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A New Approach to Aquifer Influx Calculation for Finite Aquifer System
Article · July 2011
DOI: 10.2118/150733-MS
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A New Approach to Aquifer Influx Calculation for Finite Aquifer System
*Omeke James E; SPE, Nwachukwu A.; SPE, Awo R.O; SPE, Boniface Obah; SPE, Uche.I.N; SPE (Shell Chair, Federal University of Technology Owerri)
Copyright 2011, Society of Petroleum Engineers Inc.
This paper was prepared for presentation at the 35th Annual SPE International Technical Conference and Exhibition in Abuja, Nigeria, July 30th- August 3rd, 2011.
This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435
Abstract
Among the classic models used in the evaluation of aquifer properties, the most used are Van Everdingen &
Hurst, (1949), Fetkovich approximate (1971), Carter &
Tracy (1960), Allard & Chen (1984), and Leung (1986).
Due to the inherent uncertainties in the aquifer characteristics, all the models require historical reservoir performance data to evaluate aquifer property parameters. The fact that the reservoir-aquifer boundary pressure is not constant with time gives rise to computational challenges. Most authors seek to eliminate this disadvantage by bringing up approximate models.
Fetkovich assumes a constant boundary pressure while deriving his model and required a superposition to get a proper result. This led to the introduction of some forms of pressure approximation and iterations; any error introduced is constituted by this. Higher computational ability is mostly required especially in the evaluation of the Everdingen & Hurst model due to the application of Duhamel superposition principle. More uncertainty however is also seen in the geometry and areal continuity of the aquifer itself.
In this paper, a fast and simple approach was used to develop a new aquifer influx model for a finite aquifer system that admits a pseudo-steady-state flow regime.
The partial differential equation that characterizes the flow of aquifer in the reservoir was considered. It was simplified through some basic assumption and in turn
resulted to ordinary differential equation. with an appropriate boundary condition, a solution was gotten that enables a direct calculation of cumulative water influx at a given time without the use of superposition or pressure approximations or iterative means. In other to derive a solution under varying reservoir-aquifer boundary pressure the dependency of the boundary pressure with time was described as an exponential trend, which is mostly the case.
In other to validate the developed model, comparison was made with Carter & Tracy, Fetkovich and Van Everdingen Models. Van Everdingen and Hurst Model was used as the standard for comparison since it represents the exact solution of radial diffusivity equation. Two case examples were considered. The results show a close match between the developed model and the existing models, also from the error analysis made, it was noticed that the predictions of the newly developed model were better than that of the Carter & Tracy.
Introduction
There are more uncertainties attached to water influx evaluation than to any other aspect of reservoir engineering. This is simply because one seldom drills wells into an aquifer to gain the necessary fluid and rock properties. Instead, these properties have been inferred from what has been observed in the reservoir
Several models have been developed for estimating water influx that is based on assumptions that describe the characteristics of the aquifer. Due to the inherent uncertainties in the aquifer characteristics, all of the proposed models require historical reservoir performance data to evaluate constants representing aquifer property parameters since these are rarely known from exploration and development drilling with sufficient accuracy for direct application.
The model proposed by Everdingen & Hurst is considered as one of the best in terms of an exact solution of the radial diffusivity equation, thus, requiring a bigger computational effort. Other proposed models, such as Fetkovich, and Carter & Tracy sought to eliminate the
2 Omeke J.E, Nwachukwu A, Awo R.O, Boniface O, Uche, I.N SPE disadvantage of the required computing power, and thus
became more popular in commercial flow simulators.
In this work, a comparative analysis will be performed using the Van Everdingen & Hurst, Carter & Tracy, Fetkovich and the newly developed water influx model.
The theoretical basis of Van Everndingen & Hurst model, Fetkovich and Carter and Tracy method are thus presented.
The theoretical basis of existing models Van Everdingen & Hurst Model
The dimensionless form of the diffusivity equation is basically the general mathematical equation that is designed to model the transient flow behavior in reservoirs or aquifers. In a dimensionless form, the diffusivity equation is:
+ = (1)
Applying the Laplace transformation, Van Everdingen &
Hurst solved the diffusivity equation of the reservoir- aquifer system considering as boundary condition a constant pressure in the boundary. The flow supplied by the aquifer at the point of contact with the reservoir is given by the Darcy equation:
= (2)
Where = , in radian, represents a factor describing a radial sector.
Equation 2 can be re-written as
− =
∆ ≡ ( ) (3)
Where ( ) is the dimensionless flow supplied by aquifer, calculated at the boundary reservoir-aquifer (point
= 1). The accumulated influx is the integral of the flow over time, as expressed by the Equation below
= ∫ = ∆ ∫ (4)
If ( ) is the integral of with regards to , Equation (4) becomes
= ∆ ( ) (5)
Where is the influx constant of water into the aquifer represented by Equation (6)
= 2 ∅ ℎ (7)
The authors presented the solution of equation (1) in terms of dimensionless water influx ( ) which is frequently presented in tabular form or as a set of polynomial expressions giving as a function of for a range of ratios of the aquifer to reservoir radius = for radial aquifers. Fanchi (1985) matched the Van Everdingen and Hurst tabulated values of the dimensionless pressure as a function of dimensionless time and dimensionless radius. In other to develop the polynomial form of the solution giving as a function of , for a range of ratios of the aquifer to reservoir radius
= , the Van Everdingen & Hurst tabulated value were matched.
The Fetkovich Method
Fetkovich (1971) developed a method of describing the approximate water influx behavior of a finite aquifer for radial and linear geometries. In many cases, the results of this model closely match those determined using the van Everdingen and Hurst approach. The Fetkovich theory is much simpler because it does not require the use of superposition. Hence, the application is much easier, and this method is also often utilized in numerical simulation models.
The Fetkovich model is based on the premise that the productivity index concept will adequately describe water influx from a finite aquifer into a hydrocarbon reservoir.
That is, the water influx rate is directly proportional to the pressure drop between the average aquifer pressure and the pressure at the reservoir–aquifer boundary. The method neglects the effects of any transient period. Thus, in cases where pressures are changing rapidly at the aquifer–reservoir interface, predicted results may differ somewhat from the more rigorous van Everdingen and Hurst. However, in many cases pressure changes at the waterfront are gradual and this method offers an excellent approximation to the method discussed above.
This approach begins with two simple equations. The first is the productivity index (PI) equation for the aquifer, which is analogous to the PI equation used to describe an oil or gas well.
= = ( − ) (8)
The second equation is an aquifer material balance equation for a constant compressibility, which states that the amount of pressure depletion in the aquifer is directly proportional to the amount of water influx from the aquifer, or:
= ( − ) (9)
The maximum possible water influx will occur if = 0 or:
= (10)
Combining equation (9) and (10) gives:
= 1 − = 1 − (11)
Differentiating equation (11) gives:
= −
(12)Combining equation (8) and (12) yields:
= ( − ) (13)
To use this solution in the case in which the boundary pressure is varying continuously as a function of time, the superposition technique must be applied. Rather than using superposition, Fetkovich suggested that, if the reservoir–aquifer boundary pressure history is divided into a finite number of time intervals, the incremental water influx during the nth interval is:
(∆ ) = (( ) − ( ) ) 1 − ∆
(14)
Where ( ) is the average aquifer pressure at the end of the previous time step, this average pressure is calculated from Equation (11) as:
( ) = 1 −( ) (15)
The average reservoir boundary pressure is estimated from:
( ) =( ) ( ) (16)
The productivity index J used in the calculation is a function of the geometry of the aquifer. Fetkovich calculated the productivity index from Darcy’s equation for bounded aquifers. Lee and Wattenbarger (1996) pointed out that the Fetkovich method can be extended to infinite- acting aquifers by requiring that the ratio of water influx rate to pressure drop is approximately constant throughout the productive life of the reservoir. The productivity index J of the aquifer is given in the appendix.
Carter & Tracy method
To reduce the complexity of van Everdingen and Hurst water influx calculations, Carter and Tracy (1960) proposed a calculation technique that does not require
superposition and allows direct calculation of water influx.
The primary difference between the Carter–Tracy technique and the van Everdingen and Hurst technique is that Carter-Tracy assumes constant water influx rates over each finite time interval. Using the Carter–Tracy technique, the cumulative water influx at any time, tn, can be calculated directly from the previous value obtained at tn−1, or:
( ) =
( ) + [( ) − ( ) ] × ∆ ( ) ( )
( ) ( ) ( ) 17 For an infinite-acting aquifer, Edwardson et al. (1962) developed an approximation of as a function of dimensionless time and for a given dimensionless radius . When the dimensionless time
>100, the following approximation can be used for :
= [ ( ) + 0.80907]
18 With the derivative given by:= 19
New model development
Flow within the aquifer can be described by the following equation:
∇⃗ ∇⃗ + = ∅ 20
If the effect of potential gradient is negligible, water saturation in the aquifer zone Is one i.e. totally saturated with water, flow is single phase (water) ,the inflow parameters are constant with time, i.e., The reservoir is finite (i.e. pressure response is felt by the external boundary), then the resulting equation will become:
=
∅21
The above equation is further simplified to the following form:
= ∅ , ∴ = ∅ 22
The equation above can be expressed in terms of cumulative water influx as follows:
= ∅ 23
4 Omeke J.E, Nwachukwu A, Awo R.O, Boniface O, Uche, I.N SPE ( ) = 2− 2 ∅ℎ
5.615
≡
24= ℎ ( )
Equation 23 now becomes:
=
25Just like Fetkovich, the aquifer influx rate is modeled with the aquifer productivity index equation which is given as follows:
= = − ( − ) 26
= − 27
Differentiating the above equation with respect to time gives:
= − ,
28 Substituting equation 28 into 25 gives:=
−1 2 2 29 Let =Then equation 29 becomes:
= −1
2
2 30
The major challenge in the above equation is to develop an expression that will represent the changes in Reservoir-Aquifer boundary pressure with time since it is not a constant. In other to achieve this, an exponential decline was used to model the rate at which this boundary pressure varies with time, knowing that at Pseudo-steady flow, the boundary pressure falls with time which in most cases can be fitted well with an exponential curve.
Therefore, in this study, the variation of boundary pressure with time is given as follows:
= −
31The value of the constant is determined on solution of equation 31:
( ) = − + ( ) 32 N/B = 0, =
From equation 32, a plot of ( ) against time will yield a straight line graph with slope = − and intercept ( ).
Therefore with the pressure history data, the above process can be implemented.
Substituting equation 31 into equation 30 and simplifying gives:
+ = −
33The above equation is a second-order linear ordinary differential equation. With the initial conditions:
= 0, = 0, = 0, =
The solution of equation 33 is thus given as:
( ) = 1 − −
34Substituting the value = yields the final form of the equation as:
( ) = 1 − − −
35The aquifer productivity index are given as follows
=
[. ( ). ]
,
36,
=
.[ ( )]
Comparative analysis of models Example case-1
Aquifer properties
Permeability 60 m D
porosity 0.12
water viscosity, cp 0.7 cp Total compressibility 0.000009 1/psi Reservoir radius 9200 ft Dimensionless radius 5 Aquifer radius 46000 ft Encroachment Angle 60 Degrees
thickness 120 ft
Pressure history
time(days) pressure(Pr), psi
0 2850
365 2610
730 2400
1095 2220
1460 2060
Result & discussion Table 1
Time (days)
Pressure (Pr) psi
New Model, MMBBLS
Van E.&Hurst,
MMBBLS
Fetkovich, MMBBLS
Carter Tracy, MMBBLS
0 2850 0 0 0 0
365 2610 0.53927952 0.63899090 0.55792963 1.01175758 730 2400 1.96076167 2.20129719 2.05631089 2.63182996 1095 2220 4.03420961 4.23019547 4.17754888 4.65814078 1460 2060 6.57943626 6.56363885 6.68744516 6.96523908
Fig 1: graphical summary for case 1
From the result displayed in Table 1 and fig.1 above, it is obvious that the newly developed model gave a close match to the more accurate Van Everdingen and Hurst Model as well as other models. The degree of accuracy of each model is seen when the total average of the absolute value of their percentage error is compared. From the error analysis done, Fetkovich model gave a value of 0.04481 or 4.481%, Carter and Tracy 0.1883 or 18.83%
and the newly developed model gave 0.06284 or 6.284%.
It is also obvious that the newly developed model gave a better prediction than Carter and Tracy model for the system considered.
Example case-2 Aquifer properties
Permeability 200 m D
porosity 0.25
water viscosity, cp 0.55 cp Total compressibility 0.000009 1/psi Reservoir radius 9200 ft Dimensionless radius 5 Aquifer radius 46000 ft Encroachment Angle 140 Degrees
thickness 100 ft
time(days) pressure(Pr), psi
0 2740
365 2500
730 2290
1095 2109
1460 1949
1825 1818
2190 1702
2555 1608
2920 1635
3285 1480
3650 1440
Result Table 2
time(days) pressure(Pr) psi
New model, MMBBLS
Van E.&Hurst, MMBBLS
Fetkovich, MMBBLS
Carter&
Tracy, MMBBLS
0 2740 0 0 0 0
365 2500 3.154779 3.779273 3.925246 6.10151 730 2290 10.80446 12.86882 13.55035 15.44766 1095 2109 20.97693 24.05700 25.49990 26.24490 1460 1949 32.36467 35.88860 37.9731 37.39915 1825 1818 44.39727 47.42165 49.93082 47.91984 2190 1702 56.4544 58.1560 60.87136 57.52527 2555 1608 68.61625 67.87701 70.61977 65.8536 2920 1635 86.30000 74.76752 77.34139 69.80722 3285 1480 93.81064 81.05893 83.31391 76.50875 3650 1440 107.1148 87.92163 89.94994 81.30911 0
1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000
0 500 1000 1500 2000
Pressure, psi
time(days)
Van & Hurst Fetkovitch New model Carty tracy
6 Omeke J.E, Nwachukwu A, Awo R.O, Boniface O, Uche, I.N SPE
Fig 1: graphical summary for case 2
From the error analysis done, Fetkovich model gave a value of 0.0395 or 4.481%, Carter and Tracy 0.109 or 10.9% and the newly developed model gave 0.107 or 10.7% all relative to Van Everdingen and Hurst model.
Still, the newly develop model gave a slightly better prediction than Carter Tracy Model based on the data used.
Conclusion
1. A new aquifer influx model has been developed using a different approach
2. The predictions done by the newly developed model showed good degree of accuracy
3. No iterations, pressure averaging or superposition were involved in the process of calculation
4. The model is very simple and can be attached as a subroutine in reservoir simulators
Recommendation
The accuracy of the model will strongly depend on the evaluation of the constant from a semi log plot of reservoir –aquifer boundary pressure against time or a Cartesian plot of natural logarithm to base e of boundary pressure against time. As a sequel to this work, the theoretical basis of the constant will be investigated so as to obtain an analytical expression. Also the rate at which the boundary pressure changes with time will be analyzed with a polynomial.
Nomenclature
=
=
=
= ,
= ℎ ,
= ℎ ,
=
= , 1/
=
= , /
=
= ,
= ,
= ,
= ,
= ,
∅ =
= , /
= ,
ℎ = ℎ ,
∆ = ℎ
= ℎ ,
( ) =
References
1. Van Everdingen, A. F. & Hurst, W.: The Application of the Laplace Transformation to Flow Problems in Reservoirs.
Trans. AIME, 186: 305-324, 1949.
2. Fetkovich, M. J.: A Simplified Approach to Water Influx Calculations - Finite Aquifer Systems. J.
Pet. Tech., 814- 828, July 1971.
3. Carter, R. D. & Tracy, G. W.: An Improved Method for Calculating Water Influx. J. Pet. Tech., 58-60, Dec. 1960
4. Allard, D. R. & Chen, S. M.: Calculation of Water Influx for Bottom-Water Drive Reservoirs. In: SPE Annual Technical Conference and Exhibition, 59.
0 20000000 40000000 60000000 80000000 10000000 12000000
0 1000 2000 3000 4000
Pressure,psi
time, days
New model Van-Evandigen&Hurst Fetkovich
Carter Tracy
Houston, TX, Sept. 16-19, 1984. Proceedings.
Richardson, TX, SPE, 1984. (SPE 13170) 5. Leung, W. F.: A Fast Convolution Method for
Implementing Single-Porosity Finite/Infinite Aquifer Models for Water-Influx Calculations. SPE Res. Eng., 490- 510, Sept.1986
6. Fanchi, J.: Analytical Representation of the Van Everdingen-Hurst Influence Functions. SPE J., 405–425, June, (1985)
7. Lee, J. and Wattenbarger, R.: Gas Reservoir Engineering, SPE Textbook Series, Vol. 5 (Dallas, TX: Society of Petroleum Engineers). (1996) 8. Craft, B. C. and Hawkins, M. F., Jr.: Applied
Petroleum Reservoir Engineering, Prentice-Hall, Inc., Englewood Cliffs, N. J. (1959) 126, 206, 224.
9. Dake, L. Fundamentals of Reservoir Engineering (Amsterdam: Elsevier), (1978)
10. Dake, L.P. The Practice of Reservoir Engineering (Amsterdam: Elsevier), (1994).
11. J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems, 5th ed., McGraw- Hill, New York, 1993
12. R. Haberman, Elementary Applied Partial Differential Equations, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1983
13. H. Levine, Partial Differential Equations, Studies in Advanced Mathematics, Vol. 6, American
Mathematical Society, Rhode Island, 1991
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