Modeling flow in naturally fractured reservoirs: effect of fracture aperture distribution on dominant sub-network for flow
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Fracture network connectivity and aperture (or conductivity) distribution are two crucial features controlling flow behavior of naturally fractured reservoirs. The effect of connectivity on flow properties is well documented. In this paper, however, we focus here on the influence of fracture aperture distribution. We model a two-dimensional fractured reservoir in which the matrix is impermeable and the fractures are well connected. The fractures obey a power-law length distribution, as observed in natural fracture networks. For the aperture distribution, since the information from subsurface fracture networks is limited, we test a number of cases: log-normal distributions (from narrow to broad), power-law distributions (from narrow to broad), and one case where the aperture is proportional to the fracture length. We find that even a well-connected fracture network can behave like a much sparser network when the aperture distribution is broad enough (α ≤ 2 for power-law aperture distributions and σ ≥ 0.4 for log-normal aperture distributions). Specifically, most fractures can be eliminated leaving the remaining dominant sub-network with 90% of the permeability of the original fracture network. We determine how broad the aperture distribution must be to approach this behavior and the dependence of the dominant sub-network on the parameters of the aperture distribution. We also explore whether one can identify the dominant sub-network without doing flow calculations.
KeywordsNaturally fractured reservoir Non-uniform flow Effective permeability Percolation Waterflood
A large number of oil and gas reservoirs across the world are naturally fractured and produce significant oil and gas (Saidi 1987). Efficient exploitation of these reservoirs requires accurate reservoir simulation. Naturally fractured reservoirs, like all reservoirs, are exploited in two stages: primary recovery and secondary recovery [sometimes followed by tertiary recovery, i.e., enhanced oil recovery (EOR)], with different recovery mechanisms. During primary production, the reservoir is produced by fluid expansion. In secondary production and EOR, since the fractures are much more permeable than the matrix, the injected water or EOR agent flows rapidly through the fracture network and surrounds the matrix blocks. Oil recovery then depends on efficient delivery of water or EOR agent to the matrix through the fracture network. Dual-porosity/dual-permeability models are still the most widely used methods for field-scale fractured-reservoir simulation, as they address the dual-porosity nature of fractured reservoirs and are computationally cheaper, although they are much simplified characterizations of naturally fractured reservoirs. To generate a dual-porosity/dual-permeability model, it is necessary to define average properties for each grid cell, such as porosity, permeability, matrix-fracture interaction parameters (typical fracture spacing, matrix-block size or shape factor), etc. (Dershowitz et al. 2000). Therefore, the fracture network used to generate the dual-porosity model parameters is crucial. Homogenization and other modeling approaches likewise require one to designate a typical fracture spacing (Salimi 2010). The hierarchical fracture model (Lee et al. 2001) also requires that one define effective properties of the matrix blocks and fractures which are too small to be represented explicitly.
This paper is the first part of a three-part study showing that the appropriate characterization of a fractured reservoir differs with the recovery process. In this paper, we show that even in a well-connected fracture network, far above the percolation threshold, flow may be so unequally distributed that most of the network can be excluded without significantly reducing the effective permeability of the fracture network. The implications of this finding for characterization of naturally fractured reservoirs are the subject of parts two and three. Briefly, in primary production, any fracture much more permeable than the matrix provides a path for escape of fluids, while in waterflood or EOR, the fractures that carry most injected water or EOR agent play a dominant role. In this paper, we restrict our attention to flow of injected fluids, as a first step toward modeling recovery processes that depend on contact of injected fluids with matrix.
Field studies and laboratory experiments show flow channeling in individual fractures and highly preferential flow paths in fracture networks (Neretnieks et al. 1982; Neretnieks 1993; Tsang and Neretnieks 1998). Cacas et al. (1990a, b) proposed that a broad distribution of fracture conductivities is the main cause of the high degree of flow channeling. In order to understand these phenomena, many theoretical studies have been done. The separate influences of fracture network connectivity distributions (Robinson 1983, 1984; Hestir and Long 1990; Balberg et al. 1991; Berkowitz and Balberg 1993; Berkowitz 1995; Berkowitz and Scher 1997, 1998; de Dreuzy et al. 2001a) and fracture conductivity distributions (Charlaix et al. 1987; Nordqvist et al. 1996; Tsang et al. 1988; Tsang and Tsang 1987) on flow channeling have been considered and also the interplay of these two key factors (Margolin et al. 1998; de Dreuzy et al. 2001b, 2002). Berkowitz (2002) further pointed out that even a well-connected fracture network can exhibit sparse preferential flow paths if the distribution of fracture conductivities is sufficiently broad. Katz and Thompson (1987) proposed a similar finding for pore networks. Although the “unimportant” fractures carry little flow, they still can be important to the connectivity and the preferential flow paths. It is not clear whether one can eliminate those “unimportant” fractures without significantly affecting the flow properties of the fracture network. Also, how broad must the distribution of fracture conductivities be to obtain this result is still an open question.
We propose that, for the dual-porosity/dual-permeability simulation of a waterflooding process or EOR or in homogenization, for the purpose of modeling the fluid exchange between fractures and matrix blocks, only the sub-network which carries by far most of the injected water is of primary importance in characterizing the reservoir. It is important to understand the factors that influence the sub-network. Since the effect of fracture connectivity on flow properties of fracture networks is well discussed, we focus here on the influence of fracture aperture (i.e., fracture conductivity) distribution.
As the first step in our research, in this work we systematically study the influence of the fracture aperture distribution on the dominant sub-network for flow. In this work, we define “the dominant sub-network” as the sub-network obtained by eliminating a portion of fractures while retaining 90% of the original network equivalent permeability. In other words, we are interested in how broad the aperture distribution must be that a well-connected fracture network can exhibit a sparse dominant sub-network with nearly the same permeability. The properties of the dominant sub-network are also examined. If the fracture network is poorly connected, i.e., near the percolation threshold, it is well established that only a small portion of the fractures connects the injection well and the production well. Here we focus on well-connected fracture networks. Since information on fracture apertures, especially in the subsurface, is limited, we test power-law distributions (from narrow to broad), log-normal distributions (from narrow to broad), and one case in which the aperture is proportional to the fracture length.
This report is organized as follows: In Sect. 2, we introduce the numerical model and the research process of this study. In Sect. 3, we analyze the dominant sub-network. In Sect. 4, the possibility of identifying the dominant sub-network without doing flow simulations is discussed. Our conclusions are summarized in the last section.
2 Numerical model and research process
2.1 Numerical model
For simplicity in this initial study, we examine flow in a quasi-two-dimensional fractured reservoir. We use the commercial fractured-reservoir simulator FracMan™ (Dershowitz et al. 2011) to generate fracture networks. A 3D fracture network is generated in a 10 m × 10 m × 0.01 m region. The shape of each fracture is a rectangle. Each fracture is perpendicular to the plane along the flow direction and penetrates the top and bottom boundaries of the region. The enhanced Baecher model is employed to allocate the location of fractures. Two fracture sets which are nearly orthogonal to each other are assumed, with almost equal numbers of fractures in the two sets.
For fracture apertures, we adopt two kinds of distributions which have been proposed in previous studies: power-law and log-normal. In each kind of distribution, a range of parameter values are examined. The aperture is randomly assigned to each fracture. In the case where the aperture is proportional to the fracture length, the fracture aperture follows the same power-law distribution as fracture length. The details of the aperture distribution are introduced below.
To focus on the influence of fracture aperture distributions on the dominant sub-network, except for the aperture distribution, all the other parameters remain the same for all the cases tested in this study, including fracture length, orientation, etc.
2.2 Flow simulation model
Mafic™, a companion program of FracMan™, is employed to simulate flow in the fracture networks.
As mentioned above, we believe that when the aperture distribution is broad enough, there is a dominant sub-network which approximates the permeability of the entire fracture network. Our main interest lies in examining the influence of the aperture distribution (the exponent α in a power-law distribution and the standard deviation σ in a log-normal distribution) on the dominant sub-network. Countless criteria can be used to decide which portion of fractures to remove, such as fracture length, aperture, [length × aperture], velocity, etc. Here we choose a criterion based on the flow simulation results. Mafic™ subdivides the fractures into finite elements for the flow calculations. The flow velocity at the center of each finite element and the product of flow velocity and aperture (Q nodal) can be obtained. Based on this value, we compute the average value (Q) of all the elements in each fracture. Q is then used as the criterion to eliminate fractures: Fractures are eliminated in order, starting from the one with the smallest value of Q to the one with the largest value of Q. After each step, we calculate the equivalent network permeability of the truncated network. It should be noted that the elimination of fractures is based on the flow in the original fracture network, not the truncated network.
We also describe the properties of the “backbone,” i.e., the set of fracture segments that conduct flow, specifically its aperture distribution. The backbone is determined by removing fractures which do not belong to the spanning cluster, as well as dead ends. In other words, the backbone is formed by the fracture segments with nonzero Q. The dead ends are often parts of a fracture rather than the entire fracture. In order to describe the properties of the conducting backbone, we reduce the fracture network to its backbone at the start and at each step after eliminating fractures.
Because the generation of the fracture network is a random process, an infinite number of fracture networks could be generated with the same parameter values for the density, orientation, fracture length, and the aperture distribution. In this study, for each set of parameter values, we generate one hundred realizations.
2.4 Percolation theory
Percolation theory is a powerful mathematical tool to analyze transport in complex systems (Aharony and Stauffer 2003; Sahimi 2011). It has been widely used to describe the connectivity and the conductivity of fracture networks.
Our research focuses on well-connected fracture networks, so we employ percolation theory here to analyze the connectivity of the initial fracture network, to illustrate how far above percolation threshold, and how well connected, the initial fracture network is.
The percolation threshold p c is the value at which a cluster of fractures connects the opposite sides of the region. The threshold value is affected by the position, orientation, and length distribution of fractures, the system size, etc. Masihi et al. (2008) studied the percolation threshold of fracture networks with different fracture-length distributions and different system sizes. For fracture networks generated in a 10 m × 10 m region with random orientation, when the length follows a power-law distribution with exponent α = 2, they proposed that the percolation threshold is around 0.66. In our case, the system size and the power-law exponent are consistent with their work, but the fractures are not randomly orientated, but in two perpendicular sets. As suggested by Masihi et al. (2005, 2008), the percolation threshold for a fracture network with two perpendicular fracture sets is lower than that for a model with randomly oriented fractures. Also, the truncation of the fracture-length distribution impacts the threshold value. Since the percolation threshold value is not our focus, here we consider 0.5–0.7 as a reasonable estimate of the percolation threshold. For the cases we study here, the value of the percolation parameter \(p\) of initial fracture networks is around 0.9. Considering the definition of p in Eq. (4), a value p = 1 corresponds to infinite fracture density (zero probability of not intersecting another fracture). Thus, our fracture network is far above the percolation threshold and is well connected.
3 Identifying the dominant sub-network based on flow simulation results
3.1 Models without correlation between fracture aperture and length
3.1.1 Power-law aperture distribution
In this paper, we mainly show the results for α with values 1.001, 2, and 6. The results for α with additional values examined in this study can be found elsewhere (Gong and Rossen 2015).
If we compare cases with aperture distributions from narrow to broad, we find that when the aperture distribution is broad (α ≤ 2), most of the fractures can be eliminated without significantly affecting the equivalent permeability: The fracture network behaves as a sparser sub-network. As the aperture distribution becomes narrower (α increase from 1.001 to 6), to retain a certain percent of the original fracture network permeability, more fractures are needed (Fig. 3).
For this initial study, for simplicity, we chose to study a 10 m × 10 m region with no-flow boundaries on top and bottom in Fig. 4. As a result, the region near those boundaries shows fewer fractures in the dominant sub-network. However, Fig. 4 suggests that the size of the region affected by the boundaries is limited, and that the main conclusion of our work is that most flow passes through relatively few fractures, and the remaining fractures can be eliminated without significantly affecting the network permeability. This is not dependent on finite-size limitations.
We may summarize our arguments of the cases with power-law aperture distributions as follows. For all of the cases with a power-law aperture distribution, at least a portion of fractures can be eliminated without significantly affecting the effective network permeability. The number of fractures that can be removed is strongly affected by the value of α, i.e., the breadth of the aperture distribution. The broader the aperture distribution is, the more fractures can be eliminated without significantly affecting the overall flow behavior. When the aperture distribution is broad enough (α ≤ 2), the original fracture network behaves as a sparse sub-network, and the total length of the fractures in the sub-network is much shorter than that of the original fracture network. The importance of each fracture to the flow behavior of the entire fracture network cannot be simply related to its aperture or length; some fractures with narrow aperture or short length play a more important role than others with broader aperture or greater length.
3.1.2 Log-normal aperture distribution
In this paper, we mainly show the results for σ with values 0.1, 0.4, and 0.5. The results for σ with additional values examined in this study can be found elsewhere (Gong and Rossen 2015).
The distributions of Q for fractures in the original networks with log-normal aperture distributions are similar to those with power-law aperture distributions (cf. Fig. 6). When the aperture distribution is narrow (σ = 0.1), the distribution of Q is also narrow: Most of the fractures carry a similar amount of flow. As a result, when a portion of fractures is eliminated, the equivalent network permeability is strongly affected. As the aperture distribution becomes broader, the distribution of Q is also broader, and there is a small portion of fractures which carry much more flow than the others. In other words, the fracture network shows stronger preferential flow paths when the aperture distribution becomes broader. Thus, removing a portion of fractures which carry little flow does not greatly reduce the equivalent network permeability, as the fractures that play a more important role are still in the system.
Similar to the cases of power-law aperture distributions, most fractures carry similar flow when they are in the dominant sub-network and in the original fracture network, which indicates that they behave similarly.
In summary, for the log-normal aperture distributions, we conclude that when the aperture distribution is broad enough (σ ≥ 0.4), most of the fractures can be taken out without significantly affecting the equivalent network permeability. In contrast to the cases of power-law aperture distributions, the fractures with larger aperture tend to play a more important role for the flow behavior of the fracture network, although the flow carried by each fracture cannot be simply related to the fracture aperture.
3.2 Aperture proportional to fracture length
As mentioned above, all the cases we test in this study follow a power-law length distribution with exponent α = 2, which is truncated between 0.2 and 6 m. Since in this section aperture is proportional to fracture length, the apertures also follow a power-law distribution with exponent α = 2 and lie in the range of 0.4 to 12 mm. For the case described above with α = 2 and aperture independent of fracture length, the apertures lie mostly in the range of 0.01 to 0.1 mm. Whether or not aperture is dependent on fracture length, the difference between the smallest and the largest values is nearly one order of magnitude, although the absolute values are different. The absolute value does not matter to the normalized results presented below.
Whether or not aperture is proportional to fracture length, the original fracture network behaves as a sparse network, and the cumulative length of the conducting backbone of the dominant sub-network is roughly half of the total length of the original fracture network. However, in contrast to the cases where the aperture is independent of the fracture length, the fractures with narrower aperture (shorter fractures) tend to be less important to flow in the network than those with larger aperture (longer fractures) when the aperture is proportional to the fracture length.
4 Possibility of identifying the dominant sub-network without doing flow simulation
Fracture elimination criteria
Number of intersections (n)
Flow simulation results (q)
Aperture × length (dl)
Aperture2 × length (d 2 l)
Aperture3 × length (d 3 l)
Aperture × length2 (dl 2)
Aperture3/length (d 3/l)
Aperture × number of intersections (dn)
Length × number of intersections (ln)
Aperture × length × number of intersections (dln)
We compare the effective permeability of the sub-network with a portion of fractures eliminated from the original fracture network using these criteria. Not all the cases examined above are tested here: For the cases with power-law aperture distributions, we examine α = 1.001, 2, and 6; for the cases of log-normal aperture distributions, we examine σ = 0.1, 0.2, and 0.6; we also test the cases where the aperture is proportional to the fracture length.
This work focuses on the effect of fracture aperture distribution on the dominant sub-network that by itself retains 90% of the effective permeability of the original fracture network. A number of aperture distributions are tested: log-normal and power-law distributions (from narrow to broad), and one where the aperture is proportional to the fracture length. If the aperture distribution is broad enough (α ≤ 2 for power-law aperture distributions and σ ≥ 0.4 for log-normal aperture distributions), most of the fractures can be eliminated without significantly reducing the effective permeability. As the exponent α of a power-law aperture distribution increases or the standard deviation σ of a log-normal aperture distribution decreases, fewer and fewer fractures can be removed without significantly reducing the network equivalent permeability.
The importance of each fracture to the overall flow is not simply related to aperture or length. For the cases of both the log-normal and power-law aperture distributions, and that where the aperture is proportional to the fracture length, there are some fractures with relatively narrow aperture that play a greater role in the overall flow than some others with larger aperture. It is also true that some fractures with relatively large aperture carry much less flow than most of the fractures.
Flow simulations are more effective at identifying the largest sub-network that retains 90% of the original permeability than eliminating fractures based on length, aperture, or number of intersections. Among those properties, eliminating fractures based on aperture is the most efficient choice considered here, but not as efficient as using flow calculations.
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