Efficient Finite-Difference Model for Flow in a Reservoir With Multiple Length-Scale Fractures

Abstract

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.

Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS)

“…Aligned with the EDFM approach of having independent grids for fracture and matrix [2], this work also introduced a distance-based automatic coarsening algorithm for the fracture domain. This allows the user to specify the desired (uniform) coarsening factor for the fracture domain, in similar fashion to the matrix.…”

“…Among the proposed methods, the hierarchical fracture modelling approach allows for avoiding complexities associated with the discretization and dynamic nature of fracture geometries [2]. In this approach, small-scale fractures (below the matrix grid resolution) are homogenized within the matrix rock, altering its effective permeability [3].…”

Simulation of a multistage fractured horizontal well in a tight oil reservoir using an embedded discrete fracture model

“…For d mf , Lee et al 26,27 assumed that the fluid in the matrix perpendicularly flows to the fracture. Therefore, the pressure around a fracture is linearly distributed, and the average normal distance can be computed as where dv and V represent the volumes of the microelement and grid, respectively (in m 3 ), and x n is the normal distance between the microelement and the fracture (in m).…”

“…Lee et al 26,27 proposed the embedded discrete fracture model (EDFM). The EDFM uses an orthogonal structured grid for the mesh matrix, and the fractures are divided into a series of segments based on the boundary of the matrix cell.…”

Characteristics of imbibition in tight oil reservoirs from the perspective of physical experiments and theory

“…The large surface area open to the imbibition of water in highly fractured reservoirs may result in economic production rates, even in tight oil reservoirs. The spontaneous oil‐water imbibition caused by capillary force and gravity during oilfield development was first identified in a study of a bottom water reservoir and a fractured reservoir 17,18 . Currently, almost all tight oil reservoirs are developed using hydraulic fracturing, during which imbibition occurs between the matrix and the fractures.…”