Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics

ABSTRACT

For bottom water reservoir and the reservoir with a thick oil formation, there exists partial penetration completion well and when the well products the oil flow in the porous media takes on spherical percolation. The nonlinear spheri-cal flow equation with the quadratic gradient term is deduced in detail based on the mass conservation principle, and then it is found that the linear percolation is the approximation and simplification of nonlinear percolation. The nonlinear spherical percolation physical and mathematical model under different external boundaries is established, considering the ef-fect of wellbore storage. By variable substitu-tion, the flow equation is linearized, then the Laplace space analytic solution under different external boundaries is obtained and the real space solution is also gotten by use of the nu-merical inversion, so the pressure and the pressure derivative bi-logarithmic nonlinear spherical percolation type curves are drawn up at last. The characteristics of the nonlinear spherical percolation are analyzed, and it is found that the new nonlinear percolation type curves are evidently different from linear per-colation type curves in shape and characteris-tics, the pressure curve and pressure derivative curve of nonlinear percolation deviate from those of linear percolation. The theoretical off-set of the pressure and the pressure derivative between the linear and the nonlinear solution are analyzed, and it is also found that the in-fluence of the quadratic pressure gradient is very distinct, especially for the low permeabil-ity and heavy oil reservoirs. The influence of the non-linear term upon the spreading of pressure is very distinct on the process of percolation, and the nonlinear percolation law stands for the actual oil percolation law in res-ervoir, therefore the research on nonlinear per-colation theory should be strengthened and reinforced.

For bottom water reservoir and the reservoir with a thick oil formation, there exists partial penetration completion well and when the well products the oil flow in the porous media takes on spherical percolation. The nonlinear spheri-cal flow equation with the quadratic gradient term is deduced in detail based on the mass conservation principle, and then it is found that the linear percolation is the approximation and simplification of nonlinear percolation. The nonlinear spherical percolation physical and mathematical model under different external boundaries is established, considering the ef-fect of wellbore storage. By variable substitu-tion, the flow equation is linearized, then the Laplace space analytic solution under different external boundaries is obtained and the real space solution is also gotten by use of the nu-merical inversion, so the pressure and the pressure derivative bi-logarithmic nonlinear spherical percolation type curves are drawn up at last. The characteristics of the nonlinear spherical percolation are analyzed, and it is found that the new nonlinear percolation type curves are evidently different from linear per-colation type curves in shape and characteris-tics, the pressure curve and pressure derivative curve of nonlinear percolation deviate from those of linear percolation. The theoretical off-set of the pressure and the pressure derivative between the linear and the nonlinear solution are analyzed, and it is also found that the in-fluence of the quadratic pressure gradient is very distinct, especially for the low permeabil-ity and heavy oil reservoirs. The influence of the non-linear term upon the spreading of pressure is very distinct on the process of percolation, and the nonlinear percolation law stands for the actual oil percolation law in res-ervoir, therefore the research on nonlinear per-colation theory should be strengthened and reinforced.

Cite this paper

Nie, R. and Ding, Y. (2010) Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics.*Natural Science*, **2**, 98-105. doi: 10.4236/ns.2010.22016.

Nie, R. and Ding, Y. (2010) Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics.

References

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[2] Odeh, A.S and Babu, D.K. (1998) Comprising of solu-tions for the nonlinear and linearized diffusion equations. SPE Reservoir Engineering, 3(4), 1202-1206.

[3] Bai, M.Q. and Roegiers, J.C. (1994) A nonlinear dual porosity model. Appl Math Moclellin, 18(9), 602-610.

[4] Wang, Y. and Dusseault, M.B. (1991) The effect of quad-ratic gradient terms on the borehole solution in poroelas-tic media. Water Resource Research, 27(12), 3215-3223.

[5] Chakrabarty, C., Farouq, A.S.M. and Tortike, W.S. (1993) Analytical solutions for radial pressure distribution in-cluding the effects of the quadratic gradient term. Water Resource Research, 29(4), 1171-1177.

[6] Braeuning, S., Jelmert, T.A. and Sven, A.V. (1998) The effect of the quadratic gradient term on variable-rate well-tests, Journal of Petroleum Science and Engineering, 21(2), 203-222.

[7] Tong, D.K. (2003) The fluid mechanics of nonlinear flow in porous media. Beijing: Petroleum Industry Press in Chinese, (in Chinese).

[8] Tong, D.K., Zhang, Q.H. and Wang, R.H. (2005) Exact solution and its behavior characteristic of the nonlinear dual-porosity model. Applied Mathematic and Mechanics, 26(10), 1161-1167, (in Chinese).

[9] William, E.B., James, M. and Peden, K.F.N. (1980) The analysis of spherical flow with wellbore storage. SPE 9294.

[10] Charles, A.K. and William, A.A. (1982) Application of linear spherical flow analysis techniques to field prob-lems-case studies. SPE 11088.

[11] Mark, A. and Proett, W.C.C. (1998) New exact spherical flow solution with storage and skin for early-time inter-pretation with applications to wireline formation and early-evaluation drillstem testing. SPE 49140.

[12] Joseph, J.A. and Koederitz, L.F. (1985) Unsready-state spherical flow with storage and skin. SPEJ, 25(6), 804-822.

[13] Stehfest H. (1970) Numerical inversion of Laplace transform algorithm 368, Communication of the ACM, 13(1), 47-49.

[1] Yan, B.S. and Ge, J.L. (2003) New advances of modern reservoir and fluid flow in porous media. Journal of Southwest Petroleum Institute, 25(1), 29-32, (in Chi-nese).

[2] Odeh, A.S and Babu, D.K. (1998) Comprising of solu-tions for the nonlinear and linearized diffusion equations. SPE Reservoir Engineering, 3(4), 1202-1206.

[3] Bai, M.Q. and Roegiers, J.C. (1994) A nonlinear dual porosity model. Appl Math Moclellin, 18(9), 602-610.

[4] Wang, Y. and Dusseault, M.B. (1991) The effect of quad-ratic gradient terms on the borehole solution in poroelas-tic media. Water Resource Research, 27(12), 3215-3223.

[5] Chakrabarty, C., Farouq, A.S.M. and Tortike, W.S. (1993) Analytical solutions for radial pressure distribution in-cluding the effects of the quadratic gradient term. Water Resource Research, 29(4), 1171-1177.

[6] Braeuning, S., Jelmert, T.A. and Sven, A.V. (1998) The effect of the quadratic gradient term on variable-rate well-tests, Journal of Petroleum Science and Engineering, 21(2), 203-222.

[7] Tong, D.K. (2003) The fluid mechanics of nonlinear flow in porous media. Beijing: Petroleum Industry Press in Chinese, (in Chinese).

[8] Tong, D.K., Zhang, Q.H. and Wang, R.H. (2005) Exact solution and its behavior characteristic of the nonlinear dual-porosity model. Applied Mathematic and Mechanics, 26(10), 1161-1167, (in Chinese).

[9] William, E.B., James, M. and Peden, K.F.N. (1980) The analysis of spherical flow with wellbore storage. SPE 9294.

[10] Charles, A.K. and William, A.A. (1982) Application of linear spherical flow analysis techniques to field prob-lems-case studies. SPE 11088.

[11] Mark, A. and Proett, W.C.C. (1998) New exact spherical flow solution with storage and skin for early-time inter-pretation with applications to wireline formation and early-evaluation drillstem testing. SPE 49140.

[12] Joseph, J.A. and Koederitz, L.F. (1985) Unsready-state spherical flow with storage and skin. SPEJ, 25(6), 804-822.

[13] Stehfest H. (1970) Numerical inversion of Laplace transform algorithm 368, Communication of the ACM, 13(1), 47-49.