Chapter 11 - A critique of boundary conditions, degrees of freedom, and darcy's law

Abstract

This chapter discusses some of the unsteady-state flow behavior, such as boundary conditions, degrees of freedom, and the Darcy's law. The dilemma in representing unsteady-state flow behavior is in providing solutions to partial differential equations with boundary conditions which in fact defy rigorous solution—or refer conditions for which no solutions exist. It is therefore necessary to make simplifications or accommodations, but which are suspect and must be correlated to or reconciled with experimental results. The classic unsteady-state flow relationship in terms of the Darcy's law may have several forms, depending upon whether the flowing fluid is a liquid or a gas and whether the flow geometry is assumed linear or radial. In these cases, the partial differential equations may be characterized as all of the second order and of the first degree for liquid flow, but of the second degree for gaseous flow. The representation of unsteady-state behavior by the use of partial differential forms may be dictated by maintenance of the requisite number of degrees of freedom. This refers that the assignation of boundary conditions must not be in excess of the allowable. Degrees of freedom meant the difference between the number of variables and the number of equations. To affix the solution, this difference must be made-up by assigning additional equations or statements equal in number to the degrees of freedom.