A new mathematical model of pressure buildup analysis for a well produced at constant bottomhole pressure

Abstract

This paper presents a computationally efficient and convenient method with simple procedures to forecast the pressure buildup if a well produced at constant bottomhole pressure prior to shut-in in a closed rectangular-shaped reservoir. The proposed algorithm is based on an analytical model with numerical approximation, and the pressure buildup is predicted through a series of mathematical methods such as the Laplace transform, the Green’s function and the superposition principle. The proposed model is validated by the Computer Modeling Group (CMG) simulation, the results show that the proposed model is accurate enough to predict the pressure buildup behavior of a well produced at constant bottomhole pressure prior to shut-in in a closed rectangular reservoir. The conventional models presented in the literature are mostly empirical or semi-analytical, which are not grounded in fundamental theory. Our proposed model has a solid theoretical basis, it provides a computationally efficient and convenient method for predicting pressure buildup behavior of a well produced at constant bottomhole pressure prior to shut-in in a closed rectangular reservoir. We conclude that the reservoir boundary condition, the reservoir size, the well location, the production pressure difference and the well production time prior to shut-in have significant effects on the pressure buildup behavior.

Cited as:

Lu, J., Nie, Q., Yang, E. A new mathematical model of pressure buildup analysis for a well produced at constant bottomhole pressure. Computational Energy Science, 2024, 1(3): 138-149. https://doi.org/10.46690/compes.2024.03.02

References:

Bossie-Codreanu, D.  A simple buildup analysis method to determine well drainage area and drawdown pressure for a stabilized well. Paper SPE 15977 Presented at Formation Evaluation,  September, 1989.

Camacho-V, R., Galindo-N, A., Prats, M.   Buildup solution in a system with multiple wells producing at constant wellbore pressures. Paper SPE 79104 Presented at Reservoir Evaluation and Engineering,  August, 2002.

Chen, H. K., Brigham, W. E.   Pressure buildup for a well with storage and skin in a closed square. Journal of Petroleum Technology, 1978, 30(1): 141-146.

 Clegg, M. W.  Some approximate solutions of radial flow problems associated with production at constant well pressure.  Paper SPE 1536 Presented at Society of Petroleum Engineers Journal, March, 1967.

Cobb, W. M.,   Smith, J. T. An Investigation of pressure-buildup tests in bounded Reservoirs.  Paper SPE 5133 Presented at the Fall Meeting of the Society of Petroleum Engineers of AIME, Houston, Texas,   6–9 October, 1974.

Cole, K. D., Beck, J. V.,   Haji-Sheikh, A. Heat conduction using green’s functions. New York City, USA, CRC Press Taylor and Francis Group,  2011.

Deng, Q., Nie, R., Jia, Y., et al.   A new method of pressure buildup analysis for a well in a multiwell reservoir. Paper SPE 175866 Presented  at the SPE North Africa Technical Conference and Exhibition, Cairo, Egypt,   14–16 September, 2015.

Dietz, D. N.  Determination of average reservoir pressure from build-up surveys.  Paper SPE 1156 Presented at Journal of Petroleum Technology,   August, 1965.  

Ehlig-Economides, C. A., Ramey, H. J. Jr. Pressure buildup for wells produced

at a constant pressure. Paper SPE  7985 Presented at Society of Petroleum Engineers Journal,  February, 1981.

Guppy, K. H., Cinco-Ley, H., Ramey Jr, H. J.   Pressure buildup analysis of fractured wells producing at high flow rates. Paper SPE 10178  Presented at Journal of Petroleum Technology, November, 1982.

Horner, D. R. Pressure buildup in wells. Paper WPC 4135 Presented  at the 3rd World Petroleum Congress, The Hague, the Netherlands, May 28–June 6, 1951.

Hurst, W.   Establishment of the skin effect and its impediment to fluid flow into a well bore. Petroleum Engineer, 1953, 25(11): B6-B16.

Kazemi, H.  Determining average reservoir pressure from pressure buildup tests. Paper SPE 4052 Presented at Society of Petroleum Engineers  Journal,  1974.

Kutasov, I. M.   Application of the Horner method for a well produced at a constant bottomhole pressure. Paper SPE 15143 Presented at SPE Formation Evaluation, 1989.

Lee, J., Rollins, J. B., Spivey, J. P. Pressure transient testing. Society of Petroleum Engineers, Richardson, Texas, USA, 2003.

Lin, J.,  Yang, H.  Pressure buildup analysis for a well in a closed, bounded multiwell reservoir. Chinese Journal of Chemical Engineering,  2005, 13(4):  441-450.

Lu, J., Owayed, J. F., Xu, J., et al.  An analytical model on production performance of multiple wells producing at constant bottomhole pressures. Special Topics and Reviews in Porous Media-An International Journal,  2019, 10(1): 31-48.

Lu, J., Shi, S., Rahman, M. M. New mathematical models for production performance of a well producing at constant bottomhole pressure. Special Topics and Reviews in Porous Media-An International Journal, 2018, 9(3): 261-278.

Marhaendrajana, T.,  Blasingame, T. A.   Decline curve analysis using type curves-evaluation of well performance behavior in a multiwell reservoir system.  Paper SPE 71517 Presented  at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 30–October 3, 2001.  

Matthews, C. S., Brons, F.,   Hazebroek, P.  A method for determination of average pressure in a bounded reservoir. Paper SPE 296 Presented at Transactions of the AIME,  December, 1954.

Matthews, C. S. Analysis of pressure buildup and flow test data.  Paper SPE 1631 Presented at Journal of Petroleum Technology, September, 1961.

Miller, C. C., Dyes, A. B.,   Hutchinson Jr, C. A.   The estimation of permeability and reservoir pressure from bottom hole pressure build-up characteristics. Paper SPE 950091 Presented at Journal of Petroleum Technology, December, 1950.

Muskat, M.  Use of data oil the build-up of bottom-hole pressures. Paper SPE 937044 Presented at Transactions of the AIME,  December, 1937.  

Ohaeri, C. U.  Pressure buildup analysis for a well produced at a constant pressure in a naturally fractured reservoir. Paper SPE 12009 Presented at the SPE Annual Technical Conference and Exhibition, San Francisco, California, 5–8 October, 1983.

Olarewaju, J., Lee, W. J.   Pressure buildup behavior of partially completed wells in layered reservoirs. Paper SPE 18876 Presented at the SPE Production Operations Symposium, Oklahoma City, Oklahoma, 13–14 March, 1989.

Onur, M., Serra, K.V., Reynolds, A. C.   Analysis of pressure buildup data from a

well in mutiwell system. Paper SPE  18123 Presented at SPE Formation Evaluation,  March, 1991.

Prats, M., Meneses-P, L. L., Samaneigo-V, F., et al. Pressure buildup in a well produced at constant pressure. Paper SPE 201111 Presented at SPE Journal,  December, 2020.

Ramey, H. J. Jr., Cobb, W. M. A general pressure buildup theory for a well in a

closed drainage area.  Paper SPE 3012 Presented at Journal of Petroleum Technology,   

December, 1971.

Slider, H. C.  A simplified method of pressure buildup analysis for a stabilized well. Paper SPE 3335 Presented at Journal of Petroleum Technology,   September, 1971.

Stakgold, I. Green’s functions and boundary value problems. Academic Press Inc., San Diego, California, USA, 1998.

Stehfest, H. Numerical inversion of laplace transforms. Communications of Association for Computing Machinery,  1970, 13(1): 47-49.

Streltsova, T. D. Buildup analysis for interference tests in stratified formations.   Paper SPE 10265 Presented at Journal of Petroleum Technology, February, 1984.

Theis, C. V.  The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground‐water storage. Eos, Transactions American Geophysical Union, 1935, 16(2): 519-524.

Tuma, J. J. Engineering mathematics handbook.  New York City, USA, McGraw-Hill Professional, 1997.

Umnuayponwiwat, S.,   Ozkan, E. Pressure transient behavior and inflow performance of multiple wells in closed systems. Paper SPE 62988 Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas,  1–4 October, 2000.

Uraiet, A. A., Raghavan, R. Pressure buildup analysis for a well produced at constant bottomhole pressure. Paper SPE  7984 Presented at Journal of Petroleum Technology,   October, 1980.   

Van Everdingen, A. F., Hurst, W. The application of the Laplace transformation to flow problems in reservoirs. Paper SPE  949305 Presented at Journal of Petroleum Technology,  December, 1949.

Van Everdingen, A. F. The skin effect and its influence on the productive capacity of a well.  Paper SPE 203 Presented at Journal of petroleum technology, June, 1953.

Zwillinger, D.  Standard mathematical tables and formulae.  New York City, USA, CRC Press Taylor and Francis Group, 1996.