# American Institute of Mathematical Sciences

September  2015, 20(7): 2171-2185. doi: 10.3934/dcdsb.2015.20.2171

## The modeling error of well treatment for unsteady flow in porous media

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, China

Received  August 2014 Revised  November 2014 Published  July 2015

In petroleum engineering, the well is usually treated as a point or line source, since its radius is much smaller than the scale of the whole reservoir. In this paper, we consider the modeling error of this treatment for unsteady flow in porous media.
Citation: Ting Zhang. The modeling error of well treatment for unsteady flow in porous media. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2171-2185. doi: 10.3934/dcdsb.2015.20.2171
##### References:
 [1] L. O. Aleksandrovna, S. V. Alekseevich and U. N. Nikolaevna, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968. [2] D. Alain and L. Ta-Tsien, Comportement limite des solutions de certains problèmes mixtes pour les équations paraboliques, J. Math. Pures Appl., 61 (1982), 113-130. [3] D. Alain and L. Ta-Tsien, Comportement limite des solutions de certains problèmes mixtes pour des équations paraboliques (II), Acta Math. Sci., 2 (1982), 85-104. [4] R. A. Alessandra, Hölder regularity results for solutions of parabolic equations, in Variational Analysis and Applications, Nonconvex Optim. Appl., 79, Springer, New York, 2005, 921-934. doi: 10.1007/0-387-24276-7_53. [5] L. Daqian, Z. Songmu, T. Yongji, S. Hanji, G. Ruxi and S. Weixi, Boundary value problmes with equivalued surface boundary conditions for self-adjoint elliptic differential equations I, Journal of Fudan University, (1976), 61-71. [6] L. Daqian, Z. Songmu, T. Yongji, S. Hanji, G. Ruxi and S. Weixi, Boundary value problmes with equivalued surface boundary conditions for self-adjoint elliptic differential equations II, Journal of Fudan University, (1976), 136-145. [7] L. P. K. Daniel and E. Lars, Numerical solution of first-kind Volterra equations by sequential Tikhonov regularization, SIAM J. Numer. Anal., 34 (1997), 1432-1450. doi: 10.1137/S003614299528081X. [8] L. P. K. Daniel, A survey of regularization methods for first-kind Volterra equations, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 53-82. [9] A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781420010558. [10] P. H. Valvatne, J. Serve, L. J. Durlofsky and K. Aziza, Efficient modeling of nonconventional wells with downhole inflow control devices, Journal of Petroleum Science and Engineering, 39 (2003), 99-116. doi: 10.1016/S0920-4105(03)00042-1. [11] S. Weixi, On mixed initial-boundary value problmes of second order parabolic equations with equivalued surface boundary conditions, Journal of Fudan University, (1978), 15-24. [12] C. Zhangxin, H. Guanren and M. Yuanle, Computational Science and Engineering, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006. [13] C. Zhiming and Y. Xingye, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media, Multiscale Model. Simul., 1 (2003), 260-303. doi: 10.1137/S1540345902413322.

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##### References:
 [1] L. O. Aleksandrovna, S. V. Alekseevich and U. N. Nikolaevna, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968. [2] D. Alain and L. Ta-Tsien, Comportement limite des solutions de certains problèmes mixtes pour les équations paraboliques, J. Math. Pures Appl., 61 (1982), 113-130. [3] D. Alain and L. Ta-Tsien, Comportement limite des solutions de certains problèmes mixtes pour des équations paraboliques (II), Acta Math. Sci., 2 (1982), 85-104. [4] R. A. Alessandra, Hölder regularity results for solutions of parabolic equations, in Variational Analysis and Applications, Nonconvex Optim. Appl., 79, Springer, New York, 2005, 921-934. doi: 10.1007/0-387-24276-7_53. [5] L. Daqian, Z. Songmu, T. Yongji, S. Hanji, G. Ruxi and S. Weixi, Boundary value problmes with equivalued surface boundary conditions for self-adjoint elliptic differential equations I, Journal of Fudan University, (1976), 61-71. [6] L. Daqian, Z. Songmu, T. Yongji, S. Hanji, G. Ruxi and S. Weixi, Boundary value problmes with equivalued surface boundary conditions for self-adjoint elliptic differential equations II, Journal of Fudan University, (1976), 136-145. [7] L. P. K. Daniel and E. Lars, Numerical solution of first-kind Volterra equations by sequential Tikhonov regularization, SIAM J. Numer. Anal., 34 (1997), 1432-1450. doi: 10.1137/S003614299528081X. [8] L. P. K. Daniel, A survey of regularization methods for first-kind Volterra equations, in Surveys on Solution Methods for Inverse Problems, Springer, Vienna, 2000, 53-82. [9] A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781420010558. [10] P. H. Valvatne, J. Serve, L. J. Durlofsky and K. Aziza, Efficient modeling of nonconventional wells with downhole inflow control devices, Journal of Petroleum Science and Engineering, 39 (2003), 99-116. doi: 10.1016/S0920-4105(03)00042-1. [11] S. Weixi, On mixed initial-boundary value problmes of second order parabolic equations with equivalued surface boundary conditions, Journal of Fudan University, (1978), 15-24. [12] C. Zhangxin, H. Guanren and M. Yuanle, Computational Science and Engineering, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006. [13] C. Zhiming and Y. Xingye, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media, Multiscale Model. Simul., 1 (2003), 260-303. doi: 10.1137/S1540345902413322.
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