Two-Phase Fluid Flow in Porous Media

This Demonstration shows the solution of the equations of motion for one-dimensional immiscible displacement of oil by water in a porous medium.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Consider a displacement process in which water displaces oil in a porous medium. If capillary forces are neglected, the saturation process is represented by a nonlinear hyperbolic partial differential equation known as the Buckley-Leverett equation [1]
where is the water saturation, is the fluid velocity, and are the space and time coordinates, and is the fractional flow expressed as a function of water saturation, the overall mobility,, while and are the ratios of the relative permeability to viscosity of water and of oil, respectively.
In the Buckley–Leverett solution we follow a fluid front of constant saturation, so we can write an expression for the saturation change . Substituting into the Buckley–Leverett equation, we get ; integrating in time yields an expression for the position of the water front .
In 1952 Welge [2] published an approach using the Buckley–Leverett frontal advance calculation. The method [3] consists in drawing a tangent to the curve originating at the irreducible water saturation; the point of tangency defines the water saturation at the flood front and the reciprocal of the slope is the front velocity. This Demonstration solves the Buckley–Leverett equation using the Welge method and shows that the velocity and the efficiency of the water flood depend significantly on the mobility ratio of the displacing fluid to the displaced fluid; the lower this ratio, the lower the front velocity and the more efficient the displacement.
[1] S. E. Buckley and M. C. Leverett, "Mechanism of Fluid Displacement in Sands," AIME Transactions, 146, 1942 pp. 107–116.
[2] H. J. Welge, "A Simplified Method for Computing Oil Recovery by Gas or Water Drive," AIME Transactions, 195, 1952 pp. 99–108.
[3] J. R. Franchi, Principles of Applied Reservoir Simulation, Amsterdam: Elsevier, 2006.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+