The effective placement of proppant in a fracture has a dominant effect on well productivity. Existing hydraulic fracture models simplify proppant transport calculations to varying degrees and are often found to over-predict propped or effective fracture lengths by 100 to 300%. A common assumption is that the average proppant velocity due to flow is equal to the average carrier fluid velocity, while the settling velocity calculation uses Stokes' law. To accurately determine the placement of proppant in a fracture, it is necessary to rigorously account for many effects not included in the above assumptions. In this study, the motion of particles flowing with a fluid between fracture walls has been simulated using a coupled CFD-DEM code that utilizes both particle dynamics and computational fluid dynamics calculations to rigorously account for both. These simulations determine individual particle trajectories as particle to particle and particle to wall collisions occur and include the effect of fluid flow and gravity. The results show that the proppant concentration and the ratio of proppant diameter to fracture width govern the relative velocity of proppant and fluid. Further, the dependencies of settling velocity on apparent fluid viscosity, proppant diameter and the density difference between the proppant and fluid predicted by Stokes' law were found to apply. However, additional effects have been quantified and shown to substantially alter the predictions from Stokes' law. Proppant concentration and slot flow Reynold's number were both shown to modify the settling velocity predicted by Stokes' law, as does the ratio of proppant diameter to slot width. The effect of leak-off was found to be negligible in terms of altering either the settling velocity or the relative velocity of proppant and fluid. The models developed from the direct numerical simulations have been incorporated into an existing fully 3-D hydraulic fracturing simulator. This simulator couples fracture geomechanics with fluid flow and proppant transport considerations to enable the fracture geometry and proppant distribution to be determined. Unlike all previous studies, these effects are included together and so are shown to be inter-dependent, allowing us for the first time to accurately model proppant transport. As noted above, proppant velocities have been accurately determined without simplifying approximations and with all relevant effects included, showing inter-dependence between the different effects. Two engineering fracture design parameters, injection rate and fluid rheology, have been varied to show the effect on proppant placement in a typical shale reservoir. This allows for an understanding of the relative importance of each and optimization of the treatment to a particular application.
Diagnostic Fracture Injection Tests (DFIT) are commonly interpreted with G-function derivative plots. In these plots, an upward deviation from linearity is generally attributed to fracture height recession or transverse storage. This interpretation neglects changing fracture compliance during closure. Results presented in this study show that in low permeability formations, an upward deviation of the G-function plot G*dP/dG is caused by a sharp reduction in fracture compliance due to closure. It is not necessary to invoke more complicated explanations such as height recession or transverse storage. The specific shape of the G*dP/dG curve is controlled by factors that affect fracture compliance before and after closure. These factors include the fracture height, fracture stiffness, and the residual aperture (which is affected by the microscopic and macroscopic roughness of the created hydraulic fracture). Wellbore storage broadens the post-closure peak in the G*dP/dG curve. A curve of G*dP/dG that constantly increases during the shut-in period may be caused by wellbore storage, rather than fracture tip extension. In higher permeability formations (with non-zero wellbore storage), the effect of fracture compliance is less important. As a result, changing fracture compliance during closure does not have a significant effect on the pressure transient plots, and closure decreases the rate of pressure decline because it reduces the rate at which fluid can leak off into the matrix. In all cases, as long as a single planar fracture has formed, the best pick for the minimum principal stress is at the deviation from linearity on the G*dP/dG plot, whether or not this departure is concave up or concave down. Closure picks via a tangent to near the peak in G*dP/dG (as is commonly performed) may significantly underestimate the minimum principal stress. These picks should be confirmed with other measurements, such as a subsequent injection period or a pump-in flow-back test. The fact that an extremely sharp upward trend in G*dP/dG is generally not observed in DFIT tests suggests that the actual geometry of the created fracture is rougher and more complex than is typically assumed by standard models.
Summary The effective placement of proppant in a fracture has a dominant effect on well productivity. Existing hydraulic-fracture models simplify proppant-transport calculations to varying degrees. A common assumption applied is that the average proppant velocity caused by flow is equal to the average carrier-fluid velocity, while the settling-velocity calculation uses Stokes’ law. To more accurately determine the placement of proppant in a fracture, it is necessary to account for many effects not included in previous assumptions. In this study, the motion of particles flowing with a fluid between fracture walls is simulated with a coupled computational-fluid-dynamics/discrete-element method (CFD/DEM) code that uses both particle dynamics and CFD calculations to account for both particles and fluid. These simulations (presented in metric units) determine individual particle trajectories as particle-to-particle and particle-to-wall collisions occur, and include the effect of fluid flow. The results show that the ratio of proppant diameter to fracture width governs the relative average velocity of proppant and fluid. A proppant-transport model developed from the results of the direct numerical simulations and existing correlations for particle-settling velocity has been incorporated into a fully 3D hydraulic-fracturing simulator. This simulator couples fracture geomechanics with fluid-flow and proppant-transport considerations to enable the fracture geometry and proppant distribution in the main hydraulic fracture to be determined. For two typical shale-reservoir cases, the proppant placement and width distribution have been determined, allowing comparison at the hydraulic-fracture scale, including effects observed at the particle scale. This allows for optimization of the treatment to a specific application, and the results are presented in oilfield units, considered more familiar to our readers.
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