Equations for Single-Phase Porous Media Flow

Equations for Single-Phase Porous Media Flow

• Introduction : 

The flow of a single, compressible fluid through porous, permeable rock can be described using a partial differential equation known as the diffusivity equation. Modified forms of the diffusivity equation can be used to describe gas flow. A similar equation can be derived for multiphase flow as well, and that equation is the basis for reservoir simulation. Clearly, the diffusivity equation is at the very heart of reservoir engineering and an intuitive understanding of this equation is essential to all who would do reservoir engineering.

• The Conservation Equation : 

Many physical systems – ranging from solar collectors to river deltas to flow in reservoirs – can be analyzed using the principle of conservation. This principle is closely related to the idea of a control volume in thermodynamics; it is based on the idea that the amount of “stuff” (energy, mass, whatever) entering, leaving, created, and destroyed in a given volume must be balanced. We will derive the conservation equation for a radial flow geometry, because this geometry is especially useful for well testing and inflow analysis. We could do it for any geometry we chose. 

Consider a cylindrical shell of radius r and thickness Dr (Figure 6.1). 

Equation (6.4) is the conservation equation in radial coordinates. It states that the sum of the partial differential derivatives in r and t is zero. This is also known as a divergence equation; all conservation equations (for any quantity, in any coordinate system) can be expressed in a form very similar to Equation 6.4. 

This equation must be manipulated further to be useful: it includes dependent variables r, f, and u, whereas we really want an equation in p only. We will use constitutive equations for these quantities to get the desired equation.

• Use of Darcy’s Law in the Conservation Equation

• The Case of Small and Constant Compressibility

• The Linearized Diffusivity Equation

Although Equation (6.10) is in pressure, it is nonlinear. It is very difficult to solve nonlinear partial differential equations, and we therefore seek a simplified, linear form to work with. The nonlinearity comes from two different sources.

• Dimensionless Variables

We used the concept of a dimensionless variable when we discussed the skin factor. We will extend that discussion now to better understand the linearized diffusivity equation. It seems sensible to make radius dimensionless on the wellbore radius :

• Other Coordinate Systems and Notation

• Discussion

Assumptions The steps and assumptions used to derive the linearized diffusivity equation are summarized in Table 6.1, below:

The assumptions are very important to be familiar with. Study this table! In particular, consider the following: