# Unstable Density-Driven Flow in Fractured Porous Media: The Fractured Elder Problem

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) injection in geological reservoirs [8,9,10], geothermal systems [11] and saltwater infiltration from inland sabkha or salt lakes [12,13].

## 2. Methods

#### 2.1. Conceptual Model

^{3}as in Lu et al. [26] and Post and Prommer [23].

#### 2.2. Fractured Domain Scenarios

#### 2.3. Governing Equations

**u**is the velocity [LT

^{−1}], ${\rho}_{0}$ is freshwater density [ML

^{−3}], g is the gravity [LT

^{−2}], μ is the fluid dynamic viscosity [ML

^{−1}T

^{−1}], ${k}_{m}$ is the intrinsic permeability of the porous matrix [L

^{2}], ρ is the mixed fluid density [ML

^{−3}] and ${e}_{z}$ is the unit vector in the z-direction.

^{2}T

^{−1}] (as is common for the Elder problem, the dispersion coefficients are neglected).

_{fr}) and porosity (${\epsilon}_{f}$). We should mention that Simmons et al. [35] investigated unstable DDF in fractured low-permeability porous media (vertical fractures). They studied interfracture and intrafracture convection modes. Based on a Rayleigh stability analysis, they defined conditions required for the onset of convection in each mode. Simmons et al. [36] showed that, in the low-permeable porous layer, the most likely mode is the intrafracture convection parallel to the plane of fracture. The assumptions considered in this work (2D porous domain and 1D fractures) do not allow for considering the intrafracture convection modes which require 3D simulations (3D porous domain and 3D fractures). As the aim of this work is to investigate the fractured Elder problem with dense orthogonal fracture networks that require dense grids associated to prohibitive computational costs, we adopted the discrete fracture network approach and we limited the study to the interfracture convection mode. A similar approach has been used in Graf and Therrien [32], Vujevic et al. [33], Hirthe and Graf [34] and Koohbor et al. [39]. For regular networks, Vujevic and Graf [37] have shown that the most important mode is the interfracture convection.

_{fr}[L

^{2}]) can be calculated as a function of the fracture aperture using the cubic law [35], as follows:

#### 2.4. The Rayleigh Number

#### 2.5. The Average Sherwood Number

#### 2.6. The Simulation Tool: COMSOL Multiphysics^{®}

## 3. Results and Discussion

#### 3.1. Verification of the COMSOL Model

#### 3.2. Mesh Sensitivity Analysis

#### 3.3. Effect of Fractures on the Onset of Instability

^{2})); therefore, Ra would be changed by only modifying the diffusion coefficient. The values of the Rayleigh number for our simulations are 20, 40 and 60 which correspond to the diffusion coefficients equal to $7.129\times {10}^{-5}$, $3.565\times {10}^{-5}$ and $2.376\times {10}^{-5}$ (m

^{2}s

^{−1}) for both scenarios. The corresponding concentration distributions at times of 1, 3, 5 and 10 years are plotted in Figure 7. As one can observe, for $Ra$ = 40 and 60 at scenario (A), the initial appearance of fingers are obvious, which means we have an unstable system, and it confirms the fact that if Ra exceeds a value of 40, we would have the onset of instability the in homogeneous Elder problem. The results of scenario (D) show that the convective flow patterns of scenario (D) have higher strength of free convection than scenario (A). Figure 7 indicates that the onset of free convection in the fractured Elder problem is expected to occur with lower $R{a}_{c}$, and fracture porous media would destabilize this problem. This is consistent with the results of Simmons et al. [1] and Sharp et al. [41]. To confirm these results, Figure 8 shows the plot of the variation of the Sherwood number versus time, for all simulations. We can observe that the curve corresponding to scenario (A) at Ra = 20 is quite smooth, which indicates that there is no convection in this case and the solute flux is mainly diffusive. At Ra = 40, the curve for scenario (A) is perturbed with a prompt change between 2 and 3 years. The same behavior can be observed at Ra = 60, but the curve is perturbed between 1 and 6 years. This means that the perturbations represent the onset of instability. If we follow the same analysis for scenario (D), we can conclude that, in this case, the onset on instability occurs at a lower Rayleigh number as the curves are more perturbed than scenario (A). It is relevant to note that heterogeneity can become unstable, starting at a certain time but then stabilize the system at a later time. Figure 7, which gives the concentration distributions at local times, cannot be used to indicate if this phenomenon occurs in this case. However, it is clear from Figure 8 that this phenomenon does not exist as the perturbations of the Sherwood number appears once for all cases.

#### 3.4. Effect of Fracture Aperture

#### 3.5. Effect of Fracture Density

## 4. Conclusions

- (1)
- Embedding fracture networks in the Elder problem increases the mesh sensitivity and bifurcation states of this problem. In other words, by changing the mesh size, the fractured Elder problem has more variation in both the number and shape of the plumes than the non-fractured case.
- (2)
- Fracture networks have a destabilizing impact on the Elder problem. It means that the onset of instability of fractured Elder problem occurs with the value of Rayleigh number lower than 40 which is the critical Rayleigh number of onset of instability.
- (3)
- Concerning how the structural properties of fracture networks control convective flow patterns, we explored the effect of aperture fractures and density of the fracture networks. By enlarging the aperture size in a fractured case of Elder problem, the instability increases at an early time, and since the convective flow in the fractures moves up there would be a higher number of fingers at the beginning. However, the system will be stable at the other times, and the simulation results will be the same for different aperture sizes. In addition, as the fracture density increases, various transient convective modes obtained which are different from the non-fractured case at the beginning; nonetheless, this difference exists until an optimal fracture density, and after that, the high dense fractured scenarios behave similarly to the homogeneous case in fingering processes and plume patterns.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The fractured Elder problem domain and boundary conditions. All boundaries were impermeable to flow.

**Figure 2.**Different scenarios of the fractured Elder problem: (

**A**) homogenous domain, (

**B**–

**F**) orthogonal fracture networks with increasing fracture density.

**Figure 4.**Relative concentration distribution for scenarios (A) and (C) for different levels of mesh refinement: (A1), (A2) and (A3) correspond to grids with 2509, 10,235 and 90,973 triangular elements respectively. (C1), (C2) and (C3) correspond to grids with 2671, 1,0565 and 9,1875 triangular elements respectively.

**Figure 5.**Variation of the average Sherwood number after three years (t = 3 years) versus the number of nodes used in the simulations (scenarios A and C).

**Figure 6.**Variation of the time average concentration at a local point (x = 300 m; y = 0 m) versus the number of nodes used in the simulations (scenarios A and C).

**Figure 7.**Solute distribution of half of the domain of scenarios (A) and (D) for different Rayleigh numbers at simulation times of 1, 3, 5 and 10 years.

**Figure 8.**The local Sherwood number beneath the source zone in the top layer versus time for different Rayleigh numbers for scenarios A and D.

**Figure 9.**Solute distribution of half of domain of scenario (D) for different fracture aperture at simulation times of 1, 3, 5 and 10 years. The fracture aperture from D1 to D4 increases.

**Figure 10.**Velocity fields of cases D1 and D4 at simulation time of 1 year. Blue arrows and red arrows represent velocity vectors in porous matrix and fractures, respectively.

**Figure 12.**Solute distribution of half of domain of all the scenarios at simulation times of 1, 3, 5, 10, 20 and 50 years.

**Figure 13.**The local Sherwood number beneath the source zone for all the scenarios at 2, 10, 20 and 50 years.

**Table 1.**Physical parameters used in COMSOL for the fractured Elder problem (Rayleigh number (Ra) = 400).

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

Length | L | 600 (m) |

Height | H | 150 (m) |

Freshwater density | ${\rho}_{f}$ | 1000 (kg/m^{3}) |

Saltwater density at the top boundary | ${\rho}_{s}$ | 1200 (kg/m^{3}) |

Intrinsic permeability of the medium | k_{m} | $4.845\times {10}^{-13}$ (m^{2}) |

Porosity of the medium | ${\epsilon}_{m}$ | 0.1 [-] |

Diffusion coefficient | D_{m} | $3.565\times {10}^{-6}$ (m^{2}/s) |

Dynamic viscosity | $\mu $ | ${10}^{-3}$ (kg/(m·s)) |

Gravitational acceleration | g | 9.8 (m/s^{2}) |

Porosity of the fracture | ${\epsilon}_{f}$ | 0.1 [-] |

Fracture aperture | 2b | [range of variation] |

Fracture density | 2B | [range of variation] |

Scenario | Fracture Spacing (2B) (m) | Fracture Aperture (2b) (×10^{−4} m) | Fracture Permeability (k_{fr}) (×10^{−9} m^{2}) | k_{fr}/k_{m} | Bulk Permeability (k_{b}) (×10^{−13} m^{2}) |
---|---|---|---|---|---|

Simulated scenarios parameters for effect of fracture aperture (Scenario D): | |||||

D1 | 75 | 0.8 | 0.53 | 1100 | 4.85 |

D2 | 75 | 1.6 | 2.13 | 4402 | 4.89 |

D3 | 75 | 2.4 | 4.80 | 9907 | 5 |

D4 | 75 | 3.2 | 8.53 | 17612 | 5.21 |

Simulated scenarios parameters for effect of fracture density: | |||||

A | - | - | - | - | - |

B | 300 | 5 | 20.83 | 42993 | 5.19 |

C | 150 | 3.96 | 13.07 | 26976 | 5.19 |

D | 75 | 3.14 | 8.23 | 16986 | 5.19 |

E | 37.5 | 2.49 | 5.18 | 10691 | 5.19 |

F | 18.75 | 1.98 | 3.27 | 6749 | 5.19 |

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**MDPI and ACS Style**

Shafabakhsh, P.; Fahs, M.; Ataie-Ashtiani, B.; Simmons, C.T.
Unstable Density-Driven Flow in Fractured Porous Media: The Fractured Elder Problem. *Fluids* **2019**, *4*, 168.
https://doi.org/10.3390/fluids4030168

**AMA Style**

Shafabakhsh P, Fahs M, Ataie-Ashtiani B, Simmons CT.
Unstable Density-Driven Flow in Fractured Porous Media: The Fractured Elder Problem. *Fluids*. 2019; 4(3):168.
https://doi.org/10.3390/fluids4030168

**Chicago/Turabian Style**

Shafabakhsh, Paiman, Marwan Fahs, Behzad Ataie-Ashtiani, and Craig T. Simmons.
2019. "Unstable Density-Driven Flow in Fractured Porous Media: The Fractured Elder Problem" *Fluids* 4, no. 3: 168.
https://doi.org/10.3390/fluids4030168