# Nonlinear Flow Characteristics of a System of Two Intersecting Fractures with Different Apertures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}sequestration [1,2], enhanced oil recovery [3,4], and groundwater use [5,6,7,8,9]. In tight rocks (i.e., granite and basalt), the permeability of fractures is much larger than that of the rock matrix, and as a result, the rock matrix is assumed to be impermeable [10,11,12,13]. This assumption is commonly adopted when calculating the permeability of highly fractured rock masses using modelling techniques such as discrete fracture network (DFN) models [14,15,16]. Another assumption is that the fluid flow through fractures is in the linear regime and follows the cubic law in which the flow rate is linearly proportional to the pressure difference [10,17,18,19,20]. However, in the high-pressure packer tests and/or karst systems, the fluid flow is in the nonlinear regime where the cubic law is not suitable [21,22]. In such a case, the permeability/conductivity of a fractured rock mass will be underestimated when still using the cubic law [21]. Therefore, the nonlinear flow characteristics of fractured rock masses should be investigated.

## 2. Theoretical Background

**u**is the flow velocity tensor, ρ is the fluid density, P is the hydraulic pressure,

**T**is the shear stress tensor, t is the time, and

**f**is the body force tensor. For a laminar flow in single fractures, the convective acceleration terms, $\left(\mathbf{u}\cdot \nabla \right)\mathbf{u}$, are the only source of nonlinearity, which are strongly affected by the geometries of void spaces resulting from complex surface roughness of fractures [48]. When the flow rate is sufficiently small, the Navier-Stokes equations can reduce to the cubic law, in which the flow rate is linearly proportional to the pressure gradient and is commonly adopted when simulating fluid flow in complex fracture networks [14,49,50], as follows:

## 3. Flow Testing System

#### 3.1. Specimen Preparation

_{0}) of each fracture was designed by shifting a separated glass plate. However, the shifting was handled by hand. Therefore, the accurate E

_{0}should be calculated again after the model is manufactured. The dye solute was injected into the model and then all the inlet and outlets were closed. A high-resolution charged coupled device (CCD) camera (Optem Inc., New York, NY, USA) was then utilized to capture each part of the model. Finally, the geometries of the fractures were re-plotted in AutoCAD. As a result, three experimental models (Cases 1–3) were made; their geometries included tortuous segment length (L

_{t}), aperture, and surface roughness and are shown in Table 1, in which the mechanical aperture is represented by E

_{0}and fracture surface roughness is represented by root-mean-square of first derivative of asperity height, Z

_{2}, and is calculated using the following equation [58,59]:

_{i}and z

_{i}are the coordinates of the fracture surface profile, and M is the number of sampling points along the length (i.e., x coordinate) of a fracture.

#### 3.2. Test System

^{3}at a room temperature 20 °C. The dynamic viscosity is 0.001 Pa·s. The glass is impermeable and fluid only flows along the connected fractures.

#### 3.3. Test Procedure

## 4. Numerical Models and Flow Calculation

#### 4.1. Numerical Models and Meshing

#### 4.2. Boundary Conditions and Flow Calculation

## 5. Results and Analysis

#### 5.1. Comparison between Flow Tests and Numerical Simulations

#### 5.2. Parametric Studies

#### 5.2.1. Effect of Fracture Aperture on Q~ΔP Curves

#### 5.2.2. Effect of fracture Aperture on E~ΔP Curves

#### 5.2.3. Effect of Fracture Aperture on ΔP_{c}

_{c}) that corresponds to E = 0.1 and aperture of segments 2 and 4. For the six cases considered, with the increment of aperture of segments 2 and 4 from 3 mm to 5 mm, the average ΔP

_{c}increases significantly from 38.84 Pa to 489.38 Pa. However, with continuously increasing aperture of segments 2 and 4 from 5 mm to 7 mm, the average ΔP

_{c}changes slightly, or even decreases to some extent. This indicates that ΔP

_{c}is more sensitive to a smaller aperture of segments 2 and 4. Before selecting the governing equation for modeling fluid flow in rock fractures, the applied pressure difference should be compared with ΔP

_{c}. When the applied pressure difference is less than ΔP

_{c}, the cubic law that can be simply solved should be selected. When the applied pressure difference is larger than ΔP

_{c}, the Navier-Stokes equations or the Forchheimer’s law should be adopted to solve the nonlinear flow through rock fractures/fracture networks.

#### 5.2.4. Effect of Fracture Aperture on e/E_{0}~ΔP Curves

_{0}is the mechanical aperture. Generally, to quantify the effect of parameters such as fracture surface roughness, Reynolds number and contact on the hydraulic properties of rock fractures, the ratio of hydraulic aperture to mechanical aperture is utilized [61,62,63]. Therefore, the influence of aperture of segments 2 and 4 on the variations of e/E

_{0}was analyzed in this section and the results are presented in Figure 14, Figure 15 and Figure 16. For each case, with increasing ΔP when ΔP is large, the fluid flow enters the nonlinear flow regime and energy losses due to the formation of eddies. As a result, the value of ΔQ/ΔP decreases and then the e that is back-calculated according to Equation (2) decreases. Thus, e/E

_{0}decreases, and the decreasing rate decreases as ΔP increases. In Figure 14, since the apertures of the two crossed fractures are both 3 mm, the variations of e/E

_{0}versus ΔP have the same trends and orders of magnitude. When increasing the aperture of segments 2 and 4 to 5 mm, the results show that the e/E

_{0}of segment 2, which is represented by outlet 2 in Figure 15a, decreases more significantly than that of segment 3, and the decreasing rate of e/E

_{0}of segment 2 is smaller than that of segment 3. Since the apertures of segments 2 and 4 are the same, that is 5 mm, the variations of e/E

_{0}of segments 2 and 4 are close to each other as shown in Figure 15b. Figure 15c shows the same results as that in Figure 15, in which the variations of e/E

_{0}of segment that has a larger aperture (i.e., segments 2 and 4 that have an aperture of 5 mm) are more robust than one with a smaller aperture (i.e., segment 3 that has an aperture of 3 mm). With continuously increasing the aperture of segments 2 and 4 to 7 mm, the variations of e/E

_{0}as shown in Figure 16 are close to those shown in Figure 15. The difference is that the decreasing degree of e/E

_{0}of a segment that has a larger aperture is more significant. Therefore, even though increasing aperture of segments 2 and 4 from 5 mm to 7 had a negligibly small effect on the ΔP

_{c}, its influence on e/E

_{0}cannot be negligible and should be considered.

## 6. Conclusions

## 7. Data Availability

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Comparisons of Q~P curves between simulation and test results for Case 1 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 5.**Comparisons of Q~P curves between simulation and test results for Case 2 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 6.**Comparisons of Q~P curves between simulation and test results for Case 3 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 7.**Variations of Q~P curves for Case 4 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 8.**Variations of Q~P curves for Case 5 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 9.**Variations of Q~P curves for Case 6 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 10.**Variations of E~P curves for Case 4 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 11.**Variations of E~P curves for Case 5 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 12.**Variations of E~P curves for Case 6 with different combinations of outlets: (

**a**) outlet 2, (

**b**) outlet 3, (

**c**) outlet 4, (

**d**) outlet 2 and 3, (

**e**) outlet 2 and 4, (

**f**) outlet 2 and 3 and 4.

**Figure 14.**Variations of e/E

_{0}~P curves for Case 4 with different combinations of outlets: (

**a**) outlet 2 and 3, (

**b**) outlet 2 and 4, (

**c**) outlet 2 and 3 and 4.

**Figure 15.**Variations of e/E

_{0}~P curves for Case 5 with different combinations of outlets: (

**a**) outlet 2 and 3, (

**b**) outlet 2 and 4, (

**c**) outlet 2 and 3 and 4.

**Figure 16.**Variations of e/E

_{0}~P curves for Case 6 with different combinations of outlets: (

**a**) outlet 2 and 3, (

**b**) outlet 2 and 4, (

**c**) outlet 2 and 3 and 4.

Case No. | Segment No. | L_{t} (mm) | E_{0} (mm) | Z_{2} | Number of Hexahedral Cells |
---|---|---|---|---|---|

1 | 1 | 220.27 | 2.70 | 0.42 | 212,320 |

2 | 241.97 | 2.63 | 0.25 | ||

3 | 207.33 | 2.48 | 0.18 | ||

4 | 240.53 | 2.06 | 0.15 | ||

2 | 1 | 220.27 | 2.81 | 0.42 | 243,830 |

2 | 241.97 | 3.61 | 0.25 | ||

3 | 207.33 | 2.52 | 0.18 | ||

4 | 240.53 | 3.88 | 0.15 | ||

3 | 1 | 220.27 | 2.89 | 0.42 | 305,070 |

2 | 241.97 | 6.45 | 0.25 | ||

3 | 207.33 | 3.2 | 0.18 | ||

4 | 240.53 | 6.09 | 0.15 | ||

4 | 1 | 220.27 | 3 | 0.42 | 229,140 |

2 | 241.97 | 3 | 0.25 | ||

3 | 207.33 | 3 | 0.18 | ||

4 | 240.53 | 3 | 0.15 | ||

5 | 1 | 220.27 | 3 | 0.42 | 275,720 |

2 | 241.97 | 5 | 0.25 | ||

3 | 207.33 | 3 | 0.18 | ||

4 | 240.53 | 5 | 0.15 | ||

6 | 1 | 220.27 | 3 | 0.42 | 323,060 |

2 | 241.97 | 7 | 0.25 | ||

3 | 207.33 | 3 | 0.18 | ||

4 | 240.53 | 7 | 0.15 |

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**MDPI and ACS Style**

Liu, R.; Jiang, Y.; Jing, H.; Yu, L.
Nonlinear Flow Characteristics of a System of Two Intersecting Fractures with Different Apertures. *Processes* **2018**, *6*, 94.
https://doi.org/10.3390/pr6070094

**AMA Style**

Liu R, Jiang Y, Jing H, Yu L.
Nonlinear Flow Characteristics of a System of Two Intersecting Fractures with Different Apertures. *Processes*. 2018; 6(7):94.
https://doi.org/10.3390/pr6070094

**Chicago/Turabian Style**

Liu, Richeng, Yujing Jiang, Hongwen Jing, and Liyuan Yu.
2018. "Nonlinear Flow Characteristics of a System of Two Intersecting Fractures with Different Apertures" *Processes* 6, no. 7: 94.
https://doi.org/10.3390/pr6070094