Pseudo-time is a mathematical time function that accounts for the high compressibility of gas. When pseudo-time is used, a significant improvement is achieved in the accuracy of long-term gas rate forecasting or in the analysis of gas production data.

Pseudo-time or pseudo-pressure?

Pseudo-time is often confused with pseudo-pressure. So, to start with, a discussion of the origin of each of these terms is appropriate. The equation for flow of gas in the reservoir is very similar to that for liquid flow. In Pressure Transient Analysis (PTA), or Rate Transient Analysis (RTA), the flow equation is solved after making certain assumptions. In particular two assumptions are very important. They are:

• The compressibility is constant
• The viscosity is constant

For liquids, these assumptions are reasonable, and the equation can be solved analytically. This is referred to as the "liquid flow solution", and forms the basis of all PTA and RTA analysis. The result is an analytical relationship between pressure and time, for example, transient radial flow of a liquid is described by:

Pressure (p) = constant * log (time (t)) + ….

For gas, these assumptions are appropriate only under very limited conditions. Outside the range of these conditions, severe errors can occur. The reason is that gas is highly compressible, and the compressibility (c), is not constant but changes with pressure. Gas viscosity (m) has the same problem, but not to the same degree. Another factor that complicates the solution of the flow equation for gas is the compressibility factor, z. It, too, changes with pressure (Note that compressibility (c), and compressibility factor (z), have similar sounding names but are completely different concepts).

The changing gas properties discussed above occur in two different combinations:

1. pressure (p), viscosity (m) and compressibility factor (z)
2. time (t), viscosity (m) and compressibility (c)

The first combination of problems was solved rigorously in the mid 70's by Al-Hussainy and Ramey (1), when they introduced the gas pseudo-pressure function (p_p). That is why all gas PTA and RTA equations are solved in terms of pseudo-pressure, and not pressure. The transformation of pressure to pseudo-pressure is an exact transformation, and is completely rigorous. Thus the "liquid flow" solution can be used for gas, provided that pseudo-pressure is used instead of pressure (p). For example, transient radial flow of gas is described by:

pseudo-pressure (p_p) = constant * log (time (t)) +…

The second combination of varying gas properties is not amenable to a completely rigorous solution. In other words, there is no transformation such as the one that was used for pressure, which will make the flow equation solvable analytically. For many years, as an approximation, it was assumed that the compressibility (c), and the viscosity (m) were constant and equal to the values at the initial reservoir pressure. For pressure transient analysis (PTA) of large reservoirs, this assumption works well. However it fails under two conditions:

• Low pressure at the wellbore
• Depleting reservoirs

As can be seen from the above figure, at low pressures, gas compressibility cannot be assumed constant.

In 1980, Agarwal (2) introduced a transformation called pseudo-time (t_a). The mathematical definition of pseudo-time is reminiscent of the definition of pseudo-pressure. However, unlike pseudo-pressure, it does not yield an exact solution, but only an improved solution. Agarwal was focussed on pressures in the wellbore, and he defined pseudo-time in terms of the viscosity and compressibility at wellbore conditions. This definition accounted for the large change in gas compressibility 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 90's, when the gas flow equations were being used in RTA for analyzing, or forecasting data affected by reservoir depletion, it was realized that the traditional assumption of constant initial viscosity and compressibility, while adequate for transient flow, was inappropriate for Boundary Dominated Flow (depleting systems). Moreover, the Agarwal pseudo-time definition did not solve the problem, because it was focused on the wellbore pressure. Blasingame (3) and co-workers introduced a new definition of pseudo-time to account for depletion effects. Instead of defining the pseudo-time transformation in terms of wellbore conditions like Agarwal did, they defined it in terms of the viscosity (m_av) and compressibility (c_av) at the average pressure in the reservoir (p_av).

With the introduction of 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 gas production forecasting provided pressure is replaced by pseudo-pressure (p_p), and time is replaced by pseudo-time (t_a). For example, transient radial flow of gas is described by:

pseudo-pressure (p_p) = constant * log (pseudo-time (t_a)) +…

and Boundary Dominated Flow of gas is given by:

pseudo-pressure (p_p) = constant * pseudo-time (t_a) +…

Keep in mind that the pseudo-time transformation is not exact, and as a consequence, two different pseudo-time formulations are needed, one for buildup and a different one for drawdown. They look similar, but they are radically different from each other.

Definition of pseudo-time
Pseudo-time is defined as:

Pseudo-time for use in buildup analysis (PTA), is defined in terms of pressure at the wellbore:

Pseudo-time for use in Production Data analysis (RTA), is defined in terms of average pressure in the reservoir:

APPLICATIONS OF PSEUDO-TIME

Gas Buildup Test - Low pressure, low permeability

Gas Rate Forecast - Gas rate forecasting using the traditional assumption of constant compressibility underpredicts the recoverable reserves. The pseudo-time formulation has corrected this problem and has significantly improved the quality of long-term forecasting of gas production. The use of pseudo-time gives results directly comparable to those obtained by numerical simulation. A comparison of a forecast with and without pseudo-time is shown, along with the numerical simulation. Note that because pseudo-time is an approximation, there can exist complex combinations of reservoir conditions where the correction may not be totally satisfactory.

Gas Production Data Analysis (RTA) - One of the objectives of RTA is to determine the size of the reservoir. When analyzing gas rates using RTA, the average pressure is unknown, because the reservoir size is not known in advance. This results in an iterative solution as follows:

• Assume a value for Original-Gas-in-Place;
• At each point in time, knowing the cumulative production, calculate the corresponding (depleted) average reservoir pressure;
• Calculate the pseudo-time;
• Plot the data;
• Analyze the data to obtain the Original-Gas-in-Place;
• Replace the assumed value with the calculated value;
• Iterate until converged.

Pseudo-pressure is exact. It corrects for m and z.

Pseudo-time is an approximation. It corrects for m and c.

References:

1. Al-Hussainy, R; Ramey, H.J. Jr.; "Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting," JPT (1966) 18, 624-636.
2. Agarwal, R.G.: " 'Real Gas Pseudo-Time' - A New Function for Pressure Buildup Analysis of MHF Gas Wells", paper SPE 8279 presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, 23-26 September.
3. Palacio, J.C; and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Gas Well Production Data," paper SPE 25909 presented at the 1993 Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Denver, 26-28 April.

Louis Mattar, Executive Technical Analyst, IHS Markit
Posted 17 March 2016

This article was published by S&P Global Commodity Insights and not by S&P Global Ratings, which is a separately managed division of S&P Global.