Pressure transient analysis of multistage fracturing horizontal wells with finite fracture conductivity in shale gas reservoirs

Abstract

A new analytical solution of pressure transient analysis is proposed for multistage fracturing horizontal well (MFHW) with finite-conductivity transverse hydraulic fractures in shale gas reservoirs. The effects of absorption, diffusion, viscous flowing, stress sensitivity, flow convergence, skin damage and wellbore storage are simultaneously considered as well in this paper. Laplace transformation, source sink function, perturbation method and superposition principle are, respectively, employed to solve related mathematical models of reservoir system and hydraulic fracture system. And then boundary element method (BEM) is applied to couple reservoir system and hydraulic fracture system. The transient pressure is inverted from Laplace space into real time space with Stehfest numerical inversion algorithm. Based on this new solution, the distribution of transient pseudo-pressure for MFHW with multiple finite-conductivity transverse hydraulic fractures is obtained. Different flowing regimes are identified, and the effects of relevant parameters are analyzed as well. The essence of this paper is considering the effects of transient gas flowing occurrence in finite-conductivity hydraulic fractures with BEM. Compared with some existing models and numerical simulation model of shale gas reservoirs, this proposed new model can provide a relative more accurate analysis of the relevant parameters, especially for the fracture conductivity. In conclusion, this new model provides the relative accurate and comprehensive evaluation results for multistage fracturing horizontal technology.

Appendix

The one-dimensional linear flow is assumed in the hydraulic fractures, the compressibility of fractures and gas is ignored, and the flow state in hydraulic fractures meets the continuous pseudo-state theory. Thus, the governing equation is presented as:

$$- \frac{\partial (\rho v)}{\partial x} + \rho \frac{{2k_{fi} }}{{\mu w_{F} }}\frac{{\partial p_{f} }}{\partial y}\left| {_{{y = y_{F} }} } \right. = 0,\quad (0 < x < x_{F} )$$

(A 1)

Based on the gas state equation, the gas density can be presented as follows:

$$\rho = \frac{pM}{ZRT}$$

(A 2)

Meanwhile, the gas pseudo-pressure is defined as the following formation:

$$\phi = \int_{{p_{0} }}^{p} {\frac{p}{Z\mu }} dp$$

(A 3)

Submitting Eqs. (A2) and (A3) into Eq. (A1), one can obtain:

$$\frac{{\partial^{2} \phi_{F} }}{{\partial x^{2} }} + \frac{{2k_{fi} }}{{k_{F} w_{F} }}\frac{\partial \phi }{\partial y}\left| {_{{y = y_{F} }} } \right. = 0,\quad (0 < x < x_{F} )$$

(A 4)

In this paper, we assume that the flow rate along the hydraulic fractures is variable. At arbitrary position (x _F, y _F), the flow rate from formation into hydraulic fractures can be obtained applying the Darcy’s law:

$$\frac{{p_{sc} T}}{{271.44k_{fi} hT_{sc} }}q(x_{F} ,y_{F} ) = \Delta x\frac{{\partial \phi_{f} }}{\partial y}\left| {_{{y = y_{F} }} } \right.$$

(A 5)

where Δx represents the microelements, and we make an assumption that the flux density is constant in this microelement.

In terms of the inner boundary condition, we assume that the well keeps constant production rate condition. Similarly, the inner boundary condition can be obtained with Darcy’s law:

$$w_{F} \frac{{\partial \phi_{F} }}{\partial x}\left| {_{{x_{F} = 0}} } \right. = q_{F} \frac{{p_{sc} T}}{{271.44k_{Fi} hT_{sc} }}$$

(A 6)

where q _F is the total production rate of the hydraulic fractures, sm³/d.

With the definition of these dimensionless variables in Table 1, the above equations from Eqs. (A4)–(A6) with the formula of dimensionless can be presented as follows:

$$\frac{{\partial^{2} \phi_{FD} }}{{\partial x_{D}^{2} }} + \frac{2}{{C_{FD} }}\frac{{\partial \phi_{D} }}{{\partial y_{D} }}\left| {_{{y_{D} = y_{FD} }} } \right. = 0,\quad (0 < x_{D} < 1)$$

(A 7)

And the boundary conditions [Eqs. (A5) and (A6)] also can be converted into the dimensionless formation:

$$q_{D} (x_{FD} ,y_{FD} ) = - \frac{2}{\pi }\frac{{\partial \phi_{D} }}{{\partial y_{D} }}\left| {_{{y = y_{FD} }} } \right.$$

(A 8)

$$\frac{{\partial \phi_{FD} }}{{\partial x_{D} }}\left| {_{{x_{FD} = 0}} } \right. = - \frac{{\pi q_{FD} }}{{C_{FD} }}$$

(A 9)

The region that contains the hydraulic fractures and its vicinity is supported by proppants. Therefore, the stress sensitivity can be ignored in this region. The dimensionless pseudo-pressure distributed throughout this region in Laplace domain is as follows:

$$\frac{{\partial^{2} \overline{{\eta_{FD} }} }}{{\partial x_{D}^{2} }} + \frac{2}{{C_{FD} }}\frac{{\partial \overline{{\eta_{D} }} }}{{\partial y_{D} }}\left| {_{{y_{D} = y_{FD} }} } \right. = 0,\quad (0 < x_{D} < 1)$$

(A 10)

The boundary conditions [Eqs. (A8) and (A9)] also can be converted into the dimensionless formation:

$$\overline{{q_{D} }} (x_{FD} ,y_{FD} ) = - \frac{2}{\pi }\frac{{\partial \overline{{\eta_{D} }} }}{{\partial y_{D} }}\left| {_{{y = y_{FD} }} } \right.$$

(A 11)

$$\frac{{\partial \overline{{\eta_{FD} }} }}{{\partial x_{D} }}\left| {_{{x_{FD} = 0}} } \right. = - \frac{{\pi \overline{{q_{FD} }} }}{{C_{FD} }}.$$

(A 12)