• oil-water compressible displacement;
  • three-dimensional moving boundary;
  • second-order upwind finite difference fractional steps;
  • l3 error estimate;
  • numerical simulation of energy sources


The research of the three-dimensional (3D) compressible miscible (oil and water) displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in basin evolution, as well as to the rational evaluation in prospecting and exploiting oil-gas resources, and numerical simulation of seawater intrusion. The mathematical model can be described as a 3D-coupled system of nonlinear partial differential equations with moving boundary values. For a generic case of 3D-bounded region, a kind of second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic is put forward. Some techniques, such as the change of variables, calculus of variations, and the theory of a priori estimates, are adopted. Optimal order estimates in l2 norm are derived for the errors in approximate solutions. The research is important both theoretically and practically for model analysis in the field, for model numerical method and for software development. Thus, the well-known problem has been solved.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1103–1129, 2014