Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 28

the medium, but little of the permeability (flow). In contrast, fractures may have neg-
ligible storage, but high permeability. So, the porous medium and the fractures are
envisioned as two separate but overlapping continua and fluid mass transfer between
these two continua occurs at the fracture-porous medium interface (Barenblatt et al.
1960 ; Barenblatt and Zheltov 1960 ; Warren and Root 1963 ; Lemonnier and Bour-
biaux 2010a ).
Before writing down the governing equations of compositional flow in naturally
fractured reservoirs in the framework of the dual-porosity/dual-permeability concept,
we must introduce the following definitions. First, we define the matrix porosity ˆ m

and the fracture porosity ˆ f as the pore volume of the matrix blocks and fractures,

respectively, divided by unit volume of both media. Second, we must distinguish
between quantities in the matrix and fractures. For instance, c m , k ʱ , S m ,ʱ , ˁ m ,ʱ , and

v m ,ʱ denote the concentration of component k , saturation, density, and velocity of

phase are the same variables

in the fracture system. Now, using Eq. ( 83 ), the mass balance equation for each
component k , including water, in the matrix blocks, after summing over all phases,
is expressed as

ʱ in the matrix blocks, while c f , k ʱ , S f ,ʱ , ˁ f ,ʱ , and v f ,ʱ

ˆ m
ʱ

∂
∂

S m ,ʱ ˁ m ,ʱ c m , k ʱ +∇· ˁ m ,ʱ c m , k ʱ v m ,ʱ + d m , k ʱ

t

ʱ

= I m , k ʱ − F mf , k ʱ , (120)

ʱ

while in the fracture system, the same equation will read as follows

ˆ f
ʱ

∂
∂

S f ,ʱ ˁ f ,ʱ c f , k ʱ +∇· ˁ f ,ʱ c f , k ʱ v f ,ʱ + d f , k ʱ

t

ʱ

= I f , k ʱ + F mf , k ʱ , (121)

ʱ

where the volumetric source/sink terms I m , k ʱ and I f , k ʱ are as defined by relation

( 104 ) and ʱ

per unit bulk volume of the reservoir. The molecular diffusion and dispersion flux
of component k in phase

F mf , k ʱ is the matrix-fracture mass flow rate of component k in phase

m , can be
written in Fickian form using relation ( 84 ), with the diffusion-dispersion tensor in
each medium having the same form of relation ( 85 ). As in Eq. ( 104 ), the terms

ʱ in medium M, i.e., d M , k ʱ , where M = f ,

I M , k ʱ
contain the chemical reaction rates of component k in the fluid phases and rock phase
and the injection/production rate of the same component per unit volume, which can
be written as

L M , k ʱ = ˁ M ,ʱ c M , k ʱ Q M ,ʱ , (122)