Program solves for oil well inflow performance

Roy Engineer, Gary Grillette
Bechtel Petroleum Operations Inc.
Tupman, Calif.

IPR is a critical component in analyzing oil well production. IPRs are used for designing completion equipment, estimating well deliverability, predicting when natural flow will cease, and when artificial lift will be required.

All analytical techniques for nodal analysis begin with a well's IPR, and it is a key reference.

Program concept

An oil well's IPR relates bottom hole flowing pressure, Pwf, with oil production rate, qo.

The productivity index, J, an earlier concept, is still widely used. J assumes that the production rate is inversely proportional to Pwf. However, this concept is a special case of the IPR when Pwf is above the bubble point pressure, Pb, and the rate-vs.-Pwf forms a straight line.

Muskat1 pointed out that this relationship should not be expected to hold when two-phase, gas and liquid, flow exists in the reservoir.

Gilbert2 recognized that J varied with Pwf below the bubble point pressure and proposed the use of qo-vs.-Pwf for well analysis. He termed this complete curve as the inflow performance relationship or IPR of an oil well. Vogel,3 using a computer model of a solution-gas-drive reservoir, developed a generalized IPR curve equation, Equation 1 (see equation box). The terms in the equation are as follows:

• qo = Oil production rate, b/d

• Pwf = Bottom hole flowing pressure, psi

• qm = Maximum theoretical oil rate when Pwf is lowered to atmospheric pressure, b/d

• Ps = Static reservoir pressure, psi.

Vogel's IPR curve (Fig. 1 [20197 bytes]) is limited to relationships below the bubble point and for wells with little or no water production. However, experience has shown that the "curved IPR" concept is valid for all drive mechanisms.

Standing4 also developed a procedure for predicting future IPRs from the current IPR. But the application of this method is difficult because it requires one to know the oil formation volume factor (FVF), oil viscosity, and relative permeability.

Fortunately, the industry does have some empirical techniques for quickly predicting the present and future IPRs of oil wells. These techniques incorporate Vogel's generalized curve equation and require only data from well tests. Our IPR computer program constructs oil well IPRs for five cases: A, B, C, D, and E.

Case A

In Case A, reservoir conditions at test time are that Ps Pwf Pb. The objective is to establish the current IPR curve for a well.

Many oil wells do not fall within the limits imposed by Vogel such as a solution-gas-drive reservoir with static pressure, Ps, at or below bubble point, Pb. If the static pressure is above the bubble point, then a single-phase liquid flow will occur in the reservoir until the bottom hole flowing pressure declines below Pb.

Thus, a straight line IPR, or a constant J, could be expected before the bubble point, and a Vogel-type curve after the bubble point (Fig. 2 [23671 bytes]).

If one assumes a reservoir with Ps Pwf Pb and a test at Point A on Fig. 2, the straight-line slope between Ps and Pb is the productivity index, J, above the bubble point. From the test data then, we get Equation 2; therefore, the rate qb at Pwf = Pb is expressed as Equation 3.

The IPR below Pb can be developed by slightly modifying Vogel's IPR equation. To do this, we assume that Ps = Pb, qm = qc because qb = 0 when Ps = Pb = Maximum rate for Vogel section, and qc is also equal to qm - qb. Vogel's IPR equation then becomes Equation 4, where qo is the oil rate at Pwf < pb.

Differentiating Equation 4 with respect to Pwf yields Equation 5. The slope, which is negative, or J at Pwf = Pb is expressed as Equation 6.

Because the slope of the IPR curve immediately above and below Pb is equal, then we get Equation 7. To develop the IPR in this case:

• The qb and qm from the test data are calculated with Equations 3 and 7.

• IPR between the coordinates of Pwf = Ps, qo = 0 and Pwf = Pb, qo = qb would be a straight line.

• IPR for Pwf below Pb can be developed by modifying the conventional Vogel's equation with the following substitutions: below bubble point, Ps = Pb, maximum oil rate = qm - qb, and oil rate at Psf = qo - qb. Substitution of these parameters in Vogel's conventional equation obtains Equation 8.

Case B

In Case B, reservoir conditions at test time are that Ps Pb Pwf. The objective is to generate the current IPR curve for a well.

In this case, assume that an oil well had been tested and we know qo (= q'), Ps, and Pwf. The static pressure, Ps, is above the bubble point, but the bottom hole flowing pressure, Pwf, is less than Pb (Point B in Fig. 2).

As in Case A, at pressures below Pb, Vogel's equation is modified as shown in Equation 9 (Nomenclature is the same as in Fig. 2).

Because for generating Vogel section Ps = Pb; therefore, qc = qm - qb and q' = qo - qb.

Differentiating Equation 9 with respect to Pwf, obtains Equation 10. Therefore, the slope or J at Pwf = Pb is shown by Equation 11.

Now, the slope of the IPR above Pb is qb/(Ps - Pb).

Therefore, Equation 11 can be written as Equation 12.

In Equation 12, all factors except qm and qb are known. The IPR curve at Pwf below Pb can be developed using the modified Vogel's equation as shown in Equation 8. In Equation 12, all parameters are also known except qm and qb. Therefore, a simultaneous solution of Equations 8 and 12 will solve for qm and qb.

To establish the present IPR curve for an oil well when Ps Pb Pwf using these equations, the following steps are necessary:

• The qm and qb are solved for with Equations 8 and 12.

• IPR curve above Pb is a straight line between coordinates qo = 0, Pwf = Ps and qo = qb, Pwf = Pb.

• Equation 8 can generate the IPR curve for Pwf below Pb.

Case C

In Case C, reservoir conditions at test time are that Pb Ps Pwf. The objective is to generate future IPR curves when Ps declines below Pb.

When average static reservoir pressure, Ps, is below Pb, it becomes complicated to predict future IPRs. Other than when well bore damage is evident, or where water breakthrough or coning situations are encountered, the IPR curves will primarily be a function of future reservoir pressures.

The maximum production, (qo)maxf, at some future static pressure, Psf, can be estimated from a correlation by Eickmier, Equation 13:5

In Equation 13, Ps and (qo)m are at current test conditions.

To generate the IPR curve from Equation 13, where the future pressure is Psf < pb < ps, the following steps are to be followed:

• Obtain qo, Ps, and Pwf from current test data.

• Calculate (qo)m from Vogel's Equation (Equation 1).

• Calculate (qo)maxf from Equation 13.

• Assume different values of Pwf and calculate qo from Equation 14.

Case D

In Case D, reservoir conditions at test time are that Pb Ps Pwf. The objective is to generate the current IPR curve for the well.

In such a case, Vogel's equation (Equation 1) can be directly used to establish the current IPR curve for the well.

Case E

In Case E, reservoir conditions at test time are that Pb Ps Pwf, and Sg Sgc. Two tests are available and the objective is to generate the future IPR curve for the well.

This case is also known as the "Pivot Point" method and was developed by D.C. Uhri, et al.6 This technique can quickly predict future IPRs of oil wells producing from solution-gas-drive reservoirs where gas saturation, Sg, exceeds the critical saturation, Sgc.

The technique incorporates Vogel's generalized IPR curve equation and can be solved graphically with two sets of well test data. The method becomes more accurate as the spread between Ps values associated with the well tests becomes greater.

The IPR program uses the numerical approach for IPR predictions. Because this requires many equations, details of this method are not covered in this article. However, a reader can review the original paper, Reference 6.

Program use

IPR is a user-friendly PC program. Table 1 [9546 bytes] and Table 2 [8040 bytes] show example outputs of Cases B and F. Input for the two cases are as follows:

• Case B. Ps = 2,500 psig, Pwf = 1,600 psig, Pb = 2,000 psig, qo = 400 bo/d, and qw = 50 bw/d. The objective is to generate the current IPR data.

• Case E. Test No. 1: Ps = 2,500 psig, Pwf = 1,600 psig, qo = 100 bo/d, and Pb = 3,000 psi. Test No. 2: Ps = 2,200 psig, Pwf = 1,200 psig, and qo = 85 bo/d. The objective is to generate IPR data for the well at Psf = 1,500 psig.

• Further instructions on the use of the program are provided in the Readipr file (see editor's note).

Considerations

IPR is a practical and useful program; however, its output will solely depend on the quality of the input data. Extra caution is required for Case E when two sets of well test data are needed for the input. No stimulation or workovers should have been performed between the two tests.

The IPR relationships for Cases A and B were compared with slightly different techniques developed in References 7 and 8, and identical results were obtained.

For the current IPR, the program solves the problem for the wells with water production by calculating oil production using the modified Vogel equations and using a straight line productivity index, J, for the water IPR. The two sets of IPR data are then added together for generating the total liquid IPR for the well.

Experience has shown that such an approach of estimating the water IPR is valid for all practical purposes.

Acknowledgment

Editor's note: To obtain a copy of the IPR program subscribers can send a blank 31/2 diskette formatted to MS DOS and a self-addressed, postage paid or stamped return diskette mailer to: Production Editor, Oil & Gas Journal, 3050 Post Oak Blvd., Suite 200, Houston, TX 77056, USA.

Subscribers outside the U.S. should send the diskette and return mailer without return postage to the same address. This mail offer will expire Oct. 31, 1997.

References

1. Muskat, M., and Evenger, H.H., "Calculation of theoretical Productivity Factor," Petr. Trans. Am. Inst. of Mining & Met.Engr., 1942, p. 146.

2. Gilbert, W.E., "Flowing and Gas-Lift Well Performance," Drilling and Production Practice, API, 1954, p. 126.

3. Vogel, J.V., "Inflow Performance Relationships for Solution Gas Drive Wells," JPT, January 1968.

4. Standing, M.B., "Concerning the Calculation of Inflow Performance of Wells Producing from Solution Gas Drive reservoir," JPT Forum, September 1971.

5. Eickmier , J.R., "How to Accurately Predict Future Well Productivities," World Oil, May 1968.

6. Uhri, D.C., and Blount, E.M., "Pivot Point Method Quickly Predicts Well Performance," World Oil, May 1982.

7. Brown, K.E., The Technology of Artificial Lift Methods, Vol. 1, PennWell Books, Tulsa, 1977.

8. Patton, D., "Generalized IPR Curves for Predicting Well Behavior," Petroleum Engineer Int'l., September 1980.

Bibliography

1. Nind, T.E.W., Principles of Oil Well Production, McGraw Hill, New York, 1964.