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# Method for estimating the Young's modulus of a rock using a water hammer

UDK: 622. 276.031.011.43:53.091
DOI: 10.24887/0028-2448-2019-3-70-73
Key words: hydraulic fracturing, water hammer, crack size, Young's modulus, functional minimization
Authors: A.M. Ilyasov (RN-UfaNIPIneft LLC, RF, Ufa), K.R. Kadyrova (RN-UfaNIPIneft LLC, RF, Ufa), V.A. Baikov (RN-UfaNIPIneft LLC, RF, Ufa), I.D. Latypov (RN-BashNIPIneft LLC, RF, Ufa)

The flow rate of oil after fracturing is determined by the permeable surface area of the formed crack and its width. The main geometrical parameters of fracture are height, width and length which cannot be directly measured. The only measured data for hydraulic fracturing are wellhead and bottom pressure. Another way to determine the size of the crack is to get a hydraulic fracture design. But it is necessary to know the elastic rock modules in the interval of the hydraulic fracturing operation. Experimental core studies of elastic constants can give quite high errors at least due to the unloading of core samples and the formation of cracks in the core material when lifted to the surface.

This article presents the method and algorithm of determining the size of a hydraulic fracture and the effective Young's modulus of rock within the interval of fracture development after a water hammer according to downhole gauges. The method is based on solving direct problems of the natural fluctuations of a hydraulic fracture after stopping the pump. Natural fluctuations of a hydraulic fracture are described with the use of the linearized generalized Perkins – Kern – Nordgren (PKN) model of a hyperbolic type. The inverse coefficient problem is solved by the least squares method. On the coefficients found it is possible to estimate the rigidity of a hydraulic fracture, its geometrical parameters, as well as the Young's modulus of the rock within the interval of fracture development. A comparison is carried out between geometrical parameters of hydraulic fractures found using the suggested method and the figures of the design test on substitution obtained with the help of the RN-GRID simulator, provided the Young’s modulus is derived from the new accepted method. Calculations showed fine precision.

References

1. Holzhausen C.R., Gooch, R.P., Impedance of hydraulic fracture: its measurement and use for estimating fracture closure and dimensions, SPE 13892-MS, 1985.

2. Patzek T.W., De A., Lossy transmission line model of hydrofractured well dynamics, SPE 46195-MS, 2000.

3. Wylie E.B., Streeter V.L., Fluid transients in systems, New Jersey: Englewood Cliffs, Prentice-Hall, 1993, 463 p.

4. Paige R.W., Murray L.R., Roberts J.D.M., Field application of hydraulic impedance testing for fracture measurement, SPE 26525-PA, 1995.

5. Sneddon J.N., Berry D.S., The classical theory of elasticity, Berlin: Springer, 1958.

6. Carey M.A., Mondal S., Sharma M.M., Analysis of water hammer signatures for fracture diagnostics, SPE 174866-MS, 2015.

7. Iriarte J., Merritt J., Kreyche B., Using water hammer characteristics as a fracture treatment diagnostic, SPE 185087-MS, 2017.

8. Perkins T.K., Kern L.R., Width of hydraulic fractures, Journal of Petroleum Technology, 1961, V.13, no. 4, pp. 937–949.

9. Nordgren R.P., Propogation of a vertical hydraulic fracture, SPE 3009-PA, 1972.

10. Il'yasov A.M., Bulgakova G.T., The quasi-one-dimensional hyperbolic model of hydraulic fracturing (In Russ.), Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Fiziko-matematicheskie nauki = Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2016, V. 20, no. 4, pp. 739–754.

11. Baykov V.A., Bulgakova G.T., Il'yasov A.M., Kashapov D.V., To the evaluation of the geometric parameters of hydraulic fracturing crack (In Russ.), Mekhanika zhidkosti i gaza, 2018, no. 5, pp. 64-75, DOI: 1031857/S05682810001790-0.

The flow rate of oil after fracturing is determined by the permeable surface area of the formed crack and its width. The main geometrical parameters of fracture are height, width and length which cannot be directly measured. The only measured data for hydraulic fracturing are wellhead and bottom pressure. Another way to determine the size of the crack is to get a hydraulic fracture design. But it is necessary to know the elastic rock modules in the interval of the hydraulic fracturing operation. Experimental core studies of elastic constants can give quite high errors at least due to the unloading of core samples and the formation of cracks in the core material when lifted to the surface.

This article presents the method and algorithm of determining the size of a hydraulic fracture and the effective Young's modulus of rock within the interval of fracture development after a water hammer according to downhole gauges. The method is based on solving direct problems of the natural fluctuations of a hydraulic fracture after stopping the pump. Natural fluctuations of a hydraulic fracture are described with the use of the linearized generalized Perkins – Kern – Nordgren (PKN) model of a hyperbolic type. The inverse coefficient problem is solved by the least squares method. On the coefficients found it is possible to estimate the rigidity of a hydraulic fracture, its geometrical parameters, as well as the Young's modulus of the rock within the interval of fracture development. A comparison is carried out between geometrical parameters of hydraulic fractures found using the suggested method and the figures of the design test on substitution obtained with the help of the RN-GRID simulator, provided the Young’s modulus is derived from the new accepted method. Calculations showed fine precision.

References

1. Holzhausen C.R., Gooch, R.P., Impedance of hydraulic fracture: its measurement and use for estimating fracture closure and dimensions, SPE 13892-MS, 1985.

2. Patzek T.W., De A., Lossy transmission line model of hydrofractured well dynamics, SPE 46195-MS, 2000.

3. Wylie E.B., Streeter V.L., Fluid transients in systems, New Jersey: Englewood Cliffs, Prentice-Hall, 1993, 463 p.

4. Paige R.W., Murray L.R., Roberts J.D.M., Field application of hydraulic impedance testing for fracture measurement, SPE 26525-PA, 1995.

5. Sneddon J.N., Berry D.S., The classical theory of elasticity, Berlin: Springer, 1958.

6. Carey M.A., Mondal S., Sharma M.M., Analysis of water hammer signatures for fracture diagnostics, SPE 174866-MS, 2015.

7. Iriarte J., Merritt J., Kreyche B., Using water hammer characteristics as a fracture treatment diagnostic, SPE 185087-MS, 2017.

8. Perkins T.K., Kern L.R., Width of hydraulic fractures, Journal of Petroleum Technology, 1961, V.13, no. 4, pp. 937–949.

9. Nordgren R.P., Propogation of a vertical hydraulic fracture, SPE 3009-PA, 1972.

10. Il'yasov A.M., Bulgakova G.T., The quasi-one-dimensional hyperbolic model of hydraulic fracturing (In Russ.), Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Fiziko-matematicheskie nauki = Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2016, V. 20, no. 4, pp. 739–754.

11. Baykov V.A., Bulgakova G.T., Il'yasov A.M., Kashapov D.V., To the evaluation of the geometric parameters of hydraulic fracturing crack (In Russ.), Mekhanika zhidkosti i gaza, 2018, no. 5, pp. 64-75, DOI: 1031857/S05682810001790-0.

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