## 1. Introduction

[2] Fluid flow and deformation in fractured porous media has been the subject of great interest in many engineering disciplines particularly in water resources, petroleum and mining engineering as well as the nuclear waste industry. Typical examples include: primary and tertiary enhancement of oil production, utilization of geothermal energy, remediation of contaminated sites, and isolation of hazardous wastes. The phenomenon has also bearing on process, mechanical, geotechnical and agricultural engineering. Concerted efforts have been made over the past few decades to arrive at predictive capabilities for fractured porous media associated with diverse topics such as protection of fresh water supplies, reservoir production and recovery, earthquake engineering, storage of nuclear waste, movement of pollutant plumes, etc. Several mathematical models have been proposed.

[3] The early models were based on the single porosity or continuum concept. In this approach, a fractured porous medium was grossly treated as an equivalent continuum with a single fluid constituent. The physical quantities such as potential, porosity and fluid pressure were averaged over representative blocks of the fractured porous medium containing sufficiently large number of fractures. The single porosity approach was associated with a number of drawbacks [e.g., see *Long et al.*, 1982; *Wilson et al.*, 1983] including, identification of the representative blocks and determination of the equivalent model parameters. Moreover, because of the averaging process involved, the model results, i.e., the fluid pressure and the fluid flux distribution, represented neither the situation in the pores nor in the fractures. In fact, the application of the single porosity models was only justified if one was dealing with large-scale flow fields, in which a detailed description of the field variables within the fractured porous medium was not of particular concern.

[4] A major departure from the single porosity approach was first made by *Barenblatt et al.* [1960] and *Warren and Root* [1963]. They modeled flow through rigid fractured porous media as a complex of two interacting flow regions: one representing the fracture network and the other the porous blocks. The fracture network was characterized by high permeability and low storage, and the porous blocks were characterized by low permeability and high storage. The two flow regions were in turn coupled through a leakage term controlling the transfer of fluid mass between the pores and fractures. An extension of Barenblatt's model to deformable fractured porous media was later given by *Duguid and Lee* [1977]. Also motivated by Barenblatt's work, *Aifantis* [1977, 1979, 1980] used the theory of mixtures and proposed a coupled double-porosity model for deformable fractured porous media. Alternatives to Aifantis' formulation were given by *Wilson and Aifantis* [1982], *Khaled et al.* [1984], *Valliappan and Khalili-Naghadeh* [1990], *Khalili-Naghadeh and Valliappan* [1991], *Auriault and Boutin* [1993], *Bai et al.* [1993], and *Chen and Teufel* [1997], among others. However, Aifantis' model was incomplete, in the sense that it related pore and fracture volume changes only to the overall volume change of the fractured porous medium. More specifically, it ignored the cross-coupling effects between the volume change of the pores and fractures within the system. This deficiency was eliminated in the formulations proposed by *Khalili and Valliappan* [1996], *Tuncay and Corapcioglu* [1996], *Wang and Berryman* [1996], *Khalili et al.* [1999], and *Loret and Rizzi* [1999]. The significance of the cross-coupling effects on the pore and fracture fluid pressure response of double-porosity media was highlighted by *Khalili* [2003].

[5] Recently, *Lewis and Ghafouri* [1997], *Bai et al.* [1998], *Pao and Lewis* [2002], and *Nair et al.* [2005] extended the double-porosity concept to include multiphase flow, albeit subject to a number of restrictions. *Lewis and Ghafouri* [1997] introduced the assumption that the applied loads were taken only by the porous blocks, and hence neglected the fracture deformation. *Bai et al.* [1998] and *Nair et al.* [2005] considered the fracture deformation, but treated the porous blocks and fracture network separately. They introduced two effective stress equations for the quantification of the deformation field, in contradiction of the definition of the effective stress principle. Moreover, none of the models took into account the cross-coupling effects between the fluid phases completely and/or consistently.

[6] The main objective in this paper is to provide a rigorous treatment of the theory of flow and deformation in fractured porous media saturated with two immiscible fluids. The governing differential equations are formulated using a systematic macroscopic approach based on the theory of poroelasticity satisfying the balance equations of mass and momentum. The effective stress equation of the system is derived, and the cross-coupling terms between the phases within the system are identified. Matrix displacement vector, pore air pressure, pore water pressure, fracture air pressure and fracture water pressure are introduced as seven primary variables for a three-dimensional boundary value problem. The governing equations are closed using the static equilibrium or balance of momentum of the fractured porous medium (three equations) and the balance of momentum of the fluid phases (four equations). All model parameters are identified in terms of measurable physical entities.