# Modeling the Transient Flow Behavior of Multi-Stage Fractured Horizontal Wells in the Inter-Salt Shale Oil Reservoir, Considering Stress Sensitivity

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## Abstract

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## 1. Introduction

## 2. Physical Modeling

- (1)
- The reservoir is bounded by two parallel impermeable boundaries at the top and bottom, with an infinite lateral boundary. The reservoir’s thickness is h, and the initial formation pressure is p
_{i}and is equal everywhere. - (2)
- The inter-salt shale oil reservoir is assumed to be a dual-porosity medium, which is based on the Warren–Root model. The matrix porosity and fracture porosity are ϕ
_{m}and ϕ_{f}, respectively, and the permeabilities are K_{m}and K_{f}. As K_{f}is much larger than K_{m}, pseudo-steady cross-flow occurs between these two systems. - (3)
- A fractured horizontal well can be located anywhere (represented by z
_{w}) in the formation, with the horizontal section of the well parallel to the top and bottom boundaries. A total of M transverse fractures are formed after multi-stage fracturing. - (4)
- The matrix permeability can be affected by the dissolution of salt. The fracture permeability is affected by the stress-sensitive effect, and oil flow in fracture system obeys Darcy’s law.
- (5)
- The single-phase oil is compressible with a constant viscosity and compression coefficient, but the wellbore storage effect and the skin effect are considered.
- (6)
- The oil flow in the inter-salt shale oil reservoir is at a constant reservoir temperature.

## 3. Mathematical Modeling

#### 3.1. Mathematical Model of a Line-Sink in an Inter-Salt Shale Oil Reservoir

- Governing equations

_{t}= (C

_{p}+ C

_{L}), MPa

^{−1}.

- Boundary conditions

- Initial conditions

#### 3.2. The Dimensionless Form of the Line-Sink Model

#### 3.3. Solution of the Mathematical Line-Sink Model

#### 3.3.1. Pedrosa’s Linearization

_{D}(r

_{D}, t

_{D}) is an intermediate variable, also known as the perturbation deformation function.

#### 3.3.2. Perturbation Technique

_{D}(dimensionless permeability modulus) [27,28]:

_{D}<< 1), researchers [25,26,29] have suggested that the zero-order approximate solution can satisfy the requirements of engineering accuracy, so Equation (19) becomes:

#### 3.3.3. Laplace Transformation

_{D}, so Equation (23) can be written as:

_{0}and K

_{0}are the Bessel function.

_{w},y

_{w}), r

_{D}can be calculated by

#### 3.4. Pressure Responses of the MFHW in an Inter-Salt Shale Oil Reservoir

_{i}, i = 1, 2, ···, M−1) may be equal or unequal. Considering that the length of the hydraulic fractures may be different, we assumed that the length of the right wing and the left wing of the hydraulic fracture were L

_{fRi}and L

_{fLi}, respectively.

_{i j}, Y

_{i j}), and the coordinates of the endpoint are denoted as (x

_{i j}, y

_{i j}) and (x

_{i j}

_{+1}, y

_{i j}

_{+1}).

_{i,j}at different discrete segments is different. We assumed that the flux density was q

_{i,j}for different discrete segments. Therefore, the pressure response caused by the discrete segment (i, j) at any location (x

_{D}, y

_{D}) in the inter-salt shale oil reservoir can be obtained by integrating the line-sink solution along the discrete segment:

_{D}and x

_{wD}, Equation (40) can be written as:

_{D}, y

_{D}) caused by 2N × M segments on m hydraulic fractures can be obtained as:

_{Dk,v}, Y

_{Dk,v}) as:

_{wD}in the real time domain can be obtained by Stehfest’s numerical inversion algorithm [31]. The bottom-hole pressure response of multistage fractured horizontal wells in an inter-salt shale oil reservoir, considering the salt dissolution and stress sensitivity, can be obtained by the following equation:

## 4. Results and Discussion

#### 4.1. Behaviorial Analysis of Transient Pressure

#### 4.2. Effect of Salt Dissolution

#### 4.3. Effect of Stress Sensitivity

#### 4.4. Effect of the Storativity Ratio

#### 4.5. Effect of the Interporosity Flow Coefficient

#### 4.6. Effects of the Parameters of the Hydraulic Fractures

## 5. Conclusions

- (1)
- The pressure response and corresponding pressure derivative curves of a MFHW in the inter-salt shale oil reservoir with consideration of the stress sensitivity of natural fractures were analyzed, and eight main flow periods could be observed in the type curves of transient pressure.
- (2)
- The influence of salt dissolution on the transient pressure curves of the fractured horizontal well in an inter-salt shale oil reservoir was negligible because the permeability decreased by only 5.06% when the average pressure dropped from 22.5 MPa to 7.5 MPa according to the experimental results of the effect of salt dissolution on the shale’s permeability.
- (3)
- The effect of the stress sensitivity of the fracture system on the pressure derivative curves became apparent in the radial flow period of natural fractures (Period 6). The pressure derivative curves gradually turned upward with an increase in the dimensionless permeability modulus. The stronger the stress sensitivity, the more serious the damage to the reservoir. It was therefore more difficult for the shale oil to flow, and greater drawdown pressure was required.
- (4)
- The effects of the storativity ratio, the interporosity flow coefficient and the parameters of the hydraulic fractures on the transient pressure curves were analyzed to better understand the transient flow behavior of the MFHW in an inter-salt shale oil reservoir.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Letters | |

B | fluid volume factor, m^{3}/sm^{3} |

C | wellbore storage coefficient, m^{3}/Pa |

C_{ft} | total compressibility coefficient of the fracture system, Pa^{−1} |

C_{mt} | total compressibility coefficient of the matrix system, Pa^{−1} |

C_{L} | fluid compressibility coefficient, Pa^{−1} |

C_{p} | rock compressibility coefficient, Pa^{−1} |

c_{1}, c_{2} | empirical coefficients, which can be determined by experiments |

K_{f} | permeability of the fracture, m^{2} |

K_{fi} | initial permeability of the fracture, m^{2} |

K_{m} | permeability of the matrix, m^{2} |

K_{mi} | initial permeability of the matrix, m^{2} |

L | Characteristic length, m |

L_{fLi}, L_{fRi} | Lengths of the left and right wings of the ith fracture, m |

M | Number of hydraulic fractures |

N | Number of segments on the wing of each fracture |

p | pressure, Pa |

p | reference pressure, Pa |

p_{i} | initial pressure, Pa |

p_{f} | fracture pressure, Pa |

p_{m} | matrix pressure, Pa |

q_{ex} | cross flow from the matrix to the fracture, kg/(m^{3}·s) |

q_{i,j} | flux per unit of length of a discrete segment (i, j), m^{2}/s |

q_{sc} | surface oil production rate, m^{3}/s |

r | radial distance, m |

S | skin factor, dimensionless |

t | time, s |

v_{fr} | radial velocity component of oil flow in fracture, m/s |

Greek letters | |

α | matrix block shape factor, 1/m^{2} |

β | empirical coefficient, which can be determined by experiments |

γ | permeability modulus, Pa^{−1} |

ρ_{0} | reference oil density under the reference pressure, kg/m^{3} |

ρ_{f} | Oil density in the fracture system, kg/m^{3} |

ρ_{m} | oil density in the matrix system, kg/m^{3} |

ϕ_{0} | initial porosity, dimensionless |

ϕ_{f} | fracture porosity, dimensionless |

ϕ_{m} | matrix porosity, dimensionless |

μ | oil viscosity, Pa·s |

ξ_{D} | perturbation deformation function |

ξ_{D0} | zero-order perturbation deformation function |

Superscripts | |

$\overline{}$ | Laplace transform domain |

Subscripts | |

D | dimensionless |

i | initial condition |

sc | standard state |

f | fracture system |

m | matrix system |

## Appendix A. Experimental Evaluation of the Salt Dissolution

Inlet Pressure | Outlet Pressure | Pressure Difference | Permeability |
---|---|---|---|

25 | 20 | 5 | 0.514 |

20 | 15 | 5 | 0.5 |

15 | 10 | 5 | 0.493 |

10 | 5 | 5 | 0.488 |

_{1}and c

_{2}are the empirical coefficients, which can be determined by experiments.

_{1}= β, Equation (A1) can be rewritten as follows by integrating the equation above

_{i}is the initial permeability, p

_{i}is the initial pressure.

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**Figure 3.**Bilogarithmic type curve of the dimensionless pressure and the pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 4.**The effect of salt dissolution on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 5.**The effect of stress sensitivity on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 6.**The effect of the storativity ratio on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 7.**The effect of the interporosity flow coefficient on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 8.**The effect of the number of transverse fractures on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 9.**The effect of the spacing of hydraulic fractures on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

**Figure 10.**The effect of the fracture’s half-length on the dimensionless pressure and pressure derivatives of a MFHW in an inter-salt shale oil reservoir.

Dimensionless pressure | ${p}_{l\mathrm{D}}=\frac{2\pi {K}_{\mathrm{fi}}h\left({p}_{\mathrm{i}}-{p}_{l}\right)}{{q}_{\mathrm{sc}}B\mu}$ (l = f, m) |

Dimensionless time | ${t}_{\mathrm{D}}=\frac{{K}_{\mathrm{fi}}t}{\left({\varphi}_{\mathrm{f}}{c}_{\mathrm{ft}}+{\varphi}_{\mathrm{m}}{c}_{\mathrm{mt}}\right)\mu {L}^{2}}$ |

Dimensionless radius | ${r}_{\mathrm{D}}=\frac{r}{L}$ |

Storage coefficient | $\omega =\frac{{\varphi}_{\mathrm{f}}{c}_{\mathrm{ft}}}{{\varphi}_{\mathrm{f}}{c}_{\mathrm{ft}}+{\varphi}_{\mathrm{m}}{c}_{\mathrm{mt}}}$ |

Transfer coefficient | $\lambda =\alpha \frac{{K}_{\mathrm{mi}}-F}{{K}_{\mathrm{fi}}}{L}^{2}$ |

Dimensionless permeability modulus | ${\gamma}_{\mathrm{D}}=\frac{{q}_{\mathrm{sc}}B\mu}{2\pi {K}_{\mathrm{fi}}h}\gamma $ |

Dimensionless wellbore storage coefficient | ${C}_{\mathrm{D}}=\frac{C}{\left({\varphi}_{\mathrm{f}}{C}_{\mathrm{ft}}+{\varphi}_{\mathrm{m}}{C}_{\mathrm{mt}}\right)h{L}^{2}}$ |

Dimensionless production rate | ${q}_{\mathrm{D}}=\frac{\tilde{q}L}{{q}_{\mathrm{sc}}}$ |

Parameters | Symbols | Values | Units |
---|---|---|---|

Formation thickness | h | 50 | m |

Formation pressure | p_{i} | 2.34 × 10^{7} | Pa |

Permeability modulus | γ | 0.12 | MPa^{−1} |

Matrix porosity | ϕ_{m} | 0.10 | dimensionless |

Matrix permeability | K_{m} | 2.4 × 10^{−19} | m^{2} |

Fracture porosity | ϕ_{f} | 0.039 | dimensionless |

Fracture permeability | K_{f} | 2.0 × 10^{−13} | m^{2} |

Oil viscosity | μ | 2.95 × 10^{−3} | Pa·s |

Matrix compressibility | c_{mt} | 6.2 × 10^{−11} | 1/Pa |

Fracture compressibility | c_{ft} | 4.3 × 10^{−9} | 1/Pa |

Half-length of the hydraulic fracture | X_{f} | 230 | m |

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**MDPI and ACS Style**

Huang, T.; Guo, X.; Peng, K.; Song, W.; Hu, C.
Modeling the Transient Flow Behavior of Multi-Stage Fractured Horizontal Wells in the Inter-Salt Shale Oil Reservoir, Considering Stress Sensitivity. *Processes* **2022**, *10*, 2085.
https://doi.org/10.3390/pr10102085

**AMA Style**

Huang T, Guo X, Peng K, Song W, Hu C.
Modeling the Transient Flow Behavior of Multi-Stage Fractured Horizontal Wells in the Inter-Salt Shale Oil Reservoir, Considering Stress Sensitivity. *Processes*. 2022; 10(10):2085.
https://doi.org/10.3390/pr10102085

**Chicago/Turabian Style**

Huang, Ting, Xiao Guo, Kai Peng, Wenzhi Song, and Changpeng Hu.
2022. "Modeling the Transient Flow Behavior of Multi-Stage Fractured Horizontal Wells in the Inter-Salt Shale Oil Reservoir, Considering Stress Sensitivity" *Processes* 10, no. 10: 2085.
https://doi.org/10.3390/pr10102085