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Non-Darcian flow has been well documented for fractured media, while the potential non-Darcian flow and its driven factors in field-scale discrete fracture networks (DFNs) remain obscure. This study conducts Monte Carlo simulations of water flow through DFNs to identify non-Darcian flow and non-Fickian pressure propagation in field-scale DFNs, by adjusting fracture density, matrix hydraulic conductivity, and the general hydraulic gradient. Numerical simulations and analyses show that interactions of the fracture architecture with the hydraulic gradient affect non-Darcian flow in DFNs, by generating and adjusting complex pathways for water. The fracture density affects significantly the propagation of hydraulic head/pressure in the DFN, likely due to fracture connectivity and flow channeling. The non-Darcian flow pattern may not be directly correlated to the non-Fickian pressure propagation process in the regional-scale DFNs, because they refer to different states of water flow and their controlling factors may not be the same. Findings of this study improve our understanding of the nature of flow in DFNs.

Darcy’s law proposed by Henry Darcy maintains that the specific discharge of water increases linearly with the gradient of hydraulic head along a 3.5-m-long saturated column filled with homogeneous sand [

This study aims at exploring the potential for non-Darcy flow in field-scale discrete fracture networks (DFNs). Fluid flow transition from Darcian to non-Darcian has been confirmed and broadly studied in the case of a single rock fracture [

We will also explore the possible impact of non-Darcy flow on the transient dynamics of water flow through DFNs, which has not been addressed in previous studies. Darcy or non-Darcy flow is usually defined using steady-state flows, where the asymptotic flow rate is used to build the relationship with the pressure drop. Water flow in natural aquifers is often transient, due to the change of input (such as short-term weather change) and/or output (i.e., pumping), and therefore the transient flow dynamics are practically important.

The rest of this work is organized as follows. Section 2 presents the Monte Carlo approach to simulation water flux and pressure propagation through multiple DFNs, which is one of the most efficient ways to investigate the impact of fracture properties on water flow behaviors. Results of the Monte Carlo simulations and evaluation using a standard dispersion equation (SDE) are presented in section 3. In section 4, we discuss the possible signal of non-Darcian flow and non-Fickian pressure propagation and their characterizations. The impacts of fracture density and rock matrix permeability on non-Darcian flow and non-Fickian pressure transfer are investigated. Conclusions are drawn in section 5.

There are three major steps in the Monte Carlo simulation of water flow through saturated field-scale DFNs. First, we generate equally possible but different realizations of stochastic fracture networks for DFNs for each pre-assigned fracture density, using Hydro Geo Sphere (HGS) software (v.111, Aquanty Inc., Waterloo, ON, Canada) [

The two-dimensional DFN has a dimension of 50 m (discretized into 100 blocks) along the longitudinal direction (x axis) and 25 m (50 blocks) vertically (z axis). Three scenarios of DFNs, with each containing 100 realizations of DFNs, are built, which have their own unique time-dependent seed based on the current system time to generate random fractures. Each fracture network is composed of two superimposed sets of fractures, which are orthogonal to each other (orientations are 0˚ and 90˚) as observed commonly in realistic DFNs [^{−8} to 1 × 10^{−7} m/s.

Both the steady-state and transient groundwater flow through the confined aquifer generated above were solved by HGS. Parameters for the flow model are the same as those used in [

Hydraulic head (or the propagation of the hydraulic pressure) is typically described by the well-known Boussinesq flow equation [

∂ p ∂ t = D ∂ 2 p ∂ x 2 (1)

where p(x, t) denotes the spatially and temporally varying pressure, and D represents the diffusion coefficient. Considering the initial condition: p(x, t = 0) = 0 and the constant-pressure boundary conditions (p(x = 0, t = 0) = p_{l} for the inlet boundary), we obtain the following analytical solution for model (1):

p ( x , t ) = p l [ 1 − erf ( x 4 D t ) ] , (2)

where erf(・) represents the error function.

In the following two sections, we will model the pressure propagation using the SDE(1) to investigate the mechanism behind the propagation of the pressure (head).

The ensemble average of steady-state water flux across the outlet (right) boundary for all 100 realizations is plotted in

Results show that the DFN with 100 fractures contains the largest noise in the simulated flux (especially for a relatively small hydraulic gradient: 1 × 10^{−6} < J < 1 × 10^{−3}), and the relationship between hydraulic gradient and flux transfers from non-linear (power-law) to linear gradually (

The simulated transient flux at the outlet boundary is shown in ^{−7} m/s. For illustration purposes, here we show the results with three hydraulic gradients: J = 1 × 10^{−5}, 1 × 10^{−4}, and 1 × 10^{−3}. To explore the impact of K on transient flux, we re-ran the above Monte Carlo simulations and obtain the transient flux for K = 1 × 10^{−8} m/s (

The fracture density may affect Non-Darcian flow in field-scale fractured networks by generating complex flow paths for water, which can change with the magnitude of the general hydraulic gradient J. For a dense DFN (i.e., the DFN with 100 fractures, see _{s}, the overall flow paths reach stable (or reach the capacity of connection), and the corresponding longitudinal flux is now mainly controlled by J, resulting in Darcian flow.

For a sparse DFN (such as the one with 20 fractures shown in

that observed in a fluvial setting with a small proportion of high-permeability ancient channels [_{s} to reach the connection capacity and the Darcian flow regime. Therefore, if J is less than J_{s}, both the DFN’s internal architecture and the pressure gradient affect water flux, generating strong non-Darcian flow. When J is larger than J_{s}, Darcian flow dominates all DFNs. The denser for the DFN, the larger for the threshold J_{s}. For example, Monte Carlo simulations of this study show that J_{s} is equal to 1 × 10^{−4}, 2 × 10^{−4}, and 1 × 10^{−3} for the DFN with 20, 60 and 100 fractures (see ^{−4} and 10^{−1} in natural aquifers, and hence flow in the field-scale DFN most likely follows Darcy’s law, especially for the DFN with sparse fractures.

The channeling of flow can also be found in

The sparse DFN exhibits stronger non-Fickian pressure propagation, especially at the early time, due to the following three reasons. First, the sparse DFN has a relatively small effective hydraulic conductivity, resulting in an overall slow motion for water. It therefore takes a longer time for the transient flux in the spare DFN to reach its asymptote, generating the transient flux with a delayed arriving limb at the early time (see

the SDE curve in

The delayed transfer of water at the early time is also observed for dense DFNs (

Decrease of the rock matrix hydraulic conductivity tends to retard further the propagation of pressure for all DFNs tested in this study, implying that a larger permeability contrast between fractures and matrix leads to a stronger non-Fickian propagation of the hydraulic pressure. It is also noteworthy that for the sparse DFN (with 20 fractures), the inlet boundary for some realizations may not be directly connected with the major fractures, and hence the decrease of the matrix hydraulic conductivity causes significantly slow arrival of the transient flux (by comparing

We do not find direct correlation between non-Darcian flow and non-Fickian pressure propagation in field-scale DFNs. Non-Darcian flow quantifies the steady-state flux, while non-Fickian pressure propagation focuses on the evolution dynamics of transient flux before reaching its steady-state asymptote. Hence, they need not to be directly connected. Indeed, the above Monte Carlo simulations showed that the dense DFN with strong non-Darcian flow tends to exhibit weak non-Fickian pressure propagation. Our analysis also shows that the threshold hydraulic gradient J_{s} distinguishes Darcian and non-Darcian flow, while the initial regime related to fracture connectivity and architecture affects the early-time non-Fickian pressure propagation.

This study conducts Monte Carlo simulations to identify possible non-Darcian flow and non-Fickian pressure propagation in field-scale discrete fracture networks. Multiple scenarios of DFNs are generated with different fracture densities and matrix hydraulic conductivities. Flux needed for Darcian/non-Darcian flow analysis is calculated from the steady-state flow models using HGS, and the transient motion of water is modeled to reveal possible non-Fickian propagation of hydraulic pressure. Numerical simulations and result analysis lead to the following three main conclusions.

First, both the fracture network architecture and the general hydraulic gradient J affect the Darcian/non-Darcian flow in DFNs. The fracture density may affect flow dynamics by generating complex flow paths for water, and the gradient Jcan adjust the flow field and provide the criterion for Darcian/non-Darcian flow. Strong non-Darcian flow appears for J less than the threshold J_{s}, where this threshold increases with an increasing fracture density.

Second, fracture density affects significantly the propagation of hydraulic head or pressure in the DFN. A sparse DFN can cause both delayed motion of water at the early time (likely due to the small effective hydraulic conductivity and the poor fracture connectivity) and enhanced flow at the middle/late time (likely due to the enhanced flow channeling), resulting in strong non-Fickian pressure propagation.

Third, the non-Darcian flow pattern needs not to be directly related to the non-Fickian pressure propagation process in field-scale DFNs, because they refer to different states of water flow and their controlling factors may not be exactly the same.

This work was funded partially by the National Natural Science Foundation of China under grants 41628202, 41330632 and 11572112. This paper does not necessarily reflect the views of the funding agency.

Lu, B.Q., Zhang, Y., Xia, Y., Reeves, D.M., Sun, H.G., Zhou, D.B. and Zheng, C.M. (2018) Identifying Non-Darcian Flow and Non-Fickian Pressure Propagation in Field-Scale Discrete Fracture Networks. Journal of Geoscience and Environment Protection, 6, 59-69. https://doi.org/10.4236/gep.2018.65005