Two-Phase Pseudo-Pressure
Subtopics:
Solution Gas Drive
Solution Gas Drive Theory
The most theoretically correct solution for multiphase flow in the reservoir is to use a pseudo-pressure variable called the Reservoir Integral (R.I.), defined as:
Although mathematically rigorous, this variable is totally impractical, as it requires knowledge of the distribution of saturations in the reservoir, in space and in time. It only applies when a simulation of the reservoir has been conducted.
A more practical alternative is to define a Sandface Integral (S.I.) which calculates pseudo-pressure in terms of the saturation at a specified point in the reservoir, usually the sandface, as follows:
This too is impractical to evaluate, but it forms the basis for defining a two-phase pseudo-pressure that can be evaluated, if appropriate rock and fluid properties are available. Although this definition is only an approximation to the Reservoir Integral, and is not fully rigorous, it is adequate for analysis and modeling of solution gas systems. The two-phase pseudo-pressure is defined as:
To calculate y2p, the variation of kro and krg with pressure must be known beforehand. In reality, y2p can be defined in terms of kro alone, along with a pressure-saturation relationship. In other words, the information contained in the krg(p) relationship is already contained in the combination of kro(p) and pressure-saturation. This is a two-part procedure, the first part is obtaining the relative permeability data for the reservoir rock, and the second part is assuming an appropriate model for fluid flow in the reservoir.
If this pressure-saturation relationship is used in conjunction with the two-phase pseudo-pressure definition, an alternative definition of two-phase pseudo-pressure (m(p)) results:
This definition appears to be simpler than the preceding one, which contains both the kro and krg terms. However, in actual fact, the information contained in this simpler formulation is complete, and it accounts for, implicitly, both krg and kro, through the pressure-saturation formulation. Because the relative permeability terms and multiphase effects are built into the definition of two-phase pseudo-pressure, the analysis results in the determination of the absolute permeability (not the effective permeability to oil – k not ko) and the skin due to damage (sd) (independent of gas saturation effects near the wellbore – gas block).
For analyzing buildup and drawdown tests, the two-phase pseudo-pressure procedure is recommended, provided suitable relative permeability and PVT data are available. The results of the analysis will include the absolute permeability (k), and the skin due to damage (sd). For drawdown tests, the producing GOR is used to relate the pressure and saturation terms that occur in the pseudo-pressure function. In the case of buildup tests, the pseudo-pressure function is calculated based on the producing GOR at the instant of shut-in.
Solution Gas Drive Procedure
1. From the well test data, tabulate: the shut-in time in hours, the shut-in pressure (psi), and the producing GOR (Mscf / stbbl) at the instant of shut-in.
2. From the special core analysis data, tabulate: kro, krg, So, and calculate the relationship between krg / kro and So.
3. From the laboratory PVT analysis, or from appropriate PVT correlations, tabulate PVT data for p (psi), Rs (Mscf / stbbl), mo (cp), mg (cp), Bo (Rbbl / stbbl), and Bg (Rbbl / Mscf).
4. For the full range of test pressures, calculate the relationships of shut-in pressure (pws) vs. Rs, mo, mg, Bo, and Bg.
5. From the producing GOR at the instant of shut-in and PVT data, calculate the relationship between krg / kro and pws using the following equation:
For an oil-gas system, the above relationship assumes that the Gas-Oil-Ratio (GOR) is constant throughout the two-phase region.
6. From the relationships of pws vs. krg / kro and krg / kro vs. So obtain pws vs. So.
7. From pws vs. So obtain pws vs. kro using relative permeability data.
8. Calculate two-phase pseudo-pressure m(p) vs. pws using the trapezoidal rule:
9. From the radial semilog plot of shut-in pseudo-pressure (m(p)) vs. shut-in time (t), choose a straight line which stands for the radial flow, and determine the slope (m) of this line. Calculate absolute permeability (k) and total skin (s') as follows:
Gas Condensate
Gas Condensate Theory
The most theoretically correct solution for multiphase flow in gas condensate reservoirs is to use a pseudo-pressure variable called the Reservoir Integral (R.I.), defined as:
Though mathematically rigorous, this variable is impractical to evaluate, as it requires knowledge of the distribution of saturation’s in the reservoir, in space and in time. It only applies when a simulation of the reservoir has been conducted.
A more practical alternative is to define a Sandface Integral (S.I.), as shown below, which calculates a pseudo-pressure in terms of the saturation at a specified point in the reservoir, usually the sandface as follows:
This too is impractical to evaluate, but it forms the basis for defining a two-phase pseudo-pressure that can be evaluated, if appropriate rock and fluid properties are available. This definition is only an approximation to the Reservoir Integral, and is not fully rigorous, it is adequate for analysis and modeling of gas condensate systems. This two-phase pseudo-pressure is defined as:
To calculate y2p, the variation of kro and krg with pressure must be known beforehand. In reality, y2p can be defined in terms of krg alone, along with a pressure-saturation relationship. In other words, the information contained in the kro(p) relationship is already contained in the combination of krg(p) and pressure-saturation. This is a two-part procedure, the first part is obtaining the relative permeability data for the reservoir rock, and the second part is assuming an appropriate model for fluid flow in the reservoir.
For a gas condensate system, the above relationship assumes that steady-state flow exists in the two-phase region. This is a good assumption near the wellbore, where the oil (condensate) saturation is greater than the critical oil saturation and the condensate is mobile. However, it is not valid in the bulk of the reservoir, where the oil (condensate) saturation is less than the critical oil saturation and the condensate is not mobile.
Even when the data is available for calculating y2p, it is recommended that, for gas condensate systems, y be used as the variable of analysis for calculating s2p and sd, and y2p be used to calculate kg and s'. This recommendation stems from the limitations in the assumptions underlying the y2p definition.
The literature recommends that the two-phase pseudo-pressure, y2p, be used for drawdown analysis, and single phase pseudo-pressure, y, be used for buildup analysis. However, drawdown data is usually very erratic, and in view of the discontinuity that would arise from changing from one pseudo-pressure to the other, single phase pseudo-pressure, is recommended to model both drawdown and buildup data. In this case, two-phase skin would be calculated using the single phase pseudo-pressure procedure. If the two-phase pseudo-pressure procedure is used instead, the skin calculated represents the total skin (s') at the wellbore, excluding the two-phase skin component (which is taken into account through the two-phase pseudo-pressure formulation).
If either the rock relative permeability data or the gas condensate PVT data are not available, then y2p cannot be calculated. In such cases, only s2p and sd, can be determined from the pressure transient analysis, and not kg and s'. There are two types of PVT data that can be obtained from laboratory analyses. These two types are Constant Volume Depletion (CVD) and Constant Composition Experiment (CCE) PVT data. The results from these experiments are different, and it is the CCE data that should be used when calculating y2p. It is more difficult, however, to obtain all the data from the CCE, and often the CVD data sets are more complete than the CCE. In such cases, one of two choices is recommended: either (i) use the CVD data in place of the CCE data or (ii) compute the required CCE data from equations of state.
Gas Condensate Procedure
If rock relative permeability data and condensate PVT data are available, krg is calculated as a function of pressure, as outlined below:
1. First, kro/ krg is calculated from one of the following equations shown below:
If rg and ro are molar densities:
If rg and ro are mass densities and Mo and Mg are the molecular weights (g / gmol):
If relative permeability data (kro, krg, So) are available, a tabulation of kro/ krg as a function of oil (condensate) saturation (So) is generated:
From these two tabulations, a relationship of krg as a function of pressure is generated.
2. Calculate two-phase pseudo-pressure using numerical integration (trapezoidal rule):
3. Obtain the slope (m) of the semilog plot of y2p versus log (time function), or from the derivative or regression analysis.
4. Calculate permeability, kg = kkrg from:
5. Calculate total skin (s'):
6. Drawdown:
7. Buildup:
