# Analytical model for pressure and rate analysis of multi-fractured horizontal wells in tight gas reservoirs

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## Abstract

Multi-fractured horizontal wells (MFHWs) are effective for developing unconventional reservoirs. A complex fracture network around the well and hydraulic fractures form during fracturing. Hydraulic fractures and fracture network are sensitive to the effective stress. However, most existing models do not consider the effects of stress sensitivity. In this study, a new analytical model was established for an MFHW in tight gas reservoirs based on the trilinear flow model. Fractal porosity and permeability were employed to describe the heterogeneous distribution of the complex fracture network. The stress sensitivity of fractures was also considered in the model. Pedrosa substitution and perturbation method were applied to eliminate the nonlinearity of the model. Analytical solutions in the Laplace domain were obtained using Laplace transformation. The model was then validated and applied. Finally, sensitivity analyses of pressure and rate were discussed. The presented model provides a new approach to estimate the effect of fracturing. It can also be utilized to recognize formation properties and forecast the dynamics of pressure and the production of tight gas reservoirs.

## Keywords

Tight gas Multi-fractured horizontal wells Complex fracture network Stress sensitivity## List of symbols

- \(m\left( p \right)\)
Pseudo-pressure [MPa

^{2}/(mPa·s)] \(m = 2\int_{{p_{0} }}^{p} {\frac{p}{\mu Z}{\text{d}}p}\)- \(p\)
Reservoir pressure (MPa)

- \(T\)
Temperature (K)

- \(c_{\text{g}}\)
Gas compressibility (1/MPa)

- \(c_{\phi }\)
Pore compressibility (1/MPa)

- \(c_{\text{t}}\)
Total compressibility (1/MPa)

- \(b_{3} ,\,b_{2}\)
Apparent permeability coefficient in region 3 and region 2

- \(k\)
Permeability (mD)

- \(k_{{3{\text{a}}}}\)
Apparent permeability in region 3 (mD)

- \(k_{{2{\text{a}}}}\)
Apparent permeability in matrix in region 2 (mD)

- \(k_{{2{\text{fref}}}}\)
Fracture permeability in region 2 at the boundary of the hydraulic fracture (mD)

- \(k_{{1{\text{ref}}}}\)
Permeability of region 1 at initial condition (mD)

- \(\phi\)
Porosity

- \(\phi_{{2{\text{fref}}}}\)
Fracture porosity in region 2 at the boundary of the hydraulic fracture

- \(\Lambda\)
\(\Lambda { = }\left( {\phi c_{{{\text{t}}i}} } \right)_{{_{2m} }} + \left( {\phi c_{{{\text{t}}i}} } \right)_{{_{{2{\text{fref}}}} }}\)

- \(\omega\)
Storativity ratio \(\omega = {{\left( {\phi c_{\text{t}i} } \right)_{{2{\text{fref}}}} } \mathord{\left/ {\vphantom {{\left( {\phi c_{\text{t}i} } \right)_{{2{\text{fref}}}} } \Lambda }} \right. \kern-0pt} \Lambda }\),

- \(\eta_{3}\)
Diffusivity coefficient in region 3 \(\eta_{3} = \frac{{k_{{3{\text{a}}}} }}{{\phi_{3} \mu c_{\text{t3}} }}\)

- \(\eta_{2}\)
Diffusivity coefficient in region 2 \(\eta_{2} = \frac{{k_{{2{\text{fref}}}} }}{\mu \varLambda }\)

- \(\eta_{1}\)
Diffusivity coefficient in region 1 \(\eta_{1} = \frac{{k_{{1{\text{ref}}}} }}{{\mu c_{{{\text{t}}1}} \phi_{1} }}\)

- \(\sigma\)
Shape factor (m

^{2})- \(\lambda\)
Inter-porosity coefficient \(\lambda = \frac{{\sigma k_{{2{\text{a}}}} d_{\text{ref}}^{2} }}{{k_{{ 2 {\text{fref}}}} }}\)

- \(\gamma\)
Permeability modulus (1/MPa)

- \(D_{\text{f}}\)
Fractal dimension of fracture network

- \(\theta\)
Connectivity index

- \(s\)
Laplace variables

- \(C\)
Wellbore storage coefficient (m

^{3}/MPa)- \(t\)
Time (s)

- \(q\)
Gas flow rate (m

^{3}/d)- \(\mu\)
Gas viscosity (mPa·s)

- \(Z\)
Gas deviation factor

- \(D_{\text{g}}\)
Gas diffusion coefficient (m

^{2}/s)- \(B_{\text{g}}\)
Gas volume factor (m

^{3}/m^{3})- \(L_{\text{R}}\)
Model length (m)

- \(W_{\text{R}}\)
Model width (m)

- \(h\)
Formation thickness (m)

- \(x_{e}\)
The spacing from the wellbore to the boundary (m)

- \(y_{e}\)
Half fracture spacing (m)

- \(y_{o}\)
The spacing from the exterior fracture to the boundary (m)

- \(x_{\text{f}}\)
Half fracture length (m)

- \(b_{\text{f}}\)
Half-width of the hydraulic fracture (m)

- \(w_{\text{f}}\)
Width of the hydraulic fracture (m)

- \(d_{\text{ref}}\)
Reference length (m)

## Subscript

- \({\text{D}}\)
Dimensionless

- \({\text{i}}\)
Initial condition

- \({\text{sc}}\)
Standard condition

- \(3\)
Region 3

- \(2\)
Region 2

- \(2{\text{f}}\)
Fracture in region 2

- \(2{\text{m}}\)
Matrix in region 2

- \(1\)
Region 1

- \(wf\)
Bottom hole

## Superscript

^{–}Laplace transform

## Introduction

With the continuous decrease in conventional oil and gas reserves, the development of unconventional resources, such as tight gas and shale gas, has attracted increasing attention. Tight gas reservoir features low permeability and low porosity (Huang et al. 2018), which lead to a quick decline of production for a single well. Multi-fractured horizontal wells (MFHWs) are effective for developing tight gas reservoirs. Multistage fracturing leads to the formation of a complex multi-scale coupling medium, which has complicated seepage characteristics and is composed of matrix, natural and induced fractures (fracture network), and artificial fractures.

A complex analytical model must be established to accurately characterize the complex seepage of MFHWs. Such a model can be established using two methods. One is to divide the fracture into many segments and then use the Green’s function and the point source function to solve the problem. The other one is to establish a linear flow model by simplifying the seepage process as a combination of linear flow. The advantage of the linear flow model approach is that it considers a finite conductivity of the fracture without dividing the fracture into many units. Hence, the linear flow method is more convenient and is a significant alternative for simulating the behavior in MFHW (Wang et al. 2016a). The bilinear flow model was first proposed by Cinco-Ley (1981) for studying the transient pressure behavior of a vertical fractured well with infinite conductivity in an infinite reservoir. Basing on the bilinear flow model, Wong et al. (1986) studied the pressure characteristic of a vertical well with a finite conductivity fracture. Similarly, Lee and Brockenbrough (1986) first proposed a trilinear flow model for a vertical well. The trilinear flow model was introduced into a fractured horizontal well by Brown et al. (2009). They established a multi-fractured horizontal well model by treating the simulated area around hydraulic fracture stages as a dual-porosity medium. The correctness of the model was verified by comparing it with the semi-analytical solutions obtained by Medeiros et al. (2007). Thereafter, the trilinear flow model has been widely used to investigative the dynamic characteristics of MFHWs in unconventional reservoirs (Ozcan et al. 2014; Gao 2014; Wei et al. 2015; Wang et al. 2015; Chen et al. 2016; Wang et al. 2016a, 2016b). Aside from the trilinear flow model, other multi-linear models such as five- (Zhang et al. 2016) and seven-region flow models (Yuan et al. 2015) have been proposed by some scholars. However, the accuracy of these models is not significantly improved compared with that of the trilinear flow model. Moreover, the boundaries between different regions are not easy to divide, and the parameters of each region are difficult to obtain, thereby limiting the practical applications of these models.

Due to the presence of natural fractures and induced fractures generated by hydraulic fracturing, dual-porosity assumption (Barenblatt et al. 1960; Warren and Root 1963; Kazemi et al. 1976) is generally used in the stimulated area around the hydraulic fracture. However, considering the large variations of scale in tight formation, dual-porosity assumption, which is only a first-order approximation, would inevitably lead to a deviation between simulation and actuality (Ozcan et al. 2014). Studies have shown that natural fractures obey fractal distribution in fractured reservoirs. Chang and Yortsos (1990) obtained the power law expression of permeability and porosity of fractures by introducing fractal theory. Subsequently, many scholars have applied the theory of Chang and Yortsos (1990) to seepage models of various types of fractured reservoirs (Tong and Ge 1998; Tong and Zhou 1999; Tong and Liu 2003; Tong et al. 2003; Velazquez et al. 2008; Zhao and Zhang 2011). Cossio et al. (2013) first applied this theory to the trilinear flow model and obtained a semi-analytical solution for a vertically fractured well. Wang et al. (2015) further established a fractal trilinear flow model for MFHWs in tight oil reservoirs.

The effectiveness of the complex fracture network considerably affects yield, drainage area, and final recovery (Mayerhofer et al. 2008; Warpinski et al. 2008). The pressure in the fracture drops rapidly because of the greater conductivity compared with the matrix, which will lead to a production reduction caused by fracture closure. However, previous trilinear flow models do not consider the effect of the stress sensitivity of fractures.

Based on the trilinear flow model proposed by Brown et al. (2009), a new model was established to analyze the pressure and rate responses of MFHWs in tight gas reservoirs by considering the effect of stress sensitivity of fracture. Fractal theory and dual-porosity model were considered in this model to accurately describe the complex fracture network. To obtain an analytical solution of the model, we assumed the permeability modulus of natural and induced fractures and hydraulic fracture to be equal. Although this assumption is inappropriate to some degree, some scholars (Chen et al. 2015, 2016; Teng et al. 2016; Ji et al. 2017) have already proven that this method is acceptable. With Pedrosa substitution, perturbation method, and Laplace transformation method, the analytical solution of the model in the Laplace domain was obtained. Finally, a sensitivity analysis of pressure and rate was conducted.

## Mathematical model

- 1.
The outer boundary of the rectangular tight gas reservoir is impermeable, and the length and width of the reservoir are \(L_{\text{R}}\) and \(W_{\text{R}}\), respectively.

- 2.
The height of each fracture is equal to the formation thickness. The hydraulic fractures are the same in feature and are equally spaced. The yield of each fracture is the same. No gas flow is observed at the end of the fracture, as well as at the region at the parallel fracture direction in the center of the fracture spacing.

- 3.
Region 3 is considered a single medium, and the effect of gas slippage is considered. Region 2 is considered a dual-porosity medium, and the fractal porosity and permeability coupling with stress sensitivity are employed. In region 1, the effect of stress sensitivity is considered.

- 4.
The gas flow in the reservoir is isothermal, and the effects of gravity and capillary pressure are negligible.

Definitions of dimensionless variables

Dimensionless pseudo-pressure (constant gas rate) | \(m_{jD} = \frac{{86.4k_{{2{\text{fref}}}} hT_{\text{sc}} \left( {m_{i} - m_{j} } \right)}}{{q_{\text{sc}} Tp_{\text{sc}} }},\quad j = 3,2f,2m,1\) |

Dimensionless pseudo-pressure (constant bottom hole pressure) | \(m_{{j{\text{D}}}} = \frac{{m_{i} - m_{j} }}{{m_{i} - m_{wf} }},\quad j = 3,2f,2m,1\) |

Dimensionless time | \(t_{\text{D}} = \frac{{3.6\eta_{2} }}{{d_{\text{ref}}^{2} }}t\) |

Dimensionless gas rate | \(q_{\text{D}} = \frac{{q_{\text{sc}} p_{\text{sc}} T}}{{86.4k_{{2{\text{fref}}}} T_{\text{sc}} h\left( {m_{i} - m_{wf} } \right)}}\) |

Dimensionless permeability modulus (constant gas rate) | \(\gamma_{\text{D}} = \frac{{\mu Zq_{\text{sc}} Tp_{\text{sc}} }}{{172.8p_{i} k_{{2{\text{fref}}}} hT_{\text{sc}} }}\gamma\) |

Dimensionless permeability modulus (constant bottom hole pressure) | \(\gamma_{\text{D}} = \frac{{\mu Z\left( {m_{i} - m_{wf} } \right)}}{{2p_{i} }}\gamma\) |

Dimensionless distance | \(x_{\text{D}} = {x \mathord{\left/ {\vphantom {x {d_{\text{ref}} }}} \right. \kern-0pt} {d_{\text{ref}} }}\), \(y_{\text{D}} = {y \mathord{\left/ {\vphantom {y {d_{\text{ref}} }}} \right. \kern-0pt} {d_{\text{ref}} }}\) |

Dimensionless hydraulic fracture half-width | \(b_{\text{fD}} = {{b_{\text{f}} } \mathord{\left/ {\vphantom {{b_{\text{f}} } {d_{\text{ref}} }}} \right. \kern-0pt} {d_{\text{ref}} }}\) |

Dimensionless fracture conductivity | \(F_{{1{\text{D}}}} = \frac{{b_{\text{f}} k_{{1{\text{ref}}}} }}{{d_{\text{ref}} k_{{2{\text{fref}}}} }}\),\(F_{{2{\text{D}}}} = \frac{{x_{\text{f}} k_{{2{\text{fref}}}} }}{{d_{\text{ref}} k_{{ 3 {\text{a}}}} }}\) |

Dimensionless diffusivity coefficient | \(\eta_{{1{\text{D}}}} = {{\eta_{1} } \mathord{\left/ {\vphantom {{\eta_{1} } {\eta_{2} }}} \right. \kern-0pt} {\eta_{2} }}\),\(\eta_{{3{\text{D}}}} = {{\eta_{3} } \mathord{\left/ {\vphantom {{\eta_{3} } {\eta_{2} }}} \right. \kern-0pt} {\eta_{2} }}\) |

Dimensionless wellbore storage coefficient | \(C_{\text{D}} = \frac{C}{{\varLambda hd_{\text{ref}}^{2} }}\) |

### Mathematical model in outer reservoir (region 3)

### Mathematical model in stimulated reservoir (region 2)

Similarly, governing equations in region 2 were obtained. The mathematical model in region 2 can be described as follows:

### Mathematical model in hydraulic fracture (region 1)

## Solutions

To obtain the pressure and rate solution, we performed Laplace transformation of the equations and boundary conditions. Moreover, Pedrosa substitution and perturbation method were applied to linearize the equations. Finally, analytical solutions of dimensionless bottom hole pressure and dimensionless gas rate in Laplace domain were obtained. Detailed derivation of the solution can be found in Appendix 1.

### Bottom hole pressure

### Gas rate

The gas rate in the time domain can be also obtained by using the Stehfest algorithm. \(q_{D}\) is the rate for a single symmetry element. To obtain the total rate of horizontal well, we employed Meyer’s method (Meyer et al. 2010).

## Validation and application

Relevant parameters in a real tight gas reservoir

Parameters | Values | Parameters | Values |
---|---|---|---|

\(T\) (K) | 338.9 | \(c_{\text{ti}}\) (MPa | 4.35 × 10 |

\(h\) (m) | 91.44 | \(k_{3}\) (mD) | 1.5 × 10 |

\(B_{gi}\) | 0.00509 | \(k_{2m}\) (mD) | 1.5 × 10 |

\(\mu\) (mPa·s) | 0.0201 | \(k_{{2{\text{fref}}}}\) * (mD) | 0.1 |

\(\phi_{3}\) | 0.06 | \(k_{1fref}\) *(mD) | 20 |

\(\phi_{2m}\) | 0.06 | \(b_{2} {\kern 1pt} {\kern 1pt}\),\(b_{3}\) * | 2 |

\(\phi_{{2{\text{fref}}}}\) | 0.01 | \(x_{f}\) *(m) | 62.48 |

\(\phi_{1}\) | 0.2 | \(b_{f}\) *(m) | 0.001 |

\(y_{e}\)(m) | 16.15 | \(x_{e}\) *(m) | 120 |

\(m_{i}\)(MPa | 2.84 × 10 | \(\lambda\) * | 0.5 |

\(m_{wf}\)(MPa | 9.65 × 10 | \(\gamma\) *(MPa | 0.01 |

\(L_{\text{R}}\)(m) | 905 | \(D_{f}\) * | 1.95 |

\(N_{\text{F}}\) | 28 | \(\theta\) * | 0.2 |

## Pressure and rate behavior analyses

(1) Wellbore storage stage. The pseudo-pressure curve and the pseudo-pressure derivative curve coincide at this stage, and the slopes of the two curves are 1. (2) Transitional flow. (3) Inter-porosity flow. The derivative curve of this stage is characterized by a “dip.” At this stage, the rate of decline in gas rate slows down because of inter-porosity flow. (4) Compound linear flow. The derivative curves of this stage are characterized by a slope of 1/2. At this stage, gas in outer region begins to flow linearly, and gas rate begins to decline rapidly. (5) Boundary dominated flow. The pseudo-pressure curve and the pseudo-pressure derivative curve coincide again at this stage, and the slope of the two curves is 1. The quick depletion of the formation pressure leads to a closure of the natural and induced fractures. As a result, gas rate rapidly decreases until the well stops production.

## Conclusions

- 1.
The transient pressure and rate type curves contain five flow regimes, including the wellbore storage stage, transitional flow, inter-porosity flow, compound linear flow, and boundary dominated flow.

- 2.
Inter-porosity flow coefficient is related to the density of the fracture network. The larger the inter-porosity flow coefficient is, the lower the position of pressure curves and the larger the gas rate at the inter-porosity flow stage. Storativity ratio reflects the storage capacity of the fracture network. The larger the storativity ratio is, the lower the position of pressure curves and the larger the gas rate at the early stage.

- 3.
Fractal dimension and conductivity index can fully reflect the development and connectivity of the complex fracture network. When the fractal dimension is larger and the connectivity index is smaller, more fractures are produced by fracturing process, and the connectivity between fractures is better, which leads to a lower position of pressure curves and a larger gas rate.

- 4.
Fracture conductivity considerably affects the pressure and rate at the early and middle stages. A larger fracture conductivity leads to a lower position of the pressure curves and a larger gas rate at early and middle stages.

- 5.
The effect of the stress sensitivity of the fracture is obvious and cannot be neglected. The larger the dimensionless permeability modulus, the higher the position of pressure curves and the lower the gas rate.

- 6.
The model presented here can be utilized to recognize formation properties and forecast the pressure and rate dynamics of tight gas reservoirs. In addition, the new model is recommended as an evaluation model for screening attractive tight gas reservoirs and evaluating the effect of fracturing.

## Notes

### Acknowledgements

This research was supported by the National Basic Research Program of China (2015CB250900).

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