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Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs

  • Jie Jin and Detang Lu EMAIL logo
Published/Copyright: June 13, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

The inter-porosity transfer is one of the decisive factors for gas development in fractured-porous gas reservoirs. In this article, we establish an analytical solution for the inter-porosity transfer rate and producing degree of matrix. Then, we study the law of inter-porosity transfer based on the solution. Through the Stehfest inversion transform, the typical curves of inter-porosity transfer rate and producing degree of matrix are plotted. It is found that the producing degree of matrix is close to zero in the initial period. Then, the inter-porosity transfer rate begins to increase, and the producing degree of matrix becomes larger. In the late period, the producing degree of matrix remains constant. In addition, the differences between the quasi-steady state model and the three kinds of unsteady state models are compared. It is found that the inter-porosity transfer occurs earlier in unsteady models. However, when pressure propagates to the external boundary, the transfer rate is equal between quasi-steady and unsteady models. It is also found that the inter-porosity transfer rate is slightly different in the three unsteady models, whereas in the spherical model it is largest at the intermediate period. Next, we discuss the influence of key parameters. The results reveal that gas reservoir radius, storage ratio, and inter-porosity flow factor can play an essential role in inter-porosity transfer. The findings of this study can improve our understanding of gas flow between fractures and matrix. Besides, it helps field engineers better understand the variation law of gas productivity in fractured-porous gas reservoirs, which can provide the scientific basis for making a development scheme.

Keywords: matrix; fractures; transfer rate; producing degree; dual-porosity model

Nomenclature

Variables

c

compressibility coefficient, m s 2 / kg

C

storage constant, m 4 s 2 / kg

C m

producing degree of matrix, dimensionless

h

thickness, m

K

permeability, m2

m

pseudo pressure, Pa 2 / Pa s

M

gas molecular weight, kg/mol

p

pressure, Pa

q mf

inter-porosity transfer rate, kg/m3/s

r

radius, m

S skin

skin factor, dimensionless

R

universal constant of gas, 8,314 m 2 / ( s 2 kmol K )

t

time, s

T

temperature, K

v

velocity, m/s

z

thickness of plate matrix, m

Z

gas deviation factor, dimensionless

Greek symbols

α

shape factor, m−2

ρ

density, kg/m3

ϕ

porosity, dimensionless

μ

viscosity, kg/m/s

ω

storage ratio, dimensionless

λ

inter-porosity flow factor, dimensionless

Subscripts

D

dimensionless

e

external boundary

f

fractures

i

initial state

t

total amount

sc

standard state

m

matrix

1 Introduction

The fractured-porous gas reservoirs are a crucial component of global oil and gas resources [1]. Over the years, numerous fractured-porous gas reservoirs have been explored and developed, which has greatly eased the world’s energy pressure [2]. Natural fractures and matrices make up the fractured-porous gas reservoirs. The natural fractures are caused by geological processes such as earthquakes and dissolution [3]. These natural fractures have a great influence on the gas flow [4,5]. The matrix contains many small pores, which are filled with gas. Due to the fracture network and matrix, the fractured-porous gas reservoirs have strong heterogeneity [6,7]. Generally, the permeability of fractures is greater than that of the matrix. But the porosity of the fractures is smaller than that of the matrix. Therefore, the fractures serve as the primary flow channel, and the matrix serves as the primary gas storage space [8,9,10]. The production rate of fractured-porous gas reservoirs is related to the fractures and matrix. In the initial period of production, the gas production is mainly from the gas in the fractures. With continuous production, the elastic energy in the fracture system decreases. Then the gas in the matrix begins to flow. Gas is replenished from the matrix into the fractures and then flows to the wellbore [11]. Therefore, the productivity of fractured-porous gas reservoirs is closely influenced by the transfer flow between fractures and matrix. In view of this, a precise understanding of the transfer between the matrix and fractures is necessary in addition to researching their inherent characteristics.

The rate of mass transfer per unit volume from the matrix to fractures is called the inter-porosity transfer rate. It is a critically important parameter for gas productivity in fractured-porous gas reservoirs. The accurate calculation of the inter-porosity transfer rate has always been an important research topic. Another important parameter is the producing degree of matrix. It is defined as the ratio of gas from the matrix to total gas production. It can reflect how much gas is coming from fractures and how much is coming from the matrix. For fractured-porous gas reservoirs, the matrix is the main storage space of gas. Therefore, the gas production mainly depends on the gas in the matrix. Obtaining the producing degree of matrix can help better understand the variation law of gas productivity.

So far, the dual-porosity model is still the most widely used approach to describe the gas flow between the matrix and fractures in fractured-porous gas reservoirs. Numerous well-testing and production data analyses have verified its correctness [12]. The concept of dual porosity was first proposed by Barenblatt et al. [13]. They considered the fractures and matrix as two continuous media that interact dynamically. At any point in space, there is a distribution with matrix and fractures media. In their model, there is a pressure difference and fluid transfer between the matrix and fractures. On this basis, Warren and Root [14] introduced the dual-porosity model into the field of petroleum engineering. They assumed that the fracture network is orthogonally distributed and that the matrix is divided into multiple cubes by fractures. The pressure in the matrix is equal everywhere. The inter-porosity transfer rate is proportional to the pressure difference between the fractures and matrices. Kazemi [15] first proposed the concept of shape factor. They used the finite difference method to discretize the pressure diffusivity equation and obtain the shape factor. This work was conducive to calculating the inter-porosity transfer rate. Deswaano [16] proposed that the fluid flow in the matrix is not necessarily steady. He developed an unsteady-state theory that describes the well pressure response. In his model, the inter-porosity transfer is related to the pressure variation in the matrix. In addition, he considered the matrix as plates, and the fractures were distributed between the plates. Zimmerman et al. [17] established a novel unsteady state model to simulate the fluid flow in fractured-porous reservoirs, and the matrix was considered to be spherical. In their model, the inter-porosity transfer rate is not proportional to the pressure difference between the fractures and matrix. There is pressure variation on the boundaries of the spherical matrix. Lim and Aziz [18] established a dual-porosity model with the cylindrical matrix. They took into account the pressure gradient in the matrix. Zhou et al. [19] developed a novel formula for the inter-porosity transfer rate in fractal fracture systems and heterogeneous fracture systems. They studied the effect of novel formula on bottom-hole pressure. Zheng et al. [20] developed a dual-porosity model for incompressible two-phase flow. They thought the displacement effect is the main motive force for the inter-porosity transfer. Their model is verified by four numerical examples. Besides, some scholars studied the inter-porosity transfer by experiment. Fansheng et al. [21] conducted an indoor experiment. Their result revealed that storage ratio and cross-flow factor can influence the amount and timing of inter-porosity transfer. Lian et al. [22] studied the effect of stress sensitivity on inter-porosity transfer by experiment. They found the stress sensitivity of fractures is much greater than that of matrix. They came to the conclusion that stress sensitivity makes the transfer occur earlier. Xiangguo et al. [23] studied the inter-porosity transfer in heterogeneous reservoirs. They found that the transfer rate decreases as heterogeneity increases.

Overall, scholars have done a lot of work on inter-porosity transfer between the matrix and fractures through theoretical and experimental research. However, for the theoretical research methods, existing literature focus on how the inter-porosity transfer affects the bottom hole pressure. There is no specific research on the variation law of inter-porosity transfer. For the experimental research methods, they can research the general law but cannot accurately calculate the inter-porosity transfer rate. In our study, we develop a straightforward solution for the inter-porosity transfer rate and producing degree of matrix. Then, we analyze the characteristics and variation law of inter-porosity transfer. This work is lacking in previous literature. Therefore, our work is a novelty and of practical significance.

In this article, we apply the Laplace transform and Stehfest inversion transform to obtain the analytical solution of the inter-porosity transfer rate. Then, through summing the inter-porosity transfer rate, we obtain the solution of producing degree of matrix. These methods that we used are quick and stable compared with the time-consuming numerical solution [24,25]. Moreover, by adjusting the f(s), our solution is applicable to the quasi-steady state model and the unsteady state model. It makes our model more convenient and adaptable.

The research results of this study can help better perceive the gas flow process in fractured-porous gas reservoirs. Through typical curves, scholars can identify the flow stage of inter-porosity transfer clearly. This study also reveals the specific role of gas reservoir radius, storage ratio, and inter-porosity flow factor on inter-porosity transfer.

This work is organized as follows: first, the theoretical model and analytical solution are established in Section 2. Then, the typical curves are plotted, and the curve characteristics are analyzed in Section 3. Next, the influence of key parameters is discussed in Section 4. Subsequently, a case analysis is presented in Section 5. Finally, the conclusions are summarized in Section 6. The general sketch of this work is shown in Figure 1.

Figure 1 The general sketch of this work.
Figure 1

The general sketch of this work.

2 Physical and mathematical model

2.1 Physical model

A vertical well producing from a fractured-porous gas reservoir is studied in this article. The gas reservoir is described by the dual-porosity approach, which is shown in Figure 2. The dual-porosity gas reservoir is composed of matrix and fractures. According to previous studies, the dual-porosity model can be divided into quasi-steady state model and unsteady state model. For the quasi-steady state model, it is assumed that the pressure inside the matrix system is equal everywhere. For the unsteady state model, it is assumed that there is a pressure gradient in the matrix system.

Figure 2 (a) Top view of the dual-porosity gas reservoir. (b) The quasi-steady state model.
Figure 2

(a) Top view of the dual-porosity gas reservoir. (b) The quasi-steady state model.

Figure 2(b) shows the details of the dual-porosity model. It is the quasi-steady state model. The fractures divide the matrix into same-size cubes. Figure 3 is the schematic diagram of three kinds of unsteady state models. According to the different shapes of matrix, the unsteady state models can be divided into spherical, cylindrical, and plate models.

Figure 3 The unsteady state models: (a) spherical model; (b) cylindrical model; and (c) plate model.
Figure 3

The unsteady state models: (a) spherical model; (b) cylindrical model; and (c) plate model.

The gas flow process is shown in Figure 4. When the well is producing, the gas stored in the fractures flows into the wellbore. Meanwhile, the gas stored in the matrix flows into the fractures. This flow process is called the inter-porosity transfer, which is shown by the red arrow in Figure 4. The rate of mass transfer per unit volume from the matrix to fractures is called the inter-porosity transfer rate. The primary motivation of this work is to develop an analytical solution for the inter-porosity transfer rate. Another important research parameter is the producing degree of matrix. It is defined as the ratio of gas from matrix to total gas production. Our plan is to develop the solution of producing degree of matrix on the basis of the inter-porosity transfer rate.

Figure 4 The flow process of the gas.
Figure 4

The flow process of the gas.

In addition, some assumptions are made to assist in the development of the model.

  1. The formation is an isotropic cylinder, and the thickness of the formation remains unchanged.

  2. There is a gas well with constant production at the center of the formation.

  3. The formation fluid is only gas, and the flow satisfies Darcy’s law.

  4. The capillary force and gravity are negligible.

2.2 Mathematical model

2.2.1 Quasi-steady state model

After the physical model is established, the formula of inter-porosity transfer rate can be derived by the mathematical model. Initially, the quasi-steady state model is studied. For the specific meanings of mathematical symbols, please see Nomenclature.

The continuity equation should be satisfied for the gas in the matrix and fractures, which is shown as follows:

(1) ∂ ∂ t ( ϕ f ρ ) + div ( ρ v f ) − q = 0 ,

(2) ∂ ∂ t ( ϕ m ρ ) + div ( ρ v m ) + q = 0 ,

where subscript f represents the fracture system and subscript m represents the matrix system. q represents the inter-porosity transfer rate.

The state equation of gas is as follows:

(3) p = ρ R T Z M .

Pseudo-pressure is introduced instead of pressure.

(4) m ( p ) = 2 ∫ p i p p μ ( p ) Z ( p ) d p .

According to the previous study, the matrix permeability is much lower than the fracture permeability. Therefore, the variation of fluid mass caused by seepage conduction in the matrix system is negligible compared with the inter-porosity transfer term and the elastic term. So div ( ρ v m ) can be negligible. Eqs. (1) and (2) can be transformed into

(5) μ ϕ f c f ∂ m f ∂ t = K f 1 r ∂ ∂ r r ∂ m f ∂ r + α K m ( m m − m f ) ,

(6) μ ϕ m c m ∂ m m ∂ t + α K m ( m m − m f ) = 0 .

The introduction of dimensionless parameters can simplify the mathematical formula. It can better present the characteristics of fast and convenient analytical solutions [26].

The dimensionless parameters used in this work are defined as follows:

(7) m Df = π K f h T sc Q T p sc [ m i − m f ( r , t ) ] ,

(8) m Dm = π K f h T sc Q T p sc [ m i − m m ( r , t ) ] ,

(9) t D = K f t ( c m ϕ m + c f ϕ f ) μ r w 2 ,

(10) r D = r r w ,

(11) q Dmf = 2 π h r w 2 R T sc q mf M Q p sc ,

(12) ω = c f ϕ f c f ϕ f + c m ϕ m ,

(13) λ = α K m K f r w 2 .

After the dimensionless process, Eqs. (5) and (6) can be converted to

(14) 1 r D ∂ ∂ r D r D ∂ m Df ∂ r D − ω ∂ m Df ∂ t D − ( 1 − ω ) ∂ m Dm ∂ t D = 0 ( 1 − ω ) ∂ m Dm ∂ t D − λ ( m Df − m Dm ) = 0 .

Laplace transform is performed on Eq. (14). Then, Eq. (15) can be obtained.

(15) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − ω s m ¯ Df − ( 1 − ω ) s m ¯ Dm = 0 ( 1 − ω ) s m ¯ Dm − λ ( m ¯ Df − m ¯ Dm ) = 0 ,

where m ¯ Df = ∫ 0 ∞ m df e − s t d t , m ¯ Dm = ∫ 0 ∞ m Dm e − s t d t .

From Eq. (15), we can obtain Eqs. (16) and (17) as follows:

(16) m ¯ Dm = λ ( 1 − ω ) s + λ m ¯ Df ,

(17) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − s f ( s ) m ¯ Df = 0 ,

where f ( s ) = ω ( 1 − ω ) s + λ ( 1 − ω ) s + λ

The general solution of Eq. (17) is

(18) m ¯ Df = A I 0 [ r D s f ( s ) ] + B K 0 [ r D s f ( s ) ]

Since the gas well is produced at a constant production rate, the internal boundary conditions can be obtained as follows:

(19) C D ∂ m Df ∂ t D − C D S skin ∂ ∂ t D ∂ m Df ∂ r D − ∂ m Df ∂ r D r D = 1 = 1 .

There are three kinds of external boundary conditions. Eqs. (20)–(22) represent the infinite gas reservoir boundary, closed boundary, and constant pressure boundary, respectively.

(20) m Df ( r D → ∞ ) = 0 ,

(21) ∂ m Df ∂ r D r D = r eD = 0 ,

(22) m Df ( r D = r eD ) = 0 .

Therefore, m ¯ Df can be solved by combining the internal and external boundary conditions.

In the quasi-steady state model, the inter-porosity transfer rate is related to the following factors: (i) the pressure difference between matrix and fractures; (ii) gas viscosity; (iii) shape factor; (iv) permeability of matrix; and (v) gas density.

The formula is as follows:

(23) q mf = α ρ K m μ ( p m − p f ) .

Using Eqs. (4), (7), (8), and (11) to transform Eq. (23), we can obtain

(24) q ¯ Dmf = λ ( m ¯ Df − m ¯ Dm ) .

Laplace transform is performed on Eq. (24).

(25) q ¯ Dmf = ∫ 0 ∞ q Dmf e − s t d t = λ ( m ¯ Df − m ¯ Dm ) .

In the above discussion, the relationship between m ¯ Df and m ¯ Dm is obtained, which is shown in Eq. (26) as follows:

(26) ( 1 − ω ) s m ¯ Dm + ω s m ¯ Df = f ( s ) s m ¯ Df .

So q ¯ Dmf can be calculated by m ¯ Df .

(27) q ¯ Dmf = λ ( m ¯ Df − m ¯ Dm ) = [ f ( s ) − ω ] s m ¯ Df ,

where f ( s ) = ω ( 1 − ω ) s + λ ( 1 − ω ) s + λ .

From Eq. (27), we can obtain the inter-porosity transfer rate in the Laplace domain. After that, Stehfest inversion [27] is employed to convert the inter-porosity transfer rate into physical domain.

2.2.2 Unsteady state model

The mathematical model of the unsteady state model is still composed of the seepage equation of fracture and the seepage equation of matrix. The equation of the fracture is the same as that of the quasi-steady state model. But the equation of matrix is different between the quasi-steady state model and the unsteady state model.

As mentioned above, the unsteady state models can be divided into spherical, plate, and cylindrical models. We will study the spherical model in this section. The formula derivation process for the other two models is in Appendix A.

As shown in Figure 3(a), each sphere has the same shape, and the radius is r 1 . The pressure distribution in the matrix is spherically symmetric. On the surface of a sphere, the matrix pressure p m is equal to the fracture pressure p f . The shape factor α = 15 / r 1 2 . The inter-porosity flow factor is as follows:

(28) λ = α r w 2 K m K f = 15 r w 2 r 1 2 K m K f .

The inter-porosity transfer rate can be expressed as follows:

(29) q mf = − 3 r 1 ρ K m μ ∂ p m ∂ r r = r 1 .

Define two dimensionless parameters here:

For fracture system, r D = r / r w , and for matrix system, r 1D = r / r 1 .

The dimensionless equation of the spherical model is as follows:

(30) ∂ 2 m Df ∂ r D 2 + 1 r D ∂ m Df ∂ r D − λ 5 ∂ m Dm ∂ r 1D r 1D = 1 − ω ∂ m Df ∂ t D = 0 ∂ 2 m Dm ∂ r 1 D 2 + 2 r 1 D ∂ m Dm ∂ r 1D = 15 ( 1 − ω ) λ ∂ m Dm ∂ t D ∂ m Dm ∂ r 1D r 1D = 0 = 0 m Dm ∣ r 1D = 1 = m Df ,

where W 2 = 15 ( 1 − ω ) s λ .

The Laplace transform is performed on Eq. (30).

(31) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − λ 5 ∂ m ¯ Dm ∂ r 1 D r 1D = 1 − ω s m ¯ Df = 0 ∂ 2 m ¯ Dm ∂ r 1 D 2 + 2 r 1D ∂ m ¯ Dm ∂ r 1D = W 2 m ¯ Dm ∂ m ¯ Dm ∂ r 1D r 1D = 0 = 0 m ¯ Dm ∣ r 1 D = 1 = m ¯ Df .

It can be obtained after solving Eq. (31):

(32) m ¯ Dm = 1 e W − e − W e W r 1D − e − W r 1D r 1 D m Df .

If the dimensionless transform and Laplace transform are performed on Eq. (29), then we can obtain

(33) q ¯ Dmf = 3 r w 2 r 1 2 K m K f ∂ m ¯ Dm ∂ r 1D r 1 D = 1 = λ 5 ∂ m ¯ Dm ∂ r 1D r 1D = 1 .

This is the formula of inter-porosity transfer rate in the Laplace domain. As discussed in Section 2.2.1, we need to research the relationship of q ¯ Dmf and m ¯ Df .

According to Eq. (32), we can obtain

(34) ∂ m ¯ Dm ∂ r 1D r 1D = 1 = ( W coth W − 1 ) m ¯ Df .

Substituting Eq. (34) into the first term of Eq. (31), we have

(35) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − s f ( s ) m ¯ Df = 0 ,

where f ( s ) = ω + λ 5 s 15 ( 1 − ω ) s λ coth 15 ( 1 − ω ) s λ − 1 .

Based on the above discussion, we can obtain

(36) q ¯ Dmf = λ 5 ∂ m ¯ Dm ∂ r 1D r 1D = 1 = [ f ( s ) − ω ] s m ¯ Df .

2.3 Summary of mathematical model

The inter-porosity transfer rate of both quasi-steady state model and unsteady state models can be calculated by Eq. (37). It is just that f(s) has different forms. Table 1 shows the forms of f(s) for different models.

(37) q ¯ Dmf = [ f ( s ) − ω ] s m ¯ Df .

Table 1

The f ( s ) of steady-state model and unsteady state models

Type of model f ( s )
Steady state ω ( 1 − ω ) s + λ ( 1 − ω ) s + λ
Spherical model ω + λ 5 s 15 ( 1 − ω ) s λ coth 15 ( 1 − ω ) s λ − 1
Plate model ω + λ ( 1 − ω ) 3 s tanh 3 ( 1 − ω ) s λ
Cylindrical model ω + 2 ( 1 − ω ) λ 15 s I 1 15 ( 1 − ω ) s λ I 0 15 ( 1 − ω ) s λ

The solution steps are as follows: first, calculate the fracture pseudo-pressure m ¯ Df ; then, calculate the inter-porosity transfer rate q ¯ Dmf ; and finally, use the Stehfest inversion [27] to transform the inter-porosity transfer rate from the Laplace domain to the physical domain.

2.4 The solution of producing degree of matrix

As mentioned in Section 1, the producing degree of matrix is defined as the ratio of gas from the matrix to total gas production. It can reflect the degree of gas development in the matrix. The formula is as follows:

(38) C m = Q m Q t ,

where C m is the producing degree of matrix and Q m is the gas from matrix. Q t is the total gas production.

The radius of the gas reservoir is r e, and the radius of the wellbore is r w. In order to calculate the producing degree of matrix, the gas reservoir is divided into n rings, which are shown in Figure 5. The ring width is dr. As discussed above, the inter-porosity transfer rate of each ring can be calculated. The sum of n rings is the total amount of gas from the matrix entering the fractures. The further formula to calculate the producing degree of matrix is as follows:

(39) C m = ∫ r w r e 2 π r h q mf d r Q t .

Figure 5 Schematic diagram of gas reservoir division.
Figure 5

Schematic diagram of gas reservoir division.

In order to present the work clearer, the specific steps are presented by a flowchart, which is shown in Figure 6.

Figure 6 The specific steps of the work.
Figure 6

The specific steps of the work.

3 Typical curves analysis

The main object of this article is to study the inter-porosity transfer rate and producing degree of matrix when gas reservoirs are producing. The typical curves are an appropriate tool to identify the flow stage and characteristics involved in the problem being investigated [28]. Hence, we plot the typical curves of inter-porosity transfer rate and producing degree of matrix in this section.

3.1 Typical curves of inter-porosity transfer rate

When storage ratio ω = 0.12, inter-porosity flow factor λ = 10−6, r eD = 1,000 (r eD = r e/r w), and r D = 1, under a closed boundary, plot the typical curves of inter-porosity transfer rate.

Here, r D is equal to 1, which means the inter-porosity transfer rate is near the wellbore. When r d takes different values, the inter-porosity transfer rate at different positions can be calculated.

Figure 7(a) represents the quasi-steady state model, and Figure 7(b) represents the three kinds of unsteady state models.

Figure 7 Inter-porosity transfer rate of quasi-steady state model and unsteady state models: (a) quasi-steady state model and (b) unsteady state models.
Figure 7

Inter-porosity transfer rate of quasi-steady state model and unsteady state models: (a) quasi-steady state model and (b) unsteady state models.

As shown in Figure 7(a), the typical curve of inter-porosity transfer rate near the wellbore can be divided into three stages.

Stage I is the straight line stage. At this time, the gas reservoir starts to produce. The gas production is mainly supplied by the gas originally held in the wellbore. Affected by this, the inter-porosity transfer rate curve is a straight line.

Stage II is the fracture radial flow stage and the transition stage. In this stage, the pressure in the fracture system decreases, while the pressure in the matrix system is almost constant. The pressure difference between the matrix system and the fracture system increases. Therefore, the inter-porosity transfer rate gradually increases. After that, the gas reservoir goes to the transition stage. With continuous production, the pressure in the matrix system decreases gradually. The pressure difference becomes smaller, so the inter-porosity transfer rate begins to decrease.

Stage III is the total radial flow stage. At this stage, the pressure has propagated to the external boundary. The inter-porosity transfer rate stays constant.

As shown in Figure 7(b), the inter-porosity transfer rate of unsteady state models is different from the quasi-steady state model. In the unsteady state models, the inter-porosity transfer between matrix and fractures is transient and fast, which is regarded as an unsteady process. As can be seen in Figure 7(b), the straight line stage of unsteady steady models ends earlier. The biggest difference between the quasi-steady state model and the unsteady state model happens in stage II. In the unsteady state models, the inter-porosity transfer rate first reaches its peak in a short time and then gradually decreases, which is contrary to the quasi-steady state model. The typical curves are the same in stage III between the quasi-steady state model and unsteady state models.

Comparing the three kinds of unsteady models, it can be found that their inter-porosity transfer rate is slightly different in stage II. The inter-porosity transfer rate of the spherical model is the largest. The second largest is the cylindrical model. The inter-porosity transfer rate of plate model is smaller than that of spherical model and cylindrical model.

3.2 Typical curves of producing degree of matrix

When ω = 0.12, λ = 10−6, and r eD = 1,000, plot the typical curves of producing degree of matrix. Figure 8(a) represents the producing degree of matrix for the quasi-steady state model. Since the difference in transfer rate between the three unsteady models is extremely tiny, only the spherical model is selected here for comparison with the quasi-steady model. Figure 8(b) represents the comparison between the quasi-steady state model and the spherical model.

Figure 8 Producing degree of matrix for quasi-steady state model and unsteady state models. (a) Quasi-steady state model. (b) A comparison between quasi-steady state model and spherical model.
Figure 8

Producing degree of matrix for quasi-steady state model and unsteady state models. (a) Quasi-steady state model. (b) A comparison between quasi-steady state model and spherical model.

Figure 8(a) shows that the typical curve can be divided into three stages. When t D from 0 to 103, the producing degree of matrix increases extremely slowly. When t D from 103 to 106, it increases rapidly to 0.7. After that, it remains at 0.7. It indicates that the total amount of gas from matrix entering fractures remains invariable when t D > 106. We can also draw the conclusion that 70% of the total production comes from the gas in the matrix.

It can be seen from Figure 8(b) that the variation trend of the typical curve in the unsteady state model is the same as that in the quasi-steady state model. But the typical curve of the unsteady state model begins to increase rapidly when t D = 102. It indicates that the transfer occurs earlier in the unsteady model. The growth rate of producing degree of matrix is less than that of the quasi-steady state model, so it reaches the constant state later.

3.3 Model validation

In order to verify the accuracy of our model, a comparison between our model and the model of Hassanzadeh is presented here. Hassanzadeh et al. [29] studied the shape factor in dual-porosity systems under a constant rate at the wellbore. He calculated the matrix–fracture transfer rate in infinite boundary based on their research findings. In our model, we consider the effect of wellbore constant and skin factor. But when r eD is large enough, the effects of those become small. In this situation, our model is similar to the model of Hassanzadeh. Hence, we make ω = 0.03, λ = 10−6, and r D = 1,000 and plot the typical curve of inter-porosity transfer rate in the plate model. The parameters that we choose are the same as those in Hassanzadeh’s model. The comparison result is shown in Figure 9. It can be seen that the calculation results of the two models match well. This comparison indicates our model is accurate.

Figure 9 The comparison result between our model and Hassanzadeh’s model.
Figure 9

The comparison result between our model and Hassanzadeh’s model.

4 Sensitivity analysis

Gas reservoir radius r eD, storage ratio ω, and inter-porosity flow factor λ are all key parameters for the inter-porosity transfer rate and producing degree of matrix. Different parameters will greatly change the curve’s shape. In this section, we take quasi-steady state model as an example to analyze the influence of key parameters.

4.1 The influence of gas reservoir radius

As discussed above, the inter-porosity transfer rate at various positions is different. When ω = 0.12, λ = 10−6, gas reservoir radius r eD = 1,000, and making r D = 1, 10, 100, and 1,000, under a closed boundary, plot the transfer rate curves, which are shown in Figure 10. It is found that the inter-porosity transfer rate near the wellbore is the largest. The further away from the wellbore, the smaller the inter-porosity transfer rate is. But when pressure propagates to the external boundary, the inter-porosity transfer rate is equal in the whole gas reservoir.

Figure 10 Typical curves of inter-porosity transfer rate at different positions of gas reservoirs.
Figure 10

Typical curves of inter-porosity transfer rate at different positions of gas reservoirs.

Then, we take the inter-porosity transfer rate near the wellbore as an example to study the influence of gas reservoir radius. When ω = 0.12, λ = 10−6, r D = 1, making r eD = 1,000, 1,500, and 2,000, plot the transfer rate curves, which are shown in Figure 11. It can be seen that the gas reservoir radius has no influence on the inter-porosity transfer rate in the initial and middle periods. In the late period, when the gas reservoir radius increases, the inter-porosity transfer rate becomes smaller. It can also be found that the typical curves become horizontal line later. It is because the pressure propagates to the external boundary later when the gas reservoir radius increases.

Figure 11 The influence of gas reservoir radius on inter-porosity transfer rate near the wellbore.
Figure 11

The influence of gas reservoir radius on inter-porosity transfer rate near the wellbore.

4.2 The influence of storage ratio

When λ = 10−6, r D = 1, r eD = 1,000 making ω = 0.03, 0.06 and 0.12, plot the typical curves. Figure 12 shows the influence of storage ratio on inter-porosity transfer rate. When ω becomes larger, the inter-porosity transfer rate decreases, and the curves move down. Figure 13 shows the influence of storage ratio on producing degree of matrix. When ω becomes larger, the producing degree of matrix decreases. Meanwhile, it becomes stable later. The above situation indicates that the inter-porosity transfer between the matrix and fractures will occur later when ω becomes larger. Besides, the total amount of inter-porosity transfer will decrease, which leads the gas in the matrix to be less developed.

Figure 12 The influence of ω on inter-porosity transfer rate.
Figure 12

The influence of ω on inter-porosity transfer rate.

Figure 13 The influence of ω on producing degree of matrix.
Figure 13

The influence of ω on producing degree of matrix.

4.3 The influence of inter-porosity flow factor

When ω = 0.12, r D = 1, r eD = 1,000, and making λ = 10−4, 10−5, and 10−6, plot the typical curves. Figure 14 shows the influence of λ on the inter-porosity transfer rate. Before the curves become horizontal, when λ becomes smaller, the inter-porosity transfer rate decreases. But when t D ≥ 4 × 106, the three curves are stable at the same value. At this time, λ does not influence the inter-porosity transfer rate. Meanwhile, when λ becomes smaller, the curves become horizontal later. Figure 15 shows the influence of λ on producing degree of matrix. It can be found that the shape of the three curves does not change. But the curves shift to the right with smaller λ. The above situation indicates that λ only influences the time of inter-porosity transfer. But it does not influence the total amount of inter-porosity transfer. When λ becomes smaller, it is more difficult for the matrix system to supply gas to the fracture system. The gas flow from matrix to fractures will occur later.

Figure 14 The influence of λ on inter-porosity transfer rate.
Figure 14

The influence of λ on inter-porosity transfer rate.

Figure 15 The influence of λ on producing degree of matrix.
Figure 15

The influence of λ on producing degree of matrix.

5 Case analysis

Shapingchang gas field is situated in Sichuan, western China. The main gas reservoirs in Shapingchang are carboniferous carbonate gas reservoirs. These gas reservoirs are fractured-porous gas reservoirs with medium to low porosity and permeability. Well X6 and Well Y6 in the Shapingchang gas field are exemplified in this article.

The production characteristics curves of Well X6 and Well Y6 are plotted in Figures 16 and 17. The blue curve represents the gas production per month. The red curve represents the cumulative gas production since the well was opened. The production characteristics of X6 and Y6 are analyzed through the curves.

Figure 16 The production characteristics curve of Well X6.
Figure 16

The production characteristics curve of Well X6.

Figure 17 The production characteristics curve of Well Y6.
Figure 17

The production characteristics curve of Well Y6.

According to the logging data, it is found that the fractures are relatively developed in Well X6, with 192 effective fractures in total. The average fracture density is 5.5 pieces/m, the average porosity is 6.23%, and the storage ratio is 0.11. According to Figure 16, it is found that the initial production of Well X6 is very high, about 1,500 × 104 to 2,000 × 104 m3/month. But the high production period is only 56 months, and then the production drops rapidly. After 50 months, the monthly average gas production decreased by 87.9%.

The number of fractures in Well Y6 is small, and there are 30 effective fractures in total. Its average porosity is 5.9%, and the storage ratio is 0.04. According to Figure 17, it is found that the initial production of Well Y6 is relatively low, about 780 × 104 m3/month. The high production period is about 90 months. Compared with well X6, its production decreased slowly, and the monthly average gas production decreased by 57.2% after 50 months.

According to the typical curve of producing degree of matrix in Section 3, the producing degree of matrix is very small in the early period. It indicates that the gas production mainly comes from fractures. Because the fracture conductivity is strong, the initial gas production is high. However, the gas reserve in the fractures is small. Besides, the inter-porosity transfer rate is also low in the early period. Therefore, the gas supply from the matrix to fractures is very slow, which results in a rapid reduction in gas production. During the mid-late period, the producing degree of matrix is large, demonstrating that the matrix provides the majority of the gas production. The gas well maintains a long time of low-speed production. The actual production characteristics are in line with the previous discussion.

The storage ratio of X6 is large, so the inter-porosity transfer rate is relatively small. Therefore, the gas supplied by the matrix is relatively low, which leads to gas production decreasing rapidly. The storage ratio of Y6 is small, indicating that the inter-porosity transfer rate is relatively large. So the gas supplied by the matrix is relatively high, which leads to gas production decreasing slowly. The actual production characteristics are also consistent with the sensitivity analysis.

6 Summary and conclusions

The inter-porosity transfer rate and producing degree of matrix are very essential research topics for the development of fractured-porous gas reservoirs. In this article, we establish a practical analytical solution for calculating the inter-porosity transfer rate and producing degree of matrix. Some conclusions are summarized as follows:

  1. We establish a concise formula for the inter-porosity transfer rate. By adjusting f(s), the formula is applicable to the quasi-steady state model and unsteady state model. Then, by summing the inter-porosity transfer rate at different positions of the gas reservoir, we obtain the solution for the producing degree of matrix.

  2. In the initial period of production, the producing degree of matrix is close to zero, indicating that there is almost no inter-porosity transfer. Then, the transfer rate begins to increase. In the late period, the producing degree of matrix is large, which means that most of the gas production comes from the matrix.

  3. The inter-porosity transfer rate near the wellbore is the largest. The further away from the wellbore, the smaller the inter-porosity transfer rate. In comparison to the quasi-steady state model, the inter-porosity transfer occurs earlier in the unsteady model. But when pressure propagates to the external boundary, the inter-porosity transfer rate is equal between the quasi-steady and unsteady models.

  4. The inter-porosity transfer can be significantly influenced by the gas reservoir radius, storage ratio (ω), and inter-porosity flow factor λ. When ω increases, the inter-porosity transfer rate and producing degree of matrix both decrease. Meanwhile, the transfer occurs later. When λ increases, the transfer occurs earlier. But it does not influence the total amount of inter-porosity transfer. The gas reservoir radius mainly influences the late period of the inter-porosity transfer.

  5. Two fractured-porous gas reservoirs from Shapingchang are analyzed. The producing characteristics can be explained by our findings.

The model in this article is developed based on the dual-porosity approach. The gas in the matrix that directly flows into the wellbore is negligible. However, the matrix permeability may be large in some fractured-porous gas reservoirs. In this case, our model may be inaccurate.

In this study, the fluid in the formation is single phase. In some fractured-porous gas reservoirs, there may be water in the late period during production. Hence, the fluid in the formation is two phase. Our model is not applicable to the two-phase flow. In view of this, we will develop the two-phase flow model in the future.

Acknowledgments

The authors gratefully acknowledge the funding support from the National Science and Technology Major Project (Grant No. 2017ZX05009005-002).

  1. Funding information: The funding support was granted from the National Science and Technology Major Project (Grant No. 2017ZX05009005-002).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Appendix A

A1 The inter-porosity transfer rate of cylindrical model

The shape of each cylinder is the same, and the radius is r 1 . The shape factor of cylindrical matrix is α = 15 / r 1 2 . The inter-porosity flow factor is as follows:

(A1) λ = α r w 2 K m K f = 15 r w 2 r 1 2 K m K f .

The inter-porosity transfer rate can be expressed by the following formula:

(A2) q mf = − 2 r 1 ρ K m μ ∂ p m ∂ r r = r 1 .

The dimensionless seepage equation is as follows:

(A3) ∂ 2 m Df ∂ r D 2 + 1 r D ∂ m Df ∂ r D − 2 15 λ ∂ m Dm ∂ r 1D r 1D = 1 − ω ∂ m Df ∂ t D = 0 ∂ 2 m Dm ∂ r 1D 2 + 1 r 1 D ∂ m Dm ∂ r 1D = 15 ( 1 − ω ) λ ∂ m Dm ∂ t D ∂ m Dm ∂ r 1D r 1D = 0 = 0 m Dm ∣ r 1D = 1 = m Df ,

where r D = r / r w in fracture system and r 1D = r / r 1 in matrix system.

Laplace transform is performed on the second to the fourth terms of Eq. (A3).

(A4) ∂ 2 m ¯ Dm ∂ r 1D 2 + 1 r 1D ∂ m ¯ Dm ∂ r 1D = 15 ( 1 − ω ) λ s m ¯ Dm ∂ m ¯ Dm ∂ r 1D r 1D = 0 = 0 m ¯ Dm ∣ r 1D = 1 = m ¯ Df .

The solution of Eq. (A4) is

(A5) m ¯ Dm = I 0 r D 15 ( 1 − ω ) s λ I 0 15 ( 1 − ω ) s λ m ¯ Df .

After dimensionless and Laplace transform, Eq. (A2)can be written as follows:

(A6) q ¯ Dmf = 2 15 λ ∂ m ¯ Dm ∂ r 1D r 1D = 1 .

Laplace transform is performed on the first term of Eq. (A3).

(A7) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − 2 15 λ ∂ m ¯ Dm ∂ r 1D r 1D = 1 − ω s m ¯ Df = 0 .

According to Eq. (A5), we can obtain

(A8) ∂ m ¯ Dm ∂ r 1D r 1D = 1 = 15 ( 1 − ω ) s λ I 1 15 ( 1 − ω ) s λ I 0 15 ( 1 − ω ) s λ m ¯ Df .

Substituting Eq. (A8) into Eq. (A7), we obtain

(A9) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − s f ( s ) m ¯ Df = 0 ,

where f ( s ) = ω + 2 ( 1 − ω ) λ 15 s I 1 15 ( 1 − ω ) s λ I 0 15 ( 1 − ω ) s λ .

Combining Eqs. (A6), (A7), and (A9), we can obtain

(A10) q ¯ Dmf = [ f ( s ) − ω ] s m ¯ Df .

A2 The inter-porosity transfer rate of plate model

Each plate has the same shape, and the thickness is h m . The shape factor of plate matrix is α = 12 / h m 2 . The inter-porosity flow factor is as follows:

(A11) λ = α r w 2 K m K f = 12 r w 2 h m 2 K m K f .

The inter-porosity transfer rate can be expressed by the following formula:

(A12) q mf = − 2 h m ρ K m μ ∂ p m ∂ z z = 0.5 h m .

After dimensionless and Laplace transform, Eq. (A12) can be written as follows:

(A13) q ¯ Dmf = λ 3 ∂ m ¯ Dm ∂ z D z D = 1 .

The seepage equation after Laplace transformation is as follows:

(A14) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − λ 3 ∂ m ¯ Dm ∂ z D z D = 1 − ω s m ¯ Df = 0 ∂ 2 m ¯ Dm ∂ z D 2 = 3 ( 1 − ω ) λ s m ¯ Dm ∂ m ¯ Dm ∂ z D z D = 0 = 0 m ¯ Dm ∣ z D = 1 = m ¯ Df ,

where r D = r / r w in fracture system and z D = ( z / 0.5 h m ) in matrix system.

From the second to the fourth terms of Eq. (A14), we can obtain

(A15) ∂ m ¯ Dm ∂ z D z D = 1 = 3 ( 1 − ω ) s λ tanh 3 ( 1 − ω ) s λ m ¯ Df .

Then, substitute Eq. (A15) into the first term of Eq. (14).

(A16) ∂ 2 m ¯ Df ∂ r D 2 + 1 r D ∂ m ¯ Df ∂ r D − s f ( s ) m ¯ Df = 0 ,

where f ( s ) = ω + λ ( 1 − ω ) 3 s tanh 3 ( 1 − ω ) s λ .

Combining Eqs. (A13), (A14), and (A16), the inter-porosity transfer rate of plate model is

(A17) q ¯ Dmf = [ f ( s ) − ω ] s m ¯ Df .

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Received: 2022-09-01
Revised: 2023-02-08
Accepted: 2023-04-24
Published Online: 2023-06-13

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2022-0247
Keywords for this article
matrix; fractures; transfer rate; producing degree; dual-porosity model
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