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We describe a hierarchical approach for modeling fluid flow in a naturally fractured reservoir with multiple length-scale fractures. Based on fracture length (lf) relative to the finite-difference grid size (lg), fractures are classified as belonging to one of three groups:short disconnected fractures (lf << lg),medium-length fractures (lf ~ lg), andlong fractures (lf >> lg). Effective grid-block permeabilities, associated with the short and medium length fractures, are used as input to a finite-difference simulator. We also present a separate transport equation for flow through long fractures to capture effects of large-scale high permeability pathways. Our new approach provides an improved means to include realistic and explicit fracture descriptions. Previously, Lough et al. (1997, 1998) developed a numerical method to compute the effective permeability of simulation grid-blocks with realistic fracture characterizations. Although this method handles generalized fracture geometries, it is numerically inefficient for the case where many small fractures exist. The method can also underestimate the flow contribution from long fractures. Assuming a linear potential gradient along the short fractures, we derive an analytical solution for the permeability contribution from short fractures. The solution becomes more accurate as fractures become randomly distributed and asymptotically small in length. The permeabilities from this analytical solution are used as the effective matrix permeability in computing a combined effective grid-block permeability that includes medium-length fractures. The method of Lough et al. is used for this computation. This hierarchical approach takes into account coupled flow between the rock matrix, short fractures and medium-length fractures. Long fractures are modeled explicitly in the reservoir simulator, using a transport equation that describes flow between long fractures and surrounding simulation grid-blocks. Simulation results from our new hierarchical approach are compared with those from the conventional dual porosity/permeability model. Numerical efficiency and accuracy are also examined.

Summary In this paper we describe a hierarchical approach for modeling fluid flow in a naturally fractured reservoir with multiple length-scale fractures. Based on fracture length relative to the finite-difference grid size, fractures are classified as short disconnected fractures, medium-length fractures, and long fractures. Effective grid-block permeabilities, associated with the short- and medium-length fractures, are used as input to a finite-difference simulator. We also present a separate transport equation for flow through long fractures to capture effects of large-scale high permeability pathways. Simulation results from the new hierarchical approach are compared with those from the conventional dual porosity/permeability model. Introduction The permeability of a fracture is usually much larger than that of the rock matrix. Consequently, the fluid will flow mostly through the fracture network, if the fractures are well connected. This implies that the fracture connectivities and their distribution will determine fluid transport in a naturally fractured reservoir.1 With this fact in mind, some reservoir simulators have been constructed that only model the flow through fracture systems. However, such simulators cannot be used to compute effective conductivities for fractured gridblocks because of their failure to include the flow through the permeable rock matrix. The loss of fracture connectivity occurs often and has been observed, for example, by Laubach2 and by Lorenz and Hill.3 In the model we propose, flow through both the fractures and the matrix is considered, so that losses in fracture connectivity do not automatically cause the calculated effective permeability to drop to zero. Most reservoir simulators employ dual continuum formulations for naturally fractured reservoirs, in which matrix blocks are divided by very regular fracture patterns.4,5 Part of the primary input into these simulation models is the permeability of the fracture system assigned to the individual gridblocks. Currently, this value can only be reasonably calculated if the fracture systems are regular and well connected. However, field characterization studies have shown that fracture systems are very irregular, often disconnected, and occur in swarms.2,3 Incorporating the statistical description of fractures into the characterization of simulation grids will significantly improve the modeling technology for fractured reservoirs. In our previous papers6,7 we applied periodic boundary conditions in computing the effective conductivities of a gridblock containing fractures. Fluid flow in the fractures was treated as two-dimensional (2D) potential flow governed by Darcy's law. The permeability is proportional to h2/12 where h is the fracture aperture.8 The flow in the rock matrix is a three-dimensional (3D) potential flow which also obeys Darcy's law. We developed an efficient method to solve these coupled potential problems using the boundary element method.6,7 Kamath et al.9 showed that disconnected fractures cannot be ignored because they can significantly contribute to the overall flow through the rock matrix. They also showed that the effective permeability is much larger than a simple sum of fracture and matrix permeabilities, as is commonly assumed. Jensen et al.10 applied this new method to two field cases and obtained some successful simulation results. Most naturally fractured reservoirs include fractures of multiple length scale. The effective gridblock permeability calculated by the boundary element method becomes expensive as the number of fractures increases. The calculated effective properties of gridblocks also underestimates the properties of long fractures whose length scale is much larger than the gridblock size. In this paper, we propose an efficient hierarchical modeling approach that overcomes these two unsatisfactory conditions based on the fracture length relative to the finite difference grid size. We assume that short fractures are randomly distributed and contribute to increasing the effective matrix permeability. An asymptotic solution representing the permeability contribution from short fractures is derived. With the short fracture contribution to permeability, the effective matrix permeability can be expressed in general tensor form. Thus, we also develop a boundary element method for Darcy's equation with tensor permeability. For medium-length fractures in a gridblock, a coupled system of Poisson equations with tensor permeability is solved numerically using a boundary element method. The gridblock effective permeabilities are used with a finite-difference simulator to compute flow through the fracture system. The simulator is enhanced to use a control-volume finite-difference formulation11–14 for general tensor permeability input (i.e., a 9-pt stencil for 2D and a 27-pt stencil for 3D). For simplicity of derivation, both the medium-length and long fractures considered in this paper are perpendicular to the bedding surfaces. Because the effective medium-scale calculation underestimates the permeability contribution from long fractures, we explicitly model them as fluid conduits. Thus, we derive a well-like equation for a finite reservoir simulator to model flow through these discrete long fractures. In the next section, we describe the governing equations, and then we present our new hierarchical approach to modeling flow in porous media with multiple length-scale fractures. To illuminate this new approach, we present two numerical examples: a two-dimensional model and a three-dimensional three-layer model. Using the three-layer model, we also compare our single continuum results with those from a conventional dual porosity/permeability model. Finally we conclude with a summary of our main results. Governing Equations The main premise behind our model is that the fracture aperture is small enough to allow the fractures to be treated as planar high-flow regions embedded in the homogeneous porous matrix. The flow in the matrix is governed by Darcy's law, v m = − 1.127 × k m μ ∇ ϕ m = − k ^ m ∇ ϕ m , ( 1 ) where vm (B/D/ft2) is the velocity, km (darcy) is the matrix permeability, ? is the fluid viscosity (cp), and ?m (psi) is the matrix potential or pressure. Clearly k^m=1.127×km/μ and is more convenient to work with than km.

Fractures play a critical role in storing and distributing fluid in naturally fractured reservoirs. To fully quantify the effect of fractures on reservoir performance, flow simulation is required. To model these flow effects, the fracture system needs to be well characterized, and fracture properties such as length, orientation, aperture and intensity can then be used to calculate effective simulation gridblock properties. In this paper, we show additional techniques to further improve the computational efficiency of our previously published hierarchical fracture modeling process. In the hierarchical modeling approach, we calculate an effective tensor permeability associated with the small length-scale fractures, which is then used as input to the calculation of the effective tensor permeability associated with the medium length-scale fractures. Herein, we show that an efficient analytical calculation can be correctly made for those small-scale fractures that have lengths less than 10% of the reservoir model cell-size. We have shown that a Boundary Element Method (BEM) provides an accurate effective tensor permeability, which is then used in flow simulation. However due to high computational demands, it is not practical to use the BEM technique for models much larger than one hundred thousand cells, without using parallel processing methods. To overcome this high computational demand, we use geostatistical methods to estimate effective permeabilities between cells for which the more rigorous BEM has been applied. Both hard and soft data used in the geostatistical simulation are critical to obtain a meaningful result. The hard data include effective permeabilities calculated by the BEM for about 10% of the total cells in the simulation model. Soft data include a statistical analysis of the correlation between BEM calculated tensor permeability values and fracture intensity for each simulation model-cell. Sequential Gaussian Simulation with co-Kriging is then used to estimate the effective permeability for the entire model. A field case demonstrates that the proposed methodology is efficient and practical for a large fractured reservoir model. Introduction In a previous paper (Lee et al.1), we proposed a hierarchical approach for modeling fluid flow in a naturally fractured reservoir with multiple length-scale fractures. Based on fracture length (lf) relative to the finite-difference grid size (lg), fractures are classified as belonging to one of three groups:short disconnected fractures (lflg),medium-length fractures (lf~lg), andlong fractures (lf>>lg). It was assumed that short fractures are randomly distributed and contribute to increasing the effective matrix permeability. An analytical solution representing the permeability contribution from short fractures was derived. With the short fracture contribution to permeability, the resulting effective matrix permeability could be expressed in a general tensor form. For medium-length fractures in a grid-block, a coupled system of Poisson equations with tensor permeability was solved numerically using a boundary element method. The grid-block effective permeabilities were then used with a finite difference simulator to compute flow through the coupled fracture and rock matrix system. The simulator was enhanced to use a control-volume finite difference formulation2–5 for general tensor permeability input (i.e., 9-pt stencil for 2-D and 27-pt stencil for 3-D). Long fractures are modeled explicitly in the reservoir simulator, using a transport equation that describes flow between long fractures and surrounding simulation grid-blocks. The computational time for effective permeabilities using the BEM is intensive when the number of cells in the model is very large. Whereas, the analytical method used for short fractures is very efficient. However, it is asymptotically accurate only for short fractures that do not connect to or intersect with other fractures.