Summary
This paper describes a hybrid finite volume method, designed to simulate
multiphase flow in a field-scale naturally fractured reservoir. Lee et al. (WRR
37:443-455, 2001) developed a hierarchical approach in which the permeability
contribution from short fractures is derived in an analytical expression that
from medium fractures is numerically solved using a boundary element method.
The long fractures are modeled explicitly as major fluid conduits. Reservoirs
with well-developed natural fractures include many complex fracture networks
that cannot be easily modeled by simple long fracture formulation and/or
homogenized single continuity model. We thus propose a numerically efficient
hybrid method in which small and medium fractures are modeled by effective
permeability, and large fractures are modeled by discrete fracture networks. A
simple, systematic way is devised to calculate transport parameters between
fracture networks and discretized, homogenized media. An efficient numerical
algorithm is also devised to solve the dual system of fracture network and
finite volume grid. Black oil formulation is implemented in the simulator to
demonstrate practical applications of this hybrid finite volume method.
Introduction
Many reservoirs are highly fractured due to the complex tectonic movement
and sedimentation process the formation has experienced. The permeability of a
fracture is usually much larger than that of the rock matrix; as a result, the
fluid will flow mostly through the fracture network, if the fractures are
connected. This implies that the fracture connectivities and their distribution
will determine fluid transport in a naturally fractured reservoir (Long and
Witherspoon 1985). Because of statistically complex distribution of geological
heterogeneity and multiple length and time scales in natural porous media,
three approaches (Smith and Schwartz 1993) are commonly used in describing
fluid flow and solute transport in naturally fractured formations: (1) discrete
fracture models; (2) continuum models using effective properties for discrete
grids; and (3) hybrid models that combine discrete, large features and
equivalent continuum.
Currently, most reservoir simulators use dual continuum formulations (i.e.,
dual porosity/permeability) for naturally fractured reservoirs in which matrix
blocks are divided by very regular fracture patterns (Kazemi et al. 1976, Van
Golf-Racht 1982). Part of the primary input into these simulation models is the
permeability of the fracture system assigned to the individual grid-blocks.
This value can only be reasonably calculated if the fracture systems are
regular and well connected. Field characterization studies have shown, however,
that fracture systems are very irregular, often disconnected, and occur in
swarms (Laubach 1991, Lorenz and Hill 1991, Narr et al. 2003).
Most naturally fractured reservoirs include fractures of multiple- length
scales. The effective grid-block permeability calculated by the boundary
element method becomes expensive as the number of fractures increases. The
calculated effective properties for grid-blocks also underestimates the
properties for long fractures whose length scale is much larger than the
grid-block size.
Lee et al. (2001) proposed a hierarchical method to model fluid flow in a
reservoir with multiple-length scaled fractures. They assumed that short
fractures are randomly distributed and contribute to increasing the effective
matrix permeability. An asymptotic solution representing the permeability
contribution from short fractures was derived. With the short fracture
contribution to permeability, the effective matrix permeability can be
expressed in a general tensor form. Thus, they also developed a boundary
element method for Darcy's equation with tensor permeability. 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 used with a finite difference
simulator to compute flow through the fracture system. The simulator was
enhanced to use a control-volume finite difference formulation (Lee et al.
1998, 2002) for general tensor permeability input (i.e., 9-point stencil for
2-D and 27-point stencil for 3-D). In addition, long fractures were explicitly
modeled by using the transport index between fracture and matrix in a
gridblock.
In this paper we adopt their transport index concept and extend the
hierarchical method: (1) to include networks of long fractures; (2) to model
fracture as a two-dimensional plane; and (3) to allow fractures to intersect
with well bore. This generalization allows us to model a more realistic and
complex fracture network that can be found in naturally fractured reservoirs.
To demonstrate this new method for practical reservoir applications, we
furthermore implement a black oil formulation in the simulator.
We explicitly model long fractures as a two-dimensional plane that can
penetrate several layers. The method, presented in this paper, allows a general
description of fracture orientation in space. For simplicity of computational
implementation however, both the medium-length and long fractures considered in
this paper are assumed to be perpendicular to bedding boundaries. In addition,
we derive a source/sink term to model the flux between matrix and long fracture
networks. This source/sink allows for coupling multiphase flow equations in
long fractures and matrix.
The paper is organized as follows. In Section 2 black oil formulation is
briefly summarized and the transport equations for three phase flow are
presented. The fracture characterization and hierarchical modeling approach,
based on fracture length, are discussed in Section 3. In Section 4 we review
homogenization of short and medium fractures, which is part of our hierarchical
approach to modeling flow in porous media with multiple length-scale fractures.
In Section 5 we discuss a discrete network model of long fractures. We also
derive transfer indices between fracture and effective matrix blocks. In
Section 6 we present numerical examples for tracer transport in a model with
simple fracture network, interactions of fractures and wells, and black oil
production in a reservoir with a complex fracture network system. Finally, the
summary of our main results and conclusion follows.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
21 November 2006
- Meeting paper published:
5 December 2006
- Revised manuscript received:
22 January 2008
- Manuscript approved:
27 January 2008
- Version of record:
20 August 2008
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