Fundamentals of Fluid Flow in Porous Media

 

Chapter 5

Miscible Displacement

The Equation of Continuity

Application of the principle of mass conservation of species i in a multi-component fluid mixture to an arbitrary control volume of the fluid yields the well-known equation of continuity, which, in it’s most general form, can be written as follows (Bird et al., 1960):

Where,

c_i = concentration of species i (mass per unit volume),

n_i = the mass flux vector (mass of species i per unit area per unit time)

r_i = source or sink term (mass of i per unit volume per unit time)

t = time.

In order to obtain the concentration of species i as a function of time and space from Equation 1, a constitutive equation that expresses the relationship between the fluxes and driving forces is required. Such an equation is provided by Fick’s first law of diffusion:

Where,

u = mass average velocity vector (length per unit time),

ρ = fluid (mixture) mass density (mass per unit volume),

D_o = molecular diffusion coefficient (length squared per unit time),

x_i = mass fraction of species i ( x_i = c_i / ρ ) .

Equation 5‑52) states that the flux of species i relative to stationary coordinates is the resultant of the bulk motion of the fluid and molecular diffusion. Substituting Equation (5‑53) into Equation 5‑52) gives:

Equation (5‑54) is applicable to systems with variable ρ and D_o.

When ρ and D_o are assumed constant, Equation (5‑54) simplifies to:

When Equation 4 is further simplified to represent one-dimensional flow:

where x is distance and u is velocity, both in the direction of flow.