Solution and Type Curve Analysis of Fluid Flow Model for Fractal Reservoir

ABSTRACT

Conventional pressure-transient models have been developed under the assumption of homogeneous reservoir. However, core, log and outcrop data indicate this assumption is not realistic in most cases. But in many cases, the homogeneous models are still applied to obtain an effective permeability corresponding to fictitious homogeneous reservoirs. This approach seems reasonable if the permeability variation is sufficiently small. In this paper, fractal dimension and fractal index are introduced into the seepage flow mechanism to establish the fluid flow models in fractal reservoir under three outer-boundary conditions. Exact dimensionless solutions are obtained by using the Laplace transformation assuming the well is producing at a constant rate. Combining the Stehfest’s inversion with the Vongvuthipornchai’s method, the new type curves are obtained. The sensitivities of the curve shape to fractal dimension (θ) and fractal index (d) are analyzed; the curves don’t change too much when θ is a constant and d change. For a closed reservoir, the up-curving has little to do with θ when d is a constant; but when θ is a constant, the slope of the up-curving section almost remains the same, only the pressure at the starting point decreases with the increase of d; and when d = 2 and θ = 0, the solutions and curves become those of the conventional reservoirs, the application of this solution has also been introduced at the end of this paper.

Conventional pressure-transient models have been developed under the assumption of homogeneous reservoir. However, core, log and outcrop data indicate this assumption is not realistic in most cases. But in many cases, the homogeneous models are still applied to obtain an effective permeability corresponding to fictitious homogeneous reservoirs. This approach seems reasonable if the permeability variation is sufficiently small. In this paper, fractal dimension and fractal index are introduced into the seepage flow mechanism to establish the fluid flow models in fractal reservoir under three outer-boundary conditions. Exact dimensionless solutions are obtained by using the Laplace transformation assuming the well is producing at a constant rate. Combining the Stehfest’s inversion with the Vongvuthipornchai’s method, the new type curves are obtained. The sensitivities of the curve shape to fractal dimension (θ) and fractal index (d) are analyzed; the curves don’t change too much when θ is a constant and d change. For a closed reservoir, the up-curving has little to do with θ when d is a constant; but when θ is a constant, the slope of the up-curving section almost remains the same, only the pressure at the starting point decreases with the increase of d; and when d = 2 and θ = 0, the solutions and curves become those of the conventional reservoirs, the application of this solution has also been introduced at the end of this paper.

Cite this paper

nullY. Zhao and L. Zhang, "Solution and Type Curve Analysis of Fluid Flow Model for Fractal Reservoir,"*World Journal of Mechanics*, Vol. 1 No. 5, 2011, pp. 209-216. doi: 10.4236/wjm.2011.15027.

nullY. Zhao and L. Zhang, "Solution and Type Curve Analysis of Fluid Flow Model for Fractal Reservoir,"

References

[1] [1] B. B. Mandebort, “The Fractal Geometry of Nature,” W. H. Freeman, New York, 1982.

[2] T. A. Hewett, “Fractal Distribution of Reservoir Heterogeneity and Their Influence on Fluid Transport,” Society of Petroleum Engineers, New Orlean, 5-8 October 1986, Document ID: 15386.

[3] J. A. Bakker, “A Generalized Radial Flow Model for Hydraulic Tests in Fractured Rock,” Water Resources Research, Vol. 20, No. 10, 1986, pp. 1796-1840.

[4] T. W. Doe, “Fractional Dimension Analysis of Constant Pressure Well Tests,” The 66th Annual Technical Conference and Exhibition of SPE, Dallas, 6-9 October 1991. Document ID: 22702-MS. doi:10.2118/22702-MS

[5] J. Chang and Y. C. Yortsos, “Pressure Transient Analysis of Fractal Reservoir,” SPE Formation Evaluation, Vol. 5, No. 1, 1990, pp. 31-38. doi:10.2118/18170-PA

[6] R. A. Beier, “Pressure Transient Field Data Showing Fractal Reservoir Structure,” International Technical Meeting of the SPE Calgary, Calgary, 10-13 June 1990, Document ID: 21553-MS.

[7] S. Aprilian, D. Abdassah, L. Mucharant, et al., “Application of Fractal Reservoir Mode for Interference Test Analysis in Kamojiang Geothermal Field (Indonesia),” The 68th SPE Annual Technical Conference and Exhibition, Houston, 3-6 October 1993, Document ID: 26465-MS. doi:10.2118/26465-MS

[8] D. Poon, “Transient Pressure Analysis of Fractal Reservoirs. The Petroleum Society of CIM,” The 46th Annual Technical Meeting of the Petroleum Society of CIM, Calgary, 7-9 June 1995, Document ID: 95-34.

[9] G. L. He and K. L. Xiang, “Mathematical Model and Character of Pressure Transient of Unstable Seepage Flow in Deformation Dual-Porosity Fractal Reservoirs,” Journal of Southwest Petroleum University, Vol. 24, No. 4, 2002, pp. 24-28. (in Chinese)

[10] J. E. Warrant and P. J. Root, “The Behavior of Naturally Fractured Reservoirs,” SPE Journal, Vol. 3, No. 3, 1963, pp. 245-255.

[11] D. P. Bourdet, J. A. Ayoub and Y. M. Pirard, “Use of Pressure Derivative in Well Test Interpretation,” SPE Formation Evaluation, Vol. 4, No. 2, 1989, pp. 293-302.

[12] S. C. Li, “A Solution of Fractal Dual Porosity Reservoir Model in Well Testing Analysis,” Progress in Exploration Geophysics, Vol. 25, No. 5, 2002, pp. 60-62. (in Chinese)

[13] X. Y. Kong, “Advanced Mechanics of Fluids in Porous Media,” University of Science and Technology of China Press, Hefei, 1999. (in Chinese)

[14] L. H. Zhang, J. L. Zhang and B. Q. Xu, “A Nonlinear Seepage Flow Model for Deformable Double Media Fractal Gas Reservoirs,” Chinese Journal of Conputational Physics, Vol. 24, No. 1, 2007, pp. 90-94. (in Chinese)

[15] Y. M. Hou and D. K. Tong, “Non-Steady Flow of Non-Newtonian Power-Low Permeability with Moving-Boundary in Double Porous Media and Fractal Reservoir,” Engineering Mechanics, Vol. 26, No. 8, 2009, pp. 245-250. (in Chinese)

[16] K. L. Xiang and X. Q. Tu, “The Analytical Solutions of Mathematical Model for a Fractal Composite Reservoir with Non-Newtonian Power Low Fluids Flow,” Chinese Journal of Computational Physics, Vol. 21, No. 6, 2004, pp. 558-564. (in Chinese)

[17] X. R. Deng and S. C. Li, “Solution to the Well Testing Model for Fractal Composite Reservoirs,” Journal of Xihua University Natural Science, Vol. 24, No. 2, 2005, pp. 4-7. (in Chinese)

[18] C. X. Xu, S. C. Li and W. B. Zhu, “Similar Structure of Well Test Analytical Solution in the Fractal Composite Reservoir,” Drilling Production Technology, Vol. 29, No. 5, 2006, pp. 39-43. (in Chinese)

[19] C. S. Chakrabarty, M. Farouqali and W. S. Tortlke, “Transient Flow Behavior of Non-Newtonian Power-Law Fluids in Fractal Reservoirs,” Annual Technical Meeting of the Petroleum Society of CIM, Calgary, 1993, Document ID: CIM93-06.

[20] R. R. Dyah, J. C. Kana, A. Doddy, et al., “A New Numerically Model of Single Well Radial Multiphase Flow in Naturally Fracture Reservoir System-Using Fractal Concept,” Society of Petroleum Engineers, 1999, Document ID: 57276.

[21] X. Y. Kong, D. L. Li and D. T. Lu, “Basic Formulas of Fractal Seepage and Type-Curves of Fractal Reservoirs,” Journal of Xi’an Petroleum University, Vol. 22, No. 2, 2007, pp. 1-5. (in Chinese)

[22] S. Vongvuthipornchai and R. Raghavan, “Well Test Analysis of Date Dominated by Storage and Skin: Non- newtonian Power-Law Fluid,” Society of Petroleum Engineers, 1987, Document ID: 14454.

[23] H. Stehfest, “Numerical Inversion of Laplace Transforms,” Communications ACM, Vol. 13, No. 1, 1970, pp. 47-48.

[1] [1] B. B. Mandebort, “The Fractal Geometry of Nature,” W. H. Freeman, New York, 1982.

[2] T. A. Hewett, “Fractal Distribution of Reservoir Heterogeneity and Their Influence on Fluid Transport,” Society of Petroleum Engineers, New Orlean, 5-8 October 1986, Document ID: 15386.

[3] J. A. Bakker, “A Generalized Radial Flow Model for Hydraulic Tests in Fractured Rock,” Water Resources Research, Vol. 20, No. 10, 1986, pp. 1796-1840.

[4] T. W. Doe, “Fractional Dimension Analysis of Constant Pressure Well Tests,” The 66th Annual Technical Conference and Exhibition of SPE, Dallas, 6-9 October 1991. Document ID: 22702-MS. doi:10.2118/22702-MS

[5] J. Chang and Y. C. Yortsos, “Pressure Transient Analysis of Fractal Reservoir,” SPE Formation Evaluation, Vol. 5, No. 1, 1990, pp. 31-38. doi:10.2118/18170-PA

[6] R. A. Beier, “Pressure Transient Field Data Showing Fractal Reservoir Structure,” International Technical Meeting of the SPE Calgary, Calgary, 10-13 June 1990, Document ID: 21553-MS.

[7] S. Aprilian, D. Abdassah, L. Mucharant, et al., “Application of Fractal Reservoir Mode for Interference Test Analysis in Kamojiang Geothermal Field (Indonesia),” The 68th SPE Annual Technical Conference and Exhibition, Houston, 3-6 October 1993, Document ID: 26465-MS. doi:10.2118/26465-MS

[8] D. Poon, “Transient Pressure Analysis of Fractal Reservoirs. The Petroleum Society of CIM,” The 46th Annual Technical Meeting of the Petroleum Society of CIM, Calgary, 7-9 June 1995, Document ID: 95-34.

[9] G. L. He and K. L. Xiang, “Mathematical Model and Character of Pressure Transient of Unstable Seepage Flow in Deformation Dual-Porosity Fractal Reservoirs,” Journal of Southwest Petroleum University, Vol. 24, No. 4, 2002, pp. 24-28. (in Chinese)

[10] J. E. Warrant and P. J. Root, “The Behavior of Naturally Fractured Reservoirs,” SPE Journal, Vol. 3, No. 3, 1963, pp. 245-255.

[11] D. P. Bourdet, J. A. Ayoub and Y. M. Pirard, “Use of Pressure Derivative in Well Test Interpretation,” SPE Formation Evaluation, Vol. 4, No. 2, 1989, pp. 293-302.

[12] S. C. Li, “A Solution of Fractal Dual Porosity Reservoir Model in Well Testing Analysis,” Progress in Exploration Geophysics, Vol. 25, No. 5, 2002, pp. 60-62. (in Chinese)

[13] X. Y. Kong, “Advanced Mechanics of Fluids in Porous Media,” University of Science and Technology of China Press, Hefei, 1999. (in Chinese)

[14] L. H. Zhang, J. L. Zhang and B. Q. Xu, “A Nonlinear Seepage Flow Model for Deformable Double Media Fractal Gas Reservoirs,” Chinese Journal of Conputational Physics, Vol. 24, No. 1, 2007, pp. 90-94. (in Chinese)

[15] Y. M. Hou and D. K. Tong, “Non-Steady Flow of Non-Newtonian Power-Low Permeability with Moving-Boundary in Double Porous Media and Fractal Reservoir,” Engineering Mechanics, Vol. 26, No. 8, 2009, pp. 245-250. (in Chinese)

[16] K. L. Xiang and X. Q. Tu, “The Analytical Solutions of Mathematical Model for a Fractal Composite Reservoir with Non-Newtonian Power Low Fluids Flow,” Chinese Journal of Computational Physics, Vol. 21, No. 6, 2004, pp. 558-564. (in Chinese)

[17] X. R. Deng and S. C. Li, “Solution to the Well Testing Model for Fractal Composite Reservoirs,” Journal of Xihua University Natural Science, Vol. 24, No. 2, 2005, pp. 4-7. (in Chinese)

[18] C. X. Xu, S. C. Li and W. B. Zhu, “Similar Structure of Well Test Analytical Solution in the Fractal Composite Reservoir,” Drilling Production Technology, Vol. 29, No. 5, 2006, pp. 39-43. (in Chinese)

[19] C. S. Chakrabarty, M. Farouqali and W. S. Tortlke, “Transient Flow Behavior of Non-Newtonian Power-Law Fluids in Fractal Reservoirs,” Annual Technical Meeting of the Petroleum Society of CIM, Calgary, 1993, Document ID: CIM93-06.

[20] R. R. Dyah, J. C. Kana, A. Doddy, et al., “A New Numerically Model of Single Well Radial Multiphase Flow in Naturally Fracture Reservoir System-Using Fractal Concept,” Society of Petroleum Engineers, 1999, Document ID: 57276.

[21] X. Y. Kong, D. L. Li and D. T. Lu, “Basic Formulas of Fractal Seepage and Type-Curves of Fractal Reservoirs,” Journal of Xi’an Petroleum University, Vol. 22, No. 2, 2007, pp. 1-5. (in Chinese)

[22] S. Vongvuthipornchai and R. Raghavan, “Well Test Analysis of Date Dominated by Storage and Skin: Non- newtonian Power-Law Fluid,” Society of Petroleum Engineers, 1987, Document ID: 14454.

[23] H. Stehfest, “Numerical Inversion of Laplace Transforms,” Communications ACM, Vol. 13, No. 1, 1970, pp. 47-48.