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For the problem of whether the representative elementary volume (REV) obtained in the Darcy flow is also applicable to the case of the non-Darcy flow, the study on the REV size within the non-Darcy flow is proposed tentatively. The concept of the REV in the non-Darcy flow is based on the definition of the REV. According to the determination of the REV in the Darcy flow, the intrinsic permeability

Hydraulic property of the fractured rock mass plays a vital role in on-site engineering, like the underground basement, oil storage, CO_{2} storage, and giant dams. The numerical simulation method can provide essential references for such projects, among which the Finite Element Method (FEM) is the most sophisticated numerical method. In the simulation of flow in the porous media, the equal intrinsic permeability should be determined first. However, as for the fractured porous media, before determining the equivalent intrinsic permeability, there is another critical parameter, the representative elementary volume (REV) to be determined. The REV is defined as the minimum volume of the sampling domain, beyond which the intrinsic permeability of the sampling domain remains essentially constant [

In 1982, Long et al. [

In 1960, Barenblatt et al. [

Moreover, many studies on the non-Darcy flow in the fracture have been carried out. Currently, it is considered that inertial force mainly causes the non-Darcy flow. Many experiments include it [

Throughout the previous literature, on one hand, most studies on the REV of the 2D fractured rock rely on the discrete model, and it is assumed that the rock matrix is impermeable and the fluid flow only occurs in the fracture. Considering that the pressure in the matrix would affect the velocity of flow in the breach, the REV size of fractured rock based on the dual-porosity model is investigated. Besides, the simulation method of the Darcy and non-Darcy flow in the fractured rock mass is built, which is conducive to promotion. On the other hand, almost all the studies on the REV involve the fluid flow in the fracture obeying the Cubic law, and as for the nonlinear flow, most studies focus on the single fracture and the fracture network. The survey on the REV in the non-Darcy flow has not been conducted in previous work. Based on the review of the non-linear flow behavior, the REV size of fractured rock in the non-Darcy flow is tentatively investigated.

In this paper, firstly, based on the dual-porosity model and the mathematics model [

Considering that the velocity of the fluid in the porous rock matrix is much lower than in the fracture, the fluid flow in the matrix always obeys Darcy’s law. For the single-phase flow and incompressible flow in porous media, Darcy’s law can be expressed as (in the absence of gravity)

When the flow velocity is low in the fracture, the relationship between the volumetric flow rate and the pressure gradient is linear, and it can be described as the Cubic law:

The pore pressure cohesive element (PPCE, 6-node element) is developed from a cohesive element (4-node element), which can simulate the fluid flow in fractured rock [

Fluid flow within the PPCE in 2D.

The meshing example for 2D intersecting fractures with the PPCE.

In ABAQUS, the default constitutive relationship of tangential flow is described as (

In the flow exchange of fractures and the matrix, the entire basin must follow the law of mass conservation. As shown in Figure

Fractured media.

For the standard calculation in linear FEM, when taking the theory of flow for example, the relationship can be written as follows:

If the relationship between the flow rate and pressure is linear, then the intrinsic permeability matrix is a constant matrix. However, for the non-Darcy flow, the problem is complicated. The Forchheimer flow rate

Then,

From (

Relationship between the flow rate and pore pressure.

As for the non-Darcy flow, the constitutive default of tangential flow can be adapted with a subroutine (nodes 1 and 2 in Figure

In this study, the primary incremental method is used for the calculation of the non-Darcy flow. As shown in Figure

Relationship between the flow rate and pore pressure in the incremental method.

To validate the correctness and effectiveness of the PPCE and its subroutine, several cases with the Darcy flow and the Forchheimer flow are calculated to compare with previous studies.

In this case, the steady state of fractured media is simulated. The dimensions and boundaries are shown in Figure ^{2}, and ^{2}.

2D Fractured media with applied pressure at top and bottom.

The discretization and pressure fields of this field obtained by ABAQUS-PPCE are compared with DFM (the discrete fracture model) and FM-Meshfree (fracture mapping with mesh-free), which are proposed by Lamb [

Domain discretization of each method.

ABAQUS-PPCE

DFM

FM-Meshfree

FM-Meshfree (irregular nodal distribution)

Pore pressure fields obtained by ABAQUS-PPCE and other methods.

ABAQUS-PPCE

DFM

FM-Meshfree

FM-Meshfree (irregular nodal distribution)

The pressure profile for a vertical section taken through the center of the fractured media.

In this case, ABAQUS-PPCE is used to simulate one of the most common problems in geotechnical engineering: unconfined flow analysis. The dam model’s dimensions and water elevations of the upstream and the downstream are shown in Figure

Dimensions and boundaries of the dam model.

Porous media

Fractured media

Pore pressure fields obtained by ABAQUS-PPCE.

Porous media

Fractured media

Comparison of the results obtained by different methods.

Frih et al. [

Dimensions and boundaries of the model with one fracture.

(a) The 3D pore pressure field and (b) velocity for the Forchheimer flow case.

Darcy and Forchheimer flow velocity in the fracture given by the interface model and ABAQUS-PPCE.

Above all, by comparing ABAQUS-PPCE’s results with other existing examples, it can be found that ABAQUS-PPCE is effective and accurate when simulating the Darcy flow and the non-Darcy flow (Forchheimer flow) within the fractured porous media.

In this section, a fundamental question in the field of flow in fractured rock, REV, and equivalent hydraulic properties is researched. Firstly, fracture networks are generated by Monte Carlo Simulation technique. Then REV and equivalent hydraulic properties are studies in the cases of Darcy and non-Darcy flows. Different criteria are adopted to determine whether there is inconsistency in the REV in the case of the Darcy and non-Darcy flow.

Table ^{2}. The program based on the Monte Carlo Simulation technique [

Fracture parameters used for the generation of the fracture network.

Set | Density ( | Dip (°) | Length (m) | Aperture ( | ||||
---|---|---|---|---|---|---|---|---|

Distribution | Standard deviation | Mean | Distribution | Standard deviation | Mean | |||

1 | 0.0231 | Normal | 2.08 | 108.39 | Log-normal | 2.22 | 14.51 | 85 |

2 | 0.0361 | Normal | 2.97 | 142.05 | Log-normal | 1.56 | 10.67 | 85 |

3 | 0.0247 | Normal | 2.58 | 151.91 | Log-normal | 2.26 | 13.8 | 85 |

4 | 0.0231 | Normal | 2.08 | 71.61 | Log-normal | 2.22 | 14.51 | 85 |

5 | 0.0361 | Normal | 2.97 | 37.95 | Log-normal | 1.56 | 10.67 | 85 |

6 | 0.0247 | Normal | 2.58 | 28.09 | Log-normal | 2.26 | 13.8 | 85 |

Generation of the fracture network and the rotated models.

The generated fracture network is imported into ABAQUS for modeling. The element type of the fracture is COH2D4P, and the rock matrix is CPE6MP, which is a triangle element to facilitate the process. A precise model in the side length of 2 m and its meshes are shown in Figure

A certain fracture network model inside the length of 2 m and its meshes.

When calculating the fractured rock flow, one assumption is that the fracture is the parallel smooth plane, which means the roughness of the breach is ignored. Figure

Hydraulic boundary conditions for calculation (

The values of

Before calculating the Darcy flow in a fractured network, the hydraulic conductivity of each fracture needs to be calculated by the Cubic law based on the equivalent hydraulic aperture (equal to the aperture in a smooth parallel plate model) of each fracture, and then the inherent permeability tensor

In this paper, CFD calculation software FLUENT is used for nonlinear simulation of the smooth parallel plate model with length^{−8} can be obtained with ^{−1} and all the non-Darcy flow parameters are determined.

Numerical simulation results of the smooth parallel fracture.

For the non-Darcy flow of the fractured rock mass, consistent with the non-Darcy flow of a single fracture, the slope between the flow rate and the pressure gradient decreases as the pressure gradient increases. Therefore, to determine the REV and non-Darcy coefficient tensors in the non-Darcy flow of the fractured rock mass, different hydraulic boundaries need to be applied in the same model. According to the research of Chen et al. [

Ignoring the “preferential flow” effect [

Bachmat [

In this section, firstly, the numerical results of the Darcy flow about size effect on the intrinsic permeability are presented. Then the statistical results of the non-Darcy flow are presented with size effect on inherent permeability and the non-Darcy coefficient. The graphic presentation of results is shown more clearly, with direct results of intrinsic permeability components (

Figure

The pore pressure field at the side length of 9 m and the rotation angle of 0°.

The results, Max, Min, and mean of calculated intrinsic permeability components in the Darcy flow. (a)

The values of coefficients of variation of

Figure

The results, Max, Min, and mean of calculated intrinsic permeability components in the non-Darcy flow. (a)

The results, Max, Min, and mean of calculated non-Darcy coefficient components in the non-Darcy flow. (a)

Figures

The values of coefficients of variation of

The values of coefficients of variation of

The equivalent intrinsic permeability ellipse with the increasing model size in the Darcy flow.

The intrinsic permeability tensor ellipses of the non-Darcy flow are drawn in Figure

The equivalent intrinsic permeability ellipse with the increasing model size in the non-Darcy flow.

The equivalent non-Darcy coefficient ellipse with the increasing model size in the non-Darcy flow.

According to the calculated results above, the following conclusions are mainly drawn. (I) As for the Darcy flow, if an acceptable value of “CV” of 20% is set, then the side length of 6 m can be approximated for the REV size. If the value of “CV” of 10% is accepted, then the side length of 8 m can be approximated by the REV size. (II) As for the non-Darcy flow, two parameters (the intrinsic permeability

Besides, it is worth noting that, for the flow of Darcy and non-Darcy, the variation of the CV value of the equivalent permeability coefficient

Above all, what we can conclude is that if the value of “CV” of 20% is accepted,

Above all, the pore pressure cohesive element (PPCE) with a subroutine is developed, and the simulation of the non-Darcy flow in the fracture is realized. Then the simulation method of the Darcy and non-Darcy flow in the dual-porosity model, like fractured rock, is built. Main conclusions are as follows.

(I) The PPCE can be inserted to simulate the Darcy flow in the dual-porosity model, like fractured rock, in ABAQUS. With the subroutine, the PPCE can also simulate the non-Darcy flow in fractured rock. The calculated results are compared with other existing results of the dual-porosity simulation to validate its correctness and effectiveness. A new and more convenient method is provided to simulate the flow in the dual-porosity model.

(II) The Monte Carlo Simulation technique is adopted for the generation of the fractured network. Based on the dual-porosity model with the PPCE, generated fracture networks with different side lengths and rotation angles are simulated in both Darcy and non-Darcy flow. On the one hand, the intrinsic permeability

(III) According to the study on the determination of the REV in the Darcy flow and the non-Darcy flow, the REV determined in the non-Darcy flow is a little larger than that in the Darcy flow. Thus, it is appropriate to propose that the REV should be determined by the non-Darcy coefficient

In this paper, based on the dual-medium model, the Monte Carlo fractured rock mass is studied tentatively using the non-Darcy coefficient to determine the REV value under the non-Darcy flow. The REV of the Darcy and non-Darcy flow is compared. Some new understandings have been found. For example, the REV of the non-Darcy flow is inconsistent with the REV of the Darcy flow, and the REV of the non-Darcy flow is larger than the REV of the Darcy flow.

The inconsistency between the REVs mainly results from the REV of the non-Darcy flow determined by non-Darcy coefficient

The conclusion that the REV of the non-Darcy flow is larger than that of the Darcy flow is only based on the results shown in the model calculation in this paper. Whether this conclusion is universal and whether this difference can be quantitatively described and which physical parameters of the fractured rock mass influence this outcome need to be further explored and studied in the future work.

Besides, it should also be mentioned that the PPCE has a wide range of applications in flow in fractured rock and soil, and the method proposed in this paper can also be applied in the simulation of on-site engineering. What is more, when it comes to the intersecting fractures in the non-Darcy flow, the “preference flow” is ignored in this study. It is suggested that similar studies involve this factor. For more work on the REV and its intrinsic permeability tensor of the fractured rock mass in the non-Darcy flow, factors like roughness, stress, and aperture also should be taken into consideration.

Flow velocity

Intrinsic permeability of porous media

Pressure gradient

Volumetric flow rate or discharge

Fracture aperture [L]

Aperture of the idealized parallel smooth fracture [L]

Equivalent hydraulic aperture [L]

Dynamic viscosity of fluid

Intrinsic permeability of fracture

Cross-section area

Forchheimer coefficients

Non-Darcy coefficient

Density of fluid [

Reynold’s number [-]

Fluid leak-off coefficients [

Pore pressure [

Flow rate along with

Cross-section of the model [

Intrinsic permeability tensor [

Directional cosines [-]

Calculated intrinsic permeability in each rotated model [

Constant associated with fractured rock mass [-]

Non-Darcy coefficient tensor

Modulus of the flow rate

Calculated non-Darcy coefficient in each rotated model

Coefficient of variation [-].

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the support of the National Key R&D Program of China no. 2016YFC0401804.