Pseudo-pressure is a mathematical pressure function that accounts for
the variable compressibility and viscosity of gas viscosity (
) with respect to pressure.
| Note: | If there is geomechanical reservoir behavior, see the modification for geomechanical models. |
The equation for the flow of gas in the reservoir is very similar to that for liquid flow. In well testing and rate transient analysis, analytical equations are solved after making certain assumptions. In particular, five assumptions are very important.
1. Total system compressibility (ct) is constant
2. Fluid density (ρ∝p / Z) is constant
3. Fluid viscosity (
) is constant
4. Total porosity (f) is constant
5. Fluid saturations are constant
For liquids, these assumptions are reasonable because liquid compressibility, density, 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 and rate transient analysis. The result is an analytical relationship between pressure and time. Liquid-flow solutions for different wellbore and reservoir configurations are available in the literature. For example, for the case of an infinite-acting reservoir and a vertical well producing at a constant rate, pressure can be calculated as:
Pressure ≈ Constant * log(time)
For gas, most of the assumptions listed above are no longer valid because gas density (ρ∝p / Z) can vary significantly with pressure. Gas viscosity (µg) also varies with pressure, but not to the same degree. To deal with these changing gas properties, the concept of pseudo-pressure (ψ) was developed by Al-Hussainy et al. (1966) and is defined as follows:
This transformation of pressure to pseudo-pressure is an exact transformation, which accounts for variation of gas density and viscosity. If we assume that changes of total compressibility (ct), porosity, and fluid saturations are insignificant, the gas flow equation can be written in a manner similar to the liquid equation. Therefore, the liquid-flow solution can be used for gas well test analysis and rate transient analysis provided pressure is replaced by pseudo-pressure. For the above mentioned example, pseudo-pressure can be calculated as:
Pseudo-Pressure ≈ Constant * log (time)
Modification for geomechanical models
If there is geomechanical reservoir behavior, where rock permeability is changing with pressure, you can account for this behavior in the pseudo-pressure.
Changing permeability can be expressed as 
where k is the permeability at the initial pressure, and km(p) is a (dimensionless) permeability multiplier.
To incorporate a variation of permeability with pressure into the pseudo-pressure term, the definition of pseudo-pressure is modified as follows:
| Note: | In some sources, pseudo-pressure is also referred to as the real gas potential (m(p)). |