Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 5

the sweep efficiency of a reservoir. In view of its practical importance in petroleum
engineering, Sect. 5 extends the equations of Sect. 4 to describe chemical flooding
compositional flows in a porous medium. The equations for modelling compositional
flow in a fractured porous medium are introduced in Sect. 6 . The proposed method-
ology is suitable for modelling any type of fractured reservoirs, including double-,
triple-, and other multiple-continuum conceptual models.

2 Multiphase Flow in Porous Media

In fluid mechanics, multiphase flow is treated as a generalization of the modelling
of a two-phase immiscible flow, where the two fluids are not chemically related and
coexist in contact separated by well-defined interfaces at the microscopic scale. In
reservoir simulations, we are typically interested in the simultaneous flow of two or
more fluid phases coexisting within the porous solid matrix. In principle, the physics
of such flows can be described using the framework of continuum mixture theory
for the development of the governing equations, in which the various phases are
considered as distinct fluids with individual thermodynamic and transport proper-
ties and with different flow velocities. The transport phenomena are mathematically
described by the basic principles of conservation for each phase separately and by
appropriate kinematic and dynamic conditions at the interfaces. Whereas the detailed
structure of these interfaces and the fluid volumes bounded by them are in general
inaccessible to macroscopic observation, their geometry influences the dynamics of
the multiphase mixture. To cover this difficulty, mixture theory makes use of the
volume fraction

ˆ ʱ of phase ʱ , which is defined as a scalar function of position x

and time t such that 0 ≤ ˆ ʱ ≤ 1. Therefore, for any volume V in the mixture, the

integral

V ˆ ʱ (

x , t ) d x , (1)

gives the instantaneous fraction of volume V that is occupied by the fluid phase

ʱ . Another quantity which is important in the description of multiphase flow is the

phase saturation S ʱ , defined as the fraction of void space of a porous medium that
is filled by phase ʱ ,

S ʱ = ˆ ʱ
ˆ

,

(2)

where denotes the skeletal porosity of the solid matrix, which is the sum of the

fluid volume fractions in a saturated porous medium. In other words, it is the total
fraction of void space in the material that can be occupied by the fluid phases.

ˆ