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The key issue for simulating flow in fracture rocks, however, is how to handle
fracture-matrix interaction under different conditions. For instance, under multiphase
and nonisothermal conditions, a critical aspect involves the interaction of mass and
thermal energy at fracture-matrix interfaces. In general, most mathematical mod-
els rely on continuum approaches and involve developing conceptual models. They
incorporate the geometrical information of the fracture-matrix system, define mass
and energy conservation equations for the fracture-matrix domains, and solve a num-
ber of discrete nonlinear algebraic and constitutive equations, which express relations
and constraints of physical processes, variables, and parameters as functions of pri-
mary unknowns. Conceptual models employed to represent fractured porous media
include: the discrete fracture and matrix models (DFM) (Lichtner
1988
; Steefel and
Lichtner
1998a
,
b
; Stothoff and Or
2000
), the effective-continuum method (ECM)
(Wu
2000
), the dual-continuummethods, including double- and multi-porosity, dual-
permeability models, and the more general Multiple Interacting Continuum (MINC)
approach (Barenblatt et al.
1960
; Barenblatt and Zheltov
1960
; Warren and Root
1963
; Pruess and Narasimhan
1985
; Wu and Pruess
1988
; Bai et al.
1993
). A dual
porosity model of multidimensional, compositional flow in naturally fractured reser-
voirs as derived by the mathematical theory of homogenization was presented by
Chen (
2007
). The dual-continuum model, such as the double-porosity and the dual-
permeability concept, has been the most widely used approach for modelling fluid
flow, heat transfer, and chemical transport through fractured reservoirs because of its
computational efficiency and its ability to match many types of field-observed data.
A unified scheme based on the dual-continuum method has been recently reported
by Wu and Qin (
2009
), which can be used with different fracture-matrix concep-
tual models. The mathematical formulation of dual-continuum models as used in
industrial simulators and the-state-of-the-art modelling of the physical mechanisms
driving flows and interactions/exchanges within and between fractures and matrix
media have been described in two separate papers by Lemonnier and Bourbiaux
(
2010a
,
b
).
Fluid motion in a petroleum reservoir is governed by the laws of conservation of
mass, momentum, and energy. These physical laws are often represented mathemat-
ically on the macroscopic level by a set of partial differential or integral equations,
referred to as the governing equations. As long as compressible or multiphase flow
or heat transfer is involved, these equations are inherently nonlinear. Amathematical
model for describing the flow and transport processes in fractured porous media con-
sists of these equations, together with appropriate constitutive relations and a set of
boundary and/or initial conditions. In this paper we intend to develop such a model by
steps of increasing level of complexity. We first describe in Sect.
2
the equations gov-
erning the simultaneous flow of two or more fluid phases within a porous medium. In
Sect.
3
we write down the equations used to model the transport of multicomponents
in a fluid phase in a porous medium. This model is extended in Sect.
4
to describe
compositional flow in a general fashion, where each phase may involve many compo-
nents and mass transfer between the phases is an important effect. Chemical flooding,
i.e., the injection of chemical components in production wells is an important tech-
nique employed in enhanced oil recovery to reduce the fluid mobility and increase
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