RESTORATION OF PRESSURE AT THE POOL BOUNDARY BASED ON THE SOLUTION OF INVERSE PROBLEM

S.O. Huseynzade, Cand. Sc. (Phys.-Math.)
Azerbaijan State \University of Oil and Industry
20 Azadlyg Ave., Baku, AZ 1010, Azerbaijan
e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Èlektron. model. 2018, 40(4):19-28
https://doi.org/10.15407/emodel.40.04.019

ABSTRACT

A numerical method has been proposed to solve the inverse problem of determining conditions on the outer pool boundary based on information obtained from the hole. The rectilinear-parallel flow of a single-phase fluid in a rectangular pool described by a linear parabolic equation is considered. The initial state of the pool, as well as the pressure and flow rate of liquid in the gallery of production wells are considered to be set, and the pressure on the outer boundary of the pool is unknown. This problem belongs to the class of boundary inverse problems. First, the nonlocal perturbation method of the boundary condition is applied. After that, the problem is discretized and a special representation is proposed to solve the resulting system of difference equations. As a re- sult, the problem is reduced to two difference problems and one linear equation with respect to the approximate value of pressure at the boundary of the pool. On the basis of the proposed computational algorithm, numerical calculations for model problems are carried out.

KEYWORDS

oil pool, rectilinear-parallel flow, boundary inverse problem, nonlocal perturba- tion method, difference method.

REFERENCES

  1. Alifanov, O.M., Artyukhin, E.A. and Rumyantsev, S.V. (1988), Ekstremalnyie metody resheniya nekorrektnykh zadach [Extreme methods for solution of incorrect problems], Nauka, Moscow, Russia.
  2. Basniev, S., Dmitriev, N.M. and Rosenberg, G.D. (2005), Neftegazovaya gidromekhanika [Oil and gas hydromechanics], Izhevsky institut kompyuternykh issledovaniy, Moscow, Russia.
  3. Kanevskaya, D. (2002), Matematicheskoe modelirovanie gidrodinamicheskikh protsessov razrabotki mestorozhdeniy uglevodorodov [Mathematical modeling of hydrodynamic pro- cesses in hydrocarbon field development], Izhevsky institut kompyuternykh issledovaniy, Moscow, Russia.
  4. Aziz, and E. Settari, E. (2004), Matematicheskoe modelirovanie plastovykh system [Mathematical modeling of reservoir systems], Izhevsky institut kompyuternykh issle- dovaniy, Moscow, Russia.
  5. Alifanov, M. (1988), Obratnyie zadachi teploobmena [Inverse heat transfer problems], Mashinostroenie, Moscow, USSR. Vestnik  Yugorskogo Universiteta, Vol. 46, Iss. 3, pp. 51-59.
  6. Kerimov, N.B. and Ismailov, M.I. (2012), An inverse coefficient problem for the heat equa- tion in the case of nonlocal boundary conditions, J. of Mathematical Analysis and Applica- tions, Vol. 396, no. 2, pp. 546-554. https://doi.org/10.1016/j.jmaa.2012.06.046
  7. Tikhonov N. and Arsenin, V.Ya. (1986), Metody resheniya nekorrektnykh zadach [Meth- ods for solution of incorrect problems], Nauka, Moscow, Russia.
  8. Kabanikhin, I. (2009), Obratnyie i nekorrektnyie zadachi [Inverse and incorrect prob- lems], Sibirskoe nauchnoye izdatelstvo, Novosibirsk, Russia.
  9. Samarskiy, A. and Vabischevich, P.N. (2009), Chislennyie metody resheniya obratnykh zadach matematicheskoi fiziki [Numerical methods for solution of inverse problems of mathe- matical physics], Izdatelstvo LKI, Moscow, Russia.
  10. Yaparova, M. (2013), “Numerical modeling of solutions of the inverse boundary value problem of heat conductivity”, Vestnik YuUrGU. Seriya Matematicheskoye modelirovanie i programmirovanie, Vol. 6, Iss. 3, pp. 112-124.
  11. Sabitov, B. (2016), Pryamye i obratnye zadachi dlya uravneniy smeshannogo parabolo- giperbolicheskogo tipa [Direct and inverse problems for the equations of mixed parabolic- hyperbolic type], Nauka, Moscow, Russia.
  12. Korotky, I. and Starodubtseva, Yu.V. (2015), Modelirovanie pryamykh i obratnykh gra- nichnykh zadach dlya statsionarnykh modelei teplomassoperenosa [Simulation of direct and inverse boundary value problems for stationary models of heat and mass transfer], Izda- telstvo Uralskogo Universiteta, Ekaterinburg, Russia.
  13. Verzhbitsky, A. (2017), “Inverse problems on the definition of boundary regimes”,

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