Plan de acción

Gas flow in a reservoir under transient conditions can be approximated by the combination of Darcy’s law (rate equation) and the continuity equation. In general,

which in radial coordinates reduces to

From the real gas law,

and therefore

If the permeability k is considered constant, then Equation (4-64) can be approximated further:

Performing the differentiation on the right-hand side of Equation (4-65) and assuming that Z and μ are either constant or that they change uniformly and slowly with pressure, then

Via rearrangement and remembering that

it becomes

Therefore, Equation (4-65) can be written as

For an ideal gas, c_g = 1/p and, as a result, Equation (4-55) leads to

This approximation is in the form of the diffusivity equation [see Equation (2-18) in Chapter 2] and its solution, under the assumptions listed in this section, could have the shape of the solutions of the equation for oil, presuming that p² instead of p is used. This approximation was used in earlier sections of this chapter and was developed independently. Pressure squared differences can be used as a reasonable approximation. However, the assumptions used to derive Equation (4-70) are limiting and, in fact, they can lead to large errors in high-rate wells unless using the Al-Hussainy and Ramey (1966) real gas pseudopressure function.

The real gas pseudopressure function, m(p), is defined as

where p_o is some arbitrary reference pressure (could be zero). The differential pseudopressure Δm(p), defined as m(p) – m(p_wf), is then the driving force in the reservoir.

For low pressure it can be shown that

whereas for high pressure (both p_i and p_wf higher than 3000 psi),

The real gas pseudopressure can be used instead of pressure squared difference in any gas well deliverability relationship (properly adjusted for the viscosity and the gas deviation factor). For example, Equation (4-47) would be of the form

Of course, more appropriately, the real gas pseudopressure can be used as an integrating factor for an exact solution of the diffusivity equation for a gas. Beginning with Equation (4-65), and using the definition of the real gas pseudopressure [Equation (4-71)] and the chain rule, it can be written readily that

Similarly,

Therefore, Equation (4-65) becomes

The solution of Equation (4-77) is exactly similar to the solution for the diffusivity equation in terms of pressure. Dimensionless time has (by convention) been defined as

and dimensionless pressure as

All solutions (such as the line source solution and the wellbore storage and skin effect solution) that have been developed for oil wells using the diffusivity equation in terms of pressure are applicable for a gas well using the real gas pseudopressure. For example, the logarithmic approximation to the exponential integral [compare Equations (2-19) and (2-20)] would lead to an analogous expression for a natural gas. Thus,

This expression can be used for transient IPR curves for a gas well.

Example 4-9. Transient IPR for a Gas Well

With the data for the well in Appendix C, calculate transient IPR curves for 10 days, 3 months, and 1 year.

Solution

Table 4-6 gives the viscosity, gas deviation factor, and real gas pseudopressure for the reservoir fluid described in Appendix C. For the initial condition of p_i = 4613 psi, the real gas pseudopressure, viscosity, and gas compressibility factor are 1.265 × 10⁹ psi²/cp, 0.0235 cp, and 0.968, respectively.

(These values, derived from numerical fits, are slightly different from the ones in Appendix C.)

Table 6. Calculated Viscosity, Gas Deviation Factor, and Real Gas Pseudopressure for Example 4-9

Equation (4-80) can be used for the transient IPR calculations if real gas pseudopressures are to be used. If pressure squared differences are to be used, then the denominator 1638T must be replaced by 1638μZT.

The gas compressibility can be calculated from Equation (4-35). At initial conditions,

The slope 0.045 was obtained from Figure 4-1 at T_pr = 1.69 and p_pr = 6.87. Thus the total system compressibility is

Hence for time equal to 10 days (240 hr), Equation (4-80) becomes

and finally,

Similar expressions can be developed readily for the other times.

Figure 4-8 is a graph of the transient IPR curves for 10 days, 3 months, and 1 year.

Figure 4-8. Transient IPR curves for Example 4-9.

Note: A large fraction of all natural gas-producing wells are hydraulically fractured, which makes the inflow relationships presented in this chapter, implying radial flow, sometimes invalid. If the

reservoir permeability is relatively large (i.e., k > 10 md), then the fracture lengths are relatively

short, and the well performance can be described with a fractured well skin factor and a radial flow equation is appropriate. However, in low-permeability reservoirs fracture lengths are very large and radial flow-equivalent skin effects are not appropriate. Fracture-performance models are then

indicated. In such cases, non-Darcy flow effects are likely to occur in a hydraulic fracture so these effects need to be included. Reservoir inflow to hydraulically fractured gas wells is covered in Chapters 17 and 18.

Username: & Book: Petroleum Production Systems, Second Edition. No part of any chapter or book may be reproduced or transmitted in any form by any means without the prior written permission for reprints and excerpts from the publisher of the book or chapter. Redistribution or other use that violates the fair use privilege under U.S. copyright laws (see 17 USC107) or that otherwise violates these Terms of Service is strictly prohibited. Violators will be prosecuted to the full extent of U.S.

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References

1. Al-Hussainy, R., and Ramey, H.J., Jr., “Application of Real Gas Theory to Well Testing and Deliverability Forecasting,” JPT, 637–642 (May 1966).

2. Aronofsky, J.S., and Jenkins, R., “A Simplified Analysis of Unsteady Radial Gas Flow,” Trans.

AIME, 201: 149–154 (1954).

3. Brown, G.G., Katz, D.L., Oberfell, C.G., and Alden, R.C., “Natural Gasoline and the Volatile Hydrocarbons,” NGAA Paper, Tulsa, OK, 1948.

4. Carr, N.L., Kobayashi, R., and Burrows, D.B., “Viscosity of Hydrocarbon Gases Under Pressure,”

Trans. AIME, 201: 264–272 (1954).

5. Coles, M.E., and Hartman, K.J., “Non-Darcy Measurements in Dry Core and Effect of Immobile Liquid,” SPE Gas Technology Symposium, Calgary, Alberta, March 15–18, 1998.

6. Cooke, C.E., Jr., “Conductivity of Proppants in Multiple Layers,” JPT, 1101–1107 (September 1973).

7. Dranchuk, P. M., and Abou-Kassem, J. H., “Calculation of z-Factors for Natural Gases Using Equations of State,” J. Can. Pet. Tech., 14: 34–36 (July–September 1975).

8. Geertsma, J., “Estimating the Coefficient of Inertial Resistance in Fluid Flow Through Porous Media,” SPE J., 14: 445 (1974).

9. Janicek, J.D., and Katz, D.L., “Applications of Unsteady State Gas Flow Calculations,” Paper presented at the University of Michigan Research Conference, June 20, 1955.

10. Jones, S.C., “Using the Inertial Coefficient, β, to Characterize Heterogeneity in Reservoir Rock,”

SPE Paper 16949, 1987.

11. Katz, D.L., Cornell, D., Kobayashi, R., Poettmann, F.H., Vary, J.A., Ellenbaas, J.R., and Weinang, C.F., Handbook of Natural Gas Engineering, McGraw-Hill, NY, 1959.

12. Lee, A. L., Gonzales, M. H., and Eakin, B. E., “The Viscosities of Natural Gases,” Trans. AIME, 237: 997–1000 (1966).

13. Li, D., and Engler, T. W., “Literature Review on Correlations of the Non-Darcy Coefficient,”

SPE Paper 70015, 2001.

14. Liu, X., Civan, F., and Evans, R.D., “Correlation of the non-Darcy flow coefficient,” JCPT, 43:

50 (1995).

15. McCain, William D, Jr., The Properties of Petroleum Fluids, 2nd edition, PennWell Publishing Co., Tulsa, OK, 1990.

16. Pascal, H., and Quillian, R. G., “Analysis of Vertical Fracture Length and non-Darcy Flow Coefficient Using Variable Rate Tests,” SPE Paper 9348, 1980.

17. Standing, M.B., “Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems,” SPE, Dallas (1977).

18. Standing, M.B., and Katz, D.L., “Density of Natural Gases,” Trans. AIME, 146: 140–149 (1942).

19. Tek, M.R., Coats, K.H., and Katz, D.L., “The Effect of Turbulence on Flow of Natural Gas Through Porous Reservoir,” J. Pet. Tech., 799 (July 1962).

20. Thauvin, F., and Mohanty, K.K., “Network Modeling of Non-Darcy Flow Through Porous Media,” Transport in Porous Media, 31(1): 19 (1998).

21. Wichert, E., and Aziz, K., “Calculation of Z’s for Sour Gases,” Hydrocarbon Processing, 51(5) (1972).

22. Wang, X., Thauvin, F., and Mohanty, K.K., “Non-Darcy Flow Through Anisotropic Porous Media,” Chem. Eng. Sci., 54: 1859 (1999).

23. Wang, X., “Pore-Level Modeling of Gas-Condensate Flow in Porous Media,” PhD dissertation, University of Houston (May 2000).

Username: & Book: Petroleum Production Systems, Second Edition. No part of any chapter or book may be reproduced or transmitted in any form by any means without the prior written permission for reprints and excerpts from the publisher of the book or chapter. Redistribution or other use that violates the fair use privilege under U.S. copyright laws (see 17 USC107) or that otherwise violates these Terms of Service is strictly prohibited. Violators will be prosecuted to the full extent of U.S.

Federal and Massachusetts laws.

Problems

4-1. For two natural gases, A and B, the only differences between their compositions are C₁ and C₂. Assume the composition in gas A: C₁ = 0.8, C₂ = 0.15 and gas B: C₁ = 0.85, C₂ = 0.1. The

pseudocritical pressure of gas A and gas B are 663 psi and 661.2 psi, respectively. What is the specific gravity of gas B if the gravity of gas A is 0.6815?

4-2. A closed container has 10 ft³ volume and is filled with a gas of γ_g = 0.7. The temperature of the container is 200°F and the pressure is 4000 psi. If the pressure in the container increases to 4100 psi at the same temperature, how would the volume change? What is the slope of the Standing-Katz correlation ∂Z/∂p_pr at this temperature?

4-3. For the gas compositions below and conditions in Appendix C, compute the gas density.

Methane = 0.875 Ethane = 0.075 Propane = 0.025

Case 1: Nitrogen = 0.025, other gases = 0

Case 2: Carbon dioxide = 0.025, other gases = 0 Case 3: Hydrogen sulfide = 0.025, other gases = 0

4-4. The pseudocritical temperature and pressure of a sour gas are 397°R and 720 psi, respectively.

To use the Standing-Katz graph, T_pc and p_pc are corrected to 367°R and 657.6 psi. What are the percentages of H₂S and CO₂ in this gas?

4-5. For the well described in Appendix C, plot the IPR curve for bottomhole pressure 500 < p_wf <

3500 at steady-state conditions, for both Darcy and Non-Darcy Flow [D = 7.6 × 10^–4 (Mscf/d)^–1)].

4-6. A well has stabilized gas production following Aronofsky and Jenkins’s relation. Assume the reservoir average pressure is 3000 psi, drainage area is 80 acres, and net thickness is 80 ft. The wellbore radius is 0.328 ft and the perforated thickness is 20 ft. The well produces 2 MMscf/d at p_wf

= 2000 psi and 5.2 MMscf/day at p_wf = 1000 psi. What is the permeability? Use γ_g = 0.7, μ_g = 0.01 for gas, and the skin factor is 5. Generate IPR curves for this well at the specified condition.

4-7. For a gas well at steady-state flow, use the following data to

a. generate the IPR equation in a standard quadratic format (ax² + bx + c = 0) for a gas well.

b. What is the gas flow rate if p_wf = 300 psi?

c. generate an IPR plot for this well.

d. Discussion: At what flow rate can the non-Darcy flow effect be neglected and why?

Reservoir: k = 0.17 md, h = 78 ft, p_e = 4350 psi, T = 180°F, skin factor = 5, r_e = 1000 ft, and r_w = 0.328 ft

Fluid: γ_g = 0.65, and at average condition, μ_g = 0.02 cp, Z = 0.95 Non-Darcy flow coefficient D = 10^–3

4-8. Assume a well in a gas reservoir is under pseudosteady state and the production rate is

following Equation (4-47). The reservoir average pressure decrease rate is 500 psi/year. The non-Darcy coefficient is 10^–3 (Mscf/d)^–1.

a. What is the cumulative production for 3 years if the skin factor is zero?

b. If the skin factor is 10, what is the cumulative production for 3 years?

c. At what flow rate is the non-Darcy flow effect on production reduction equal to the effect of skin factor of 10?

Use data in Appendix C. The bottomhole pressure is 3000 psi. r_e = 1490 ft. Ignore the transient to pseudosteady state process.

4-9. Calculate a cumulative production curve for the gas well in Appendix C, assuming infinite-acting behavior [Equation (4-80)] for 1 year. Bottomhole pressure is 2000 psi. What fraction of the original gas-in-place will be produced if A = 4000 acres?