This study presents the stochastic Monte Carlo
simulation (MCS) to assess the uncertainty of flow and conservative transport
in 3-D discrete fracture networks (DFNs). The MCS modeling workflow involves
a number of developed modules, including a DFN generator, a DFN mesh
generator, and a finite element model for solving steady-state flow and
conservative transport in 3-D DFN realizations. The verification of the
transport model relies on the comparison of transport solutions obtained from
HYDROGEOCHEM model and an analytical model. Based on 500 DFN realizations in
the MCS, the study assesses the effects of fracture intensities on the
variation of equivalent hydraulic conductivity and the exhibited behaviors of
concentration breakthrough curves (BTCs) in fractured networks. Results of
the MCS show high variations in head and Darcy velocity near the specified
head boundaries. There is no clear stationary region obtained for the head
variance. However, the transition zones of nonstationarity for

Successful characterizations of flow and contaminant transport in fractured geologic formations depend on adequate descriptions of complex geometrical structures, which comprise a wide variety of fractures and their connections (Ahmed et al., 2015; Pichot et al., 2012; Weng et al., 2014). The fracture characteristics can be quantified by using various statistical parameters, including the fracture orientation, length, shape, and permeability alongside the fracture intensity and connectivity (Bonnet et al., 2001; Botros et al., 2008; Bour et al., 2002; Koike et al., 2015; Stephens et al., 2015). These commonly used parameters represent fracture networks at sites of interest and bridge gaps between limited field observations and flow and transport implementations for site-specific issues.

Using the discrete fracture network (DFN) approach to characterize the flow and transport in fractured media is a challenging task for practical applications. Intensive research over the past 3 decades has led to the development of numerous models that are based on the DFN approach to model the flow or transport in fractured formations (Cacas et al., 1990a, b; de Dreuzy et al., 2013; Hyman et al., 2015a; Liu and Neretnieks, 2006; Long et al., 1985; Pichot et al., 2012; Xu and Dowd, 2010). Advanced 3-D DFN approaches typically include procedures of fracture generation, DFN meshing, flow and transport, or particle tracking (de Dreuzy et al., 2013; Erhel et al., 2009; Hyman et al., 2014; Pichot et al., 2012; Xu and Dowd, 2010; Zhang, 2015; Trinchero et al., 2016; Fourno et al., 2019). Particle tracking algorithms are usually preferred to simulate DFN transport and have recently been widely implemented to evaluate the time resistance of contaminants for fractured formations (Hyman et al., 2015a, b; Makedonska et al., 2015; Painter et al., 2008; Stalgorova and Babadagli, 2015; Wang and Cardenas, 2015). The objective of the Lagrangian approach is to avoid numerical difficulties in solving the advection dispersion equation (ADE) in complex DFN domains. Such DFN transport models use the released particles to represent the contaminant with a specified mass or concentration. Many previous studies have discussed the issues in treating particle movement in fracture networks (Hyman et al., 2015a, b; Johnson et al., 2006; Makedonska et al., 2015; Painter et al., 2008; Park et al., 2003; Wang and Cardenas, 2015; Zafarani and Detwiler, 2013).

Over the years, many studies have focused on developing flow and transport models and integrating DFN simulation workflows for 3-D fracture networks (Hyman et al., 2014, 2015a; Lee and Ni, 2015). Specifically, the DFN transport was mainly modeled based on Lagrangian approaches such as particle tracking and random walk algorithms (e.g., Makedonska et al., 2015; Painter et al., 2008; Stalgorova and Babadagli, 2015; Wang and Cardenas, 2015). Numerical solutions to the ADE based on the Eulerian approach have not been widely implemented because of computational issues, such as numerical dispersion and convergence in the model for complex fracture connections (Odling, 1997; Berrone et al., 2018).

With the advantages of computational technologies, the stochastic modeling of flow and Eulerian-based transport in 3-D DFNs has become a feasible task. It is an important issue to quantify flow and transport uncertainties based on available DFN properties. The objectives of this study are to develop and implement numerical models for stochastic modeling of flow and conservative transport in 3-D DFNs. The stochastic Monte Carlo simulation (MCS) is conducted to assess the flow and transport uncertainty induced by the 3-D DFNs. In this study, we first assess the developed ADE model by comparing the solutions of simple porous fractures with those from the HYDROGEOCHEM finite element model (Yeh et al., 2004) and the analytical model developed by Wexler (1992). Then, we use the MCS to evaluate the equivalent hydraulic conductivity for specified DFN statistical parameters. The collected flow and transport realizations enable the analyses of flow and transport uncertainties in the fractured simulation domain. The simulation results are expected to provide general insight into the evaluations of flow and transport uncertainty based on the available DFN geometrical properties.

In this study, the fractures in a DFN are considered to be porous media with impermeable surfaces that are connected to the formation matrix. The two impermeable surfaces of a fracture are considered to be two rough parallel plates that enable fluids to pass through the fracture at a relatively high velocity (e.g., Kwicklis and Healy, 1993; Lee and Ni, 2015; Pruess and Tsang, 1990). The following presents the mathematical formulas, a brief description of the mesh generation, and the finite element models for simulating the 3-D DFN flow and transport.

The mathematical formulation for the DFN consists of flow and transport in a
set of 2-D porous fracture plates connected in a 3-D domain. The coupling of
flow and transport in porous media has been widely investigated in fields
that are related to groundwater hydrology (Dagan, 1989; Hartley and Joyce,
2013; Yeh et al., 2004; Zheng and Bennett, 2002). Based on the concept of
mass conservation and Darcy's law, the equations for solving the steady-state
and depth-averaged hydraulic head for 2-D porous fractures can be expressed
as

Similar to the flow simulation, the depth-averaged conservative solute
transport equation for saturated fractured porous media is governed by the
ADE and can be written as (e.g., Dagan, 1989; Ni et al., 2009; Zheng and
Bennett, 2002)

Definitions of fracture orientation for a fracture in a 3-D DFNe. In this study, the positive trend and plunge angles were clockwise from the north and downward from the horizontal plane, respectively.

The information of fracture orientations enables the direct simulation of flow and transport in a series of 2-D fractures and the reproduction of the flow and transport behaviors for a 3-D DFN system. This study defines a DFN without isolated fractures as an effective discrete fracture network (DFNe). Figure 1 shows the definitions of the individual fracture in a 3-D DFNe. Based on the long axis of an elliptical porous fracture, the positive trend and plunge angles are defined as clockwise from the north and downward from the horizontal plane, respectively. In this study, the intersections of a fracture and the simulation boundaries have to be identified (Fig. 1) before the mesh generation is implemented. We generate the fracture length for the long and short axes of each fracture in the 3-D DFN based on the uniform distribution. The larger value of the two generated radii is used to obtain the long axis of the elliptical fractures. In addition, isolated fractures that are not connected to other fractures and simulation boundaries are removed for computational efficiency.

Example of a generated mesh for two intersected fractures in

Figure 2 shows an example of a DFNe connection for two intersected fractures. Figure 2a shows the generated mesh in the 3-D domain for two intersecting fractures. Mesh generation begins with the generation of initial fracture meshes for each fracture plate (i.e., Fractures 1 and 2 in Fig. 2c and e, respectively). Figure 2b displays two intersecting fractures that were individually rotated back to the 2-D horizontal plane. In Fig. 2b the plunge and trend values differ for each fracture in a 3-D DFNe, so the fractures in a 2-D coordinate system might not overlap. The intersections for each fracture are also located in different areas (see Fig. 2b). However, the length of the intersection should be identical to that of the intersecting fractures. The fracture intersections and simulation boundaries are recorded for our unstructured mesh generation model.

The mesh generation starts with generating initial mesh for each fracture.
The mesh generator allows users to define intervals of mesh boundary nodes.
In this study, the Delaunay triangulation algorithm is used for generating
the initial meshes. The special treatment of fracture intersections rely on
the boundary recovery algorithm. In the process of boundary recovery for
fracture intersections, we allow the node interval be reduced with a defined
ratio according to the smallest node interval along the edges of the
connected fractures. The ratio can be

To solve the governing equations of flow and transport for the DFNe
framework, we employ the Galerkin finite element method and the biconjugate
gradient matrix solver to solve Eqs. (1) and (4). The linear function for
hydraulic heads and the concentration at the nodes surrounding an element of
the triangular mesh system can be represented with

The features of the HYDROGEOCHEM model are not for DFN flow and transport modeling. For simple cases such as a single horizontal fracture plate or cross-shaped porous fracture, it is possible to simulate a fracture and matrix system using the HYDROGEOCHEM model if one can use small mesh sizes to resolve the fracture apertures and matrix system. In addition, the differences of the hydraulic conductivity between the fracture and matrix need to be large enough to minimize the influence of the matrix. Because the HYDROGEOCHEM was developed based on the finite element method, the numerical dispersion might be similar to the developed model in this study. This study further uses a two-dimensional analytical solution proposed by Wexler (1992) to conduct verification of the developed model. The comparison is limited to the case with advection and dispersion in a horizontal porous fracture plate.

Based on the verified DFN flow and transport model, we then conducted 500 MCS
realizations to assess the upscaled flow behaviors with various fracture
intensities for 2-D profiles (i.e.,

Conceptual model for the verification of a 2-D horizontal fracture
plate and a cross-shaped fracture network:

This study employs two cases in a 2 m

The flow and transport parameters that were used for the transport verification cases.

The study of Wexler (1992) considers a horizontal two-dimensional domain. The
simulation domain is similar to the first case in Sect. 3.1. However, the
study of Wexler (1992) considers an

In the test example (Case 3), the horizontal porous fracture plate has the
size of 2 m

The equivalent hydraulic conductivity for a specified representative
elementary volume (REV) is the basis for conducting flow upscaling for
practical problems that cover simulation domains on the order of hundreds of
meters to several kilometers. Similar to the fractured rock volume in the
test case, we generate 500 DFN realizations to assess the flow and upscaling
behaviors for various fracture intensities and the associated fracture
properties (e.g., fracture locations, plunges and trends, and sizes). Table 2
lists the parameters for generating the DFN realizations. In this study, the

Conceptual model, a DFNe realization, and the associated flow field for the numerical example.

Parameters that were used to generate 3-D DFNs for the flow example.

The calculation of the equivalent hydraulic conductivity for a rock system
considers the concept of mass conservation applied to a REV. The flow passing
through the 3-D control volume can be represented by the following formula:

Parameters that were used for the transport simulations in the 3-D DFNe realizations.

In this numerical example, we investigate the effects of averaging strategies
(i.e., along vertical lines or on profiles perpendicular to flows) on the
observations of breakthrough curves (BTCs) in fractured formations. This numerical example involves
using the random DFNe flow realizations from the example in Sect. 3.3.
Table 3 lists the parameters applied to the transport simulations. Figure 5
shows the boundary conditions for the transport simulations. Constant
concentration values of 1.0 and 0.0 (kg m

Conceptual model and the specified well and profile locations for the calculations of the flow and transport uncertainties.

The

Comparison of the concentration distributions for the developed DFN model (dashed lines) and the HYDROGEOCHEM model (solid lines) for a horizontal fracture plate.

Comparison of the concentration distributions for the developed DFN
model and the HYDROGEOCHEM model for a cross-shaped fracture network:

We further define four profiles to assess the flow and transport
uncertainties. The profiles can be considered as a series of wells installed
along the profiles. The vertical profile (

Three profiles perpendicular to the flow direction have the same

This study focused on a relatively small fractured rock volume that was
2 m

The comparison of solute transport for a continuous inlet source in
a 2-D horizontal porous fracture at

Comparisons of the DFN properties and equivalent hydraulic conductivity for the 500 generated DFN realizations. The superscripts e and t represent the effective and total fracture intensities.

Figures 6 and 7 show a comparison of the concentration distributions for the horizontal fracture plate and cross-shaped fractures. The results in Figs. 6 and 7 show that identical solutions were obtained from both the developed DFN model and the HYDROGEOCHEM model. All the temporal and spatial variations in the plume were determined, and the solutions from the developed and HYDROGEOCHEM models were found to be identical. Figure 7 shows the concentration distributions after 3.0 days (Fig. 7a and b) and 5.0 days (Fig. 7c and d) when using the HYDROGEOCHEM and developed models for a cross-shaped fracture system. With a specified small upward flow applied to the vertical fracture, portions of the concentrations moved upward near the fracture line of intersection (Fig. 7b and d). This slightly upward flow relied on the constant head of 9.01 m that was applied on the top side of the vertical fracture. Again, the developed and HYDROGEOCHEM models were found to be identical for the cross-shaped fracture network.

A realization of the transport simulation based on the DFNe flow field in Fig. 4.

Mean concentration BTCs (solid lines) and associated SD intervals (dashed lines) at the sampling wells for

Figure 8 shows the comparison of the solute concentration obtained from
analytical (dashed lines) and the developed (solid lines) models. Figure 8a–d show the concentration distribution at time 5.0, 7.5, 10.0, and
15.0 days, respectively. The concentration for the contour lines are for the
relative concentration (

Comparison of flow uncertainties for vertical profile along the flow
direction.

Figure 9 shows the results of the flow simulations when applied to the
500 DFN realizations. Figure 9a shows that the number of fractures increased
with the effective fracture intensity

Transport uncertainties for the vertical profile along the flow
direction.

Figure 10 shows a realization of the transport simulation based on the DFNe flow field in Fig. 4. These results revealed that the continuous concentration released on the left boundary gradually migrated along the connected fractures. The spatial distributions of the concentration on the fracture plates were highly variable. This finding validated the concept that was proposed by Park et al. (2003), who stated that local flow cells contribute less to flow and contaminant transport in fracture formations.

Figure 11 shows the mean concentration BTCs (solid lines) and the associated
SD intervals (i.e.,

The acceptable number of the MCS realizations was decided based on the
comparison of statistical moments for different numbers of MC realizations.
With the realization numbers up to 500, the overall trends of head and
velocity variances were obvious, except for some variations along the
selected profiles. Figure 12 shows the distribution flow uncertainties along
the flow direction. Please note that the results were based on collections of
vertical averaged heads and Darcy velocities at each

Figure 12c and d show the distribution of velocity variance in

Figure 12e–h show the means and variances of the Darcy velocities in the

Figure 13 shows the distributions of transport uncertainty at different times
along the centerline profile (

Numerical solutions to the ADE based on the Eulerian approach have not been widely implemented because of heavy computational costs. This study developed and implemented numerical models for stochastic modeling of flow and conservative transport in 3-D DFNs. The developed ADE model was validated by comparing the solutions of simple porous fractures with those from the HYDROGEOCHEM finite element model and an analytical model. When testing the transport model, identical temporal and spatial solutions were obtained from the developed model and the HYDROGEOCHEM model based on a Gaussian-type initial plume that was released in the porous fracture plate. For a simplified case, an analytical 2-D transport solution exists. The developed model accurately produced the results of concentration distributions in a horizontal fracture plate.

The MCS flow simulations showed that different fracture intensities can result in variations in the equivalent hydraulic conductivity that were 2 to 3 orders of magnitude lower than the fracture hydraulic conductivity values.

Simulations of transport in 3-D DFNs revealed that the maximum concentration of mean BTCs for different averaging strategies might not have reached the concentration. The sampling strategies along the wells and profiles yielded similar BTCs patterns. Based on the MCS, the means and SD for the two sampling strategies were observed at different sampling locations. MCS results showed that a smaller sampling volume can lead to relatively large values of mean concentrations and concentrations of SD for specified times.

MCS flow and transport showed that the distribution of the head variance
exhibited high variations at the inlet boundary, and the head variance
gradually decreased to a small value at the outlet boundary. The extremely
high head variation near the DFN inlet boundary could be induced by the generated
DFN realizations. No stationary zone for the head variance was obtained based
on collected MCS realizations. In the study, the value of the highest

The data that support the models in this study are available from the corresponding author upon request.

CFN and IHL conceived and designed the main idea of paper. IHL, CFN and FPL developed the model and analyzed the results. CPL and CCK contributed the data preparation and figures. IHL and CFN wrote the paper. All Authors read and approved the final draft.

The authors declare that they have no conflict of interest.

This research was partially supported by the Ministry of Science and Technology of the Republic of China under grants MOST 103-2221-E-008-049-MY3, NSC 102-2116-M-008-010, and NSC 100-2625-M-008-005-MY3 as well as by the Institute of Nuclear Energy Research, the Atomic Energy Council of the Executive Yuan, under grant NL1030099, and by the Soil and Groundwater Pollution Remediation Fund in 2017. Edited by: Philippe Ackerer Reviewed by: two anonymous referees