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Flow (in Porous Media) with Mathematica

Peter Valkó
Organization: Texas A&M University
Department: Harold Vance Department of Petroleum Engineering
Education level


The course teaches modeling flow in porous media using analytical, semi-analytical and numerical methods. It relies heavily on the services of Mathematica and therefore it contains an introductory part dealing with the software itself. Fluid flow in porous media concepts are introduced and revisited in the context of problem solving approach.

The primary goal is to equip the students with tools to conduct engineering research.

A web site is dedicated to the course. The index file contains a "live" calendar with links to Mathematica notebooks.

Mathematica can support effectively many of the computations in petroleum engineering research by providing an unprecedented insight into the theoretical basis and everyday practice of modeling flow in porous media. In this course analytical, semi-analytical and numerical methods are developed and illustrated. Mathematica is used exclusively as a medium to derive, illustrate, and present the material.

The classes are held in a computerized classroom with enough computers to accommodate every student. A workshop atmosphere is maintained thorough the course.

  • Numbers, symbolics, plots
  • Palettes, typesetting, styles
  • Variables and expressions
  • Lists, tables, matrices
  • Graphs
  • Animation
  • More visualization in 2D
  • Algebraic solve
  • More visualization in 3D
  • Derivatives and integrals
  • Differential equations
  • Data handling
  • Add-ons

    Single phase flow in homogenous and isotropic porous media (6 hours)
  • Reservoir-well systems
  • Driving mechanisms, under-saturated, gas cap, water drive
  • Outer boundary conditions
  • Inner boundary conditions / well models well
  • Flow regimes and time invariant characteristics

    Analytical and semi-analytical methods (6 hours)
  • Solutions in Laplace space
  • Numerical inversion of Laplace transform

    Modeling numerical methods in Mathematica (4 hours)
  • Finite difference / finite volume element
  • Handling boundary conditions and wells
  • Matrix formalism and linear algebra solution methods
  • Streamline simulation
  • Collocation/weighted residuals
  • Finite element/boundary element

    Outlook (4 hours)
  • Numerical well test analysis
  • Horizontal, deviated, multi-lateral and fracture treated wells
  • Heterogeneous and anisotropic media / fractured reservoirs
  • Layers, flow units, flow compartments
  • Coupling reservoir simulation with geomechanical modeling
  • Subjects

    *Applied Mathematics > Numerical Methods
    *Science > Physics > Fluid Mechanics