# Flow Simulation in Fractures Using Non-Linear Flux Approximation Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{−}to fracture unit C

_{+}.

_{+}to grid C

_{−}through unit surface f can be expressed as:

_{+}and T

_{−}is the conductivity of C

_{+}to C

_{−}and C

_{−}to C

_{+}, respectively. In the two-point flow format, the values of the two are equal:

_{f}and t

_{f}is large, resulting in a large error in the results. Only when n

_{f}and t

_{f}are relatively close can this format obtain accurate solutions.

- Process 1: Unit C
_{+}to surface f; - Process 2: Surface f to Unit C
_{−}.

_{+}to f as an example) into the sum of three adjacent unit to surface subflows of C

_{+}, where adjacent units refer to units that have at least one common edge with C

_{+}.

_{+}, if the vector Kn

_{f}passes through the surface T

_{1}-T

_{2}-T

_{3}, then Kn

_{f}can be represented as:

_{+}to f can be expressed as:

_{+}to f can be expressed as:

_{+}to C

_{−}as the sum of the weights of the flow rate from C

_{+}to f and the flow rate from f to C

_{−}, as follows:

_{+}and C

_{−}pressure:

_{+}and T

_{−}depend on the pressure of the auxiliary units in the grid. Therefore, during the simulation process, conductivity is not a fixed value and changes with the change of pressure. Under the fully implicit solution strategy, each Newton step requires the derivative of the conductivity. In this case, the format is actually a multi-point flow format, and the Jacobi matrix is also more dense. In fact, in seepage mechanics problems, the variation of conductivity in each Newtonian step is not significant. Based on this, Nikitin proposed an explicit solution framework, where the conductivity is fixed within each time step and updated outside the time step. Nikitin and other scholars have shown through numerical experiments that there is no difference in accuracy between the two methods, and both exhibit excellent stability. But it shows that the Jacobi matrix under the solving framework is more sparse and the solving efficiency is higher.

_{+}and C1

_{−}, the same layer grids of C1

_{+}are all adjacent grids of C1

_{−}, and vice versa.

_{1}, Tg

_{1}, Tp

_{2}, and Tg

_{2}are the conductivity calculated based on pressure for grid C1, gravity for grid C1, pressure for grid C2, and gravity for grid C2. z

_{1}and z

_{2}represent the unit centroid heights of C1 and C2.

## 3. Transmissibility for EDFM

_{i}and y

_{i}are the molar fractions of component i in the oil and gas phases, respectively; μ

_{o}and μ

_{g}are the viscosity of the oil and gas phases, respectively; k

_{ro}and k

_{rg}are the relative permeability of the oil and gas phases, respectively; K is the absolute permeability tensor; q

_{i}is the mass flow rate of component i; q

^{NNC}

_{i}is the mass flow rate of component i passing through NNC. The calculation method for the mass flow rate term is similar to that for the flow term, which can be expressed as:

_{n}and d

_{n}represent the characteristic area and characteristic distance, respectively. These two parameters can be understood as the flow characteristic parameters of two control volumes connected through NNC. The calculation method for T varies for different types of NNC.

_{n}is the intersection area between the fracture and the background grid, and K is the harmonic average of the permeability between the fracture and the matrix. Assuming that the pressure changes linearly along the normal direction of the fracture, d

_{n}can be calculated as follows:

_{1}and T

_{2}are the conductivity of the two fractures, which can be calculated by the following equation:

_{int}is the length of the intersection line of two fractures; w

_{f}is the fracture opening; K

_{f}is the permeability of the fracture. d

_{f}is the distance between the fracture center and the intersection line, and d

_{f}is the weighted average distance between two fracture segments. For the third type of NNC, the conductivity can be directly calculated using Equation (19), simply replacing d

_{n}with the distance between the centroids of two fracture segments.

_{F1−F2}of F1 and F2 is calculated.

_{f}starting from the centroid of F1 falls within the angle range of vectors t1 and t2 composed of the centroid of F1 and its two adjacent unit centroids, then Kn

_{f}can be expressed as:

_{+}to C

_{−}as the sum of the weights of the flow rate from C

_{+}to f and the flow rate from f to C

_{−}, as follows:

## 4. Verification Examples

#### 4.1. Application Effect of EDFM Single Fracture Sheet

#### 4.2. Application Effect of DFM Single Fracture

## 5. Application Cases

#### 5.1. Model Establishment

^{3}/d. At the same time, the minimum wellbore pressure is set to 10 Bar. When the fixed oil production volume reaches the minimum pressure at the bottom of the well, it is converted to fixed bottom pressure production.

#### 5.2. Optimization of Transformation Parameters

^{3}. The soaking time is set to 0, 50, and 100 days. According to the principle of orthogonal design, a total of nine sets of orthogonal experimental plans were designed (see Table 1).

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

^{2}; ${\overline{d}}_{\mathrm{f}}$—The average distance from all elements within a polygon to the intersection line, m; ${d}_{\mathrm{I}}$, ${d}_{\mathrm{II}}$—The distance from the center of gravity of two sub-polygons to the intersection line, m; H—fracture opening, m; K

_{0}—Permeability at the root of the fracture, μm

^{2}; k

_{f}, k

_{m}—Permeability of fractures and matrix, μm

^{2}; ${K}_{m}$—Permeability tensor of matrix; ${K}_{\mathrm{ro}}$, ${K}_{\mathrm{rg}}$, ${K}_{\mathrm{rw}}$—Relative permeability of Oil, Gas and Water; L—The length of the intersection line between two fractures, m; ${S}_{\mathrm{I}}$, ${S}_{\mathrm{II}}$—The area of two sub-polygons, m

^{2}; V—Volume of matrix grid, m

^{3}; w(x)—The opening of the fracture at point x, m; w

_{0}—The opening of the fracture root, m; ${\lambda}_{\mathrm{o}}$, ${\lambda}_{\mathrm{g}}$, ${\lambda}_{\mathrm{w}}$—Fluidity of oil, gas, and water three-phase flow, kg∙m

^{−3}∙(mPa∙s)

^{−1}; ${\mu}_{\mathrm{o}}$, ${\mu}_{\mathrm{g}}$, ${\mu}_{\mathrm{w}}$—Viscosity of oil, gas, and water phases, mPa∙s; $\mathsf{\Omega}$—The area of a polygon within a fracture sheet, m

^{2}.

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**Figure 1.**Schematic diagram of non-orthogonal grids in DFM and EDFM, (

**a**) Non-orthogonal grids in DFM. C2 block is a grid block with inclined fractures. Since the grid block should coincide with the fracture and stratum at the same time, orthogonality is lost. (

**b**) Non-orthogonal grids in EDFM. In order to calculate the flow rate within the same fracture, it is necessary to divide the fracture polygon into several two-dimensional polygon grids.

**Figure 3.**Schematic diagram of three types of NNC in EDFM. (

**a**) the NNC between the fracture element and adjacent background grid blocks. (

**b**) the NNC between two units of a single fracture. (

**c**) the NNC between two intersecting fractures.

**Figure 8.**DFM grid, where the fracture grid has an inclination angle, and the enlarged blue grid represents the nonorthogonal grid around the fracture grid. It can be seen that the DFM method generates a large number of nonorthogonal grids around it in order to maintain the layered structure and simulate the inclined shape of the fracture.

Number | Horizontal Section Length/ m | Number of Segments | Daily Injection Volume/ m ^{3} | Soaking Time/ Day |
---|---|---|---|---|

1 | 300 | 6 | 14 | 0 |

2 | 300 | 7 | 28 | 50 |

3 | 300 | 8 | 42 | 100 |

4 | 400 | 6 | 28 | 100 |

5 | 400 | 7 | 42 | 0 |

6 | 400 | 8 | 14 | 50 |

7 | 500 | 6 | 42 | 50 |

8 | 500 | 7 | 14 | 100 |

9 | 500 | 8 | 28 | 0 |

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**MDPI and ACS Style**

Wang, T.; Li, X.; Liu, F.; Zhang, L.; Xie, H.; Bai, Y.
Flow Simulation in Fractures Using Non-Linear Flux Approximation Method. *Water* **2023**, *15*, 4281.
https://doi.org/10.3390/w15244281

**AMA Style**

Wang T, Li X, Liu F, Zhang L, Xie H, Bai Y.
Flow Simulation in Fractures Using Non-Linear Flux Approximation Method. *Water*. 2023; 15(24):4281.
https://doi.org/10.3390/w15244281

**Chicago/Turabian Style**

Wang, Taichao, Xin Li, Fengming Liu, Lijun Zhang, Haojun Xie, and Yuting Bai.
2023. "Flow Simulation in Fractures Using Non-Linear Flux Approximation Method" *Water* 15, no. 24: 4281.
https://doi.org/10.3390/w15244281