When analyzing well performance, resource-intensive hydrodynamic models are often used, the alternative to which are simple analytical models. To build an accurate hydrodynamic model, numerical calculations require correct initial data, which may not be available, and large computing power, so the use of such a model is not always justified. On the other hand, the analytical approach, having a high speed of calculation, does not consider a few parameters of the system under study. In the simplest cases, a homogeneous isotropic reservoir with single-phase filtration is considered. The Green's function for an infinite flat homogeneous isotropic reservoir can be given as an example of solving a homogeneous problem. This approach is not always acceptable from the point of view of practical application; at least it is necessary to model a finite heterogeneous reservoir. There is also a class of inverse problems of well hydrodynamic studies, dynamics adaptation and similar tasks, where both high speed of calculations and consideration of many peculiarities of the considered domain are required, but existing commercial software and analytical approaches cannot always satisfy these conditions for the reasons mentioned above.
The article consider an approach that incorporates the advantages of both numerical and analytical approaches in modeling filtration and well performance. The idea is to numerically search for a correction term to the simplest analytical models of wells and fractures to account for the inhomogeneity of the filtration region. The correction term includes the physical and capacitive properties of the formation and considers the boundary conditions, which allows us to significantly accelerate complex hydrodynamic calculations. Based on this approach, a program is implemented that promptly calculates well productivity in heterogeneous formations and calculates the matrix of mutual productivities to evaluate well performance.
References
1. Basquet R. et al., A semi-analytical approach for productivity evaluation of wells with complex geometry in multilayered reservoirs, SPE 49232-MS, 1998,
DOI: https://doi.org/10.2118/49232-MS
2. Blasingame T., Shahram A., Rushing J., Evaluation of the elliptical flow period for hydraulically-fractured wells in tight gas sands - Theoretical aspects and practical considerations // SPE-106308-MS, 2007, DOI: http://doi.org/10.2118/106308-MS
3. Henk A. vander Vorst, Iterative Krylov methods for large linear systems, Cambridge University Press, 2003, 230 p.
4. Kikani J., Modeling pressure-transient behavior of sectionally homogeneous reservoirs by boundary-element method, SPE-19778-PA, 1993, DOI: https://doi.org/10.2118/19778-PA
5. Kuchuk F.J., Habashy T., Pressure behavior of laterally composite reservoirs, SPE-24678-PA, 1998, DOI: https://doi.org/10.2118/24678-PA
6. Levitan M.M., Crawford G.E., General heterogeneous radial and linear models for well-test analysis, SPE-78598-PA, 2002, DOI: http://doi.org/10.2118/78598-PA
7. Jin Y., Chen K.P., Chen M., Analytical solution and mechanisms of fluid production from hydraulically fractured wells with finite fracture conductivity, Journal of Engineering Mathematics, 2015, V. 92, pp. 103–122, DOI: http://doi.org/10.1007/s10665-014-9754-x
8. Yudin E., Gubanova A., Krasnov V., The method of express estimation of pore pressure map distribution in reservoirs with faults and wedging zones, SPE-191582-18RPTC-MS, 2018, DOI: http://doi.org/10.2118/191582-18RPTC-MS
9. Yudin E., Lubnin A. et al., Differential approach to determination of compartmentalized reservoir properties, SPE-161969-MS, 2012, http://doi.org/10.2118/161969-MS
10. Yudin E., Poroshin P., Korikov D. et al., Analysis and prediction of well performance in heterogeneous reservoirs based on field theory methods, SPE-201955-MS, 2020, http://doi.org/10.2118/201955-MS
11. Oliver D.S., The averaging process in permeability estimation from well test data, SPE-19845-PA, 1990, DOI: https://doi.org/10.2118/19845-PA
12. Il’in A.M., A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. The two-dimensional case (In Russ.), Matematicheskiy sbornik = Mathematics of the USSR-Sbornik, 1976, V. 99(141), DOI: https://doi.org/10.1070/sm1976v028n04abeh001663
13. Il’in E.M., Features of weak solutions of elliptic theory with discontinuous leading coefficients. II. Corner points of the break line (In Russ.), Zapiski nauchnogo seminara LOMI, 1974, V. 47, pp. 166–169.
14. Ladyzhenskaya O.A., Ural’tseva N.N., Lineynye i kvazilineynye uravneniya ellipticheskogo tipa (Linear and quasilinear equations of elliptic type), Moscow: Nauka Publ., 1973, 576 p.
15. Nazarov S.A., Plamenevskiy B.A., Ellipticheskie zadachi v oblastyakh s kusochno-gladkoy granitsey (Elliptic problems in domains with piecewise smooth boundaries), Moscow: Nauka Publ., 1991, 335 p.
16. Oganesyan L.A., Rukhovets L.A., Variational-difference schemes for linear second-order elliptic equations in a two-dimensional region with piecewise smooth boundary (In Russ.), Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki = USSR Computational Mathematics and Mathematical Physics, 1968, no. 8:1, pp. 97–114,
DOI: https://doi.org/10.1016/0041-5553(68)90008-6
17. Izmailov A.F., Solodov M.V., Chislennye metody optimizatsii (Numerical optimization methods), Moscow: FIZMATLIT Publ., 2005. - 304 с.
18. Prats M., Hazebroek P., Strickler W.R., Effect of vertical fractures on reservoir behavior – Compressible-fluid case, SPE-98-PA, 1962,
DOI: https://doi.org/10.2118/98-PA
19. Ramey H.J., Approximate solutions for unsteady liquid flow in composite reservoirs, JCPT, 1970, 70-01-04, DOI: https://doi.org/10.2118/70-01-04
20. Rosa A.J. et al., Pressure transient behavior in reservoirs with an internal circular discontinuity, SPE-26455-PA, 1996,DOI: https://doi.org/10.2118/26455-PA
