Click to see subtopics:Pseudo-time is a mathematical time function that accounts for the variable total compressibility (ct) and viscosity (µg) of gas, as well as the variable porosity (φ) with respect to time and pressure.
The equation for 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, 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 and rate transient analysis. The result is an analytical relationship between pressure and time. 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For gas, most of the assumptions listed above are no longer valid, since gas density (ρ∝p/Z), viscosity (µg) and total compressibility (ct) can vary significantly with pressure. Pseudo-pressure (ψ) and pseudo-time (ta) are used to deal with these changing properties and linearize the flow equations for gas. With the introduction of pseudo-pressure and pseudo-time, 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 forecasting provided pressure is replaced by pseudo-pressure, and time is replaced by pseudo-time. For the above mentioned example, pseudo-pressure can be calculated as:
Pseudo-pressure ≈ Constant * log(pseudo-time)
Development of pseudo-time
It should be noted that the concept of pseudo-time is not amenable to a completely rigorous solution, as is the case for pseudo-pressure, because the gas properties change with pressure, not time.
Pseudo-time was developed by Agarwal (1980) and he defined pseudo-time in terms of the viscosity (µg) and compressibility (ct) at the wellbore (see buildup pseudo-time). This definition accounted for the large change in total compressibility (ct) that occurs at low pressures (early time in a buildup). It had little effect on late time data, and was generally used for buildups only.
In the 90s, when the gas flow equations were being used for analyzing or forecasting data affected by reservoir depletion, it was realized that the Agarwal definition of pseudo-time for buildups, was inappropriate for boundary dominated flow (depleting systems). Palacio and Blasingame (1993) introduced a new definition of pseudo-time to account for the depletion effects. Instead of defining the pseudo-time transformation in terms of wellbore conditions like Agarwal did, they defined it in terms of the average reservoir pressure (see drawdown pseudo-time). This pseudo-time is appropriate for boundary dominated flow.
Anderson and Mattar (2005) found that, in a reservoir with significant transient flow, it was more appropriate to define pseudo-time in terms of the average pressure within the region of investigation rather than the average reservoir pressure (see corrected pseudo-time).
Different pseudo-time formulations are needed for buildup, for drawdown in a boundary dominated flow regime, and for a transient flow regime. The equations look similar, but they are radically different from each other.
Buildup pseudo-time
Pseudo-time for use in buildup analysis is defined in terms of pressure at the wellbore:
If the wellbore has significant storage, a difference in the early time behavior will be noted, since pressure and thus compressibility and viscosity are changing most near the wellbore. Late time behavior is not typically affected by buildup pseudo-time since the pressure, viscosity, and compressibility become constant at the wellbore and in the reservoir.
Drawdown pseudo-time
Pseudo-time for use in drawdown analysis is defined in terms of average reservoir pressure:
Note that the gas in place (GIP) must be known in order to perform the material balance calculations necessary to determine the average reservoir pressure.
Corrected pseudo-time
In the conventional definition of pseudo-time, the compressibility and viscosity terms are evaluated at average reservoir pressure conditions. Clearly, the average reservoir pressure is only a function of Original-Gas-In-Place (OGIP) and cumulative production. During the transient flow period, before any boundary effects are observed, the flow behavior of two different-sized reservoirs should be similar, and independent of the OGIP – they are both infinite-acting reservoirs. However using the conventional definition of pseudo-time, the result would be that the producing rates would be different because their average reservoir pressures are different due to the different OGIPs.
If a well is producing under boundary-dominated conditions, the average reservoir pressure is a very reasonable datum at which to establish fluid properties such as cg. However, if the well production is still in transient flow and no reservoir boundaries have been observed, the average reservoir pressure based on total reservoir volume is not an appropriate datum to use. Consequently, pseudo-time can cause anomalous model responses under certain conditions.
Anderson and Mattar (2005) proposed that the average reservoir pressure used in the pseudo-time calculation during the transient flow period should be calculated based on the gas-in-place of the investigated volume at that time.
This way, during transient flow, pseudo-time is independent of OGIP. As soon as the reservoir enters boundary-dominated flow, the conventional definition of pseudo-time is automatically resumed, because at that time the region of investigation is the whole reservoir.
The volume of investigation is calculated based on the radius of investigation formula:
This volume is adjusted for the effect of the reservoir boundaries and the coalescence of regions of influence caused by interference between wells.
Modification for geomechanical models
If there is geomechanical reservoir behavior, where rock permeability and formation compressibility are changing with pressure, you can account for this behavior in the pseudo-time.
Changing permeability can be expressed as keffective(p)=k⋅km(p) 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-time term, the definition of pseudo-time is modified as follows:
| Note: | Variation of the formation compressibility with pressure is incorporated in the ct(p) term. |