Interpretation of well testing on unsteady state is based on the idealized case of a single well operating at a constant rate. However, in practice, the flow rate is not constant, in particular due to the difficulty of maintaining a constant rate. If do not take into account the changes in the flow rate before the research, this will cause inaccuracies in the calculations and errors in the analysis of the well test results. Traditionally, the transition from modeling the well production process with a constant flow rate to modeling with variable rates is carried out using the principle of superposition over time. The algorithm for calculating pressure using the superposition principle has a linear complexity, where the most resource-intensive element is the calculation of the differential pressure function. In addition, the calculation can take considerable time when calculating a complex analytical model ‘well – reservoir – boundary’ for cases of well production history with tens and hundreds rates. Such cases make these calculations unsuitable for engineering practice.
The method is proposed to speed up of a bottomhole well flowing pressure calculation with variable rates according to the production history in this paper. The proposed method is based on the use of the pressure drop approximation function, which is a fourth-order polynomial of the logarithm of time. The coefficients of the polynomial are found by using the Levenberg – Marquardt optimization method. The comparative analysis of speed and quality of bottomhole well flowing pressure calculation for three models is given by the number of analytical model function calls, the calculation time, the maximum relative deviation, the average quadratic deviation of the relative error. It is shown that the calculated pressure values obtained by the traditional and proposed method (using an approximating polynomial function) for models of a vertical well, a horizontal well and a horizontal well with multi-stage hydraulic fracturing in an infinite homogeneous reservoir are identical. The proposed method allows to significantly increase the calculation speed with minimal calculation error.
References
1. Walker A.C., Estimating reservoir pressure using the principle of superposition, SPE-2324-MS, 1968, DOI: https://doi.org/10.2118/2324-MS
2. Cinco-Ley H., Samaniego F., Use and misuse of the superposition time function in well test analysis, SPE-19817-MS, 1989, DOI: https://doi.org/10.2118/19817-MS
3. Earlougher R.C.Jr., Advances in well test analysis, SPE Monograph Series, 1977, V. 5, 264 p.
4. Stewart G., Well test design and analysis, PennWell Corp., 2011, pp. 52–82.
5. Van Everdingen A.F., Hurst W., The application of the Laplace transformation to flow problems in reservoir, AIME, 1949, V. 186, pp. 305–324.
6. Collins R.E., Flow of fluids through porous materials, Reinhold Publishing Corp., 1961, pp. 108–123.
7. Roweis S., Levenberg-Marquardt optimization, URL: https://cs.nyu.edu/~roweis/
8. Ozkan E., Raghavan R., New solutions for well-test analysis problems, Parts 1 & 2, SPE-28424-MS, 1991, DOI:10.2118/28424-MS
9. Yao Shanshan, Wang Xiangzeng, Li Min, Ju Ning, A composite model for multi-stage fractured horizontal wells in heterogeneous reservoirs, SPE-182016-MS, 2016, DOI: 10.2118/182016-MS
