Proppant transport in the well is an intriguing topic that has attracted a lot of interest in recent years [1-8]. Proppant transport processes determine the uniformity of proppant placement, which has a direct impact on productivity. Cipolla et al. [9] estimate that the difference between poor and excellent proppant uniformity is about $2.5M in net present value per well.

Traditionally, engineers have relied on the simple assumption that proppant travels one-to-one with the fluid in the wellbore, and thus the proppant distribution is proportional to the distribution of the fluid, which is controlled predominantly by limited entry. This assumption is, however, only partially correct. While the proppant is indeed affected by fluid flow, there are other factors that contribute to the final answer. The three primary physical phenomena here are proppant inertia, particle settling, and perforation erosion.

Proppant grains are usually much denser than the fracturing fluid. As a result, during the 90-degree turn from the wellbore into a perforation, some of the particles are unable to complete the turn and ‘miss the hole,’ and remain in the wellbore. This process promotes a toe bias – i.e. more proppant flowing into the toe (or downstream) clusters of the stage.

In practice, toe-bias is usually not observed in field data. This occurs because several other physical processes, such as erosion, counteract the effect of particle inertia. Particles that are barely able to make the turn tend to hit the downstream side of the perforation and cause a higher rate of erosion in the downstream direction. This process leads to large, elongated perforations that can be observed using downhole imaging techniques after the treatment. As they erode, a positive feedback loop develops as bigger perforations tend to accept more fluid and thus attract more proppant. Downstream erosion is more rapid when there is a higher particle velocity in the wellbore, and so this proppant velocity effect is stronger towards the heel of the well [5]. This process often dominates over the aforementioned inertial effects and leads to a heel bias in field data.

Finally, particle settling in the wellbore is a very important and often overlooked phenomenon. As the flow rate decreases along the stage, the fluid velocity becomes too low to fully suspend particles, which leads to settling. Consequently, perforation phasing starts to play a significant role since perforations that are located in the upper part of the well receive less proppant, while the perforations that are located in the lower part of the well receive more proppant. This effect, combined with the non-uniform erosion, proppant inertia, and variations of the initial perforation diameter versus phasing (or gun clearance), leads to complicated coupled mechanisms that determine the final fluid and proppant distribution between perforation clusters.

To address this complex problem, we developed a standalone wellbore simulator called StageOpt. It captures perforation and wellbore scale phenomena without explicitly modeling hydraulic fractures. This leads to rapid computations and enables users to perform analysis and optimization quickly. The primary drawback of StageOpt, however, is the lack of fractures and the inability to evaluate production. Thus, to provide a more general solution, the proppant transport physics has been implemented into our flagship simulator ResFrac Pro [10].

The remainder of the article is focused on comparison between the StageOpt and ResFrac Pro for several test cases. The same identical scenarios are simulated with both StageOpt and ResFrac Pro. Then, the results are compared and similarities and differences are highlighted.

The simulations have 8 clusters with 3 shots per cluster, 80 bpm maximum injection rate, 1 ppa proppant concentration of 40/70 mesh, and a total injection time of 1.5 hours, as shown in the table above.

The first simulation imitates an ‘unoriented’ perforation design, in which the actual perforation angles relative to the gun are 0, 120, and 240 degrees. While StageOpt can model randomness of phasing associated with the unoriented gun by performing hundreds of Monte Carlo simulations, this is not feasible in ResFrac Pro, and therefore a defined phasing is used.

The figure below shows the simulation result assuming no stress shadow interaction between the fractures. Black markers correspond to StageOpt, while the red markers correspond to ResFrac Pro. The panels in the figure show phasing, equivalent perforation diameter after treatment, and slurry volume versus perforation number (1 – heel, 24 – toe). Proppant mass and slurry volume are plotted for each cluster (1 – heel, 8 – toe). The equivalent perforation diameter is simply $\sqrt{4 A⁄π}$, where A is perforation area. This definition is useful when perforation holes are not circular, which is true in our case due to the directionality of erosion.

As we can see from the figure, there is an excellent agreement between StageOpt and ResFrac Pro. Perforation diameters vary noticeably within each individual cluster due to different initial diameters (because shots along the top of the well are further from the perforation gun due to gravity, and so have smaller initial diameter). There is an overall heel bias due to perforation erosion being affected by the velocity in the wellbore.

Although this particular scenario has heel bias, in real data, a variety of different heel/toe bias trends are observed. Sometimes, wells exhibit a toe bias, sometimes a heel bias, sometimes a heel/toe bias (least in the middle), and sometimes there is a flat distribution. For model calibration in a particular setting, the model has a handful of tuning parameters that increase or weaken the effect of various physical processes, allowing the model to match the diversity of behaviors that are observed in reality.

The slurry distribution correlates with the diameter distribution. The proppant distribution is affected by the inertial effect and has a smaller heel bias than the slurry distribution. In addition, there is some toe bias in the final cluster that is associated with the accumulation of proppant concentration in the wellbore due inertial effects. Comparison between ResFrac Pro and StageOpt shows only very minor differences. These are associated with details of implementation (such as discretization of the wellbore in ResFrac Pro), and also some differences in the physics, such as the presence of explicitly modeled fractures, fluid compressibility, and wellbore friction.

The next figure plots a similar comparison, but with stress shadow turned on. There are some visible discrepancies in this case. The perforation friction was not much higher than the net pressure at the end of the simulation, which resulted in somewhat altered flow distribution. Specifically, the first fracture received less fluid and proppant. This in turn affected proppant mass distribution and erosion, as highlighted in the figure. These differences between StageOpt and ResFrac Pro are not unexpected. StageOpt has a simplified ‘fracture stress shadow’ model, while ResFrac Pro has a much more detailed representation of fracture growth and stress interaction.

The figure below compares StageOpt and ResFrac results for a similar scenario, but for oriented vertical (or zero degree) perforations. Consistent with the previous comparisons, there are some minor differences between StageOpt and ResFrac Pro associated with stress shadow, but the overall trends are very similar. Note the ‘inline effect’ in the results, whereby the second and third perforations within each cluster receive more proppant than the first. This occurs because particles that miss the first perforation tend to accumulate near the azimuth of this perforation. As a result, if the following perforations have the same orientation, the particles can more easily enter the subsequent perforations.

To summarize, the effects associated with proppant transport and distribution in a perforated wellbore have been implemented in ResFrac Pro. This enables users to investigate the effect of particle distribution on fracture growth and production. ResFrac Pro can be used in conjunction with StageOpt. For instance, users can run a series of rapid simulations to optimize perforation design with StageOpt and then run a broader set of simulation with ResFrac Pro to evaluate additional design parameters, such as well spacing.

Here are a few caveats to keep in mind. There are inherent differences between StageOpt and ResFrac Pro, so that the results will never be identical. For instance, ResFrac Pro considers fluid compressibility. For typical problems, this is not the issue, but it can play a role for highly compressible fluids. There is pipe friction in ResFrac Pro, which is not included in StageOpt. Again, for typical problems, this should not cause a big discrepancy, but it can for some exotic cases. Stress shadow and the effect of fractures is perhaps the biggest influencer on the difference between the two algorithms, as discussed in this blog post. Specifically, stress shadow in conjunction with relatively low perforation friction (e.g. caused by high rate of erosion) is able to alter the fluid distribution so that fractures with higher levels of stress shadow receive less fluid. In addition, the possibility of proppant screen-out in some of the fractures as well as strong fluid redistribution if one of the fractures reaches a depleted area are also not captured in StageOpt. Finally, there are some other small differences that need to be considered, such as surface injection schedule is used in ResFrac Pro and downhole schedule is used in StageOpt, and fluid rheology models are slightly different in two algorithms. Nevertheless, as is shown in this blog post, the results are nearly identical for the considered case that represents reasonable parameters that are encountered in the field. This enables users to employ StageOpt and ResFrac Pro together to solve complicated optimization problems.

For more information on StageOpt, readers are encouraged to read this blog post. Those who are interested in mathematical foundations of the algorithm can find them in [3].

The new capability will be released in our next ResFrac update, which will be within a few weeks. Once it is out, please give it a try, and let us know what you think! Here is a brief summary of the user-experience in the app.

In the Simulation Builder, under the ‘Wells and Perforations’ panel, there is a radio button that allows the user to select a perforation phasing option. The ‘NoEffect’ option corresponds to the original ResFrac implementation, in which there is no effect of phasing on proppant distribution (i.e. the amount of proppant that flows into a perforation is proportional to the slurry flow rate and proppant concentration in the wellbore). There are two more options. The ‘BuiltIn’ option allows users to select one of the typical designs that are common in field applications. These include ‘Top’, ‘Bottom’, ‘TopBottom’, ‘Side’, ‘Unoriented60’, ‘Unoriented90’, and ‘Unoriented120’. The ‘Custom’ option allows users to define and use any other perforation design, as shown in the example below.

There are a variety of other options that are related to adjusting perforation erosion (such as the effect of lateral wellbore velocity on perforation erosion), initial perforation diameter and its variation with gun clearance, as well as various uncertainties that are customary in the field.

On the output side, we have added the ‘proppant distribution’ panel, as shown in the top right panel in the figure below.

The new proppant transport plots have been added in addition to our standard plots that can be either: (a) line plots (showing data such as WHP or production), (b) 3D plots (showing the distribution of various properties in the well, matrix, and/or fractures), (c) plots of fracture and production properties versus depth, or (d) coming soon, heat maps of proppant transport and production looking down the gun barrel.

In the example above, the ‘proppant distribution’ panel shows the cumulative proppant mass outflow for each perforation for three consecutive stages along a well. Alternatively, the plotting tool provides options to plot other quantities of interest, such as perforation diameters, perforation areas, slurry volume, proppant and slurry flow rates. The properties can be plotted on a per-cluster or per-shot basis. It is possible to observe the temporal evolution of these properties by cycling through temporal snapshots. Finally, for any particular perforation shots or cluster, properties can be plotted versus time using the visualization tool’s existing line plot functionality.

To sum up, the effect of phasing on proppant distribution between perforations and clusters has finally been implemented in ResFrac Pro! It has been tested and fine-tuned to ensure a consistent answer between ResFrac Pro and StageOpt, thus enabling synergy between the two being used to solve practical problems. StageOpt can be used for relatively quick evaluations, while ResFrac Pro can be used to accurately account for the effect of hydraulic fractures as well as to evaluate the effect of perforation design on production.

Once it is released, we look forward to hearing your feedback on the user-experience in the simulation builder and visualization tool, and most importantly, we hope that you find it useful in optimizing your perforation designs and maximizing recovery.

References

[1] Ahmad, F.A., and Miskimins, J.L. 2019a. Proppant transport and behavior in horizontal wellbores using low viscosity fluids. In Proceedings of the Hydraulic Fracturing Technology Conference and Exhibition, Houston, TX, SPE-194379-MS.

[2] Ahmad, F.A. and Miskimins, J.L. 2019b. An experimental investigation of proppant transport in high loading friction-reduced systems utilizing a horizontal wellbore apparatus. In Proceedings of the Unconventional Resources Technology Conference, Denver, CO, URTEC-2019-414-MS.

[3] Dontsov, E.V., 2023. A model for proppant dynamics in a perforated wellbore. International Journal of Multiphase Flow 167, 104552.

[4] Dontsov, E., Hewson, C., and McClure, M. 2023b. Analysis of uniformity of proppant distribution between clusters based on the proppant wellbore dynamics model. In Proceedings of the Unconventional Resources Technology Conference, Houston, TX, URTEC-3854538-MS.

[5] Dontsov, E., Ponners, C., Torbert, K., and McClure, M. 2024. Practical Optimization of Perforation Design with a General Correlation for Proppant and Slurry Transport from the Wellbore. In Proceedings of the Hydraulic Fracturing Technology Conference and Exhibition, Houston, TX, SPE-217771-MS.

[6] Liu, X., Wang, J., Singh, A., Rijken, M., Wehunt, D., Chrusch, L., Ahmad, F., and Miskimins, J. 2021. Achieving near-uniform fluid and proppant placement in multistage fractured horizontal wells: A computational fluid dynamics modeling approach. SPE Production & Operations, 36: 926–945.

[7] Snider, P., Baumgartner, S., Mayerhofer, M., and Woltz, M. 2022. Execution and learnings from the first two surface tests replicating unconventional fracturing and proppant transport. In Proceedings of the Hydraulic Fracturing Technology Conference, The Woodlands, TX, SPE-209141-MS.

[8] Wu, C.-H. and Sharma, M.M. 2016. Effect of perforation geometry and orientation on proppant placement in perforation clusters in a horizontal well. In Proceedings of the Hydraulic Fracturing Technology Conference, The Woodlands, TX, SPE-179117-MS.

[9] Cipolla, C.,  Singh, A., McClure, M. McKimmy, M. and Lassek, J. 2024. The Perfect Frac Stage, What’s the value? In Proceedings of the Unconventional Resources Technology Conference, Houston, TX, URTEC- 4044071-MS.

[10] McClure, M., Kang, C., Hewson, C.,  Medam, S., Dontsov, E., and Singh, A.. 2023b. ResFrac Technical Writeup. 14th Edition. ArXiv:1804.02092.