### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Multiple-Fracture Flow Experiment
- 3. GeoFlow Simulation
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[1] A fluid flow experiment was conducted on a granite sample containing two intersecting fractures. At constant confining pressure, water was supplied to the sample via a single inlet port, and the effluent was collected using four isolated outlet ports. The flow rate varied widely among these ports, indicating the formation of 3-D preferential flow paths (channeling flow), which likely occur in fractured rocks but have been considerably difficult to identify by existing methods. The novel concept of GeoFlow, a discrete fracture network model simulator in which fractures have a heterogeneous aperture distribution, has been developed to analyze such complex fluid flow. A fluid flow simulation was conducted using GeoFlow with aperture distributions within the two fractures, as determined using fracture surface topography data. Despite the simplicity of the simulation, GeoFlow revealed a 3-D channeling flow within the sample, which explains the general trend of the uneven outflows in the experiment.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Multiple-Fracture Flow Experiment
- 3. GeoFlow Simulation
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[2] Fractures in rocks are recognized as the predominant pathways of resources and hazardous materials, such as groundwater, hydrocarbons, geothermal fluids, and the high-level nuclear wastes, because fractures usually have much greater permeability than the matrix permeability [*Watanabe et al.*, 2008; *Nemoto et al.*, 2009]. It has been suggested for some time that a fluid flow within a fracture should be characterized by formation of preferential flow paths (i.e., channeling flow) due to the heterogeneous aperture distribution created by the rough surfaces contacting, in part, each other [*Brown*, 1987; *Pyrak-Nolte et al.*, 1988; *Tsang and Tsang*, 1989; *Durham*, 1997; *Brown et al.*, 1998]. Indeed, according to field investigations at depths of up to 600 m in the Stripa mine in Sweden, the area in which more than 90% of the water flow was thought to occur in channels was estimated to be approximately 5% to 20% of the fracture plane depending on the confining pressure [*Abelin et al.*, 1985]. Moreover, laboratory investigations have also shown that the area in which flowing fluid (water) exists because of the channeling flow is expected to be similar percentages of the fracture plane, at a wide range of confining pressures of up to 100 MPa [*Watanabe et al.*, 2009].

[3] On the basis of previous studies, including a number of studies that are not cited above, the formation of the preferential flow paths in three dimensions (3-D channeling flow) in a fracture network should be addressed to clarify fluid migration. In analyzing the 3-D fluid flow in a fracture network, numerical methods may be more practical and effective than experimental methods, because it is difficult to conduct experiments using a multiple-fracture sample. With respect to numerical methods, modeling approaches for a fractured rock have traditionally been divided into two rough classes: continuum models and discrete fracture network (DFN) models [*Long et al.*, 1982; *Vesselinov et al.*, 2001; *Berkowitz*, 2002; *Ando et al.*, 2003; *Neuman*, 2005; *Illman et al.*, 2009]. Furthermore, each class can be formulated in deterministic and stochastic frameworks. Continuum models are applied mainly for the prediction of fluid flow behavior averaged over a large domain. Single values of hydraulic parameters are defined at each point throughout the domain of interest in deterministic continuum models, while the interior bulk flow and transport properties of dominant fractured rock features, and remaining rock mass, are represented as separate correlated random function of space in stochastic continuum models. On the other hand, DFN models are best treated in a stochastic framework by considering Monte Carlo analyses based on multiple realizations of a fractured system, because complete field data for deterministic models are usually difficult to obtain. DFN models can naturally incorporate geometrical properties in fractures (fracture size, aperture, location, orientation, and density), and as a result can account explicitly for the contribution of individual fractures on fluid flow, which cannot be considered properly in continuum models. However, in the conventional DFN models, individual fractures were characterized by a single aperture value [*Jing et al.*, 2000], and the effects of the heterogeneity of aperture distributions on 3-D fluid flow were still not considered.

[4] Therefore, the authors have developed GeoFlow, a novel discrete fracture network (DFN) model simulator, in which fractures are characterized by aperture distribution, rather than a single aperture value in the conventional DFN model simulator. Although numerical methods for a 3-D fluid flow simulation in a network of fractures with aperture distributions have been reported [*Morris et al.*, 1999; *Stockman et al.*, 2001; *Johnson et al.*, 2006], these methods focus only on fracture flow and so would not be suitable for DFN model simulations on natural fractured rocks with nonzero matrix permeabilities. In contrast, one novel aspect of GeoFlow is the combination of fractures having aperture distributions with a matrix, and this aspect was reached by applying a hybrid DFN and continuum model. The present paper first describes a fluid flow experiment on a cylindrical granite sample containing two intersecting fractures to demonstrate that 3-D channeling flow must be considered to interpret fluid flow even in the simplest fracture network. A fluid flow simulation of the sample by GeoFlow is then conducted, and the potential to predict 3-D channeling flows in fracture networks is demonstrated.

### 2. Multiple-Fracture Flow Experiment

- Top of page
- Abstract
- 1. Introduction
- 2. Multiple-Fracture Flow Experiment
- 3. GeoFlow Simulation
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[5] The fluid flow experiment was conducted using a cylindrical granite sample (diameter: 100 mm, length: 147 mm) having two intersecting tensile fractures (fractures A and B; Figure 1). A multiple-fracture sample (Figure 1a) was prepared as follows. First, two tensile fractures were sequentially induced within a cube of Inada granite, quarried in Ibaraki, Japan. The fractured granite was then fixed with mortar so that the fractures were mated fractures. The granite was then cored and cut to the prescribed dimensions. Before the experiment, the surface topography of the entire fracture plane was measured in a 0.25 mm^{2}grid system using the laser-scanning equipment reported by*Watanabe et al.* [2008] to determine the aperture distributions of fractures A and B in the fluid flow simulation by GeoFlow.

[6] A custom experimental system (Figure 1b) was used to obtain data for comparison with the results obtained by GeoFlow. The experimental system is equipped with a water pump, a pressure gauge, and a confining pressure vessel similar to that reported by *Watanabe et al.* [2008]. The confining pressure vessel enabled the evaluation of uneven outflow from the sample at a prescribed confining pressure. Using the water pump, room temperature water is injected into the top of the confining pressure vessel through a single inlet port. Water flows through the sample and out of the bottom of the confining pressure vessel through five outlet ports (ports 1–5), which are isolated from each other by a port separator of stainless steel. The port separator is attached with a silicone rubber sealant. Note that the sample is hydraulically saturated in advance by injecting water from the bottom through ports 1–4 and that port 5 is used only to confirm the isolation of the ports. The pressure gauge is used to measure the hydraulic pressure at the inlet side (the atmospheric pressure at the outlet side), where the outlet tube is placed at the same elevation as the inlet tube to cancel the effect of gravity and maintain the saturated condition.

[7] The present experiment was conducted at a constant confining pressure of 36 MPa and a hydraulic pressure difference of 0.6 MPa (Figure 2). The confining pressure resulted in normal stresses of 35 and 30 MPa for fractures A and B, respectively. During the experiment, flow rates from the outlet ports (ports 1 through 4) were measured and analyzed. Ports 1 and 2 were assigned to fracture A, and ports 3 and 4 were assigned to fracture B. On the basis of the confirmation of the total flow rate (sum of flow rates from all ports) being proportional to the hydraulic pressure difference of ≤0.6 MPa for the sample at the given confining pressure, although a deviation from linearity occurred at >0.6 MPa, probably because of the fracture opening as a result of the higher pore pressure, a GeoFlow fluid flow simulation based on Darcy's law would be appropriate for the present experiment.

### 3. GeoFlow Simulation

- Top of page
- Abstract
- 1. Introduction
- 2. Multiple-Fracture Flow Experiment
- 3. GeoFlow Simulation
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[8] A GeoFlow fluid flow simulation (GeoFlow simulation) involves the following two main steps (Figure 3). A fracture network is first created in a 3-D matrix by mapping 2-D fractures with aperture distributions. The Darcy flow through a matrix element interface is then calculated for an equivalent permeability continuum reflecting contributions of both the matrix and fracture permeabilities.

[9] The fracture is a rectangular plane having dimensions of *L*_{i} × *L*_{j} in the *i*-*j* coordinate system, whereas the matrix was a rectangular parallelepiped space having dimensions of *L*_{x} × *L*_{y} × *L*_{z} in the *x*-*y*-*z* coordinate system (Figure 3a). The fracture is divided into *N*_{i} × *N*_{j} elements, where the fracture element is a rectangular cell having dimensions of *L*_{i}/*N*_{i} × *L*_{j}/*N*_{j}, and each cell has an aperture value, *a*(*i*, *j*). The aperture value of each cell was determined on the basis of the results of surface topography measurement in the present study, as described later. However, the aperture determination method is not limited to the present method. The fracture is mapped in the *x-y-z* coordinate system by determining the coordinates of the centroid and the direction of a specific pair of normal and tangent vectors for the fracture plane.

[10] The matrix containing fractures is divided into *N*_{x} × *N*_{y} × *N*_{z} elements so that the matrix element is a rectangular parallelepiped cell having dimensions of *L*_{x}/*N*_{x} × *L*_{y}/*N*_{y} × *L*_{z}/*N*_{z} (Figure 3b), and a steady state laminar flow of a viscous, incompressible fluid is calculated for an equivalent permeability continuum. In our preliminary study, it was empirically confirmed that the ratio of the matrix element to the fracture element should be less than two, so that fluid flow through a specific fracture simulated by a 2-D Reynolds equation can be reproduced in a 3-D GeoFlow simulation with preserving characteristics of the original aperture distribution. For the equivalent permeability continuum reflecting contributions of both the matrix and fracture permeabilities, the fluid flow model based on Darcy's law is as follows:

where *A*_{x}, *A*_{y}, and *A*_{z}are the cross-sectional areas,*k*_{x}, *k*_{y}, and *k*_{z} are the permeabilities in different directions, and *μ* and *P* are the viscosity and pressure of the fluid, respectively. In order to determine the Darcy flow through the matrix element interface, a finite difference form of equation (1) is solved under given boundary conditions with the incomplete Cholesky conjugate gradient (ICCG) method, in which the use of the product, *A*_{i}k_{i} (*i = x*, *y*, *z*), represented by the following equation, characterizes the GeoFlow simulation:

where *w*_{f}_{,n} and *a*_{f}_{,n} are the width and aperture, respectively, of the *n*th fracture element intersecting the matrix element interface, and *A*_{m} and *k*_{m} are the area and permeability, respectively, of the matrix part (Figure 3b). As expressed by equation (2), equivalent permeabilities were determined at all matrix element interfaces in GeoFlow simulation. Fundamental concept of this conversion technique for equivalent permeability was previously reported by *Jing et al.* [2000], and quite different from the concept of *Snow* [1969] or *Long et al.* [1982], which determines single isotropic equivalent permeabilities for the matrix elements. The first term of the right side of equation (2) is based on the local cubic law assumption [*Ge*, 1997; *Oron and Berkowitz*, 1998; *Brush and Thomson*, 2003; *Konzuk and Kueper*, 2004].

[11] A fracture network consisting of two fractures, which had aperture distributions of fractures A and B under the given stress conditions, was created in the matrix of 140 mm × 140 mm × 140 mm by imitating the spatial configuration of fractures in the sample. The aperture distributions were determined using the surface topography data for fractures A and B by closing two opposite surfaces so that the total flow rate in the GeoFlow simulation was approximately equal to the experimental value. Aperture determination was based on the work of *Watanabe et al.* [2008], where the fracture closure under normal stress was simulated simply by uniform reduction of all apertures in conjunction with replacing overlapping asperities (negative apertures) with zero apertures. Ideally, it is desirable to determine the aperture distribution for each fracture using the surface topography and permeability data for that fracture. However, the permeability of each fracture could not be measured separately in the present experiment. Therefore, the two fractures were assumed to have the same permeabilities (2.9 × 10^{−13} m^{2}) or hydraulic apertures (1.87 μm) at the given normal stresses because the difference in the applied normal stress between fractures A and B was small (35 MPa for fracture A and 30 MPa for fracture B). Since the surface topography was measured in the 0.25 mm^{2} grid system, the fracture element had dimensions of 0.25 mm × 0.25 mm. The matrix was divided into matrix elements of 280 × 280 × 280 so that the dimensions of the matrix element are 0.5 mm × 0.5 mm × 0.5 mm. In order to simulate the fluid flow experiment, the unidirectional water flow was analyzed by GeoFlow using a matrix permeability of 1 × 10^{−19} m^{2}, a constant viscosity of 1 × 10^{−3} Pa s, and a hydraulic pressure difference of 0.6 MPa. (The boundary conditions are shown in Figure 4.) The matrix permeability was based on the permeability of Inada granite [*Takahashi et al.*, 1990].

### 4. Results and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Multiple-Fracture Flow Experiment
- 3. GeoFlow Simulation
- 4. Results and Discussion
- 5. Conclusions
- Acknowledgments
- References

[12] The fluid flow experiment provided flow rates of 0.13 cm^{3} h^{−1} for port 1, 0.38 cm^{3} h^{−1} for port 2, 0.05 cm^{3} h^{−1} for port 3, and 0.58 cm^{3} h^{−1} for port 4 (total flow rate: 1.14 cm^{3} h^{−1}). In the experiment, ports 1 and 2 (or ports 3 and 4) were both assigned to fracture A (or fracture B). However, the flow rates differed significantly between the two ports. In the case of fracture A, the flow rate for port 2 was approximately three times greater than that for port 1. Moreover, in case of fracture B, the difference in the flow rate exceeded one order of magnitude, indicating a considerably uneven flow within the sample.

[13] Calculating the flow rate ratio as a flow rate for the port over the total flow rate, the flow rate ratio was 11% for port 1, 33% for port 2, 5% for port 3, and 51% of port 4 (see Figure 5). Surprisingly, considering the total flow rate in the fractured rock sample, the flow rate for port 3 was almost negligible, whereas the flow rate ratio for port 4, assigned to an identical fracture, accounted for approximately half of the total flow rate. This was the most remarkable experimental finding.

[14] The experimental results indicated 3-D channeling flow, which was not predicted by the conventional DFN model simulations but should be predicted to clarify fluid migration, especially in the field of reservoir engineering. Reservoir engineers have encountered a large difference in productivity between wells when developing fractured reservoirs.*Tamagawa et al.* [2010]attempted to determine the reason for a three-orders-of-magnitude difference in productivity between two wells in the Yufutsu oil/gas field in Hokkaido, Japan. However, it was not possible to reproduce the large difference in productivity by the conventional DFN model simulation, despite a reliable fracture network model based on well logging, acoustic emission, and stress measurements. Considering the nature of the conventional DFN simulation, which ignores the 3-D channeling flow in a fracture network, this may be a logical result. In order to address this concern, we must first investigate 3-D channeling flow in laboratory-scaled fracture networks, as described herein.

[15] The GeoFlow simulation clearly exhibited 3-D channeling flow in the equivalent permeability continuum of the multiple-fracture sample, which had an equivalent permeability distribution generated by the combination of the aperture distributions of fractures A and B at the given stresses with the matrix permeability (Figure 4). The average permeability at each element in the equivalent permeability continuum was calculated using *k*_{i} in equation (2) of the six element interfaces (*A*_{i} is the area of those element interfaces), and elements having a permeability of ≥1 × 10^{−14} m^{2} are shown in different colors depending on the value (Figure 4a). Uncolored points are elements with permeabilities of <1 × 10^{−14} m^{2}because they have interfaces with no fracture or small-aperture fracture(s). The gray points indicate the locations of two fractures. Since the topographies of two opposite fracture surfaces were not identical, the aperture was not uniform throughout the fracture plane, but rather an aperture distribution and corresponding permeability distribution was produced in each fracture plane. The present GeoFlow simulation for the equivalent permeability continuum provided a flow rate distribution in a unidirectional flow, from top to bottom inFigure 4a. The flow rate at each element was normalized by the maximum value for all of the flow rates. Figure 4b shows the elements that have a normalized flow rate of ≥0.001 as different colors, depending on the value, while visualizing remarkable localization of the fluid flow in the fracture network. Uncolored points are elements having flow rates of <0.001, and gray points show the locations of two fractures. Since the fracture part had much higher permeability than the matrix part (Figure 4a), only the fracture part conducted the fluid flow, and preferential flow paths were formed because of the permeability distribution in the fracture network. Even visual observation revealed that the flow rate for port 4 was dominant, as observed in the fluid flow experiment.

[16] The GeoFlow simulation provided flow rates of 0.07 cm^{3} h^{−1} for port 1, 0.05 cm^{3} h^{−1} for port 2, 0.02 cm^{3} h^{−1} for port 3, and 0.87 cm^{3} h^{−1} for port 4 (total flow rate: 1.01 cm^{3} h^{−1}). The flow rate ratio was 7% for port 1, 5% for port 2, 2% for port 3, and 86% for port 4 (Figure 5). As shown in Figure 5, it was possible to reproduce the most remarkable experimental finding that the flow rate ratio of port 4 for fracture B was dominant, whereas the flow rate ratio of port 3 for the identical fracture was considerably small. However, the numerical results did not exactly match the experimental results. This may have been caused by a difficulty in the aperture determination. Since the present method involved dismantling of the sample, the apertures within the sample, particularly at the intersection of the fractures, cannot be exactly the same as the real values. A more accurate aperture determination would be possible using a method that does not involve dismantling the sample, which is beyond the scope of the present study. Therefore, the flow paths predicted by the present GeoFlow simulation cannot be exactly the same as the real flow paths, and, consequently, the present level of disagreement between the numerical and experimental results is reasonable. Nevertheless, despite the simplicity, the most remarkable experimental finding could be reproduced, demonstrating that the GeoFlow has significant potential to predict 3-D channeling flow in a fracture network.

[17] As described herein, GeoFlow allows novel techniques for the investigation of fluid flows, especially in laboratory-scaled fracture networks. On the basis of previously reported aperture distribution data, the investigation of fluid flows in various types of fracture network can be conducted systematically with no technical difficulty. Moreover, using the aperture distribution determined by X-ray CT for naturally fractured core samples [*Watanabe et al.*, 2011a, 2011b] should be effective when considering an application of GeoFlow simulation in a specific field. However, the final goal in GeoFlow studies is the investigation of fluid flows in field-scale fracture networks. For this purpose, the aperture distributions of fractures of various size must be predicted from information of laboratory scale fractures. Although multiscaled modeling of the fracture surface is possible using its fractal nature [*Brown*, 1995; *Matsuki et al.*, 2006], the prediction of aperture distributions under stress has not yet been achieved. Moreover, even if aperture distributions can be predicted, an upscaling method is necessary in creating an equivalent permeability continuum for field-scale investigations. These considerations will be investigated in the future.