The Equation of Continuity

The Equation of Continuity 2016-10-25T11:54:44+00:00

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):
Equation of Continuity in general form


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

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

ri = 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:
Fick's first law of diffusion


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

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

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

xi = mass fraction of species i ( xi = ci / 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

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

When ρ and Do are assumed constant, Equation (5‑54) simplifies to:
Equation 5-55

When Equation 4 is further simplified to represent one-dimensional flow:
Equation 5-56

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




If you have any questions at all, please feel free to ask PERM!  We are here to help the community.


Sign up and NOW to receive the latest news, updates and technological advancements made for the Special Core Analysis & Enhanced Oil Recovery Industry

If you don't sign up, you won't know when the next breakthrough occurs!

Sign up for our FREE Newsletter!

We respect your privacy!