Broad application of hydraulic fracturing techniques in oil industry since the 1950s led to the emergence of a large number of studies devoted to both production forecast methods and reservoir performance evaluation techniques to interpret the results of various well testing procedures. It is remarkable that analytical investigations devoted to the postfracturing analysis are focused primarily on the early-time unsteady flow regime, while for the production forecast it is commonly used the approximation of steady-state or pseudosteady-state flow models. The focus in oil engineering has recently turned towards unconventional reserves. Unsteady-state flow plays crucial role for reservoirs with low-mobility oil by making a major contribution to the cumulative oil production. With the appearance of the available computational techniques, the emphasis has shifted to the numerical simulation of flows, and the search for analytic approximations gone by the wayside.

A number of articles is devoted to discussing the critical issues associated with numerical simulations. Among others, it should be noted that the large-scale grid cannot adequately simulate transient flow regime of the fluids with low mobility owing to the fact that the cell size is much larger than the characteristic scale of the variations of physical parameters, particularly pressure. Changing the sizes of cells, including local refinement, always results in time-consuming model and adversely affect the convergence of the numerical scheme.

The study presents an approach to the analytical modeling of the production rate of the fractured vertical well during the unsteady flow regime. Asymptotic Laplace space solution based on trilinear flow model is developed to describe the flow at early times. The authors propose an asymptotic solution, which describes the flow rate towards vertical fracture under the assumption of an infinite reservoir, using the desuperposition concept to couple the trilinear and pseudoradial flow solutions.

Verification of the proposed model was carried out by comparison with the solution given by the finite-difference hydrodynamic commercial simulator. The model allows for quick and accurate assessment of the hydraulically fractured well production, avoiding errors associated with the convergence of numerical methods at the early times, as well as significantly reducing the time of calculation.References

1. Dyes A.B., Kemp C.E., Caudle B.H., Effect of fractures on sweep-out pattern,

Petroleum transactions, AIME, 1958, V. 213, pp. 245–249.

2. Gringarten A.C., Ramey H.J.Jr., The use of source and Green’s functions in

solving unsteady flow problems in reservoirs, Society of Petroleum Engineers

Journal, 1973, V. 13, no. 5, pp. 285–296, SPE 3818-PA.

3. Gringarten A.C., Ramey H.J.Jr., Raghavan R., Unsteady-state pressure distributions

created by a well with a single infinite-conductivity vertical fracture,

Society of Petroleum Engineers Journal, 1974, V. 14, no. 4, pp. 347–360,

SPE 4051-PA.

4. Lee S.-T., Brockenbrough J.R., A new approximate analytic solution for finiteconductivity

vertical fractures, SPE Formation Evaluation, 1986, V. 1, no. 1,

pp. 75–88, SPE 12013-PA.

5. Azari M., Wooden W.O., Coble L.E., A complete set of Laplace transforms

for finite-conductivity vertical fractures under bilinear and trilinear flows, SPE

20556, 1990.

6. Ozkan E., Raghavan R., New solutions for well-test-analysis problems: Part 1:

Analytical considerations, SPE 18615-PA, 1991.

7. Ozkan E., Raghavan R., New solutions for well-test-analysis problems: Part 2:

Computational considerations and applications, SPE 18616-PA, 1991.

8. Lefevre D., Pellissier G., Sabathier J.C., A new reservoir simulation system for

a better reservoir management, SPE 25604, 1993.

9. Elahmady M., Wattenberger R.A., Coarse scale simulation in tight gas reservoirs,

Journal of Canadian Petroleum Technology, 2006, V. 45, no. 12,

pp. 67–71.

10. Durlofsky L.J., Upscaling and gridding of fine scale geological models for

flow simulation, Proceedings of 8th International Forum on Reservoir Simulation

Iles Borromees, 2005, V. 2024.

11. Burgoyne M.W., Little A.L., From high perm oil to tight gas - A practical approach

to model hydraulically fractured well performance in coarse grid

reservoir simulators, SPE-156610, 2012.

12. Van Everdingen A.F., Hurst W., The application of the Laplace transformation

to flow problems in reservoirs, Journal of Petroleum Technology, 1949, V. 1,

no. 12, pp. 305–324.

13. Ibrahim M.H., Wattenbarger R.A., Rate dependence of transient linear

flow in tight gas wells, Journal of Canadian Petroleum Technology, 2006, V. 45,

no. 10.

13. Blasingame T.A., Poe B.D. Jr., Semianalytic solutions for a well with a single

finite-conductivity vertical fracture, SPE 26424, 1993.

14. Brown M., Ozkan E., Raghavan R., Kazemi H., Practical solutions for pressure-

transient responses of fractured horizontal wells in unconventional shale

reservoirs, SPE Reservoir Evaluation and Engineering, 2011, V. 14, no. 6

Broad application of hydraulic fracturing techniques in oil industry since the 1950s led to the emergence of a large number of studies devoted to both production forecast methods and reservoir performance evaluation techniques to interpret the results of various well testing procedures. It is remarkable that analytical investigations devoted to the postfracturing analysis are focused primarily on the early-time unsteady flow regime, while for the production forecast it is commonly used the approximation of steady-state or pseudosteady-state flow models. The focus in oil engineering has recently turned towards unconventional reserves. Unsteady-state flow plays crucial role for reservoirs with low-mobility oil by making a major contribution to the cumulative oil production. With the appearance of the available computational techniques, the emphasis has shifted to the numerical simulation of flows, and the search for analytic approximations gone by the wayside.

A number of articles is devoted to discussing the critical issues associated with numerical simulations. Among others, it should be noted that the large-scale grid cannot adequately simulate transient flow regime of the fluids with low mobility owing to the fact that the cell size is much larger than the characteristic scale of the variations of physical parameters, particularly pressure. Changing the sizes of cells, including local refinement, always results in time-consuming model and adversely affect the convergence of the numerical scheme.

The study presents an approach to the analytical modeling of the production rate of the fractured vertical well during the unsteady flow regime. Asymptotic Laplace space solution based on trilinear flow model is developed to describe the flow at early times. The authors propose an asymptotic solution, which describes the flow rate towards vertical fracture under the assumption of an infinite reservoir, using the desuperposition concept to couple the trilinear and pseudoradial flow solutions.

Verification of the proposed model was carried out by comparison with the solution given by the finite-difference hydrodynamic commercial simulator. The model allows for quick and accurate assessment of the hydraulically fractured well production, avoiding errors associated with the convergence of numerical methods at the early times, as well as significantly reducing the time of calculation.References

1. Dyes A.B., Kemp C.E., Caudle B.H., Effect of fractures on sweep-out pattern,

Petroleum transactions, AIME, 1958, V. 213, pp. 245–249.

2. Gringarten A.C., Ramey H.J.Jr., The use of source and Green’s functions in

solving unsteady flow problems in reservoirs, Society of Petroleum Engineers

Journal, 1973, V. 13, no. 5, pp. 285–296, SPE 3818-PA.

3. Gringarten A.C., Ramey H.J.Jr., Raghavan R., Unsteady-state pressure distributions

created by a well with a single infinite-conductivity vertical fracture,

Society of Petroleum Engineers Journal, 1974, V. 14, no. 4, pp. 347–360,

SPE 4051-PA.

4. Lee S.-T., Brockenbrough J.R., A new approximate analytic solution for finiteconductivity

vertical fractures, SPE Formation Evaluation, 1986, V. 1, no. 1,

pp. 75–88, SPE 12013-PA.

5. Azari M., Wooden W.O., Coble L.E., A complete set of Laplace transforms

for finite-conductivity vertical fractures under bilinear and trilinear flows, SPE

20556, 1990.

6. Ozkan E., Raghavan R., New solutions for well-test-analysis problems: Part 1:

Analytical considerations, SPE 18615-PA, 1991.

7. Ozkan E., Raghavan R., New solutions for well-test-analysis problems: Part 2:

Computational considerations and applications, SPE 18616-PA, 1991.

8. Lefevre D., Pellissier G., Sabathier J.C., A new reservoir simulation system for

a better reservoir management, SPE 25604, 1993.

9. Elahmady M., Wattenberger R.A., Coarse scale simulation in tight gas reservoirs,

Journal of Canadian Petroleum Technology, 2006, V. 45, no. 12,

pp. 67–71.

10. Durlofsky L.J., Upscaling and gridding of fine scale geological models for

flow simulation, Proceedings of 8th International Forum on Reservoir Simulation

Iles Borromees, 2005, V. 2024.

11. Burgoyne M.W., Little A.L., From high perm oil to tight gas - A practical approach

to model hydraulically fractured well performance in coarse grid

reservoir simulators, SPE-156610, 2012.

12. Van Everdingen A.F., Hurst W., The application of the Laplace transformation

to flow problems in reservoirs, Journal of Petroleum Technology, 1949, V. 1,

no. 12, pp. 305–324.

13. Ibrahim M.H., Wattenbarger R.A., Rate dependence of transient linear

flow in tight gas wells, Journal of Canadian Petroleum Technology, 2006, V. 45,

no. 10.

13. Blasingame T.A., Poe B.D. Jr., Semianalytic solutions for a well with a single

finite-conductivity vertical fracture, SPE 26424, 1993.

14. Brown M., Ozkan E., Raghavan R., Kazemi H., Practical solutions for pressure-

transient responses of fractured horizontal wells in unconventional shale

reservoirs, SPE Reservoir Evaluation and Engineering, 2011, V. 14, no. 6