Pseudo-Pressure (y)

Pseudo-pressure is a mathematical pressure function that accounts for the variable compressibility and viscosity of gas with respect to pressure. It is often confused with pseudo-time.

The equation for the flow of gas in the reservoir is very similar to that for liquid flow. In well testing, analytical equations are solved after making certain assumptions. In particular, four assumptions are very important:

1. Total system compressibility (ct) is constant

2. Gas viscosity (mg) is constant

3. Total porosity (f) is constant

4. Fluid saturations (Sw and Sg) are constant, if Swi is not equal to 0

For liquids, these assumptions are reasonable, since liquid compressibility and viscosity do not vary significantly with pressure, and the equations can be solved analytically. These analytical solutions are referred to as the liquid flow solutions, and form the basis of all well test analysis. The result is an analytical relationship between pressure and time, which, for an infinite-acting reservoir, can be written as:

Pressure = Constant * log (time) + ...

For gas, the assumptions listed above are no longer valid, since gas compressibility (cg) and thus the gas compressibility factor (z) can vary significantly with pressure. Gas viscosity (mg) also varies with pressure, but not to the same degree. To deal with these changing gas properties, the concept of pseudo-pressure (y) was developed by Al-Hussainy et al. (1966), which is defined as follows:

This transformation of pressure to pseudo-pressure is an exact transformation, and is completely rigorous. Thus, the liquid flow solution given above can still be applied for gas, provided that pseudo-pressure is used instead of pressure. By substituting pseudo-pressure for pressure, the gas flow equation becomes:

Pseudo-Pressure = Constant * log (time) + ...

Note:    In some sources, pseudo-pressure is also referred to as the real gas potential (m(p)).