# Study on the Temporal and Spatial Multiscale Coupling Flow of Shale Oil

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}is defined as the time for the shale oil diffusion and transfer along the porous medium. The pressure field is divided into several parts. Different parts correspond to different regions. The changing amplitude of each part is the time scale contribution α

_{i}. Based on the peak value of α

_{i}, the main factor that influences the mass transfer is identified. The development system is optimized based on this factor. Afterward, a field study for the validation of the addressed model is conducted. The effect of the size, porosity, permeability and Fick’s diffusion coefficient of unstimulated region on the shale oil flow temporal scale contribution is discussed. Lastly, the spatial distribution of time scales is evaluated to reveal the coupling between the spatial and temporal scales. The results from the study would be helpful in selecting a more effective working system for developing shale reservoirs.

## 2. Shale Oil Flow Model

#### 2.1. Physical Model

_{e}(m), while that parallel to the x direction was represented by x

_{e}(m). Moreover, the boundary beside the wellbore and artificial fracture was the inner boundary, whereas the other boundary was the outer boundary. The artificial fracture had a length of l

_{f}(m) and a width of w (m). The size parallel to the y axis of the Stimulated Region was x

_{s}(m). The reservoir pressure is represented by p

_{0}(Pa), while the wellbore pressure is given by p

_{w}(Pa).

#### 2.2. Coupling between the Fick’s Diffusion in Kerogen and the Fluid Flow in the Matrix

_{sn}(s

^{−1}·Pa

^{−1}) and σ

_{ns}(s

^{−1}·Pa

^{−1}) between the kerogen surface and the organic pore depend on the dynamic absorption and desorption effect of shale oil component [13,14,15]. In addition, the cross flow coefficients, σ

_{mn}(s

^{−1}·Pa

^{−1}), σ

_{nm}(s

^{−1}·Pa

^{−1}), σ

_{mf}(s

^{−1}·Pa

^{−1}), and σ

_{fm}(s

^{−1}·Pa

^{−1}) between the natural fracture network and the organic and inorganic pores in the matrix depend on the microstructure of the matrix and the size and the shape of the fracture network. In short, different microstructures of porous media in the shale reservoir, and the absorption, diffusion and desorption on the kerogen were upscaled as different values of cross flow coefficients, which is known as dynamic coupling.

_{ns}was a circle with the radius R

_{ns}(m) and spherical angle 2θ

_{n}. The oil flowed out of S

_{ns}through its normal direction, where r

_{ns}(m) is the radius of the contact surface.

_{n}(mol/m

^{3}) is the oil component concentration in the kerogen pore, n

_{ϕ}(1/m

^{2}) is the number of kerogen pores adhering to a unit inorganic matrix surface, V

_{ϕ}(m

^{3}) and V

_{o}(m

^{3}) are the organic pore and oil component volumes, respectively, p

_{n}(Pa) is the pressure in the kerogen, C (Pa

^{−1}) is the fluid compressibility, $\overrightarrow{{j}_{nm}}$ (mol/m

^{2}/s) is the diffusion flux from kerogen to the matrix, and μ (Pa·s) is the viscosity. The momentum balance is given by Equation (2).

_{ns}is the function of spherical angle θ. In order to obtain the value of $\overrightarrow{{j}_{nm}}$, the natural coordinate in Equations (5) and (8) is turned into the real coordinate (see Equation (12)).

_{nm}(mol/(m

^{3}·s)), is derived and given by Equation (14).

_{nm}(Pa

^{−1}·s

^{−1}) is given by Equation (16).

_{m}(m) is the matrix throat length, c

_{m}(mol/m

^{3}) is the shale oil component concentration in the inorganic matrix, and $\overrightarrow{{j}_{mn}}$ (mol/m

^{2}/s) is the diffusion flux from matrix to the kerogen. The mass transfer rate, q

_{mn}(mol/(m

^{3}·s)), between the kerogen pore and the inorganic matrix is given by Equation (19).

_{mn}(Pa

^{−1}·s

^{−1}), is given by Equation (21).

#### 2.3. Shale Oil Flow Model in Region 1

^{3}/s). Moreover, the coupling between Regions 1 and 2 conformed to non-equilibrium effect.

_{lf}(m

^{2}) and ϕ

_{lf}are the permeability and porosity of the hydraulic fracture, and p

_{lf}(Pa) is the hydraulic fracture pressure.

#### 2.4. Shale Oil Flow Model in Region 2

_{n}(m) and D

_{s}(m) are the Fick’s and surface diffusion coefficients, respectively, λ

_{ns}(s

^{−1}) and λ

_{sn}(s

^{−1}) are the absorption and desorption coefficients of the organic pore surface, respectively, and p

_{m}(Pa) and p

_{s}(Pa) are the pressures in the inorganic pore and on the organic pore surface, respectively.

_{n}(mol/(m

^{3}·s)), is given by Equation (24).

^{2}/s) is the diffusion flux. The solution of Equation (2) is given by Equation (25).

_{n}(m) is given by Equation (28).

_{f}and k

_{f}(m

^{2}) are the porosity and permeability of the natural fracture network, respectively, and ϕ

_{m}and k

_{m}(m

^{2}) are the porosity and permeability of the inorganic pore, respectively.

#### 2.5. Shale Oil Flow Model in Region 3

#### 2.6. Definition and the Estimation of Temporal Scale

_{i}corresponding to Darcy flow inside the natural fracture network is given by Equation (33).

## 3. Case Study

#### 3.1. Comparison with Eagle Ford Oilfield Production

^{−6}corresponded to the hydraulic fracture. The peak around the temporal scale 2 × 10

^{−5}corresponded to the shale oil flow along the hydraulic fracture. A small valley existed on the right-hand side of the peak, which was due to the sharp decline of permeability at the boundary of Region 2. The peak around the time scale 8 × 10

^{−5}corresponded to the matrix and kerogen. The peaks were higher since more fluid flowed out of Region 2. The peak around the time scale 4 × 10

^{−4}corresponded to Region 3. Since more oil flowed from Region 3, the peak was wider.

#### 3.2. Effect of the Size of the Unstimulated Region on the Shale Oil Flow

_{s}is shown in Figure 4. First, the production mainly included four stages. During Stage 1, the oil mainly flowed from the hydraulic and natural fracture network into the wellbore. Therefore, the production rate decreased slowly. Stage 1 was quite short. During Stage 2, the pressure propagated to the intersection between the stimulated and unstimulated regions, while the intersection performed as a fault. The well rate decreased sharply. During Stage 3, the oil flowed from the kerogen pore to the inorganic matrix and natural fracture. More oil flowed through the boundary of Region 2. The production declined more slowly than Stage 2. However, Stage 3 was quite short. Afterwards, more oil diffused and desorbed inside the kerogen. At Stage 4, the well rate decreased more rapidly. Eventually, the oil concentration in the kerogen decreased with time. The diffusion turned slow. At Stage 5, the well rate decreased even more rapidly than Stage 4.

_{s}, the fluid flowed through a longer Region 2, which is why Stage 1 was longer. At Stage 2, the production rate was also higher. Due to longer intersection between the stimulated and unstimulated regions, Stage 2 was shorter. During Stage 3, the production was as high as the case with shorter x

_{s}. Region 3 was smaller. Therefore, Stage 3 was shorter. The oil in the kerogen flowed more easily to the wellbore. The production rate in Stage 4 was higher. Meanwhile, desorption happened more easily than the surface diffusion. More gas entered the kerogen pore. Due to this reason, during Stage 5, the production rate decreased more slowly.

_{s}= 2/14, due to the longer Region 2, the peak around the time scale 10

^{−5}was higher and wider. In Region 2, the cross-flow was easier than the fluid flow through the outer boundary of Region 2. Less fluid flowed out of Region 3. More smaller peaks and valleys appeared on the right-hand side of the high peak. The valleys corresponded to the intersection between the stimulated and unstimulated regions. While the small peaks corresponded to the cross-flow in Region 2, the pressure at the intersection became lower. Additionally, the peaks corresponding to Region 3 became narrower and shifted leftwards. More oil in the matrix and kerogen flowed into Region 2. The peak around the time scale 8 × 10

^{−5}was higher on its right. Another peak appeared on the right-hand side of the time scale 4 × 10

^{−4}and corresponded to the desorption of oil. The peak was higher than the case with the smaller region. For the case when x

_{s}= 3/14, the peak near the temporal scale 0.005 corresponded to the desorption of oil in Region 3. The valley between the two peaks indicated the low surface diffusion rate in organic pore. Moreover, the peaks corresponded to the natural fracture network and connected with each other.

#### 3.3. Effect of Unstimulated Region’s Porosity, Diffusion Coefficient and Permeability on Shale Oil Flow

^{−5}vanished. Additionally, the peaks on the right-hand side of the temporal scale 2 × 10

^{−5}were lower. In the inorganic matrix and organic pore, fluid flow and diffusion through the boundary of region 2 were easier than the case with tighter Region 3. The left-hand side of the peaks corresponding to Region 3, the matrix and the kerogen were less steep. Moreover, the peak around the time scale 8 × 10

^{−4}was higher on the right-hand side, which indicates that the kerogen in Region 3 was exploited.

#### 3.4. Effect of the Lagged Time on Shale Oil Flow

## 4. Spatial Distribution of Time Scale

^{−6}. Only the hydraulic fracture and natural fracture were affected by the temporal scale. The shale oil flowed through the artificial fracture more easily. The oil component concentration in the end point of the hydraulic fracture could easily be maintained due to flow from Region 3 than the area beside the wellbore. In addition, the outer boundary of Region 2 performed as a fault. Therefore, the contribution was higher in the area far from the wellbore and lower beside the wellbore.

^{−5}. Due to low oil concentration in the hydraulic fracture, more fluid flowed from Region 3 into the hydraulic fracture than from the natural fracture network. The time scale contribution was lower. In the intersection between the stimulated and unstimulated regions, the transfer rate was mainly composed of the fluid flow out of the natural fracture. The contribution was higher in the area far from the wellbore and lower beside the wellbore. However, far from the hydraulic fracture, the oil flowed from the matrix to the natural fracture. Consequently, the temporal scale distribution stripe was along the hydraulic fracture. In the inorganic pore, the contribution in Region 3 was larger than the one beside the wellbore. Additionally, the fluid in the left-hand side of Region 3 flowed to Region 2. Therefore, the contribution in the left-hand side of Region 3 was higher. In the area 0 < x

_{D}< 0.05, since the oil mainly flowed in the direction parallel to the wellbore, the drainage area was as strip parallel to the wellbore. Meanwhile, the time scale contribution was higher. The area with low time scale contribution penetrated the area 0 < x

_{D}< 0.05. The time scale contribution distribution in the kerogen was similar to the matrix. However, since the pressure propagated into the kerogen later than to the matrix, the contribution was lower than the matrix. The strip with high time scale contribution was narrower than in the matrix. The fluid flow in the matrix exerted little effect to the kerogen surface.

^{−5}. In the area beside the hydraulic fracture, since there was low oil concentration in the natural fracture, more shale oil flowed out of the inorganic matrix and organic kerogen. The temporal scale contribution was higher than the area far from the hydraulic fracture. However, more oil entered the natural fracture network due to the dynamic coupling and the Fick’s diffusion rather than the mass transfer inside the natural fracture network. Some of the desorbed oil entered Region 2 through surface diffusion. The contribution was higher at the outer boundary of Region 2. However, the contribution was lower than that beside the hydraulic fracture. In the area 0 < x

_{D}< 0.05 and 1 < y

_{D}< 2, since the oil in the matrix entered the end point of hydraulic fracture rather than the oil in the area x

_{D}= 0.05, the time scale contribution was higher. In the kerogen pore or the area far from the wellbore, the area with larger contribution was narrower. The oil flow beside the outer boundary of Region 3 was quite difficult. The time scale contribution in the matrix and kerogen was lower. However, in the area 0.05 < x

_{D}< 1 and 0 < y

_{D}< 1, Fick’s diffusion in organic pore occurred later than the mass transfer in the inorganic pore. Therefore, the temporal scale contribution was higher. The contribution in the area x

_{D}= 0.7 was lower, which corresponded to the dynamic coupling between the kerogen and the inorganic pore. The oil in inorganic pore reached the natural fracture network earlier than the one in the kerogen. The diffusion in the area 0.1 < x

_{D}< 0.7 was mainly affected by the matrix, while the diffusion in the area x

_{D}> 0.7 was mainly affected by the kerogen. On the kerogen surface, more oil was desorbed, and the surface diffusion coefficients in Regions 2 and 3 were equal to each other, which promoted the surface diffusion. Therefore, the contribution in Region 3 was larger. In the area 0.05 < x

_{D}< 1 and 0 < y

_{D}< 1, the oil concentration was lower, while the surface diffusion was parallel to the wellbore. The area with higher contribution of time scale 8 × 10

^{−5}was distributed as strips.

^{−4}. Beside the wellbore, the oil flowed through a longer distance from Region 3. In the area beside the hydraulic fracture, the time scale contribution was lower. Most of the oil supply to the natural fracture was from Region 3 at the intersection between the stimulated and unstimulated regions. The temporal scale contribution was higher far from the hydraulic fracture. Most of the fluid inside the matrix and kerogen flowed through Region 3. The time scale contribution was higher in the matrix and kerogen. Additionally, in the left-hand side of Region 3, more fluid flowed to the hydraulic fracture, and therefore, the contribution was higher. However, more oil remained in the right-hand side of Region 3 and flowed through the direction parallel to the hydraulic fracture. The time scale contribution was consequently high. In the area y

_{D}= 1, the oil diffusion into Region 2 was more evident than the cross-flow, and therefore, the area with low contribution was narrower.

## 5. Conclusions

- (1)
- The production was composed of five stages. Stage 1 was prolonged, while Stage 3 was shortened in the case with a larger stimulated region and time lag effect. The well rate in Stage 5 was lower when the unstimulated region had a lower diffusion coefficient, permeability, porosity or a larger size. The time lag effect reflected the production fluctuation during Stages 1–3.
- (2)
- The larger time scale had a larger contribution in the kerogen and unstimulated region, while the smaller time scale had a larger contribution in the natural fracture and stimulated region. In short, the time scale corresponded to different mediums. The oil mainly flowed through the outer boundary of the stimulated region through surface diffusion. The time scale spatial distribution diagram also showed that the oil flowed into the end point of the hydraulic fracture at an early time. Moreover, the outer boundary needed a longer time to be exploited.
- (3)
- In the time scale contribution diagram, the left-hand side peaks corresponded to the natural and hydraulic fracture network, whereas the right-hand side peaks corresponded to the unstimulated region and the surface of the kerogen pore. In the case with a tight and larger unstimulated region, small valleys existed on the left of the time scale distribution diagram, while the peaks at the right of the diagram were higher. In the case with the larger stimulated region, both the surface diffusion and the desorption of oil were promoted, and therefore, the well rate was enhanced.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Symbol | Definition |

B | Migration rate of shale oil component, s/m |

c | Concentration, mol/m^{3} |

C | Gas compressibility coefficient, Pa^{−1} |

D | Diffusion coefficient, m^{2}/s |

h | Reservoir height, m |

k | Permeability, m^{2} |

J | Diffusion flux, mol/m^{3}/s |

M | Molecular mass |

n | Number |

p | Pressure, Pa |

q | Mass transfer rate, m^{3}/s |

Q | Production, m^{3}/s |

R | Conventional gas constant |

T | Temperature |

t | Time, s |

t_{j} | Temporal scale |

w | Width, m |

x | Length along the hydraulic fracture, m |

y | Length along the wellbore, m |

Z | Compression factor |

Greek letters | |

α_{i} | Contribution of each temporal scale |

μ | Viscosity, Pa·s |

σ | Transfer coefficient, s^{−1}·Pa^{−1} |

θ | Angle, ° |

ϕ | Porosity |

Subscripts | |

0 | Shale reservoir |

D | Dimensionless |

e | Model size |

f | Natural fracture |

fm | From the natural fracture to the matrix |

i | The index of temporal scales |

lf | Hydraulic fracture |

m | Matrix |

mf | From the matrix to the natural fracture |

o | Oil |

s | Stimulated region |

w | Wellbore |

ϕ | Pore |

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**Figure 3.**Comparison between the results of the temporal scale analysis and the field data. (

**a**) Dimensionless well rate. (

**b**) Time scale distribution.

**Figure 5.**Effect of diffusion coefficient, porosity and permeability of the unstimulated region on well rate. (

**a**) Dimensionless well rate. (

**b**) Time scale distribution.

**Figure 8.**Spatial distribution of time scale 2 × 10

^{−5}contribution. (

**a**) Natural fracture. (

**b**) Matrix. (

**c**) Kerogen pore.

**Figure 9.**Spatial distribution of time scale 8 × 10

^{−5}contribution. (

**a**) Natural fracture. (

**b**) Matrix. (

**c**) Kerogen pore. (

**d**) Kerogen surface.

**Figure 10.**Spatial distribution of time scale 8 × 10

^{−4}contribution. (

**a**) Natural fracture. (

**b**) Matrix. (

**c**) Kerogen pore. (

**d**) Kerogen surface.

Parameter | Value | Parameter | Value |
---|---|---|---|

k_{f} | 3.5 × 10^{−3} μm^{2} | D_{n} | 3.8 × 10^{−2} m^{2}/s |

ϕ_{n} | 10% | Z | 1.1 |

p_{0} | 21 MPa | h | 20 m |

p_{w} | 14 MPa | x_{e} | 100 m |

ϕ_{f} | 5% | y_{e} | 200 m |

w | 2 × 10^{−3} m | D_{s} | 10^{−4} m^{2}/s |

k_{m} | 10^{−7} μm^{2} | μ | 18 mPa·s |

ϕ_{m} | 15% | λ_{ns} | 1.17 × 10^{2} s^{−1} |

l_{f} | 100 m | λ_{sn} | 2.66 × 10^{−2} s^{−1} |

C | 5 × 10^{−10} Pa^{−1} | σ_{fm} | 1.357 × 10^{−10} Pa ^{−1}·s^{−1} |

ε_{s} | 45% | σ_{mf} | 1.293 × 10^{−10} Pa ^{−1}·s^{−1} |

k_{lf} | 1.05 × 10^{−2} μm^{2} | σ_{nm} | 2.715 × 10^{−14} Pa ^{−1}·s^{−1} |

ϕ_{lf} | 2 × 10^{−2}% | σ_{mn} | 2.687 × 10^{−14} Pa ^{−1}·s^{−1} |

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Li, B.; Su, Y.; Lu, M.
Study on the Temporal and Spatial Multiscale Coupling Flow of Shale Oil. *Processes* **2022**, *10*, 939.
https://doi.org/10.3390/pr10050939

**AMA Style**

Li B, Su Y, Lu M.
Study on the Temporal and Spatial Multiscale Coupling Flow of Shale Oil. *Processes*. 2022; 10(5):939.
https://doi.org/10.3390/pr10050939

**Chicago/Turabian Style**

Li, Binglin, Yuliang Su, and Mingjing Lu.
2022. "Study on the Temporal and Spatial Multiscale Coupling Flow of Shale Oil" *Processes* 10, no. 5: 939.
https://doi.org/10.3390/pr10050939