The Construction Method for Solving Radial Flow Problem through the Homogeneous Reservoir

Affiliation(s)

Institute of Applied Mathematics, Xihua University, Chengdu, China.

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China.

Institute of Applied Mathematics, Xihua University, Chengdu, China.

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China.

ABSTRACT

On the basis of similar structure of solutions of ordinary differential equation (ODE) boundary value problem, the similar construction method was put forward by solving problems of fluid flow in porous media through the homogeneous reservoir. It is indicate that the pressure distribution of dimensionless reservoir and bottom hole in Laplace space, which take on the radial flow, also shows similar structure, and the internal relationship between the above solutions were illustrated in detail.

On the basis of similar structure of solutions of ordinary differential equation (ODE) boundary value problem, the similar construction method was put forward by solving problems of fluid flow in porous media through the homogeneous reservoir. It is indicate that the pressure distribution of dimensionless reservoir and bottom hole in Laplace space, which take on the radial flow, also shows similar structure, and the internal relationship between the above solutions were illustrated in detail.

Cite this paper

S. Li, W. Li, X. Li and L. Xu, "The Construction Method for Solving Radial Flow Problem through the Homogeneous Reservoir,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 517-522. doi: 10.4236/am.2012.36078.

S. Li, W. Li, X. Li and L. Xu, "The Construction Method for Solving Radial Flow Problem through the Homogeneous Reservoir,"

References

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[2] M. H. Jia and S. C. Li, “The Similar Structure of Solution Differential Equation on Boundary Value Problem,” College Mathematics, Vol. 21, No. 5, 2005, pp. 37-39.

[3] R. P.Agarwal and D. O’Regan, “Ordinary and Partial Differential Equations,” Springer, New York, 2009.

[4] D. L. Powers, “Boundary Value Problems,” 5th Edition, Academic Press, Elsevier, New York, 1999.

[5] M. A. Khamsi, “Homogeneous Linear Equations with Constant Coefficients,” 1992. http://www.sosmath.com/diffeq/second/constantcof/constantcof.html.

[6] P. S. Zheng, S. C. Li and Y. F. Zhang, “The Formal Similarity of Solutions on the Class of Ordinary Differential Equation System,” Journal of Jilin University (Information Science Edition), Vol. 23, No. 8, 2005, pp. 56-60.

[7] P. S. Zheng, S. C. Li and Y. F. Zhang, “The Solution’s Structure of a Type of Ordinary Differential Equation System with Closed Right Boundary Conditions,” Journal of Xihua University (Natural Science Edition), Vol. 24, No. 6, 2005, pp. 88-90.

[8] Z. C. Chen, P. H. Liu and S. C. Li, “The Similar Structure of Composite Bessel Equation on Fixed Solution Problem,” Journal of Chongqing Techno Business University (Natural Science Edition), Vol. 23, No. 1, 2006, pp. 1-4.

[9] P. H. Liu, Z. C. Chen and S. C. Li, “Similar Structure of Fixed Solution Problems for Composite Abnormal Bessel Equations,”Journal of Xihua University (Natural Science Edition), Vol. 25, No. 2, 2006, pp. 2326.

[10] S. C. Li, “The Similar Structure of Solution of Second-Order Linear Homogeneous Differential Equations with Constant Coefficients on the Boundary Value Problem,”Journal of Xihua University (Natural Science Edition), Vol. 26, No. 1, 2007, pp. 84-85.

[11] S. C. Li, “The Similar Structure of Solution to the Boundary Value Problem for Second-Order Linear Homogeneous Differential Equations,” Journal of Xihua University (Natural Science Edition), Vol. 28, No. 5, 2009, pp. 40-41.

[12] J. Yan, S. C. Li and C. L. Xing, “The Similar Structures of Solutions to First Class Boundary Value Problem of Second-Order Euler’s Equation,” Journal of Xihua University (Natural Science Edition), Vol. 28, No. 8, 2009, pp. 86-88.

[13] Q. Y. Li, S. C. Li and Y. Chi, “The Similar Structure of Solution of Linear Homogeneous Differential Equations with Constant Coefficients on the Boundary Value Problem,” Sichuan Ordnance Journal, Vol. 31, No. 4, 2010, pp. 126-129.

[14] S. C. Li, “Preliminary Exploration and Prospects of the Similar Structure of Solutions of Differential Equations,” Journal of Xihua University (Natural Science Edition), Vol. 29, No. 2, 2010, pp. 223-226.

[15] J. Kevorkian, “Partial Differential Equations: Analytical Solution Techniques,” 2nd edition, Springer, New York, 2000.

[16] A. D. Polyanin, “Handbook of Linear Partial Differential Equations for Engineers and Scientists,” Chapman & Hall/CRC Press, Boca Raton, 2002.

[17] A. Sommerfeld, “Partial Differential Equations in Physics,” Academic Press, New York, 1949.

[18] J. D. Tian and S. C. Li, “The Formal Similarity of Solutions in the Laplace Space on the Class of Quasilinear Partial Differential Equation,” Mathematical Theory and Applications, Vol. 24, No. 2, 2004, pp. 66-73.

[19] M. H. Jia and S. C. Li, “The Formal Similarity of Solutions in the Laplace Space on the Class of Fluid Flow Differential Equation,” Journal of Electronic Science and Technology of China, Vol. 3, No. 2, 2005, pp. 172-174.

[20] S. C. Li, “The Formal Similarity of Solutions in the Laplace Space on the Class of Partial Differential Equation System,” Journal of Xihua University (Natural Science Edition), Vol. 26, No. 4, 2007, pp. 83-86.

[21] J. P. Su, S. C. Li and C. J. Li, “The Similar of Solutions in the Laplace Space of Composite Parabolic Partial Differential Equation,” Journal of Zaozhuang University, Vol. 26, No. 2, 2009, pp. 6-11.

[22] S. C. Li, P. S. Zheng and Y. F. Zhang, “The Similar Structure of Pressure Distribution in the Homogenous Reservoir,” Pure and Applied Mathematics, Vol. 22, No. 4, 2006, pp. 459-463.

[23] W. Z. Xu, S. C. Li and P. S. Zheng, “The Structure of the Solution of Pressure Distribution in the Fractal Homogenous Reservoir and Analytic Graph,” Mathematics and its Applications, Atomic Energy Publishing Company, Beijing, 2007, pp. 541-544.

[24] S. C. Li, P. S. Zheng and Y. F. Zhang, “The Similar Structure of Pressure Distribution in the Composite Reservoir,” Journal of Mathematics in Practice and Theory, Vol. 38, No. 3, 2008, pp. 23-28.

[25] S. C. Li, P. S. Zheng and Y. F. Zhang, “Similar Structure of Pressure Distribution in the Multilayer Reservoir,” Applied Mathematics A Journal of Chinese Universities, Vol. 24, No. 2, 2009, pp. 234-238.

[26] S. C. Li, X. P. Li and B. G. Huang, “Research on Homogeneous Reservoir Pressure Performance,” Drilling & Production Technology, Vol. 25, No. 1, 2002, pp. 50-51.

[27] S. C. Li, D. H. Liu and P. Z. Zhang, “Pressure Distribution of the Homogeneous Reservoir with Different Boundary Conditions,” Journal of Jianghan Petroleum Institute, Vol. 23, No. 2, 2001, pp. 16-17.

[28] S. C. Li, P. S. Zheng and Y. F. Zhang, “The Similar Structure of Pressure Distribution in the Fractal Reservoir,” Resources, Environment & Porous Flow Mechanics, Proceedings of 8th Research Conference of Porous Flow Mechanics of China, China Scientific & Technical Publishers, Beijing, 2005, pp. 79-83.

[29] S. C. Li and J. J. Zhang, “Similar Structure of Pressure Distribution in the Fractal Dual Porosity Reservoir,” Journal of Xihua University (Natural Science Edition), Vol. 25, No. 1, 2006, pp. 40-43.

[30] C. X. Xu, S. C. Li and W. B. Zhu, “The Similar Structure of Pressure Distribution in the Dual Porosity Reservoir,” Drilling & Production Technology, Vol. 29, No. 4, 2006, pp. 28-30.

[31] C. X. Xu, S. C. Li and W. B. Zhu, “The Similar Structure of Pressure Distribution in the Fractal Composite Reservoir,” Drilling & Production Technology, Vol. 29, No. 5, 2006, pp. 39-42.

[32] P. S. Zheng, C. X. Xu and W. Z. Xu, “Well Analusis Method Based on the Similar Structure of Pressure Distribution in the Composite Reservior,” Drilling & Production Technology, Vol. 30, No. 3, 2007, pp. 49-50.

[33] W. B. Zhu, S. C. Li and C. X. Xu, “The Similar Structure of Pressure Distribution in the Fractal Multilayer Reservoir,” Drilling & Production Technology, Vol. 31, No. 3, 2008, pp. 67-69.

[34] P. S. Zheng, S. C. Li and W. B. Zhu, “The Similar Structure of Pressure Distribution in the Double Porosity Composite Reservoir,” Drilling & Production Technology, Vol. 31, No. 4, 2008, pp. 80-81.

[35] S. C. Li and B. G. Huang, “Laplace Transform and Bessel Functions and the Theoretical Basis of Well Test Analysis,” Petroleum Industry Press, Beijing, 2000.

[1] S. C. Li, L. Z. Yi and P. S. Zheng, “The Similar Structure of Differential Equations on Fixed Solution Problem,” Journal of Sichuan University (Natural Science Edition), Vol. 43, No. 4, 2006, pp. 933-934.

[2] M. H. Jia and S. C. Li, “The Similar Structure of Solution Differential Equation on Boundary Value Problem,” College Mathematics, Vol. 21, No. 5, 2005, pp. 37-39.

[3] R. P.Agarwal and D. O’Regan, “Ordinary and Partial Differential Equations,” Springer, New York, 2009.

[4] D. L. Powers, “Boundary Value Problems,” 5th Edition, Academic Press, Elsevier, New York, 1999.

[5] M. A. Khamsi, “Homogeneous Linear Equations with Constant Coefficients,” 1992. http://www.sosmath.com/diffeq/second/constantcof/constantcof.html.

[6] P. S. Zheng, S. C. Li and Y. F. Zhang, “The Formal Similarity of Solutions on the Class of Ordinary Differential Equation System,” Journal of Jilin University (Information Science Edition), Vol. 23, No. 8, 2005, pp. 56-60.

[7] P. S. Zheng, S. C. Li and Y. F. Zhang, “The Solution’s Structure of a Type of Ordinary Differential Equation System with Closed Right Boundary Conditions,” Journal of Xihua University (Natural Science Edition), Vol. 24, No. 6, 2005, pp. 88-90.

[8] Z. C. Chen, P. H. Liu and S. C. Li, “The Similar Structure of Composite Bessel Equation on Fixed Solution Problem,” Journal of Chongqing Techno Business University (Natural Science Edition), Vol. 23, No. 1, 2006, pp. 1-4.

[9] P. H. Liu, Z. C. Chen and S. C. Li, “Similar Structure of Fixed Solution Problems for Composite Abnormal Bessel Equations,”Journal of Xihua University (Natural Science Edition), Vol. 25, No. 2, 2006, pp. 2326.

[10] S. C. Li, “The Similar Structure of Solution of Second-Order Linear Homogeneous Differential Equations with Constant Coefficients on the Boundary Value Problem,”Journal of Xihua University (Natural Science Edition), Vol. 26, No. 1, 2007, pp. 84-85.

[11] S. C. Li, “The Similar Structure of Solution to the Boundary Value Problem for Second-Order Linear Homogeneous Differential Equations,” Journal of Xihua University (Natural Science Edition), Vol. 28, No. 5, 2009, pp. 40-41.

[12] J. Yan, S. C. Li and C. L. Xing, “The Similar Structures of Solutions to First Class Boundary Value Problem of Second-Order Euler’s Equation,” Journal of Xihua University (Natural Science Edition), Vol. 28, No. 8, 2009, pp. 86-88.

[13] Q. Y. Li, S. C. Li and Y. Chi, “The Similar Structure of Solution of Linear Homogeneous Differential Equations with Constant Coefficients on the Boundary Value Problem,” Sichuan Ordnance Journal, Vol. 31, No. 4, 2010, pp. 126-129.

[14] S. C. Li, “Preliminary Exploration and Prospects of the Similar Structure of Solutions of Differential Equations,” Journal of Xihua University (Natural Science Edition), Vol. 29, No. 2, 2010, pp. 223-226.

[15] J. Kevorkian, “Partial Differential Equations: Analytical Solution Techniques,” 2nd edition, Springer, New York, 2000.

[16] A. D. Polyanin, “Handbook of Linear Partial Differential Equations for Engineers and Scientists,” Chapman & Hall/CRC Press, Boca Raton, 2002.

[17] A. Sommerfeld, “Partial Differential Equations in Physics,” Academic Press, New York, 1949.

[18] J. D. Tian and S. C. Li, “The Formal Similarity of Solutions in the Laplace Space on the Class of Quasilinear Partial Differential Equation,” Mathematical Theory and Applications, Vol. 24, No. 2, 2004, pp. 66-73.

[19] M. H. Jia and S. C. Li, “The Formal Similarity of Solutions in the Laplace Space on the Class of Fluid Flow Differential Equation,” Journal of Electronic Science and Technology of China, Vol. 3, No. 2, 2005, pp. 172-174.

[20] S. C. Li, “The Formal Similarity of Solutions in the Laplace Space on the Class of Partial Differential Equation System,” Journal of Xihua University (Natural Science Edition), Vol. 26, No. 4, 2007, pp. 83-86.

[21] J. P. Su, S. C. Li and C. J. Li, “The Similar of Solutions in the Laplace Space of Composite Parabolic Partial Differential Equation,” Journal of Zaozhuang University, Vol. 26, No. 2, 2009, pp. 6-11.

[22] S. C. Li, P. S. Zheng and Y. F. Zhang, “The Similar Structure of Pressure Distribution in the Homogenous Reservoir,” Pure and Applied Mathematics, Vol. 22, No. 4, 2006, pp. 459-463.

[23] W. Z. Xu, S. C. Li and P. S. Zheng, “The Structure of the Solution of Pressure Distribution in the Fractal Homogenous Reservoir and Analytic Graph,” Mathematics and its Applications, Atomic Energy Publishing Company, Beijing, 2007, pp. 541-544.

[24] S. C. Li, P. S. Zheng and Y. F. Zhang, “The Similar Structure of Pressure Distribution in the Composite Reservoir,” Journal of Mathematics in Practice and Theory, Vol. 38, No. 3, 2008, pp. 23-28.

[25] S. C. Li, P. S. Zheng and Y. F. Zhang, “Similar Structure of Pressure Distribution in the Multilayer Reservoir,” Applied Mathematics A Journal of Chinese Universities, Vol. 24, No. 2, 2009, pp. 234-238.

[26] S. C. Li, X. P. Li and B. G. Huang, “Research on Homogeneous Reservoir Pressure Performance,” Drilling & Production Technology, Vol. 25, No. 1, 2002, pp. 50-51.

[27] S. C. Li, D. H. Liu and P. Z. Zhang, “Pressure Distribution of the Homogeneous Reservoir with Different Boundary Conditions,” Journal of Jianghan Petroleum Institute, Vol. 23, No. 2, 2001, pp. 16-17.

[28] S. C. Li, P. S. Zheng and Y. F. Zhang, “The Similar Structure of Pressure Distribution in the Fractal Reservoir,” Resources, Environment & Porous Flow Mechanics, Proceedings of 8th Research Conference of Porous Flow Mechanics of China, China Scientific & Technical Publishers, Beijing, 2005, pp. 79-83.

[29] S. C. Li and J. J. Zhang, “Similar Structure of Pressure Distribution in the Fractal Dual Porosity Reservoir,” Journal of Xihua University (Natural Science Edition), Vol. 25, No. 1, 2006, pp. 40-43.

[30] C. X. Xu, S. C. Li and W. B. Zhu, “The Similar Structure of Pressure Distribution in the Dual Porosity Reservoir,” Drilling & Production Technology, Vol. 29, No. 4, 2006, pp. 28-30.

[31] C. X. Xu, S. C. Li and W. B. Zhu, “The Similar Structure of Pressure Distribution in the Fractal Composite Reservoir,” Drilling & Production Technology, Vol. 29, No. 5, 2006, pp. 39-42.

[32] P. S. Zheng, C. X. Xu and W. Z. Xu, “Well Analusis Method Based on the Similar Structure of Pressure Distribution in the Composite Reservior,” Drilling & Production Technology, Vol. 30, No. 3, 2007, pp. 49-50.

[33] W. B. Zhu, S. C. Li and C. X. Xu, “The Similar Structure of Pressure Distribution in the Fractal Multilayer Reservoir,” Drilling & Production Technology, Vol. 31, No. 3, 2008, pp. 67-69.

[34] P. S. Zheng, S. C. Li and W. B. Zhu, “The Similar Structure of Pressure Distribution in the Double Porosity Composite Reservoir,” Drilling & Production Technology, Vol. 31, No. 4, 2008, pp. 80-81.

[35] S. C. Li and B. G. Huang, “Laplace Transform and Bessel Functions and the Theoretical Basis of Well Test Analysis,” Petroleum Industry Press, Beijing, 2000.