Reservoir limit testing is a powerful tool used to estimate the reservoir volumes of finite systems. The procedure was first introduced by Jone's(1) which is restricted to only depletion drive reservoirs due to the boundary conditions assumed to solve the redial pseudo-steady-state flow diffusivity equation. But in water and/or gas cap drive reservoirs, water influx and gas intrusion affects test results and then the conventional procedure of Jone's method for bounded systems is not applicable.
An analytical method has been developed to estimate the boundary of the reservoirs producing under moving boundary systems (bottom water and/or gas cap drive mechanisms) by applying the principle of superposition theorem to reflect the reservoir pressure drop pulses at the free levels of the displacing fluids to estimate the time at which the pseudo-steady-state prevail.
The reservoir limit testing analyzing reservoir performance to estimate the original hydrocarbon in place is one of the fundamental tasks that reservoir engineers perform. The testing procedure is performed by flowing a well either at constant or variable flow rates (1, 2) until pseudo-steady-state flow condition start to prevail. During this period of the flow regime, the bottom hole flowing pressure exhibit a linear function of time and the slope is indicative for reservoir pore volume.
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This method of analysis was first introduced by Jone's (1) 1956 for bounded systems. While, in water and/or gas cap drive systems, the effect of water influx and/or gas intrusion towards the oil zone will be to increase the average reservoir pressure, this increase in pressure is felt throughout the conventional test and affects the pressure measurements.
Kaczorowsk (3) 1993, use the concept of Jone's definition for pseudo-steady-state flow regime to develop an empirical method to determine the movable hydrocarbon volumes in water drive systems. The procedure require to achieve two reservoir limit tests performed using two different flow rates and require to reach the pseudo-steady-state flow regime for each. This method is applicable to determine the remaining movable hydrocarbon volumes in water drive gas reservoirs of small to moderate size and oil reservoirs that are producing above the bubble point pressure.
Reservoir limits testing in moving boundary systems.
New analytical methodology has been developed to estimate the boundary of the reservoirs producing under bottom water and/or gas cap drive mechanisms (moving boundary systems) by assuming the fluids interfaces levels between oil-water and / or oil-gas as a no-flow boundary during the early period of production in which the transient flow regime prevail. During this period of the flow, the fluids interfaces still not offer any response to that pressure drop pulses created in the reservoir during the test.
Meanwhile, the principle of superposition (image) theorem used to reflect the created pressure drop pulses at the free levels of the displacing fluids to estimate the time at which the reservoir pressure drop pulses reach the interfaces positions.
The principle of superposition could be applied at the fluids interfaces levels of zero capillary pressure at which the pressure gradient is zero. This methodology suggests reflecting the transient pressure drop pulses behavior that is created in the oil zone as a results of oil production activities at the free levels of the displacing fluids interfaces (water and gas) to estimate the fluids boundaries rather than reflecting a boundary to predict the pressure drop behavior that is conventionally used in bounded systems as shown in schematic drawing (1).
Figure (1): Schematic drawing shows the created and the superimposed reflected pressure drop pulse at the fluids interfaces.
The created pressure drop pulse in the reservoir due to oil production operation will travel at specific rate depending on the displaced fluid (oil) and the reservoir zone characteristics to reach the existing displacing fluids interfaces in contact with oil zone(4&5); hence the existing displacing fluids zones will response to that created pressure drop pulse by a reflected pressure drop pulse depends on its own fluids and zone properties, this response of the displacing fluids start as soon as the reservoir pressure drop pulse reach the fluids interface. Thus, the fluids interfaces could be considered as a no-flow boundaries and the reservoir may behave as an infinite acting reservoir as well as these interfaces still not affected by the reservoir pressure drop pulses. While, the reservoir may behave as a pseudo-steady-state when these interfaces start to response for pressure drop pulses created in the oil zone due to oil production activity.
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Since the oil is slightly compressible fluid; it is normally takes a short period of time for the created pressure drop pulse in oil reservoir zone to be felt at the displacing fluids interfaces.
Oil reservoirs under aquifer and/or gas cap drive mechanisms
The equation describe the reservoir pressure drop behavior at the fluids interfaces under aquifer drive and/or gas cap drive mechanisms exhibits both vertical and horizontal components of flow at the contacted fluids interface that is already given by Al-Sudani J.A.(5) 2009 which is written in field units as follows:
Where, ; The dimensionless reservoir pressure drop obtained by joining the pressure drop equations written in reservoir and aquifer zones at the free water level which could be written as follows;
Where, ; the dimensionless time that is obtained by the following equation.
Where, ( the dimensionless reservoir radius that is calculated as follows;
Where, ; the effective vertical and horizontal aquifer permeability components respectively.
Equation (1) has been drawn in semi-log graph as presented in figure (2),
Hence, converting figure (2) to a linear scale graph will yield to generate the graph as shown in figure (3) at which it could be notice that the last parts for all curves are approximately straight lines. During this part of the test, the reservoir pressure drop rate with time or becomes constant which is defined by the pseudo-steady-state flow period.
Moreover, it could be notice that the point at which the straight lines start to separate from the entire curves (Beginning of the straight line-PSS) usually deflect at point of dimensionless time () regardless of the values of dimensionless reservoir radius .
This could be simplifying the test to determine the reservoir radius (reservoir limit test) by substituting the value of () in the equation used to estimate the dimensionless reservoir time (), as follows:
Solving equation (7) for reservoir radius, it could be written as follows:
While, the time to reach the pseudo-steady state for bottom water drive reservoir in which the pressure drop pulse reach the reservoir boundary could be obtained by arranging equation (7) for time (t) and could be written as follows:
Where, ; the time to reach the pseudo-steady-state in (hours).
It could be notice the required time to reach the reservoir boundary under water and/or gas cap drive mechanisms usually takes time greater than that in depletion drive reservoirs, this may reflect the additional size of aquifer-water or gas-cap zones to the original reservoir size, in addition to the pressure support that could be provided by the such displacing fluids zones that tends to reduce the magnitude of the pressure drawdown pulse created in the oil zone and then reduce its traveling rate to reach the reservoir boundaries.
The dimensionless pressure drop behavior with dimensionless time figure (3) shows a straight line trend in the part of pseudo steady state period at () of constant slope equal to ( for any ratios of reservoir vertical to horizontal permeability (). Since, using this property it could be possible to derive an equation for estimating reservoir radius if the ratio of pressure drop with time is recorded, as follows;
Where, ; the reservoir pore volume.
Thus, equation (14) could be written as follows.
However, the reservoir fluid initial volume (oil or gas) could be calculated using the following expression:
Since, equation (16) could be used to predict the pore volume of reservoirs producing under combined drive mechanisms when the gradient of pressure with time ( could be calculated.
In this section, the two simulated and field examples illustrated by Ksczorowski (3) for water drive system have been used to show the capability for the presented model to estimate the reservoir limit testing using only initial slope of pressure versus time, rather than using two elongated tests that should be reached the pseudo-steady-state condition which is very difficult to be recognized in external drive systems due to the continuous pressure support provided outer boundary for such reservoir systems. Moreover, another oil field analyzed using the presented model and compared with the results obtained by MBCAL, Blasingame water drive decline curves and water drive model plot using advance software technology.
Example-1: Reservoir simulation case:
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Initial reservoir pore volume = 12.4 Bcf. The reservoir produces 2.5 Bcf of gas during the last six months. The gas saturation is 0.75,
Initial slope = 0.01213 Psi/hr for gas production of 11,096 MMSCF/D.
Gas Compressibility = 0.0001688 Psi^-1
Applying eq. (16) and solving for ,
Thus, the reservoir pore volume related to gas in place could be calculated using eq. (17).
The GIP value could be converted to an original gas in place OGIP by adding the produced value of (2.5 Bcf) during the last six month period, this result agree within 0.02% with that of simulated value.
Example-2: Field case:
Originally gas in place volume = 7.9 Bcf.
Initial slope = 0.09694 Psi/hr.
Gas Compressibility = 0.00050233 Psi^-1
This value is within 0.03% of the value determined by material balance estimation which was determine of OGIP 7.9 Bcf and 7800 RB/D of water influx.
Example-3: Field case-oil reservoir:
Small oil field of reservoir radius (112.5) and formation thickness (26.25 ft) with the following data; oil flow rate =2.5 Stb/D, Total compressibility= 3.171*10^-4, oil formation volume factor Bo=1.116 bbl/Stb, the slope of the average reservoir pressure drop with time about (0.062 psi/hr), oil saturation (0.65).
The new technique provides generalize analytical estimation for the hydrocarbon in place in water and/or gas drive reservoirs. This technique is similar to that presented by Jones which is restricted only for depletion drive systems.
The presented analytical method shows high accuracy with that obtained from material balance calculation and reservoir simulation process.
The methodology is very simple in performing the reservoir limit testing in moving boundary systems which is require only one limit test, thus it could provide much effective time to reduce the cost of tests than that procedure presented by Ksczorowski(2) that is require two limit tests with two different flow rates.
SYMBOL DIFINITION UNIT
Reservoir fluid formation volume factor RB/STB
Total compressibility in the reservoir fluid zone Psi-1
Total fluids and formation system compressibility Psi-1
Reservoir fluid column thickness ft
Absolute reservoir permeability md
Horizontal flow component permeability of the aquifer md
Vertical to horizontal permeability ratio
Vertical flow component permeability md
Initial reservoir pressure Psi
Pressure at the interface of water and/or gas zone Psi
Dimensionless pressure drop
Reservoir fluid flow rate STB/Day
Reservoir radius ft
Dimensionless reservoir radius
Production time Hours
Time of pseudo-steady-state condition Hours
Z Dimensionless constant depend on reservoir radius
ÎÂ¼f Oil viscosity Cp