After multistage fracturing, the flowback of fracturing fluid will cause two-phase flow through hydraulic fractures in shale gas reservoirs. With the consideration of two-phase flow and desorbed gas transient diffusion in shale gas reservoirs, a two-phase transient flow model of multistage fractured horizontal well in shale gas reservoirs was created. Accurate solution to this flow model is obtained by the use of source function theory, Laplace transform, three-dimensional eigenvalue method, and orthogonal transformation. According to the model’s solution, the bilogarithmic type curves of the two-phase model are illustrated, and the production decline performance under the effects of hydraulic fractures and shale gas reservoir properties are discussed. The result obtained in this paper has important significance to understand pressure response characteristics and production decline law of two-phase flow in shale gas reservoirs. Moreover, it provides the theoretical basis for exploiting this reservoir efficiently.

In order to obtain industrial gas flow, horizontal drilling and multistage hydraulic fracturing techniques have been widely used in SGR (shale gas reservoirs) development. Compared to the conventional reservoirs, SGR have their own uniqueness; for example, the shale is a self-sourcing reservoir in which the gas is stored either by compression or by adsorption on the surface of the solid material. Transport of gas in SGR is a complex multiscale flow, which is from macroscale to the molecular scale. Numerous researchers have studied migration mechanisms in SGR, but the two-phase flow caused by multistage hydraulic fracturing is usually ignored for analyzing and forecasting a shale gas well.

According to Hill and Nelson [

Some scholars employed new methods to analyze production decline of SGR. Medeiros et al. [

Until now, the studies of a multistage hydraulic fractured horizontal in SGR are mainly concentrated on single phase gas flow; the two-phase flow caused by flowback after hydrofracture in SGR does not attract much attention. In order to understand the effect of two-phase flow in SGR, a two-phase flow transient analysis model of multistage fractured horizontal well in SGR is presented, and the accurate solution to the flow model is obtained using source function theory [

Figure

The SGR are dual mechanism and with sealed boundary.

Two-phase flow exists in fractures and horizontal well.

All fractures are assumed to be equally spaced, to penetrate the formation completely, and to be perpendicular to the horizontal wellbore. Both the horizontal wellbore and the fractures are assumed to be infinitely conductive.

Ignoring the impact of gravity and capillary forces.

Two-phase flow in multistage hydraulic fractured horizontal well in SGR.

Governing flow equations of two-phase flow in fractures under three-dimensional Cartesian coordinates can be written as follows.

Gas flow equation in fractures is

Liquid flow equation in fractures is

The transient diffusion rate of the shale gas from the shale matrix

The two-phase flow pseudopressure is employed and defined as follows:

With (

Employing Taylor expansion to simplify (

Employing source function into (

To simplify the mathematical model, dimensionless parameters are defined as follows.

The dimensionless two-phase pseudopressure is

The dimensionless wellbore storage coefficient is

The dimensionless time is

The dimensionless coordinate value of

The dimensionless coordinate value of

The dimensionless coordinate value of

The dimensionless relative mass concentration is

The dimensionless radial distance in matrix system is

Using the dimensionless parameters, (

The two-phase flow mathematical model in Laplace space is obtained by taking the Laplace transformation [

With the condition of sealed boundary, the dimensionless form of the initial condition is

The dimensionless forms of the outer boundary conditions are

With Fick’s second law and the assumption of spherical matrix blocks, the transient diffusion rate equation can be written as follows [

Using dimensionless parameters and Laplace transform to rewrite (

For transient diffusion case, the center of the spherical matrix block can be treated as a no-flow boundary, so the inner boundary condition can be written as follows:

The concentration on the external surface of matrix block is in equilibrium with the pressure in the fracture. The outer boundary condition can be written as follows:

The general solution of (

With the boundary condition and general solution expression, the accurate solution of the dimensionless mathematical model of matrix system for transient diffusion case can be obtained as follows:

With the definition of hyperbolic functions, (

Equation (

Inserting (

Inserting (

To simplify, (

Employing three-dimensional eigenvalue method and orthogonal transformation to solve the accurate solution of the mathematical model,

With (

In the same way,

Use the one-dimensional eigenvalues and eigenfunctions to obtain three-dimensional eigenvalue and eigenfunction as follows:

The eigenfunction in (

Take the eigenvalues and eigenfunctions into (

Employing

With the definition of orthogonal inverse transformation and norm, there are equations as follows.

Orthogonal inverse transformation is as follows:

Norm is as follows:

The value of norm can be obtained as follows:

Taking (

Based on the assumptions, all the fractures fully penetrated the SGR and are infinitely conductive. According to the Laplace transform,

There are transform equations as follows:

The

Two-phase dimensionless characteristic production rate and other two parameters can be defined as follows:

Thus, (

Integrating (

As we can see in Figure

Schematic of multiple transverse fractures along a horizontal well.

Integrating (

The dimensionless pseudopressure response for

Fluid flow in wellbore and fractures is under the assumption of infinite conductivity, so the pressure at each point within the fractures and the horizontal wellbore is identical to bottom hole pressure, and there is an expression as follows:

With the assumption of constant flow rate, there is an expression as follows:

With (

The solution in Laplace space to the two-phase transient flow model at constant wellbore pressure production can be expressed as

The type curves of a two-phase multifractured horizontal well in SGR can be plotted by Stehfest [

In the condition of sealed boundary, the two-phase pseudopressure-transient pressure response of multistage fractured horizontal well in SGR, which has six flow regimes, can be clearly shown in Figure

Type curve of pressure transient response for two-phase flow in multistage hydraulic fractured horizontal well in SGR.

As we can see in Figure

Stage 1: wellbore storage which has the same characteristics as the conventional reservoir;

Stage 2: formation linear flow period, which is characterized by a horizontal line on both dimensionless production integral and its derivative curves;

Stage 3: interporosity flow period, which is characterized by a V shape in the derivative curve, and in this period free gas has exhausted, and desorbed gas from the shale matrix surface diffuses to factures by pressure difference;

Stage 4: fracture linear flow period, which is characterized by a drop line in type curves;

Stage 5: compound radial flow period, which is characterized by a drop line in dimensionless production integral curve and a rising line in dimensionless production derivative curve;

Stage 6: boundary control flow period, which is characterized by a rapid drop line on both dimensionless production integral and its derivative curves for the reason of sealed boundary condition.

Type curve of production decline for two-phase flow in multistage hydraulic fractured horizontal well in SGR.

Figures

Effect of initial saturation on type curves.

Effect of skin factor on type curves.

Effect of absorption index on type curves.

Effect of different fracture stages on type curves.

Effect of horizontal well lateral length on type curves.

Figure

Figure

Figure

Figure

Figure

The transient flow model of two-phase multifractured horizontal well and desorbed gas transient diffusion in SGR is established and solved, the type curves of pressure response and production decline are illustrated, and the production decline performance under the effects of hydraulic fractures and SGR properties are discussed. The following conclusions were obtained.

The two-phase multifractured horizontal well model can be solved using the eigenvalue method and orthogonal transformation. The desorbed gas transient diffusion rate can be solved using Langmuir’s equilibrium sorption isotherm and Fick’s second law.

The type curves of production decline are dominated by initial saturation, skin factor, and fracture stages. The curves of the dimensionless production rate integral and its derivative curves go down with the increase of gas initial saturation. The skin factor has primarily effect on the early two stages; the larger the skin factor is, the lower the derivative curves are. More fracture stages can improve the flow condition near a horizontal well, and the number of stages indicate the value of fracture spacing; larger fracture spacing increases the duration of fluid flow around individual fractures, and the more the fracture stages are, the more the dimensionless production rate is.

The absorption index has a primary effect on the starting time of the interporosity flow period and the depth of the trough. The trough in the type curves is an indication of desorbed gas diffusion in SGR; the more the absorption index is, the later the decline appears, and, in addition, the trough becomes more significant with larger value of absorption index.

The horizontal well lateral length has a primary effect on the early two flow stages but does not affect the intermediate and later flow behavior; the smaller the horizontal well lateral length is, the faster the decline appears.

Length of SGR (m)

Width of SGR (m)

Total mass concentration (kg/m^{3})

Wellbore storage coefficient (kg/MPa)

System compressibility (kg/(m^{3}·MPa))

Diffusion coefficient (m^{2}/s)

Characteristic function, dimensionless

Geometric factor, dimensionless

Thickness of SGR (m)

Absolute permeability (^{2})

Gas permeability in fractures, dimensionless

Liquid permeability in fractures, dimensionless

Horizontal well lateral length (m)

Half length of fractures (m)

Initial pseudopressure (kg/(m^{3}·s))

Pseudopressure (kg/(m^{3}·s))

Langmuir constant (kg/(m^{3}·s))

Orthogonal factor in

Fracture stages, dimensionless

Pressure in fractures (MPa)

Langmuir pseudopressure (MPa)

Standard state pressure (MPa)

Production of gas flow (m^{3}/s)

Two-phase characteristic production rate (m^{3}/s)

Transient diffusion rate (kg/(m^{5}·s))

Total mass flow rate (kg/s)

Production of water flow (m^{3}/s)

Radial distance in matrix block (m)

Gas constant (0.008314 MPa·m^{3}/(kmol·K))

Laplace transform parameter, dimensionless

Skin factor, dimensionless

Time (s)

Standard state temperature (K)

Langmuir volume (m^{3})

Space coordinates in Cartesian coordinates (m).

Absorption index, dimensionless

Orthogonal factor in

Orthogonal factor in

External radius of matrix block (m)

Eigenvalue, dimensionless

Gas viscosity (MPa·s)

Liquid viscosity (MPa·s)

Gas density (kg/m^{3})

Liquid density (kg/m^{3})

Porosity of fractures, dimensionless.

Shale gas reservoir.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This study was supported by National Science Fund for Distinguished Young Scholars of China (Grant no. 51125019), National Natural Science Foundation of China (Grant No. 51304165), and the 2014 Australia China Natural Gas Technology Partnership Fund Top Up Scholarships.