fractal characteristics. Li and Horne[4]observed that the predictions of a general model developed from the fractal modeling of a porous medium agree satisfactorily with the capillary pressure curve values of geysers rocks. With theoretical and experimental analyses, they demonstrated that the heterogeneity of geysers rocks could be quantitatively described by using fractal dimension values. Velazquez et al.[5]investigated the production decline behavior in a naturally fractured reservoir exhibiting single and double porosities using a fractal geometry model. They presented a combined analysis methodology using a transient well test and the boundary-dominated decline production data to characterize naturally fractured reservoir with scale-related fractures. The feasibility of the method was shown through field data.

Zhang and Tong[6]built a fractal reservoir model with consideration of a stress-sensitive coefficient and obtained solutions using methods of self-similarity and regular perturbation. In addition, the results could be used to analyze the pressure transient response of the fractal medium in stress-sensitive reservoirs. Jafari and Babadagli[7,8]indicated that the three-dimensionalpermeability distribution could be obtained by using the well log data, the outcrop, and the well test data. They provided a foundation for applying a fractal network in the reservoir numerical simulation. Using the finite element method, Zhang et al.[9]obtained the numerical solution of a nonlinear flow model in a deformable dual media fractal reservoir.

In those studies, the fractal structures embedded into a matrix network include both straight and intersecting lines, which could not adequately simulate the radial flow tending to the well bottom. However, the radial flow regime tending to the well bottom is of great significance in the underground seepage and oil exploitation field. Wechsatol et al.[10]demonstrated a tree-shaped fractal structure connecting the center origin and the points in different circles according to the bifurcation structure of the plant lamina, as shown in Fig.1. They also formulated a construction method and an optimization rule for tree-shaped fractal structures[11,12]. Such a fractal network can be used to simulate the fluid flow from the formation to the well bottom.

Xu and Yu[13]proposed a tree-shaped fractal model suitable for the fluid flow by considering the dynamic behavior of branching tubes in a tree-shaped fractal network. They analyzed the model’s transport properties and mass transfer capabilities[14,15]. According to the fractal characteristics of the pores and the capillary pressure effect, Yun et al.[16]presented a fractal model to describe the Bingham fluid flow in porous media with consideration of the starting pressure gradient influence. Based on Yun et al.’s research, Wang et al.[17,18]demonstrated a tree-shaped fractal model to consider the influence of the starting pressure gradient on the Bingham fluid seepage in a porous medium.

This paper simulates fractures using a tree-shaped fractal network, selected for its capability to simulate radial flow tending to the well bottom accurately. A transient flow model of the tree-shaped fractal reservoirs is then proposed by embedding the fracture network simulated by the tree-shaped fractal network into a matrix system. The factors influencing the dynamic characteristics of transient pressure response in treeshaped fractal reservoirs are then analyzed. The results provide a new method for the reservoir dynamic description and the well test analysis.

1. Physical model

The tree-shaped fractal network is composed of sets of branch structures. As shown in Fig.2, due to the network symmetry, only one unit of the network needs to be analyzed. The generation of the network must ensure that the ends of each level’s branches are on the same circle. Circles of different levels are concentric with the origin Oas the center. The tree-shaped fractal network is established usingNtubes starting at the point of the originO, where the initial length of the tube is l0, and the initial diameter is d0. This network contains double branches(n =2), with a branch angle of θ(θ＜π/2), and with a total network branch level ofM. Two scale factors are used for this fractal network: the length ratio αand the diameter ratio β. The branch tube is assumed to be smooth, whereas the wall thickness is neglected.

As shown in Fig.3, the reservoir thickness ish,whereas the well is located in the circle center Owith radiusrw. The mathematical model is based on the following assumptions: (1) The reservoir is divided into Mannular sections in a tree-shaped fractal network. The properties of different fracture sections are different, but those of the matrix system as well as those of the fluid are identical. (2) The fracture permeability is far greater than that of the matrixes, and the fluid only flows to the wellbore through the fracture system. (3) The formation fluid is a single-phase, slightly compressible fluid, and the seepage is regarded as an isothermal flow. (4) The capillary pressure and the gravity effects are neglected, and the permeability and the porosity do not vary with the pressure. (5) The fluid flow in the matrix and the fracture system of each section satisfies the linear flow rule.

2. Mathematical model

The parameters of the fracture system are directly generated by using the tree-shaped fractal network, as follows:

The permeability and the porosity of a fracture system,kfand φf, do not vary with the radial distance,r, in the transient flow models of the traditional double porosity (fracture and matrix system) reservoir. In order to compare the tree-shaped fractal reservoir transient flow models with the double porosity reservoir transient flow models, we have to clarify how to keep kfand φfindependent of rin the treeshaped fractal reservoir transient flow models. Underthe condition of double branches (n=2), the permeability,kfk, and the porosity,φfk, of every section in a fracture system are identical, i.e.,

Whenβis less than 0.707, a larger radius will cause a lower fracture system permeability of each section of the tree-shaped fractal reservoir. Whenβis larger than 0.707, a larger radius will cause a higher fracture system permeability of each section of the tree-shaped fractal reservoir.

According to the above analysis, the flow mathematical model of the tree-shaped fractal reservoir is expressed as follows:

3. Mathematical model solution

To simplify the mathematical model and its solution, we define the following dimensionless variables:

The dimensionless fracture system pressure of the kt hsection is

In Eq.(26), the dimensionless effective interface radius rDek, the permeability ratio kf(k+1)/kfk, and the functionSk(z)can be directly expressed by the parameters of the tree-shaped fractal network.

By substituting Eq.(3) into Eq.(16), we obtain the expression of the dimensionless effective interface radius.

The permeability ratio can be obtained by using Eq.(6)

The fluid capacitance coefficient ωkcan be obtained by substituting Eqs.(10) and (11) into Eq.(20).

4. Analysis of type curve characteristics

The dimensionless bottom hole pressure pwfDin the Laplace space is obtained by solving thelinear equations of Eq.(26), and the Stehfest numerical inversion method is used to convert pwfDto pwfD. The bilogarithmic type curves of the tree-shaped fractal reservoirs can be obtained.

Figure 4 shows the dimensionless pressure responses for a double-branched tree-shaped fractal network in a tree-shaped fractal reservoir. The fracture permeability and the fracture volume of each section are identical, thus satisfying the conditions of Eq.(13). The parameter values of the fracture system are as follows:α=1,β=0.707,θ=1,N =4,M =10, l0=10 m and d0=0.05 m. Other parameters are unrelated to the fracture system, which are rw=0.1 m,S =–1,CDe=0.0001,km=5×10–3µm2,h =10 m, φmp=0.1,Cmt=2.2×10–5MPa–1and Cft=10–4MPa.

Five flow regimes can be identified in Fig.4. Regime 1 is a pure wellbore storage regime, where both curves of the pressure and its derivative are an upward straight line with the slope of unity. Regime 2 is the transition regime influenced by the wellbore storage coefficient and the skin factor. The shape of the derivative curve is similar to a “hump,” where a greaterCDecauses a higher position of the “hump.” Given that the skin factor Sis in the index position, its value significantly affects the “hump” maximum value. Regime 3 is the early radial flow regime, which is also called the fracture system radial flow regime. If in this flow regime, there is a radial flow, the derivative curve is parallel to the horizontal ordinate, and its value is 0.5. However, this regime is hardly observed because it is often covered by the wellbore storage effect. Regime 4 is the inter-porosity flow regime, where the pressure derivative curve is V-shaped. The shape and the location of the V-shaped curves are mainly governed by the initial branch numberN, the length ratioα, and the branch angleθ. Regime 5 is the total system radial flow regime, which is observed when the transfer between the matrix and the fracture reaches a dynamic balance state. The slope of the pressure derivative curve is zero, and the curve converges to the “0.5 line.”

Figure 5 shows the effect of the initial branch number Non the type curves of the tree-shaped fractal reservoirs. The initial branch numberNmainly influences the inter-porosity flow regime. With all other parameters constant, the initial branch number Ndetermines the location, the width, and the depth of the V-shape. Three important effects of a decreasingNare illustrated in Fig.5. First, the V-shape in the derivative curve becomes wider and deeper and shifts left when the initial branch numberN decreases. The second consequence of a decrease inN is that the inter-porosity flow regime occurs earlier. Third, the fracture system radial flow regime becomes shorter and may even be dominated by the wellbore storage effect.

Figure 6 shows the effect of the length ratioα on the type curves of the tree-shaped fractal reservoirs. The length ratioαaffects the V-shape width and depth of the derivative curve. With all other parameters constant, a largerαleads to a wider and deeper V-shape. The inter-porosity flow regime lasts longer with an increase inα. On the other hand, the fracture system radial flow regime lasts shorter, and may even be covered by the pure wellbore storage regime. These results indicate that length ratioαhas a primary effect on the inter-porosity flow regime.

Figure 7 shows the effect of the branch angle θ on the type curves of the tree-shaped fractal reservoirs. The branch angleθhas a significant effect on the fracture system radial and inter-porosity flow regimes. When the branch angleθincreases, the inter-porosity flow regime lasts longer, whereas the fracture system radial flow regime becomes shorter, and may even be covered by the pure wellbore storage effect. The position of the V-shape in the pressure derivative curve moves downward with an increase inθ. In addition, the V-shape becomes wider with a larger branch angle.

Figure 8 shows the effect of the diameter ratio βon the type curves of the tree-shaped fractal reservoirs. The diameter ratioβdetermines the position of the pressure derivative curve. A smaller value ofβ causes an upward movement of the derivative curve. This phenomenon also shows the characteristics of the pressure transient responses like a closed boundary condition. When βincreases, the slope of the derivative curve decreases until it reaches zero. The critical value ofβis 0.707 when the facture system permeability and the fracture volume of each section is the same as those of the double-branched structure in the tree-shaped fractal reservoirs(n =2). Whenβis greater than 0.707, the pressure derivative curve representing the total system radial flow regime assumes a horizontal line with a value smaller than 0.5. The value of the horizontal line decreases asβincreases. Inversely, the total system radial flow regime’s pressure derivative curve assumes a horizontal line with a value larger than 0.5, even not as a horizontal line. These results indicate that the diameter ratio has a significant effect on the fracture system radial flow, the inter-porosity flow, and the total system radial flow regimes.

Figures 9 and 10, respectively, show the effects of the total branch level Mof the network on the type curves of the tree-shaped fractal reservoirs when the diameter ratio βis 0.65 and 0.75. The total branch level of the network mainly controls the total system radial flow regime. As shown in Fig.9, when the diameter ratio is smaller than 0.707, the horizontal line section representing the total system radial flow regime in the derivative curve moves upward asM decreases. The derivative curve may respond like a closed boundary condition. As shown in Fig.10, when the diameter ratio is greater than 0.707, the type curve section representing the total system radial flow regime assumes a horizontal line. This curve section also moves downward as the total branch levelM of the network increases.

5. Fluid capacitance coefficient and inter-porosity flow coefficient in different sections

Figure 11 shows the fluid capacitance coefficient λkin different sections when the diameter ratioβis 0.65, 0.707 and 0.75. When the diameter ratio βis smaller than 0.707, the fluid capacitance coefficient λkincreases with an increase of the section number k. And whenβis greater than 0.707,λkdecreases with the increase ofk. Additionally,λkdoes not vary withk, whenβis 0.707.

Figure 12 shows the inter-porosity flow coefficient ωkin different sections when the diameter ratio βis 0.65, 0.707 and 0.75. From Fig.12 we can see that when the section numberk is less than 6, the inter-porosity flow coefficient ωkdecreases dramatically with the increase of k. However, when it is greater than 6, the changing trends of ωkversuskdepend on the diameter ratioβ. When the diameter ratioβ is smaller than 0.707, the inter-porosity flow coefficient ωkincreases with the increase of the section number k(k＞6). And whenβis greater than 0.707,ωkdecreases with the increase of k(k＞6). Additionally, ωkdoes not vary withk , whenβis 0.707(k＞6).

6. Conclusions

In this paper, a transient flow model for the treeshaped fractal reservoirs is proposed. The pressure transient responses under different fractal factors are discussed. The model presented in this paper provides a new methodology for analyzing and predicting the pressure dynamic characteristics of naturally fractured reservoirs.

(1) A transient flow model of the tree-shaped fractal reservoirs is built by embedding a fracture network simulated by a tree-shaped fractal network into a matrix system. The model is solved using the Laplace conversion method. The dimensionless bottom hole pressure is obtained using the Stehfest numerical inversion method. Therefore, the bi-logarithmic type curves of tree-shaped fractal reservoirs are obtained.

(2) The factors that have a primary effect on the inter-porosity flow regime are the initial branch numberN , the length ratioα, and the branch angleθ. WhenNis smaller, the inter-porosity flow regime occurs earlier, whereas the fracture radial flow regime becomes shorter and may be covered by the well storage effect. Whenαincreases andθdecreases, the inter-porosity flow regime lasts longer, whereas the fracture radial flow regime becomes shorter, and may be covered by the well storage effect.

(3) The diameter ratioβhas a significant effect on the fracture radial flow, the inter-porosity and the total system radial flow regimes. When the fracture system permeability of each section and the fracture volume of the tree-shaped fractal reservoirs exhibit a double-branched condition (n=2), the critical value of the diameter ratioβis 0.707. The pressure derivative curve moves upward with the decrease ofβand even responds like a closed boundary condition. When βis greater than 0.707, the total system radial flow regime curve assumes a horizontal line with a value less than 0.5. Inversely, the total system radial flow regime curve assumes a horizontal line with a value greater than 0.5, even not as a horizontal line.

(4) The total branch levelM of the network mainly influences the total system radial flow regime. Whenβis greater than 0.707, the pressure derivative curves in the total system radial flow regime moves upward asM decreases, and the curve may respond like a closed boundary condition. Whenβis less than 0.707, a largerM leads to a lower position of the pressure derivative curves in the total system radial flow regime.

Acknowledgement

This work was supported by the 2014 Australia China National Gas Technology Partnership Fund Top Up Scholoarship.

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* Project supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 51125019).

Biography: TAN Xiao-hua (1986-), Male, Ph. D. Candidate

LI Xiao-ping,

E-mail: lixiaoping@swpu.edu.cn