Chapter 2: Mathematics of Fluid Flow

The purpose of this chapter is to review the mathematics of fluid flow. We limit our review to essential aspects of partial differential equations, vector analysis, numerical methods, matrices, and linear algebra. These topics are discussed in the context of two fluid flow applications: analysis of the convection/dispersion equation and diagonalization of the permeability tensor. For more details about the mathematics presented here, consult Refs. 1 through 4 .

Partial differential equations (PDEs) are frequently encountered in petroleum engineering. We review basic concepts of PDEs by considering the relevant mathematical properties of the continuity equation.

Fluid flow through a volume can be described mathematically by the continuity equation. The continuity equation has many uses, and its derivation is provided to illustrate the construction of a partial differential equation from physical reasoning. ⁵  We begin by considering the flow illustrated in Fig. 2.1 . The block in Fig. 2.1 has length (Δ x ), width (Δ y ), and depth (Δ z ). Fluid flux ( J ) is the rate of flow of mass per unit cross-sectional area normal to the direction of flow. The notation ( J _x ) _x denotes fluid flux in the x direction at location x. The cross-sectional area perpendicular to the flux direction is Δ y Δ z . Fluid flows into the block at x with fluid flux J _x and out of the block at x + Δ x with fluid flux J _x+Δx . Applying the principle of conservation of mass, we have the mass balance, which is written as