## Summary

Equal proppant distribution between perforation clusters within a stage is one of the goals of hydraulic fracturing design. However, as we know from multiple experiments and field observations, the proppant distribution is not always uniform. Therefore, the natural question is whether it is possible to use a specific design of perforations to achieve a more uniform proppant distribution. This blog post attempts to answer this question using a recently developed model for slurry flow in a perforated wellbore. The primary findings are surprising and demonstrate that a variable perforation phasing transitioning gradually from 180$^\circ$ (bottom of the well) at the heel to approximately 100$^\circ$ at the toe leads to optimal proppant placement. Under some circumstances, such as the reduced injection rate and/or increased perforation diameter, it is possible to have precisely uniform distribution of proppant for all perforation holes, according to the model. If the same perforation orientation is required for all clusters, then the optimal orientation is in the range from 110$^\circ$ to 120$^\circ$, i.e. slightly below the middle of the well. The primary reason for such result lies in the balancing of two opposing phenomena, namely, particle settling in the wellbore whereby there is always higher particle concentration in the lower part of the well, and proppant inertia that causes some particles to miss the perforation. When the aforementioned phenomena are perfectly balanced, then the amount of proppant that goes into the perforation is equal to the average concentration in the wellbore, which, once applied for all perforations, leads to uniform proppant intake for every hole. Despite the mechanisms and trends are clearly captured by the model, the results should also be tested in future laboratory studies for quantitative confirmation. In addition, we are implementing the model into ResFrac to explore potential value of the result for practical cases.

## Problem statement

This blog post is a follow-up to my previous effort and summarizes the findings from the upcoming paper [1]. Previously, a mathematical model for the problem of slurry flow in a perforated wellbore was described and the underlying physical mechanisms were discussed. The purpose of this blog post, on the other hand, is to couple the model with an optimization algorithm to investigate optimal perforation orientations that lead to the desired uniform proppant distribution between perforations. A brief description of the model is added at the beginning to cater for readers who are not familiar with the previous blog post.

Fig. 1 shows schematics for the problem and specifies the two sub-problems that are relevant for the optimization. Fig. 1$(a)$ shows the flow of suspension in a horizontal well and variation of particle volume fraction inside the well is schematically shown. In the heel or upstream part of the stage the flow velocity is sufficiently high to fully suspend particles. However, as the slurry slows down towards the heel of the stage, particles settle, which leads to higher particle concentration in the lower part of the well. Fig. 1$(b)$ shows schematics for the second sub-problem related to proppant turning. Fluid streamlines located within distance $l_f$ from the perforation enter the hole. Due to higher mass density of particles, only particles that are located within a smaller distance $l_p$ are able to enter the perforation.

As follows from [2], the first sub-problem is characterized primarily by the following two parameters
\tag{1}
G = \dfrac{8\phi_m(\rho_p\!-\!\rho_f)g d_w}{f_D\rho_f v_w^2},\qquad t_0 = \dfrac{9\mu_a d_w}{2(\rho_p\!-\!\rho_f)g a^2}.

The parameter $G$ is dimensionless gravity and it quantifies the degree of asymmetry of the particle distribution in the wellbore. At the same time, $t_0$ is the characteristic settling time that provides the time scale for reaching steady-state particle distribution. In the above equation $\phi_m=0.585$ is the maximum volume fraction of particles, $\rho_p$ is particle mass density, $\rho_f$ is fluid mass density, $g=9.8$ m/s$^2$ is gravitational constant, $d_w$ is wellbore diameter, $f_D=0.04$ is fitting parameter that can also be interpreted as a friction factor in the pipe, $v_w$ is the average wellbore velocity, $a$ is particle radius, and $\mu_a$ is apparent viscosity. Note that the parameter $G$ is inversely proportional to Shields number that is commonly used to describe sediment flow. The apparent viscosity accounts for the small scale motion caused by turbulent flow, which in turn triggers non-linear turbulent drag and effectively increases viscosity, see [2] for more details.

To illustrate the variability of particle volume fraction inside the wellbore for various values of $G$, Fig. 2 plots the solution for $G=\{0.1,1,10,10^2,10^3\}$ for the average normalized volume fraction $\langle \bar\phi\rangle = \langle \phi\rangle/\phi_m=0.1$. Such concentration corresponds to approximately 1.4 ppg. Panel $(a)$ shows the variation of the normalized particle volume fraction $\bar\phi=\phi/\phi_m$ along the vertical line passing through the center of the wellbore. Panel $(b)$ shows the variation of the normalized flow velocity $\bar v_x=v_x/v_w$, also along the vertical line passing through the center of the wellbore. Panel $(c)$ shows the variation of the normalized particle volume fraction (the same as in panel $(a)$) within the wellbore’s cross-section. The black color corresponds to the maximum allowable concentration, while the while color corresponds to no particles. Results clearly show that the parameter $G$ significantly affects the particle distribution inside the wellbore. The particles are suspended for small $G$ (or high velocity) and settle at the bottom of the well for large $G$ (or low velocity). The lower left panel also schematically shows potential perforation orientations that illustrate how the orientation can lead to accessing various particle concentrations, especially for large values of $G$.

The proppant turning model can be summarized as follows. First, the value for $l_f$ is calculated by requiring that all the fluid streamlines that are located within $l_f$ enter the perforation, see Fig.  1 $(b)$. The total flow rate in these streamlines is made equal to the flow rate through the perforation. As the fluid slows down in order to enter the perforation, particles start to slip and accumulate the total slip $s$ by the time the horizontal fluid velocity is zero, i.e. by the time the particle reaches the perforation. The magnitude of this slip depends on many parameters, but is driven by the density mismatch between the phases and the drag force that can be either laminar or turbulent. The exact expression for the slip is lengthy and is omitted here, see [2] for more details. What is important is the dimensionless slip, defined as $\bar s = s/d_p$, i.e. the slip normalized by the perforation diameter. Since $s$ does not depend on the perforation diameter, this shows that bigger perforations lead to smaller dimensionless slip and better ability of particles to turn into the perforation. Finally, the value of the dimensionless slip is used to find $l_p$ by requiring that all the streamlines that are located within $l_p$ can have a normalized horizontal slip of up to $\bar s$. The value of $l_p$ is then used to calculate the proppant flow rate through the perforation.

It is instructive to introduce a single dimensionless parameter that quantifies the degree of particle slippage past perforation. If the particle volume fraction $\phi$ is approximately constant within the zone outlined by $l_p$, then the turning efficiency can be defined as
\tag{2}
\eta=\dfrac{\langle\phi\rangle_p}{\langle\phi\rangle_{l_p}}=\dfrac{q^p_p}{\langle\phi\rangle_{l_p}q^s_p}.

In the above equation, the quantity $\langle\phi\rangle_p$ denotes the average volume fraction of slurry that enters the perforation, while $\langle\phi\rangle_{l_p}$ is the average particle volume fraction in the zone outlined by $l_p$. Therefore, the ratio $\eta={\langle\phi\rangle_p}/{\langle\phi\rangle_{l_p}}$ represents the apparent decrease of the particle volume fraction or, in other words, turning efficiency. The smaller the dimensionless slip, the higher the turning efficiency.

## Optimization results

What is interesting about the two sub-problems is that there are situations in which they can compensate each other. In particular, the turning efficiency is always $\eta<1$, which reduces the amount of proppant that enters the perforation. At the same time, with the reference to Fig. 2, particle settling can increase the amount of proppant that enters the perforation if the latter is located in the lower part of the well. This observation can be used to define the “uniformity” lines as
\tag{3}
\eta \phi(\theta)=\langle \phi\rangle_w.

In simple words, this equation states that the volume fraction of particles entering the perforation with orientation $\theta$ should be equal to the average volume fraction in the well. In this case, the volume fraction of particles after the perforation is not going to change, which is the required step towards achieving the uniform proppant distribution. Note the effect of finite $l_p$ is ignored, i.e. it is assumed that $l_p\ll d_w$.

The parametric space for the problem consists of two dimensionless parameters, the turning efficiency $\eta$ defined in (2) and the dimensionless gravity $G$ defined in (1). As was discussed in detail in [cite previous blog post], there are three limits that correspond to the guaranteed uniform distribution $U$, the situation when the last cluster takes all proppant $L$, and the situation when the sensitivity to perforation phasing is the strongest $P$. The uniformity lines are now added and are shown by the grey lines in Fig.3 for different values of $\theta$ and $\langle \bar\phi\rangle=0.1$ (the result depends on the average volume fraction). The most left line corresponds to the perforation oriented downwards or $\theta=180^\circ$. If the data points are located on the left from this line, then it becomes impossible to balance the loss due to turning efficiency by the perforation orientation. At the same time, on the right form this line, it is always possible to find an orientation that would lead to the preservation of volume fraction in the wellbore. This plot clearly demonstrates that perforations located in the lower part of the well can lead to uniform proppant placement. Also, comparisons in [2] indicate that $\eta$ varies from approximately 0.6 to 0.9, while $G$ from under 1 to up to 100. This allows to conclude right away that the optimal perforation orientations are from approximately $100^\circ$ to $180^\circ$ with the majority of points likely falling between $110^\circ$ and $120^\circ$.

##### Variable phasing

To demonstrate the application of optimal phasing, two test cases are considered. The first case is taken from the field scale experiments [3,4], while the second case is taken from the laboratory scale experiments from [5]. The field scale case is called PTST2 and used 100 mesh proppant. The stage consisted of 13 perforation clusters, each having 3 holes with $120^\circ$ phasing. The injection rate is 90 bpm, while the perforation diameter is $d_p=0.33$ in. High Viscosity Friction Reducer (HVFR) was used in the carrier fluid, see complete set of input parameters in [3,4,2].

Fig. 3$(a)$ shows the comparison of proppant volume fraction for each cluster. The red lines with circular markers correspond to the results of experiments, while the black solid lines correspond to model prediction. The amount of proppant for each individual perforation is shown by the square markers. Note that different shades of grey are used to distinguish between odd and even clusters for visualization purposes. Finally, the asterisk markers indicate the average proppant volume fraction in the wellbore before a perforation. Panel $(b)$ shows the azimuth of every perforation. Note that orientations with angles exceeding $180^\circ$ are shown as $360^\circ-\theta$ to have a smaller vertical axis. What is interesting is that the amount of proppant received by each perforation varies drastically towards the end of the stage. This is because the particle settle in that part of the wellbore and, as a result, the perforations located above receive much less proppant than that located in the lower part of the well. Panel $(c)$ shows $(G,\eta)$ parametric space for the field case and markers indicate the trajectories for three different scenarios. The crosses correspond to the original case, the triangles correspond to the case with the reduced rate of 70 bpm, while the circular markers indicate the case with 2 perforations per cluster with an increased diameter $d_p$. Note that the first or heel cluster has smaller $G$, while the toe clusters have larger $G$, therefore the trajectories follow the path from left to right as the observer moves downstream.

The reasoning behind choosing these three cases is the following. The original case is slightly outside of the $180^\circ$ uniformity line. As a result, it will not be possible to achieve completely uniform proppant distribution. Two practical ways to move the first cluster on the right from the $180^\circ$ uniformity line is to either increase the dimensionless gravity $G$, or to increase the turning efficiency $\eta$. By looking at the expression for $G$(1), one simple way is to reduce the average flow velocity by reducing the rate. That’s why the first alternative case has the rate of 70 bpm and the corresponding triangles in Fig. 4 are now all on the right from the $180^\circ$ uniformity line. One of the easiest ways to increase the turning efficiency is to increase perforation diameter. Recall from above that the dimensionless slip is inversely proportional to perforation diameter and thus increasing the diameter reduces the normalized slip and increases the turning efficiency. Changing the particle size can also help the efficiency, but this case already has 100 mesh proppant and therefore further reduction of particle size can negatively impact hydraulic conductivity of fractures and may reduce production. Using lightweight proppant can also be useful. But, as can be seen from (1), the value of $G$ actually decreases, even though the turning efficiency increases. Thus, the movement in the parametric space is in the up-left direction, which approximately follows the $180^\circ$ uniformity line and thus is less efficient. Of course, it is possible to use very lightweight proppant and end up in the $U$ limit, in which the result is always uniform, but this will probably not be economically profitable.

Fig. 4 shows various optimization results for the field scale case: $(a)$ the original case; $(b)$ the same original case, but with $10^\circ$ uncertainty added to the optimal perforation orientation; $(c)$ the case with the reduced rate; $(d)$ the case with an increased perforation diameter. The optimization results for the original case demonstrate that the optimal orientation for the first half of the stage is downwards, while after that the optimal azimuth gradually declines towards approximately $100^\circ$. The early behavior cannot be perfectly uniform since the first several clusters are located outside of the $180^\circ$ uniformity line, but nevertheless, the overall result is more uniform than the original design. Also, which is probably even more important, there is no strong variation of the amount of proppant per individual perforation, which ensures that each individual perforation is used more effectively. The addition of randomness is made to investigate the sensitivity of the result to uncertainties. The proppant distribution is still fairly uniform, but it deteriorates for the last several clusters, for which the value of $G$ is large. The reduction of rate allows to have a perfectly uniform proppant distribution, according to the model and without uncertainties. The optimal orientation gradually descends from $180^\circ$ to approximately $100^\circ$. This is of course possible because the whole trajectory in the parametric space shown in Fig. 3 is on the right from the $180^\circ$ uniformity line. The increase of perforation diameter also significantly improves the optimal result, even though the optimal proppant distribution is not perfectly uniform.

It is also interesting to examine the laboratory scale experiment, available in [5]. This is one of the rare cases, for which the amount of proppant was measured for each individual perforation. There were 3 perforation clusters with 4 holes per cluster with $90^\circ$ phasing. Tap water with 40/70 mesh proppant are pumped with the rate of 79 gal/min, see the complete list of parameters in [5]. Fig. 5 shows the results of comparisons with the model. The results for each individual perforation are compared, as well as the results per cluster are also compared. There is an overall good agreement between the model and the measurement. But what is remarkable, is that there is a strong variability of the amount of proppant received for each individual perforation, which can be sub-optimal. Fig. 5$(c)$ shows the parametric space for this laboratory scale case and crosses show the trajectory inside the parametric space. Clearly, all the points are on the right from the $180^\circ$ uniformity line and therefore it is possible to find optimal perforation orientations to have perfectly uniform proppant distribution.

Fig. 6$(a)$ shows the optimal result when each individual perforation is optimized and the distribution is indeed uniform. The optimal angle varies in a narrow range from $113^\circ$ to $122^\circ$, which is consistent with the location of the points inside the parametric space between $110^\circ$ and $120^\circ$ lines. Recall that the position of these lines shown in Fig. 5 depend on particle volume fraction and are therefore not perfectly universal. Fig. 6$(b)$ shows the effect of $10^\circ$ uncertainty. It is noticeable, especially for the last cluster.

##### Constant phasing

While the optimization of each individual perforation provides the best results in terms of the uniformity of proppant distribution, there can be operational limitations or perhaps other considerations why this may not be the best solution overall. For instance, fracture initiation pressure can be very different for different perforation orientations, or near wellbore pressure drop can be very different for various perforation orientations. It is therefore instructive to consider the optimization that is restricted to having the same perforation orientation for all clusters. Fig. 7 shows such results for the field scale example. The same four cases are considered, original, original with uncertainty, reduced rate, and increased perforation diameter. Results demonstrate that the optimal perforation orientation is approximately $110^\circ$ for all cases. There is a noticeable variability caused by using the same orientation. The result is much less sensitive to the uncertainty and becomes only slightly better if the rate is reduced or the perforation diameter is increased. The actual proppant distribution is quantitatively similar to the original result shown in Fig. 3. The average perforation orientation within the cluster is $100^\circ$ for the $120^\circ$ phasing shown in Fig.3, which is close to the optimal value of approximately $110^\circ$. The major difference, however, is the distribution of proppant for each individual perforation. In the original case, there is a very strong variation per hole, while the optimal results show a much milder variation of the amount of proppant received by each perforation.

Fig. 8 considers the laboratory scale example and employs constant phasing optimization. Fig. 8$(a)$ shows the optimal proppant distribution for the case of optimal azimuth is $118^\circ$. The use of constant phasing for all clusters introduces some variability of the resultant proppant distribution, but it is much more mild compared to the field scale case. Finally, Fig. 8$(b)$ adds $10^\circ$ uncertainty to the latter case, which provides an estimate of the sensitivity of the result to perturbations. These cases as well as the ones shown in Fig. 6 are very close to each other and the optimal perforation orientation is also almost the same. Based on the parametric space shown in Fig. 5$(c)$, this three-cluster laboratory design has relatively large values of $G$, which closely corresponds to the last three clusters within the stage. Recall that for the field case the last clusters also have the optimal perforation azimuth between $110^\circ$ and $120^\circ$.

## Conclusions

This blog post addresses the problem of proppant distribution between perforation clusters for hydraulic fracturing applications. An optimization algorithm is developed based on the recently developed model for slurry flow in a perforated wellbore. It is shown that it is possible to adjust orientation of each individual perforation to achieve more uniform proppant distribution between the clusters. Under some conditions, it is even possible to reach a fully uniform distribution, according to the model. In general, the optimal perforation placement is in the lower part of the well from the bottom at $180^\circ$ (heel clusters) to approximately middle at $100^\circ$ (toe clusters). If the same orientation is required for all perforations, then the range becomes from $110^\circ$ to $120^\circ$, at least for the examples considered. Optimization for the laboratory scale experiment shows a similar trend that the optimal perforation azimuth is between $110^\circ$ and $120^\circ$.

While the obtained result seems almost universal, care must be taken before applying it. First of all, the field scale cases can have quite different parameters, which can shift the optimal azimuths. The second reason is much more significant. The model does not consider the effect of perforation erosion and stress shadow from the previous stage. These effects can significantly change the slurry distribution between the clusters and thus affect the resultant proppant distribution. Nevertheless, the developed model significantly enhances the understanding of the processes occurring in the wellbore and can be used as a building block towards solving the problem with erosion and fractures.

Note that all the perforation angles discussed here are calculated relative to the wellbore center, rather than the center of the perforation gun (located below the wellbore center). The conversion is not difficult and requires the ratio between the perforation gun diameter and the wellbore diameter.

## References

[1] E.V. Dontsov, C.W. Hewson, and M.W. McClure. A model for optimizing proppant distribution between perforations. In American Rock Mechanics Association, Atlanta, GA, 2023.

[2] E. Dontsov. A model for proppant dynamics in a perforated wellbore. arXiv:2301.10855, 2023.

[3] P. Snider, S. Baumgartner, M. Mayerhofer, and M. Woltz. Execution and learnings from the first two surface tests replicating unconventional fracturing and proppant transport. In Proceedings of Hydraulic Fracturing Technology Resources Conference, 1-3 February 2022, Houston, Texas, USA, SPE-209141-MS, 2022.

[4] J. Kolle, A. Mueller, S. Baumgartner, and D. Cuthill. Modeling proppant transport in casing and perforations based on proppant transport surface tests. In Proceedings of Hydraulic Fracturing Technology Resources Conference, 1-3 February 2022, Houston, Texas, USA, SPE-209178-MS, 2022.

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