Introduction

With the growing shortage of domestic and foreign energy industry, producing gas from shale gas reservoirs is currently received great attention due to their potential to supply the entire world with sufficient energy for the decades to come (Wang and Krupnick 2013). However, low gas recovery rate from unconventional shale resources remains the main technical difficulty. It is estimated that only 10–30 % of GIP can be recovered from these unconventional reservoirs. Most of the gas production wells can be low or no production capacity without hydraulic fracturing or horizontal well drilling. A shale gas reservoir is characterized as an organic-rich deposition with extremely low matrix permeability and clusters of mineral-filled “natural” fractures. Through core experiment analysis on 152 cores of nine reservoirs in North America, Javadpour found that the permeability of shale bedrock is mostly 54 nd (nano-Darcy) and about 90 % are less than 150 nd (Javadpour et al. 2007; Javadpour 2009). Most of the pore diameters are in the range of 4–200 nm (10−9 m). There are three forms for the gas to store in the shale: (1) free gas stored in the fractures and pores; (2) adsorption gas stored on the surface of the bedrock, in which Hill and Nelson estimated that 20–85 % of gas in shale is stored under this condition (Hill and Nelson 2000); (3) dissolved gas stored in the kerogen (Javadpour 2009).

Hydraulic fracturing has been used to create high conductivity flow path to improve the productivity of low permeability shale reservoirs (Waters et al. 2009; Rahm 2011; Arthur et al. 2009; Vermylen and Zoback 2011). Well test is an efficient method to understand gas flow mechanisms in unconventional shale and for better fracturing design (Prat 1990; Serra 1981; Bumb and McKee 1988; Brown et al. 2011; Bello and Wattenbarger 2010; Bello 2009). Bumb and McKee (1988) have considered the effect of desorption for shale gas well test. Brown et al. (2011) have studied PTA for fractured horizontal wells in unconventional shale reservoirs. Bello and Wattenbarger have studied RDA for multi-stage hydraulically fractured horizontal shale gas well (Bello and Wattenbarger 2010; Bello 2009).

However, there are few studies on well test and rate decline analysis for fractured vertical wells in shale gas reservoirs. Most of the research focused on the pressure response of infinite conductivity fractured wells in infinite formation (Standford 1974; Rosa and Carvalho 1989; Cinco-Ley and Samaniego 1981). So, it is necessary to study the pressure response for hydraulic fractured vertical wells, which is meaningful for effective shale gas production and deep understanding about gas flow mechanisms in unconventional shale reservoirs.

In this paper, based on shale gas flow mechanisms, such as desorption, diffusion, and flow in the fractures, the mathematical model which considers the effect of outer boundary effects for hydraulic fractured vertical wells with finite conductivity has been constructed. This model also considers Fick’s first law of diffusion, Langmuir isothermal adsorption (Yin 1991), and stress sensitivity of natural fracture. Through point-source function method (Wei et al. 1999), the single-phase shale gas flow in matrix and fracture has been studied based on Fick’s pseudo-steady state diffusion model (Cohen and Murray 1981). Sensitivity analysis has been carried out for significant factors, such as desorption constant, fracture conductivity coefficient, storage capacity ratio, and inter-porosity flow coefficient. The effect of the above factors on the PTA and RDA has been presented.

Mathematical model

Assumptions

  1. 1.

    The shale gas formation consists of matrix system and fracture system. And there exists heterogeneity in the fracture system: the horizontal permeability and vertical permeability of the fracture system are different \(k_{\text{fh}} \ne k_{\text{fv}}\);

  2. 2.

    Free shale gas exists in the fracture system and can be described using Darcy law;

  3. 3.

    Free gas and adsorption gas coexist in the matrix system;

  4. 4.

    The gas flow in the matrix system can be ignored. The gas in the matrix system first will desorb and then diffuse into the fracture system, which can be characterized using pseudo-steady state diffusion;

  5. 5.

    Single layer Langmuir isothermal curve can be used to describe the gas desorption from the matrix system;

  6. 6.

    Dynamic balance exists between adsorption gas and free gas;

  7. 7.

    The gas well is at constant production rate q sc in standard condition;

  8. 8.

    The effect of gravity and capillary pressure can be ignored.

Mathematical model

Based on the above assumptions, combined with Darcy law, gas equation of state (EOS), and mass balance equation, the mathematical model can be established, which consists of desorption of shale gas, diffusion, and flow characteristics.

$$\frac{\partial }{\partial x}\left( {k_{\text{fh}} \frac{{p_{\text{f}} }}{\mu Z}\frac{{\partial p_{\text{f}} }}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {k_{\text{fh}} \frac{{p_{\text{f}} }}{\mu Z}\frac{{\partial p_{\text{f}} }}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {k_{\text{fv}} \frac{{p_{\text{f}} }}{\mu Z}\frac{{\partial p_{\text{f}} }}{\partial z}} \right) = \varphi_{\text{f}} c_{\text{gf}} \frac{{p_{\text{f}} }}{Z}\frac{{\partial p_{\text{f}} }}{\partial t} + \frac{{p_{\text{sc}} }}{{T_{\text{sc}} }}\frac{\partial V}{\partial t}$$
(1)

In the above Eq. (1), the second term in the right hand reflected the effect of the gas desorption.

The pseudo-pressure can be defined as follows:

$$\psi_{\text{f}} \left( {p_{\text{f}} } \right) = 2\int_{{p_{0} }}^{{p_{\text{f}} }} {\frac{p}{\mu Z}} dp$$
(2)

Substitute Eqs. (2) into (1), we can obtain:

$$\frac{\partial }{\partial x}\left( {k_{\text{fh}} \frac{{\partial \psi_{\text{f}} }}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {k_{\text{fh}} \frac{{\partial \psi_{\text{f}} }}{\partial y}} \right) + \frac{\partial }{\partial z}\left( {k_{\text{fv}} \frac{{\partial \psi_{\text{f}} }}{\partial z}} \right) = \varphi_{f} \mu_{i} c_{{{\text{gf}}i}} \frac{{\partial \psi_{\text{f}} }}{\partial t} + \frac{{2p_{\text{sc}} T}}{{T_{\text{sc}} }}\frac{\partial V}{\partial t}$$
(3)

According to Fick’s first law of diffusion, if the matrix is assumed as sphere, the gas diffusion rate of shale matrix per unit time and per unit volume can be expressed as follows:

$$\frac{\partial V}{\partial t} = \frac{{6D\pi^{2} }}{{R^{2} }}\left[ {V_{\text{E}} \left( {\psi_{\text{f}} } \right) - V} \right]$$
(4)

Also, we defined dimensionless variables as shown in Table 1.

Table 1 Definitions of the dimensionless variables
Full size table

The dimensionless forms for Eqs. (3) and (4) can be given as follows:

$$\frac{{\partial^{2} \psi_{\text{fD}} }}{{\partial x_{\text{D}}^{2} }} + \frac{{\partial^{2} \psi_{\text{fD}} }}{{\partial y_{\text{D}}^{2} }} + \frac{{\partial^{2} \psi_{\text{fD}} }}{{\partial z_{\text{D}}^{2} }} = \omega \frac{{\partial \psi_{\text{fD}} }}{{\partial t_{\text{D}} }} - \left( {1 - \omega } \right)\frac{{\partial V_{\text{D}} }}{{\partial t_{\text{D}} }}$$
(5)
$$\frac{{\partial V_{\text{D}} }}{{\partial t_{\text{D}} }} = \frac{{V_{\text{ED}} \left( {\psi_{\text{fD}} } \right) - V_{\text{D}} }}{\lambda }$$
(6)

Taking Laplace transformation to t D, Eqs. (5) and (6) become:

Then, Eqs. (5) and (6) can be changed to:

$$\frac{{\partial^{2} \overline{\psi }_{\text{fD}} }}{{\partial x_{\text{D}}^{2} }} + \frac{{\partial^{2} \overline{\psi }_{\text{fD}} }}{{\partial y_{\text{D}}^{2} }} + \frac{{\partial^{2} \overline{\psi }_{\text{fD}} }}{{\partial z_{\text{D}}^{2} }} = \omega u\overline{\psi }_{\text{fD}} - \left( {1 - \omega } \right)u\overline{V}_{\text{D}}$$
(7)
$$u\overline{V}_{\text{D}} { = }\frac{{\overline{V}_{{\text{ED}}} \left( {\psi_{{\text{fD}}} } \right) - \overline{V}_{\text{D}} }}{\lambda }$$
(8)

According to Langmuir isotherm adsorption equation:

$$\overline{V}_{{\text{ED}}} \left( {\psi_{{\text{fD}}} } \right) = L\left[ {V_{\text{E}} - V_{\text{i}} } \right] = L\left[ {\frac{{V_{\text{L}} p_{\text{f}} }}{{p_{\text{L}} + p_{\text{f}} }} - \frac{{V_{\text{L}} p_{i} }}{{p_{\text{L}} + p_{i} }}} \right]$$
(9)

Taking Laplace transformation to t D , Eq. (9) becomes:

$$\overline{V}_{{\text{ED}}} \left( {\psi_{{\text{fD}}} } \right) = - \frac{{V_{\text{L}} p_{\text{L}} }}{{\left( {p_{\text{L}} + p_{\text{f}} } \right)\left( {p_{\text{L}} + p_{i} } \right)}}\frac{{p_{{\text{sc}}} q_{{\text{sc}}} T}}{{\pi k_{{\text{fh}}} hT_{{\text{sc}}} }}\frac{{\mu_{i} Z_{i} }}{{2p_{i} }}\overline{\psi }_{{\text{fD}}}$$
(10)

With the definition of Adsorption index in Table 1, we can get:

$$V_{\text{ED}} \left( {\psi_{\text{fD}} } \right) = - \alpha \psi_{\text{fD}}$$
(11)

α is pseudo-pressure related, which can be assumed as a constant in the discussion range. And it is equal to the value in the initial reservoir state (Ertekin and Sung 1989; Anbarci and Ertekin 1990).

According to Eqs. (7), (8) and (11), we can obtain:

$$\frac{{\partial^{2} \overline{\psi }_{{\text{fD}}} }}{{\partial x_{\text{D}}^{2} }} + \frac{{\partial^{2} \overline{\psi }_{{\text{fD}}} }}{{\partial y_{\text{D}}^{2} }} + \frac{{\partial^{2} \overline{\psi }_{{\text{fD}}} }}{{\partial z_{\text{D}}^{2} }} = f(u)\overline{\psi }_{{\text{fD}}}$$
(12)
$$f\left( u \right) = \omega u + \frac{{\alpha u\lambda \left( {1 - \omega } \right)}}{u + \lambda }$$
(13)

Converting to spherical coordinates:

$$\frac{1}{{r_{\text{D}}^{2} }}\frac{\partial }{{\partial r_{\text{D}} }}\left( {r_{\text{D}}^{2} \frac{{\partial \overline{\psi }_{\text{fD}} }}{{\partial r_{\text{D}} }}} \right) = f\left( u \right)\overline{\psi }_{\text{fD}}$$
(14)

Inner boundary conditions:

$$\mathop {\lim }\limits_{{r_{\text{D}} \to 0}} r_{\text{D}} \frac{{\partial \overline{\psi }_{\text{fD}} }}{{\partial r_{\text{D}} }} = - 1$$
(15)

Three forms of outer boundary conditions have been considered in this paper:

Infinite outer boundary (IOB):

$$\mathop {\overline{\psi }_{\text{fD}} }\limits_{{r_{\text{D}} \to 0}} = 0$$
(16)

Constant pressure outer boundary (CPOB):

$$\psi_{\text{fD}} = 0$$
(17)

Closed outer boundary (COB):

$$\frac{{\partial \psi_{\text{fD}} }}{{\partial r_{\text{D}} }} = 0$$
(18)

Apply the Laplace transform to both sides of the above equation from Eqs. (15)–(18). Then combining Eq. (14), we can obtain the line source solution under constant production rate condition:

$$\overline{\psi }_{\text{f}} = \left\{ \begin{gathered} \frac{{p_{\text{sc}} T\mathop q\limits^{\sim } }}{{T_{\text{sc}} \pi k_{\text{fh}} h}}\frac{1}{u}K_{0} \left( {r_{\text{D}} \sqrt {f\left( u \right)} } \right)\begin{array}{*{20}c} {} & {} \\ \end{array} {\text{IOB}} \hfill \\ \frac{{p_{\text{sc}} T\mathop q\limits^{\sim } }}{{T_{\text{sc}} \pi k_{\text{fh}} h}}\frac{1}{u}\left[ {K_{0} \left( {r_{\text{D}} \sqrt {f\left( u \right)} } \right) - \frac{{K_{0} \left( {r_{\text{eD}} \sqrt {f\left( u \right)} } \right)}}{{I_{0} \left( {r_{\text{eD}} \sqrt {f\left( u \right)} } \right)}}I_{0} \left( {r_{\text{D}} \sqrt {f\left( u \right)} } \right)} \right]\begin{array}{*{20}c} {} & {} \\ \end{array} {\text{CPOB}} \hfill \\ \frac{{p_{\text{sc}} T\mathop q\limits^{\sim } }}{{T_{\text{sc}} \pi k_{\text{fh}} h}}\frac{1}{u}\left[ {K_{0} \left( {r_{\text{D}} \sqrt {f\left( u \right)} } \right) + \frac{{K_{1} \left( {r_{\text{eD}} \sqrt {f\left( u \right)} } \right)}}{{I_{1} \left( {r_{\text{eD}} \sqrt {f\left( u \right)} } \right)}}I_{0} \left( {r_{\text{D}} \sqrt {f\left( u \right)} } \right)} \right]\begin{array}{*{20}c} {} & {} \\ \end{array} {\text{COB}} \hfill \\ \end{gathered} \right.$$
(19)

Solution

The schematic diagram of hydraulic fractured vertical well in shale gas reservoirs is shown in Fig. 1. There is one vertical well with finite conductivity after hydraulic fractured in a shale gas reservoir with top and bottom boundary closed. The well is produced at a constant rate q sc. The reservoir pay zone is h. The hydraulic fracture totally fractured the whole reservoir, which means that the fracture height is h; The vertical cracks after hydraulic fracturing are symmetric against the wellbore. The fracture half length is x f and fracture width is W f. Fracture permeability is k f1; Assume that the gas only flows into the wellbore through the hydraulic fractures. The fracture tips are closed with no gas flow.

Fig. 1
figure 1

Schematic diagram of hydraulic fractured vertical well. The top and bottom boundary are closed

Full size image

Taking the case with IOB as an example, the pressure response can be obtained by superimposing line source solutions for shale gas reservoirs with top and bottom boundary closed:

$$\overline{\psi }_{\text{fD}} = \frac{1}{2}\int_{ - 1}^{1} {\overline{q}_{\text{fD}} \left( u \right)} K_{0} \left( {\sqrt {\left( {x_{\text{D}} - \xi } \right)^{2} + y_{\text{D}}^{2} } \sqrt {f\left( u \right)} } \right)d\xi$$
(20)

Here \(\overline{q}_{\text{fD}} = \frac{{q_{\text{fD}} }}{u}\)

Since the fracture is symmetric, following flow rate relationship can be obtained:

$$\int_{0}^{1} {\overline{q}_{\text{fD}} } {\text{d}}x{}_{\text{D}} = \frac{1}{u}$$
(21)

Consider the skin effect by hydraulic fracturing, the fracture pseudo-pressure \(\overline{\psi }_{\text{fD}}\) can be related to formation pseudo-pressure \(\overline{\psi }_{\text{D}}\) using following equation:

$$\overline{\psi }_{\text{fD}} \left( {x_{\text{D}} ,u} \right) = \overline{\psi }_{\text{D}} \left( {x_{\text{D}} ,y_{\text{D}} = u} \right) + \overline{q}_{\text{fD}} \left( {x_{\text{D}} ,u} \right) \times S_{\text{f}}$$
(22)

Ignoring the compressibility of fluids in the fracture, the following relationship can be obtained after combining boundary condition:

$$\overline{\psi }_{\text{wD}} - \overline{\psi }_{\text{f1D}} \left( {x_{\text{D}} ,y_{\text{D}} = 0,u} \right) = \frac{\pi }{{uC_{\text{fD}} }}\left[ {x_{\text{D}} - u\int_{0}^{{x_{\text{D}} }} {\int_{0}^{\eta } {\overline{q}_{\text{fD}} {\text{d}}x_{\text{D}} {\text{d}}\eta } } } \right]$$
(23)

According to Eqs. (20) and (22), the above equation can be written in following form:

$$\begin{gathered} \overline{\psi }_{\text{wD}} - \frac{1}{2}\int_{0}^{1} {\overline{q}_{\text{fD}} \left( {\xi ,u} \right)\left[ {K_{0} \left( {\left| {x_{\text{D}} + \xi } \right|\sqrt {f\left( u \right)} } \right) + K_{0} \left( {\left| {x_{\text{D}} - \xi } \right|\sqrt {f\left( u \right)} } \right)} \right]} {\text{d}}\xi \hfill \\ - S_{\text{f}} \times \overline{q}_{\text{fD}} + \frac{\pi }{{C_{\text{fD}} }}\int_{0}^{{x_{\text{D}} }} {\int_{0}^{\eta } {\overline{q}_{\text{fD}} {\text{d}}x_{\text{D}} {\text{d}}\eta } } = \frac{\pi }{{uC_{\text{fD}} }}x_{\text{D}} \hfill \\ \end{gathered}$$
(24)

Equation (24) can be solved using discrete numerical methods. As shown in Fig. 1, the dimensionless fracture half length can be divided into n fractions with step length Δx D. \(\overline{x}_{{{\text{D}}i}}\) is the center point of the ith discrete unit. x Di is the end point of the ith discrete unit. As the discrete unit is small enough, the flow rate can be assumed to be uniformly distributed. Equation (24) can be discretilized into following form:

$$\begin{gathered} \overline{\psi }_{\text{wD}} - \frac{1}{2}\sum\limits_{i = 1}^{n} {\overline{q}_{{{\text{fD}}i}} \int_{{x_{{{\text{D}}i}} }}^{{x_{{{\text{D}}i + 1}} }} {\left[ {K_{0} \left( {\left| {\overline{x}_{{{\text{D}}j}} + \xi } \right|\sqrt {f\left( u \right)} } \right) + K_{0} \left( {\left| {\overline{x}_{{{\text{D}}j}} - \xi } \right|\sqrt {f\left( u \right)} } \right)} \right]} {\text{d}}\xi } - S_{\text{f}} \times \overline{q}_{{{\text{fD}}j}} \hfill \\ + \frac{\pi }{{C_{\text{fD}} }}\left\{ {\sum\limits_{i = 1}^{j} {\overline{q}_{{{\text{fD}}i}} \left[ {0.5\Delta x_{\text{D}}^{ 2} + \Delta x_{\text{D}} \left( {\overline{x}_{{{\text{D}}j}} - i\Delta x_{\text{D}} } \right) + \overline{q}_{{{\text{fD}}j}} \frac{{\Delta x_{\text{D}}^{2} }}{8}} \right]} } \right\} = \frac{\pi }{{uC_{\text{fD}} }}x_{\text{D}} \hfill \\ \end{gathered}$$
(25)

And the flow rate equation can be discretilized as follows:

$$\Delta x_{\text{D}} \sum\limits_{i = 1}^{n} {\overline{q}_{{{\text{fD}}i}} \left( u \right)} = \frac{1}{u}$$
(26)

There are n + 1 unknown variables in Eqs. (25) and (26), that is \(\overline{\psi }_{\text{wD}}\) and \(\overline{q}_{{{\text{fD}}i}}\)(with \(i = 1,2, \ldots ,n\)). By solving these equations, in Laplace space, we can obtain the bottomhole pseudo-pressure \(\overline{\psi }_{\text{wD}}\) and the flow rate distribution in discrete units \(\overline{q}_{{{\text{fD}}i}}\).

Consider the effect of wellbore storage, according to Duhamel’s principle, the bottomhole pressure response is

$$\overline{\psi }_{\text{wD}} = \frac{{\overline{\psi }_{\text{wD}} }}{{1 + u^{2} C_{\text{D}} \overline{\psi }_{\text{wD}} }}$$
(27)

Using Stehfest inversion technique, we can obtain the bottomhole pseudo-pressure response in the real space.

Type curves and analysis

Type curves

Figure 2 is type curve for finite conductivity fractured vertical well in shale gas reservoirs with consideration of outer boundary effect in the log–log scale. The type curve can be divided into 8 stages: (1) the first stage is early afterflow period; the pseudo-pressure (PP) curve and pseudo-pressure-derivative (PPD) curve coincide with each other with the slope equal to 1; (2) the second stage is afterflow transition period; (3) the third stage is the bilinear flow period with slope of PPD curve equal to ¼; (4) the fourth stage is formation linear flow period with slope of PPD equal to ½; (5) the fifth stage is pseudo radial flow of natural fractures with PPD equal to 0.5; (6) the sixth stage is the channeling period from matrix to fracture, and there is a concave in the PPD curve; (7) the seventh stage is the formation pseudo radial flow period with PPD equal to 0.5; (8) the eighth period is boundary response period. These scenarios will appear for different kinds of boundary conditions: PP curve and PPD curve will become upturned when the boundary is closed; otherwise, they will become downturned when the boundary is at constant pressure.

Fig. 2
figure 2

Type curve for hydraulic fractured vertical wells with finite conductivity in shale gas reservoirs under different kinds of boundary conditions

Full size image

Sensitivity analysis

Effect of adsorption coefficient

Figure 3 shows the effect of adsorption coefficient on pseudo-pressure response (PPR) and rate decline response (RDR). Other parameters used are listed in the figure. From Fig. 3, it can be found that adsorption coefficient mainly affects the gas diffusion period from matrix to fracture. The larger the value, the concave in the PPD curve is wider and deeper. This suggested that the more gas desorbed, the more apparent is the diffusion. It can be also found that the larger the adsorption coefficient, the higher is the gas production rate. This suggests that the more gas desorbed, the more apparent is the diffusion. Also, from the definition of the adsorption coefficient, we know that the larger the Langmuir Volume, the larger is the adsorption coefficient, which also denotes the effect of the Langmuir volume.

Fig. 3
figure 3

Effect of adsorption coefficient on pseudo-pressure response (PPR) and rate decline response (RDR)

Full size image

Effect of storage capacity ratio

Figure 4 shows the effect of storage capacity ratio on PPR and RDR. Other parameters used are listed in the figure. From Fig. 4, we can find that storage capacity ratio not only affect the channeling period, but also has effect on transition period, linear flow period and formation linear flow period. The smaller the value, in the PPD curve, the wider and deeper are the concave. However, the position of PP and PPD curve in the transition period, linear flow period and formation linear flow period is more upper. Also, it can be found that the smaller the value, the smaller gas flow rate will be.

Fig. 4
figure 4

Effect of storage capacity ratio on PPR and RDR

Full size image

Effect of channeling factor

Figure 5 shows the effect of channeling factor on PPR and RDR. Other parameters used are listed in the figure. From Fig. 5, we can find that channeling factor mainly affects the diffusion time from matrix to fracture. The larger the channeling factor, the earlier is the channeling start time, and the earlier is the appearance of the concave in the PPD curve. The larger the inter-porosity factor, the earlier is the matrix-fracture diffusion start time, and there will be a higher production rate.

Fig. 5
figure 5

Effect of channeling factor on PPR and RDR

Full size image

Effect of fracture conductivity

Figure 6 shows the effect of fracture conductivity on PPR and RDR. Other parameters used are listed in the figure. From Fig. 6, we can find that the fracture conductivity mainly affects the transition period and early bilinear flow period. The larger the fracture conductivity, the characteristic of the early bilinear flow period is less clear. The PP and PPD curve are in lower position. When the value is larger enough, we cannot catch the early bilinear flow period. And the curve is close to the well-testing curve for infinite conductivity fractured well. Also, it can be found that the higher the fracture conductivity, the higher is the gas production rate.

Fig. 6
figure 6

Effect of fracture conductivity on PPR and RDR

Full size image

Effect of fracture skin factor

Figure 7 shows the effect of fracture skin factor on PPR and RDR. Other parameters used are listed in the figure. From Fig. 7, we can find that the fracture skin factor mainly affects the flow from formation to fracture. The larger the value, the higher is the PP curve and more apparent is the hump effect. The larger the fracture skin factor, the more serious damage is during the hydraulic fracturing, and the lower the gas production rate is.

Fig. 7
figure 7

Effect of fracture skin factor on PPR and RDR

Full size image

Effect of stress sensitivity

Figure 8 shows the effect of fracture stress sensitivity on PPR and RDR. Other parameters used are listed in the figure. From Fig. 8, we can find that the fracture stress sensitivity mainly affects the early period. The larger the value, the higher is the PP curve and the horizontal line of pressure curve is missing. And the larger the stress sensitivity, the higher is the production rate.

Fig. 8
figure 8

Effect of stress sensitivity on PPR and RDR

Full size image

Conclusions

In this paper, the pressure transient model and rate decline model for fractured vertical wells in shale gas reservoir have been presented, and the parameters’ sensitivity analysis has been studied. The following conclusions can be summarized:

  1. 1.

    A mathematical model describing fluid flow of hydraulic fractured well in shale gas reservoirs is established and the solutions, transient pressure and rate decline curves are analyzed.

  2. 2.

    When the well is produced at a constant rate, a bigger Langmuir volume (V L) will lead to a larger adsorption coefficient, which will cause a deeper depth of the trough during this period because big value of V L indicates that more gas will be desorbed during the interporosity flow period.

  3. 3.

    The number of fractures (M F) mainly affects the early flow regimes of the hydraulic fractured shale gas well, especially for the early linear and elliptical flow regimes. The more the M F, the smaller are the pressure drop and the pressure decreasing rate.

  4. 4.

    When the well is produced at a constant bottom-hole pressure, the bigger the Langmuir volume (V L) and well length, the greater the production rate will be in the later flow period.