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Pseudo-skin factor and optimal conductivity of a hydraulic fracture in a circular reservoir

UDK: 622.276.66.004.58
DOI: 10.24887/0028-2448-2019-3-74-77
Key words: hydraulic fracture, circular reservoir, productivity, pseudo-skin factor, effective radius, optimal conductivity
Authors: P.E. Morozov (Institute of Mechanics and Engineering, Kazan Scientific Center of RAS, RF, Kazan)

One of the most effective ways to increase well productivity is hydraulic fracturing. As a result of hydraulic fracturing the high conductivity fractures in the reservoir are formed, which reduce the filtration resistance of the bottomhole zone and increase the effective wellbore radius. In the works of I.V. Krivonosov, I.A. Charny and M. Prats, it was found that an “ideal” fracture (infinite conductivity fracture) is equivalent to a well whose diameter is equal to half the fracture length. Earlier, a similar conclusion was made in the fundamental works of F. Forchheimer and N.E. Zhukovsky, where steady-state water flow to the slit and gallery of finite length was investigated. An important step in planning a hydraulic fracturing operation is determining the optimal fracture parameters (length, width and conductivity), which are able to provide the maximum production rate at a fixed fracture volume and known values of the reservoir thickness, drainage radius, reservoir and proppant permeability’s.

The article presents a theoretical analysis of the pseudo-skin factor for rectangular and elliptical fractures and expressions for the optimal fracture half-length and width. It is shown that the effective radius of fracture with optimal conductivity is half the effective radius of an “ideal” fracture. A system of integral equations is obtained for determining the steady-state fluid flow to a finite conductivity fracture in a circular reservoir. Based on the numerical solution of a system of integral equations, graphs of flux and pressure distribution along the fracture wings were presented for different values of the dimensionless fracture conductivity, which are in good agreement with the results of M. Prats and H. Cinco-Ley. It is shown that the pseudo-skin factor is explicitly expressed through the flux distribution along the fracture.

References

1. Economides M.J., Nolte K.G., Reservoir stimulation, J. Wiley Sons, 2000, 856 p.

2. Kanevskaya R.D., Matemeticheskoe modelirovanie razrabotki mestorozhdeniy nefti i gaza s primeneniem gidravlicheskogo razryva plasta (Mathematical modeling of the development of oil and gas using hydraulic fracturing), Moscow: Nedra-Biznestsentr Publ., 1999, 212 p.

3. Prats M., Effect of vertical fractures on reservoir behavior – incompressible fluid case, SPE 1575-G, 1961.

4. Krivonosov I.V., Charnyy I.A., Calculation of flow rates of wells with fractured bottomhole formation zone (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 1955, no. 4, pp. 40–47.

5. Economides M., Oligney R., Valko P., Unified fracture design. Bridging the gap between theory and practice, Orsa Press, Alvin, Texas, 2002, 262 p.

6. Meyer B.R., Jacot R.H., Pseudosteady-state analysis of finite-conductivity vertical fractures, SPE 95941-MS, 2005.

7. Astaf'ev V.I., Fedorchenko G.D., Simulation of fluid filtration in the presence of hydraulic fracturing (In Russ.), Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Ser. Fiziko-matematicheskie nauki = Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2007, no. 2(15), pp. 128–132.

8. Riley M.F., Brigham W.E., Horne R.N., Analytic solutions for elliptical finite-conductivity fractures, SPE 22656-MS, 1991.

9. Lu Y., Chen K.P., Productivity-index optimization for hydraulically fractured vertical wells in a circular reservoir: a comparative study with analytical solutions, SPE 180929-PA, 2016.

10. Sinso-Ley N., Meng H.-Z., Pressure-transient analysis of wells with finite-conductivity vertical fractures in double porosity reservoirs, SPE 18172-MS, 1988.

11. Morozov P.E., Psevdoskin-faktor i optimal'naya provodimost' vertikal'noy treshchiny gidravlicheskogo razryva plasta (Pseudoskin factor factor and optimal conductivity of vertical induced hydraulic fracture), Proceedings of Sci International Scientific and Practical Conference “Innovatsii v razvedke i razrabotke neftyanykh i gazovykh mestorozhdeniy” (Innovations in exploration and development of oil and gas fields), Kazan, 2016, Part 2, pp. 53-56. (In Russ.)

12. Zazovskiy A.F., Todua G.T., About the stationary fluid inflow to the well with a large long vertical fracture (In Russ.), Izvestiya Akademii nauk SSSR. Mekhanika zhidkosti i gaza = Fluid Dynamics, 1990, no. 4, pp. 107–116.

13. Cinco–Ley H., Samaniego V.F., Dominguez A.N., Transient pressure behavior for a well with a finite–conductivity vertical fracture, SPE 6014-PA, 1978.

14. Barenblatt G.I., Entov V.M., Ryzhik V.M., Teoriya nestatsionarnoy fil'tratsii zhidkosti i gaza (The theory of non-stationary filtration of liquid and gas), Moscow: Nedra Publ., 1972, 288 p.

One of the most effective ways to increase well productivity is hydraulic fracturing. As a result of hydraulic fracturing the high conductivity fractures in the reservoir are formed, which reduce the filtration resistance of the bottomhole zone and increase the effective wellbore radius. In the works of I.V. Krivonosov, I.A. Charny and M. Prats, it was found that an “ideal” fracture (infinite conductivity fracture) is equivalent to a well whose diameter is equal to half the fracture length. Earlier, a similar conclusion was made in the fundamental works of F. Forchheimer and N.E. Zhukovsky, where steady-state water flow to the slit and gallery of finite length was investigated. An important step in planning a hydraulic fracturing operation is determining the optimal fracture parameters (length, width and conductivity), which are able to provide the maximum production rate at a fixed fracture volume and known values of the reservoir thickness, drainage radius, reservoir and proppant permeability’s.

The article presents a theoretical analysis of the pseudo-skin factor for rectangular and elliptical fractures and expressions for the optimal fracture half-length and width. It is shown that the effective radius of fracture with optimal conductivity is half the effective radius of an “ideal” fracture. A system of integral equations is obtained for determining the steady-state fluid flow to a finite conductivity fracture in a circular reservoir. Based on the numerical solution of a system of integral equations, graphs of flux and pressure distribution along the fracture wings were presented for different values of the dimensionless fracture conductivity, which are in good agreement with the results of M. Prats and H. Cinco-Ley. It is shown that the pseudo-skin factor is explicitly expressed through the flux distribution along the fracture.

References

1. Economides M.J., Nolte K.G., Reservoir stimulation, J. Wiley Sons, 2000, 856 p.

2. Kanevskaya R.D., Matemeticheskoe modelirovanie razrabotki mestorozhdeniy nefti i gaza s primeneniem gidravlicheskogo razryva plasta (Mathematical modeling of the development of oil and gas using hydraulic fracturing), Moscow: Nedra-Biznestsentr Publ., 1999, 212 p.

3. Prats M., Effect of vertical fractures on reservoir behavior – incompressible fluid case, SPE 1575-G, 1961.

4. Krivonosov I.V., Charnyy I.A., Calculation of flow rates of wells with fractured bottomhole formation zone (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 1955, no. 4, pp. 40–47.

5. Economides M., Oligney R., Valko P., Unified fracture design. Bridging the gap between theory and practice, Orsa Press, Alvin, Texas, 2002, 262 p.

6. Meyer B.R., Jacot R.H., Pseudosteady-state analysis of finite-conductivity vertical fractures, SPE 95941-MS, 2005.

7. Astaf'ev V.I., Fedorchenko G.D., Simulation of fluid filtration in the presence of hydraulic fracturing (In Russ.), Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Ser. Fiziko-matematicheskie nauki = Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2007, no. 2(15), pp. 128–132.

8. Riley M.F., Brigham W.E., Horne R.N., Analytic solutions for elliptical finite-conductivity fractures, SPE 22656-MS, 1991.

9. Lu Y., Chen K.P., Productivity-index optimization for hydraulically fractured vertical wells in a circular reservoir: a comparative study with analytical solutions, SPE 180929-PA, 2016.

10. Sinso-Ley N., Meng H.-Z., Pressure-transient analysis of wells with finite-conductivity vertical fractures in double porosity reservoirs, SPE 18172-MS, 1988.

11. Morozov P.E., Psevdoskin-faktor i optimal'naya provodimost' vertikal'noy treshchiny gidravlicheskogo razryva plasta (Pseudoskin factor factor and optimal conductivity of vertical induced hydraulic fracture), Proceedings of Sci International Scientific and Practical Conference “Innovatsii v razvedke i razrabotke neftyanykh i gazovykh mestorozhdeniy” (Innovations in exploration and development of oil and gas fields), Kazan, 2016, Part 2, pp. 53-56. (In Russ.)

12. Zazovskiy A.F., Todua G.T., About the stationary fluid inflow to the well with a large long vertical fracture (In Russ.), Izvestiya Akademii nauk SSSR. Mekhanika zhidkosti i gaza = Fluid Dynamics, 1990, no. 4, pp. 107–116.

13. Cinco–Ley H., Samaniego V.F., Dominguez A.N., Transient pressure behavior for a well with a finite–conductivity vertical fracture, SPE 6014-PA, 1978.

14. Barenblatt G.I., Entov V.M., Ryzhik V.M., Teoriya nestatsionarnoy fil'tratsii zhidkosti i gaza (The theory of non-stationary filtration of liquid and gas), Moscow: Nedra Publ., 1972, 288 p.


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