Adding Depth: The Superiority of Volumetric Perforation Erosion Analysis for Evaluating Stimulation Performance and Optimizing Completion Designs Free

When downhole perf imaging was initially introduced, the oil and gas industry used high-resolution acoustic imaging tools to assess perf holes as basic shapes with an equivalent circular diameter (Robinson et al. 2020). Over time, data began revealing anomalies like elongated perfs resembling keyholes in the heel clusters of many stages, along with significant outliers in terms of growth (Littleford et al. 2021). These observations led to a transition from simple circular metrics to more advanced measurements that accurately capture the two-dimensional (2D) areal profile of a perf at the casing’s outer diameter (OD), greatly enhancing the accuracy and reliability of post-frac performance evaluations.

As more data was accumulated, an evaluation of several hundred thousand perfs highlighted scenarios that were inadequately addressed by this approach. These included cases where exit holes did not appreciably grow, which is commonly observed in completion designs with low perf friction pressure and situations where perforating charges performed differently than indicated by surface tests. To address the phenomenon of minimal exit hole growth, additional measurements along the innermost surface of the perf at the casing’s inner diameter (ID) became necessary. These entry hole measurements clearly demonstrated that perf growth was a 3D issue rather than a 2D one, requiring the consideration of volumes instead of areas. With advancements in software for analyzing perfs and the integration of machine learning models, comprehensive volumetric growth analysis is now achievable, completing the evolutionary journey (Fig. 1).

The objective of this paper is to evaluate the validity and practical application of volume-based perf erosion metrics. Prior to presenting an overall evaluation, it reviews existing literature on perf design concepts for hydraulically fractured wells, explores the theory behind perf erosion, and highlights key metrics used to assess stimulation performance. Following these elements, a case study conducted jointly by Oxy and DarkVision is presented, validating volumetric perf erosion data against permanent fiber optics in a horizontal Niobrara well of the Powder River Basin (PRB).

The significance of perf design in hydraulic fracturing treatments is not a novel concept. As a matter of historical fact, perf friction pressure is fundamentally based on Bernoulli’s Theorem, which was formulated in 1738 and concerns the conservation of energy. As detailed by Brittanica, it states that if fluid flows horizontally without a change in gravitational potential energy yet experiences an increase in velocity, there must be a corresponding decrease in pressure. A modified version of Bernoulli’s original equation adapted for orifice flow, which is presented below in Eq. 1, is the version most frequently used in oil and gas applications at this time.

where:

∆P_perf is the frictional pressure drop associated with fluid flow through a restriction,

ρ_f is the density of the fluid,

Q is the volumetric flow rate of the fluid,

N_p is the number of holes open to flow,

D_p is the average or nominal hole diameter, and

C_d is the discharge coefficient.

The work done by Murphy and Juch (1960) is perhaps one of the earliest references in technical literature to the engineered application of perf friction pressure in downhole treatments. They concluded that multiple hydrocarbon-bearing zones in a single wellbore could be effectively treated simultaneously by optimizing slurry placement through the use of strategic perf design. While they do not explicitly list the equation for perf friction pressure in their paper, the attention to hole size and density within zones alludes to the importance of these elements. Although the term "pin-point" fracturing is used, their work actually serves as the foundation for the term "limited entry" that is commonly used today.

As hydraulic fracturing treatments became more prevalent for improving well performance in lower-permeability formations, greater focus was placed on perf design to optimize production rates and decrease completion costs. In one of the most well-known and referenced studies, Cramer (1987) identified a correlation between the amount of proppant pumped through a perf and the increase in its hydraulic diameter. This diameter is determined by both the hole size and a discharge coefficient (Eq. 1), which relates uphole and downhole flow stream diameters while functioning as a correction factor to indicate the extent of erosion during treatment. It is worth mentioning that the relationship outlined by Cramer in the study forms the basis for virtually all perf erosion analyses performed following its publication. Whether or not increases in hydraulic diameter are truly a function of proppant throughput rather than inertial effects is an ongoing subject of investigation, but that is beyond the scope of this paper.

Over the past three decades, additional studies have been performed that continue to emphasize the importance of perf friction pressure as it pertains to hydraulic fracturing treatments. Some of this work has been performed in a laboratory setting to more intimately understand the relationship between hole size and treatment design parameters in a controlled environment (Willingham et al. 1993). Others have been performed using actual field data acquired during stimulation operations with the intent of quickly and easily optimizing designs (Weijers et al. 2000).

As the capabilities of downhole technology improved, the complexity of completion design variable testing increased as well (Ugueto et al. 2016). Today, robust simulators capable of predicting proppant placement among clusters in the stage of a horizontal well as a function of many different parameters are commercially available (Dontsov et al. 2023). Suffice it to say that the oil and gas industry’s understanding of perf design and the role it plays in well performance has increased exponentially over the past 65 years.

While the importance and influence of these historical studies cannot be overstated, it is important to note that they represent just a small portion of all the research conducted on this subject so far. Nevertheless, concerning perf friction pressure specifically, these studies are limited to a 2D, or, at best, a pseudo-3D, analysis because they use a single value along with a correction factor to represent dynamic hole size. In cased and cemented completions using plug-and-perf technology, where exactly should the diameter of these holes be measured? Are they on the casing’s ID, the casing’s OD, or somewhere in between? Although this may seem like a rhetorical question, it highlights the core issue and the focus of this paper. With the availability of 3D measurements and a clear understanding of their implications, there is no longer a need to rely on simplified analyses using a singular diameter or area.

As previously noted, Cramer (1987) theorized that perf erosion occurs as a function of proppant placement in two distinct phases, which are entry hole rounding followed by linear growth along the perf tunnel. In that study, a section of N-80 casing was perforated, and measurements were gathered manually using handheld calipers. As detailed by Robinson et al. (2020), high-resolution acoustic imaging technology now allows for submillimetric perf measurements to be taken before and after a hydraulic fracturing treatment is performed in a true downhole environment. Fig. 2 illustrates the stages of perf tunnel growth during the stimulation process. Each stage is depicted as a radial cross section of the perf tunnel with the casing ID on top and the casing OD on bottom.

Although the visualization is fairly straightforward, a brief explanation of each stage during the growth process depicted in Fig. 2 is provided below for additional context:

Stage 0 – This stage represents an unstimulated perf. Here, a conical frustum, which is essentially a cone with its top removed, can be observed. The narrowest section of the tunnel corresponds to the exit hole and is located at the edge of the casing’s OD. The entry hole on the casing’s ID is marginally larger than the exit hole.

Stage 1 – This stage corresponds to the rounding phase proposed by Cramer (1987). At this point in the proppant placement process, the entry hole on the casing’s ID has slightly enlarged and become beveled due to a modest amount of proppant passing through the perf at high velocity. The exit hole on the casing’s OD has experienced minimal or no growth.

Stage 2 – This stage signifies the linear growth phase also proposed by Cramer (1987). At this juncture in the proppant placement process, increased proppant passage through the perf causes both the entry hole on the casing’s ID and the exit hole on the casing’s OD to expand proportionally. This is also the stage in which an effectively stimulated perf will reside at the end of a treatment according to traditional standards.

Stage 3 – This is the last and most extreme phase in the growth process. It usually occurs with "runaway" perfs that have accepted much larger quantities of proppant than initially planned in the completion design. At this stage, the linear growth proportional to the entry hole on the casing’s ID and the exit hole on the casing’s OD stops, making further growth entirely unpredictable. This is also observed in scenarios where perfs are shot through a casing connection, in which case the entry and exit hole areas become highly eccentric.

Erosional growth tendencies depend on many factors. The most notable is perf friction pressure, which, as illustrated in Eq. 1, depends on fluid density, volumetric flow rate, the number of holes open to flow, and the average or nominal hole diameter. Perf friction pressure is also contingent on the discharge coefficient, but this is dynamic and linked to the conceptual growth stages shown in Fig. 2. An important observation from analyzing numerous datasets is that completion designs with low perf friction pressure typically result in perfs settling in Stage 1 at the end of treatment, characterized by mostly entry hole growth with minimal exit hole growth. Conversely, designs with high perf friction pressure tend to result in perfs settling in Stage 2, and sometimes Stage 3, by the end of treatment. Other factors influencing erosional growth tendencies include proppant size, proppant type, fluid properties, and pipe hardness.

This observation, coupled with the earlier discussed notion that entry holes on the casing’s ID will bevel and grow before any growth occurs in exit holes on the casing’s OD, indicates that traditional perf erosion analyses relying solely on exit hole data are fundamentally flawed. The limitation of exit hole growth due to the perf friction inherent to the completion design should not skew the interpretation of results, which can potentially lead operators to make incorrect decisions on future designs based on incomplete information. Incorporating entry hole measurements helps address this issue, but a more effective analysis method involves combining both entry and exit hole measurements to evaluate perfs volumetrically.

Armed with these concepts of perf erosion during the proppant placement process, the following section of this paper examines how to evaluate entry and exit hole measurements while also emphasizing the significance of the volume of metal removed due to erosion. As mentioned previously, perf erosion analyses have historically been performed using exit hole measurements on the casing’s OD exclusively. The addition of entry hole measurements along the casing’s ID is a relatively recent advancement. However, it has quickly become apparent that exit hole measurements alone do not fully account for all the subtle nuances associated with proppant placement and erosional phenomena. To demonstrate this, consider the theoretical perf tunnel growth matrix presented in Fig. 3.

The first column of the matrix serves as the starting point for each case, with the assumption that the entry and exit hole areas are exactly the same with a ratio of 1. Although entry and exit hole dimensions always differ in reality, even for uneroded perfs, this condition represents a perfectly uniform perf tunnel in the theoretical sense. As the columns of the matrix progress from left to right, the entry-to-exit hole area ratio increases monotonically, which indicates more growth in the entrance hole relative to the exit hole.

The first row of the matrix assumes the exit hole area remains constant, regardless of the entry-to-exit hole area ratio. This represents a scenario in which all growth due to erosion occurs exclusively at the entry hole. It is worth noting that traditional perf erosion analyses using exit hole measurements only would indicate no perf erosion has occurred in this scenario, despite obvious changes to the entry hole area. As the rows of the matrix descend from top to bottom, the rate of change for the exit hole diameter continually becomes more extreme, starting initially at 0% and eventually increasing to 30% of the change in the entry-to-exit hole area ratio.

Many physical mechanisms and completion design factors can cause these different types of perf behavior, some of which were discussed in the previous section. However, the important concepts to understand for the sake of diagnostics are the relationships between entry and exit hole areas and what they imply concerning perf growth tendencies. An easy way to visualize this is by creating a plot of entry and exit hole area ratio versus difference.

Fig. 4 shows this plot with cases of entry hole growth only while a constant exit hole area is maintained for different initial exit hole diameters. Notice that this condition generates a straight line for every case, and the slope of each line is contingent on the initial exit hole diameter. Shallower slopes correspond to larger perfs while steeper slopes correspond to smaller perfs. In fact, the linear slope of entry and exit hole area ratio versus difference for any case of entrance hole growth only can easily be determined from a perfect power law correlation with initial exit hole diameter, which is also demonstrated in Fig. 4.

While the condition of entry hole growth only is straightforward and easy to understand, it is unlikely in practical applications that there will be situations with absolutely no exit hole growth. To account for variable growth conditions, Fig. 5 shows the same plot as before but with constant and variable exit hole behavior for two different initial exit hole diameters. Notice that the degree of curvature and departure from the straight line case, which, as shown in Fig. 4, represents entry hole growth only, is contingent on the rate of change for the exit hole area.

With these theoretical cases in mind, it becomes clear that perf growth tendencies can be quite predictable when entry and exit hole measurements are made available. On a plot of entry and exit hole area ratio versus difference, a straight line represents entry hole growth only while a constant exit hole area is maintained, with the slope of that line contingent on the initial exit hole diameter. The degree of curvature and departure from the straight line is contingent on the rate of change with respect to the exit hole area. These realizations allow for the creation of a novel yet simple diagnostic plot, which is shown in Fig. 6.

To demonstrate the value, validity, and veracity of this new diagnostic plot, perf erosion data from a horizontal well completed by Oxy in the Niobrara formation of PRB is presented in Fig. 7. This well was equipped with permanent fiber optics, which required in-line perforating with orientations dictated by the position of the fiber on the outside of the production casing. The data is from perfs shot in Sextant #1, which resides at the top of the borehole between 30-330 degrees clockwise (CW).

The dashed line anchored at the origin of the plot has a slope of 6.576 in^-2. According to the power law relationship demonstrated in Fig. 4, this implies that the initial exit hole diameter for perfs shot in Sextant #1 is 0.44 in. Notice that some datapoints lie to the left and right of this line, but many are relatively close to it, including those with large magnitudes of ratio and difference. This suggests that much of the erosion during proppant placement has occurred in entry holes rather than exit holes.

To the right of the diagnostic plot in Fig. 7 are submillimetric acoustic intensity images for several perfs of interest that were generated using the high-resolution acoustic imaging technology discussed previously. The position of each perf within the plot is marked as A, B, C, D, or E on the acoustic intensity images for ease of identification. A description of each datapoint along with some observable trends is provided below:

Perf A has an entry hole diameter of 0.47 in. and an exit hole diameter of 0.44 in. The entry-to-exit hole area ratio is 1.15 (low), with a difference of 0.02 in² (low). The datapoint for this perf is in the lower left corner of the plot near the origin, which is the region where uneroded perfs are expected to reside. This perf was shot in a stage that was not stimulated due to operational issues related to the fiber optic cable. The datapoint is intersected by the dashed line, which helps validate an initial exit hole diameter of 0.44 in. for perfs shot in Sextant #1. This perf corresponds to Stage 0 in Fig. 2.

Perf B has an entry hole diameter of 0.48 in. and an exit hole diameter of 0.29 in. The entry-to-exit hole area ratio is 2.62 (moderate), with a difference of 0.11 in² (low). The datapoint for this perf is in the left portion of the plot above the dashed line, which is the region where perfs with smaller exit hole diameters are expected to reside. Perfs in this region are typically uneroded, or substantially less eroded, than others in the same dataset. This perf also corresponds to Stage 0 in Fig. 2, but it clearly has a smaller exit hole diameter than Perf A, which is likely the result of atypical perforating charge performance.

Perf C has an entry hole diameter of 0.83 in. and an exit hole diameter of 0.44 in. The entry-to-exit hole area ratio is 3.58 (high), with a difference of 0.39 in² (moderate). The datapoint for this perf resides along the dashed line, but it has higher magnitudes of ratio and difference. Based on the theoretical cases discussed previously, this implies that the perf has been significantly eroded with respect to the entry hole, but not with respect to the exit hole. The image very clearly shows this to be the case, and this perf corresponds to Stage 1 in Fig. 2.

Perf D has an entry hole diameter of 1.08 in. and an exit hole diameter of 0.61 in. The entry-to-exit hole area ratio is 3.17 (high), with a difference of 0.63 in² (high). The datapoint for this perf is located in the upper right section of the plot and resides below the dashed line. This region typically includes perfs with moderate exit hole growth that matches entry hole growth. Despite the minor elongation observed in the growth of the exit hole, the acoustic intensity image confirms this to be the case. This perf aligns with Stage 2 in Fig. 2.

Perf E has an entry hole diameter of 1.67 in. and an exit hole diameter of 1.39 in. The entry-to-exit hole area ratio is 1.44 (low), with a difference of 0.67 in² (high). It is clear from the image that this perf has a high degree of eccentricity, so the quoted values are effective diameters calculated from the measured areas. The datapoint for this perf is in the lower right corner of the plot, which is the region where more extreme exit hole growth is expected, and this is evident. This is also the region of the plot most commonly associated with perfs shot in casing connections, which causes abnormal erosion due to flow through threads (Littleford et al. 2021). This perf corresponds to Stage 3 in Fig. 2.

One of the most impactful outcomes of this new diagnostic plot is the ability to empirically determine uneroded perf measurements without having access to a dedicated set of unstimulated reference perfs. If the initial exit hole diameter controls the slope of the line, as clearly shown from the theoretical cases, and a minimum entry-to-exit hole area ratio is reasonably determined from the dataset of stimulated perfs, then the initial entry hole diameter can easily be calculated. This approach can be extremely beneficial for perforating charges that have a high coefficient of variation with respect to initial exit hole diameters, which can severely impact growth calculations and interpretation of the data.

While this empirical determination method was validated using an internal dataset with well-known completion details, its accuracy was tested using anonymous high-resolution acoustic imaging data from a blind test. The operator owning the data and the completion design specifics were not disclosed to the analyst. The only information provided was that a standard deep penetrator charge with 60 degree phasing was used. Fig. 8 illustrates the empirical determination of the initial exit hole diameter compared to the unstimulated reference perf measurements for Sextants #3 and #5. To clarify, Sextant #3 resides in the lower right portion of the borehole from 90-150 degrees CW, and Sextant #5 resides in the lower left portion of the borehole from 210-270 degrees CW. It is appropriate to assume that perfs shot in these regions will likely have the same initial dimensions as the magnitude of gun clearance is the same for both. As shown, the empirical determination indicates an initial exit hole diameter of 0.34 in., which matches the value obtained from the unstimulated reference perf measurements.

While this alone demonstrates the analytical improvements afforded by the availability of entry and exit hole measurements, the true benefit of capturing high-fidelity 3D data is the ability to represent perfs volumetrically rather than as simple areas or diameters. Fig. 9 shows the equation for calculating the volume of a perf tunnel, which is represented as a conical frustum.

Although the benefits of upgrading from a basic 2D analysis to a more robust 3D analysis are intuitive and discussed in previous sections, it is important to validate this conceptually. The perf matrix presented in Fig. 3 provides the framework for different perf growth scenarios, primarily as a function of the rate of change for the exit hole diameter during the proppant placement process. Based on this, Fig. 10 plots the pre-to post-stimulation volumetric growth ratio versus difference, which is similar in form to the diagnostic plot introduced in Fig. 6, for three different rates of exit hole growth.

As the plots clearly indicate, volumetric growth ratios and differences exhibit a linear relationship, with the slope of that line once again contingent on the initial exit hole diameter. Notice that, while the magnitudes of ratio and difference correspond to the rate of change for the exit hole, the slope is the same, no matter the scenario. This linearity in the presence of nonunique growth conditions serves as the foundation for why volumetric perf erosion analysis is superior to traditional analyses based on exit hole measurements exclusively. To further validate this, Fig. 11 shows a plot of perf tunnel volume growth ratio versus difference for the same data presented in Fig. 7.

Equipped with a thorough understanding of how perfs grow as a result of erosion during the proppant placement process, and how baseline perf measurements can be empirically determined from measured entry and exit hole areas, the following section explains three distinct metrics derived from the volumetric data and their application in evaluating stimulation performance.

The volume of casing material eroded during hydraulic fracturing, referred to in this paper as Delta Volume (DV), can provide valuable insights. While perf-and-cluster level DV values offer more precise metrics for evaluating stimulation performance as a function of completion design parameters, stage-level DV values are also highly informative. Ideally, stages with identical completion designs and proppant pumping schedules should show similar post-frac DV values. However, numerous datasets show that DV values can vary significantly from stage to stage along the lateral of a horizontal well. This variation is particularly prominent in stages experiencing a plug failure, which causes a loss of isolation from previously treated intervals (Greer et al. 2023; Pehlke et al. 2024). Wien et al. (2024) highlighted the negative impact this can have on the over-or under-treatment of stages, underscoring the value of volumetric perf measurements, even when they are aggregated at the stage level.

Stimulation Distribution Efficiency (SDE) metrics, like Uniformity Index (UI), calculated using DAS-based algorithms from permanent fiber optic data, have been in use for some time (Barhaug et al. 2022; Huckabee et al. 2022; Zakhour et al. 2021). UI in these applications is typically calculated at the cluster level rather than the perf level due to spatial resolution limitations. Eq. 2 below shows how this is calculated for each individual stage:

where:

PUI is the proppant uniformity index,

σ_prop,% is the standard deviation of proppant allocation percentages of the clusters in a stage, and

μ_prop,% is the mean of proppant allocation percentages of the clusters in a stage.

To ensure a fair comparison, it is important to discuss how a similar metric using volumetric perf measurements can be calculated. For every perf in a stage, the initial uneroded volume determined from unstimulated reference perf measurements, or using the empirical approach discussed earlier, can be subtracted from the final eroded volume to calculate the volume of growth at the perf level. The growth volume for every perf in a cluster can then be summed to determine the volume of growth at the cluster level. These values can then be used to compute a Volumetric Uniformity Index (VUI) for each stage on a percent growth basis. Eqs. 3-5 below show how this process is performed:

where:

∆V_clstr,n is the volume of growth for a cluster within a stage,

V_fperf,n is the final eroded volume of a perf within the cluster, and

V_iperf,n is the initial uneroded volume of a perf within the cluster from a predetermined baseline value.

where:

∆V_clstr,n% is the percentage of volumetric growth for a cluster relative to the total growth within a stage, and

∆V_clstr,n is the volume of growth for a cluster within a stage.

where:

VUI is the volumetric uniformity index,

σ_∆Vclstr,% is the standard deviation of volumetric growth percentages of the clusters in a stage, and

μ_∆Vclstr,% is the mean of volumetric growth percentages of the clusters in a stage.

It is easy to see that the equations for PUI and VUI are very comparable, and both are intended to help operators determine which completion design is the most effective in a testing program. The cluster-level proppant allocation percentages from fiber and the cluster-level volumetric growth percentages from high-resolution acoustic imaging can also both be used to generate hydraulic fracture profiles (HFPs) that provide insights regarding proppant placement and treatment bias for each stage.

Under ideal conditions, every perf within a stimulated stage would experience the same magnitude of volumetric growth, which would result in a VUI value of 1. Even though it is the goal of virtually every operator, this, of course, is unrealistic and hardly ever happens in practical applications. Analysis of numerous datasets has shown large standard deviations with respect to volumetric growth. In some extreme cases, when the standard deviation is larger than the mean, VUI values can actually be negative. Negative VUI values are more commonly seen when excessive plug isolation issues occur, but they can also come about if an analyst chooses to calculate VUI at the perf level rather than the cluster level, which, as discussed previously, is an accurate and fair comparison to PUI values from permanent fiber. To illustrate the dilemma of perf level versus cluster-level VUI values, consider the bar charts presented in Fig. 12 and Fig. 13 that correspond to the same stage. Fig. 12 shows volumetric growth at the cluster level with a VUI of 0.85, and Fig. 13 shows volumetric growth at the perf level with a VUI of 0.68.

While this disparity in VUI values may seem counterintuitive at first, it is important to note that standard deviation is heavily influenced by sample size, which is larger for calculations performed at the perf level than calculations performed at the cluster level, at least for multi-perf cluster stage configurations. Ultimately, the analyst determines whether VUI values are calculated at the perf level or the cluster level, but it is the opinion of the authors that it should be calculated at the cluster level to maintain similarity with PUI values calculated from permanent fiber. At a minimum, the calculation method should be kept consistent across datasets.

UI, whether with respect to exit hole areas or perf tunnel volumes, can also serve as a valuable pre-stimulation metric. The computed UI for a set of unstimulated reference perfs is an excellent predictor of the ultimate post-stimulation UI potential. For instance, if the unstimulated perfs of a stage exhibit a pre-stimulation UI of 0.75, it is highly improbable that the post-stimulation UI will surpass this initial value after significant erosion has taken place. Simply put, the pre-stimulation UI indicates the maximum achievable UI value for a stage post-stimulation. To illustrate this concept, Fig. 14 below displays the exit hole diameters both before and after stimulation for the same stage. For clarity, the orange dots indicate pre-stimulation diameters while the blue dots represent post-stimulation diameters.

In this instance, the pre-stimulation UI with respect to exit hole areas is 0.964, demonstrating a very low Coefficient of Variation (CV) at just 3.6%. For context, perforating charges with initial exit hole area CVs between 5-7% are regarded as high performers, thus the perforating charge used here is exceptionally effective. The post-stimulation UI with respect to exit hole areas stands at 0.948. As previously mentioned, it is highly unlikely that post-stimulation UI values will surpass pre-stimulation UI values, which holds true in this scenario despite the exceptional charge performance. Nonetheless, the minimal difference between the initial and final UI values underscores the significant influence that charge performance has on stimulation performance and how it merits attention in the overall completion design strategy.

A variety of analytical metrics is essential to better evaluate and compare designs. While VUI explains the relationship between the standard deviation and mean of perf tunnel growth volumes in a stage due to erosional effects, it does not necessarily indicate how stages diverged from the initial completion design regarding proppant placement. A commonly used metric for this purpose is Cluster Efficiency (CE), which asserts that if the exit hole area of a single perf within a cluster increases by more than 20% of its original size, then the entire cluster is deemed "effective". However, this does not accurately reflect the performance and effectiveness of a cluster compared to the intended completion design. For example, consider a stage with only two clusters, each containing six perfs. If one cluster has a single perf with an exit hole area growth of 21%, while the other cluster exhibits an 80% growth in all six perfs, then the stage would erroneously have a CE of 100%, which is clearly misleading.

To address the ambiguity, a new efficiency definition was introduced that compares the intended proppant distribution with the actual or final distribution. Proppant Distribution Efficiency (PDE) relates the planned DV to the actual DV at the cluster level for a stage, defining an efficient cluster as one that grows within +/- 20% of the designed DV. For illustration, PDE can be computed for the stage depicted in Fig. 12. With 10 clusters in the stage, each should have an ideal DV value of 10% of the total stage DV, theoretically meaning each cluster would receive 10% of the total proppant. The acceptable DV range for each cluster in this scenario is 8-12%. Therefore, clusters growing less than 80% of the design, or 8% in this scenario, are termed "donor" clusters and are undercapitalized. Clusters growing beyond 120%, or 12% in this example, are termed "thieves" and are overcapitalized.

Fig. 15 illustrates the DV percentage for each of the ten clusters. The horizontal blue and red dashed lines indicate the +/-20% efficiency range at 8% and 12%, respectively, identifying donor and thief clusters. Ideally, each cluster should exhibit a DV of 10%, but the plot shows this is not the case. Unfortunately, only eight out of the ten stimulated clusters fall within the 8-12% range. Cluster #10, positioned at the heel end of the stage, is identified as a thief, while Cluster #7 is its corresponding donor. Since eight out of the ten clusters are within the +/-20% efficiency range, it suggests that the PDE value for this stage is 0.8 or 80%.

To effectively show how VUI and PDE can be used together to assess stimulation performance, Fig. 16 below presents a cross-plot of both metrics from an anonymous dataset. As mentioned earlier, VUI values can occasionally be negative, particularly in stages with plug isolation problems. However, because the lowest possible value for PDE is 0%, stages with negative VUI values have been excluded to keep the range of values between 0-100%.

Recall from earlier that VUI illustrates the connection between the standard deviation and mean of perf tunnel growth volumes in a stage as a result of erosion, and PDE relates the planned DV to the actual DV at the cluster level for a stage. In other words, VUI shows how consistently the perfs grow within a stage, but PDE indicates whether the intended completion design was effectively implemented. With this understanding, here is a brief discussion on the interpretation of the data presented in Fig. 16:

Overall, the dashed line anchored at the origin illustrates a linear relationship between VUI and PDE. Ideally, and aside from stimulation performance, all data points should reside along this line. This alignment would suggest that cluster-level variations in volumetric perf growth directly mirror how accurately stages were stimulated according to their intended design. However, the majority of data points fall below this line, implying that VUI values could be slightly overestimating stimulation performance in most cases.

Quadrant 1, the upper right section of the plot, shows VUI and PDE values both range from 0.5-1.0, indicating two primary findings. First, there is significant alignment between VUI and PDE values, suggesting that variations in volumetric perf growth at the cluster level directly reflects stimulation performance. Second, stages routinely found in this region have an optimized design as proppant is consistently being placed among the clusters at the intended volumes, which is the goal for which operators strive.

Quadrant 2, the lower right section of the plot, has VUI values spanning from 0.5-1.0 and PDE values ranging from 0.0-0.5, indicating that VUI might not directly correspond with stimulation performance. Despite stages in this quadrant showing a high level of consistency regarding cluster-level volumetric perf growth, it does not necessarily imply that proppant volumes are being distributed among clusters as planned. While it is certainly preferable to have higher VUI values rather than lower values, this indicates there is still room for improvement with respect to completion design.

Quadrant 3, the lower left section of the plot, shows both VUI and PDE values between 0.5-1.0. Typically, a strong correlation between VUI and PDE values in this region indicates that the VUI directly reflects stimulation performance. Nonetheless, here it implies that proppant is not being efficiently distributed among the clusters within stages, suggesting that modifications to the completion design are needed to prevent negative effects on well performance.

Quadrant 4, the upper left section of the plot, shows VUI values spanning from 0.0-0.5 and PDE values ranging from 0.5-1.0. Ideally, data points should rarely, if ever, appear in this quadrant, as it suggests an implausible scenario. If cluster-level volumetric perf growth is indicative of stimulation performance, how could a stage with an extremely low VUI value simultaneously exhibit a high degree of proppant placement among clusters nearing the intended design? One potential explanation might be completion designs that implement tapered perfs, which would inherently skew the results since equal growth can no longer be assumed or expected. Alternatively, it could indicate that some external factor has not been adequately accounted for, rendering the data inaccurate.

As shown earlier in Fig. 9, perf tunnel volumes can easily be calculated assuming the geometry of a simple conical frustum. The only measurements required are the entry and exit hole areas, and the height is assumed to be consistent with the nominal wall thickness of the casing. However, perf tunnel volumes can be calculated using different tools and methods, depending on the complexity and accuracy required.

In this section, two high-resolution acoustic data volume calculation methods are presented. The first method requires cross-sectional area measurements to be taken at multiple depths through the perf tunnel. Using these measurements, the linear conical frustum volumes between each area measurement can be calculated and summed to approximate the total volume. This method will be referred to as the Linear Approximation method. Alternatively, the second method uses a Machine Learning (ML) model that creates a point cloud representation of the perf’s profile in three dimensions and encloses the point cloud in a surface, from which a volume can be calculated.

Linear approximations using conical frustums have historically been used to model perf tunnel volumes and only require entry and exit hole area measurements to do so. The consistency and analysis of the model between perfs is a distinct advantage, which was clearly demonstrated in the section detailing theoretical growth cases and concepts. However, as seen in Fig. 17, approximating perf tunnel volumes in this manner will systemically overestimate the volume of metal eroded from the casing.

To more effectively capture the volumetric profile of a perf tunnel, additional cross-sectional area measurements are required between the casing’s ID and OD. For example, if area measurements are taken at the entry hole, the exit hole, and halfway between, the volume of the perf can be approximated by summing the volume of the frustum between the entry hole and halfway (50%) and between halfway (50%) and the exit hole. This will reduce the eroded volume overestimation, as illustrated in Fig. 18.

Therefore, it is logical that measuring additional cross-sectional areas through the perf tunnel and summing the frustum volumes will improve the accuracy of the calculations. For example, Fig. 19 shows how the accuracy of the calculation improves significantly by adding three cross-sectional area measurements at 25% increments between the entry hole and the exit hole.

To optimize the accuracy of a linear volumetric approximation using three additional area measurements, aggregate data was assessed to determine the ideal radial distances through the casing thickness to measure cross-sectional areas. Evenly spaced contours placed at the ID, 25%, 50%, 75%, and OD of the casing wall were found to be insufficient at capturing the change in the cross-sectional area along the perf tunnel. This was mainly due to most of erosion occurring between the ID and 50% of the casing wall combined with the presence of disproportionate amounts of excess volume near the base of the perf.

In the aggregate data analysis, it was found that the majority of stimulated perf tunnels become approximately perpendicular to the radius of the casing at around the halfway point (50%) of the perf tunnel. At this point of perpendicularity, the contour area will remain approximately consistent until the exit hole is reached, and further contours between these two points would not provide meaningful improvements to the volumetric approximation. Therefore, a contour measured at the halfway point (50%) of the casing wall is sufficient to provide an excellent approximation for the volume of metal eroded between it and the exit hole.

Furthermore, the aggregate analysis determined that stimulated perfs experience the greatest rate of change in the cross-sectional area between the entry hole and the 20% position of the casing wall. To best approximate the volume in this dynamic region of the perf tunnel, two cross-sectional areas measured at 10% and 20% positions of the casing wall were found to sufficiently supplement the other contours to provide an accurate conical frustum linear approximation of the perf tunnel volume. Fig. 20 shows an example of this approximation method.

Combining the submillimetric point clouds of data captured using high-resolution acoustic imaging technology along with advanced machine learning models can determine perf tunnel volumes even more accurately. To do this, a four-step perf tunnel volume generation process was developed. Fig. 21 shows the process, followed by a brief description of each step.

• ML Point Cloud Generation – The initial step involves generating a point cloud representation of the perf tunnel’s internal surface. A custom ML network is employed to map the coordinates of points along the perf tunnel, extending from the entry hole to the exit hole through the casing wall, for each cross-sectional image acquired from the high-resolution acoustic imaging technology. For each cross-sectional image, the ML network outputs a series of points that accurately represent the perf tunnel’s geometry. By aggregating the points from all cross sections corresponding to a specific perf, a detailed and comprehensive point cloud is constructed that captures the fully 3D structure of the perf tunnel.

• Surface Generation – After generating the point cloud, the next step involves surface reconstruction to convert the data into a continuous 2D surface. This is achieved using a unique, multi-variable surface reconstruction algorithm that is designed to reconstruct smooth surfaces from complex point cloud datasets, making it ideal for the current application. In this process, normalized point positions are first computed for each point in the cloud to provide the necessary orientation information to reconstruct the surface. The surface-fitting algorithm is then applied, resulting in a high-fidelity 2D surface that accurately represents the perf tunnel’s geometry.

• Surface Pruning – To calculate an accurate volume, a surface pruning task is performed using a point cloud vertex as a reference to remove any unwanted surfaces. The perf is an open structure containing two distinct openings corresponding to the entry and exit holes. To enable volume calculations, these openings must be closed. The technique proposed by Liepa (2003) for filling holes in meshes is employed to perform this task. This method effectively fills the gaps by generating a surface that seamlessly integrates with the existing geometry, ensuring a watertight mesh. The resulting closed surface defines the boundaries of the perf tunnel, allowing for the conversion from a 2D surface to a 3D volume representation.

• Volume Generation – The final step involves calculating the 3D volume enclosed by the surface. This is accomplished using the method described by Mirtich (1996), which provides a fast and accurate computation of polyhedral mass properties, including volume. By applying this method to the closed surface mesh, the total volume of the perf tunnel is determined, providing a critical parameter for evaluating the structural integrity of the casing.

With the understanding of the different methods available to determine a perf tunnel’s volume, the following section compares the accuracy of these methods. Specifically, the five cross-sectional area volume calculation method, referred to as C5 going forward (Fig. 20), is compared to the ML volume generation method (Fig. 21). Five cross-sectional areas are used as opposed to more due to the time required to accurately measure and validate the number of measurements throughout an entire wellbore, which, in some cases, can contain upwards of a thousand perfs.

Based on the aggregate study, five slices can be reliably used to accurately calculate a perf tunnel’s volume. Although simply calculating volume based on entry and exit hole areas, referred to as C2 going forward (Fig. 17), is faster, the frustum edges can deviate significantly from the true profile, especially near the mid-point. This may lead to a consistent overestimation of the perf tunnel’s volume relative to the C5 and ML methods, which is apparent by comparing Fig. 17 to Fig. 20. However, in cases where only entry and exit hole area measurements are available, analysts can attempt to compensate by assuming a curved perf profile. The excess volume should be proportional to the difference between the entry and exit hole areas, and to the thickness of the casing.

To compare perf tunnel volumes obtained from the C5 and ML methods, a sample of eight stages was selected from a single horizontal Niobrara well completed by Oxy in the PRB of Wyoming (Jones et al. 2025). This well was completed with permanent fiber optics, which provides a baseline against which the volumetric calculation methods discussed in this paper will be validated. The DAS signals captured by the fiber during stimulation can be used to estimate cluster-level proppant distribution, which should be highly correlated to cluster-level volume increases. The eight stages selected from this well were chosen based on the very low number of plugged perfs present. This maximizes the association between the high-resolution acoustic imaging technology’s data and the DAS data.

Overall, the volumes obtained from the C5 and ML methods align very well. Across the eight stages analyzed, the average absolute difference between the C5 and ML volumes was 0.01 in³ with the C5 values being larger. This translates to an average absolute error of approximately 10%. While this disparity may seem large, it is worth noting that the higher dimensionality of a volume results in error compounding with each additional dimension compared to an area or a diameter.

Taking the non-absolute average difference between the ML and C5 volumes produces a very small negative value of -2.8 × 10^-5 in³. This supports the postulation that the cross-sectional measurement method, which uses frustums to approximate the volume, tends to overestimate perf tunnel volume only slightly. Notably, a comparison of the ML volumes against the simpler volume calculation using only entry and exit hole areas (C2) yields an average non-absolute difference of -0.02 in³. This larger magnitude in disparity aligns with the expectation that more frustums used to approximate the volume provide higher accuracy.

A stage-level analysis of the differences between the C5 and ML volumes provides additional insight into the relationship between the two methods. The box and whisker plots in Fig. 22 show the absolute and non-absolute median difference between the C5 and ML volumes as a percentage of the C5 volume for each stage in the sample set.

With a non-absolute average difference very close to zero, it can be concluded that the C5 and ML volume calculation methods produce very similar results when analyzing populations of approximately ten perfs or more. Stages 9 and 28 have the highest mean absolute volume differences between the C5 and ML methods. A key reason for this could be the elevated levels of debris in the perfs of those stages. Inspecting the raw acoustic data for Stage 9 revealed that 14 of the 22 inspected perfs (~64%) have abnormally high signal returns from debris in the perf tunnels. Likewise, Stage 28 had 7 of the 11 inspected perfs (~64%) slightly obscured by debris. Conversely, the stages with the lowest mean absolute differences, Stages 19 and 33, have zero perfs obscured by debris.

Since the ML model used for these volume measurements was not trained specifically for the task of measuring perfs with debris, its deviation from the C5 measurement is not unexpected. Further training, or the development of a task-specific model, could improve the alignment between cross-sectional analysis and ML results. Fig. 23 shows the same box and whisker plots but with differences between the ML and C2 methods. For this comparison, the non-absolute difference is negative for every stage, which supports the postulation that volume calculations based on a single frustum method (C2) result in the over-estimation of perf tunnel volume, even more so than the C5 method.

A detailed perf-level comparison of the C2, C5, and ML volumes is displayed in Fig. 24. The y-axis for each plot is fixed, and, as expected, the C2 volume is greater than the C5 volume for every perf. Table 1 following Fig. 24 summarizes the mean non-absolute volumes calculated for the ML and C5 methods for each stage.

Laboratory testing was performed to validate the accuracy of the high-resolution acoustic imaging tool, specifically its ability to accurately measure and calculate the volume of perf tunnel erosion. To do this, a sample containing five perfs was machined and subsequently sandblasted to mimic erosion during the proppant placement process. Once prepared, the sample was imaged using the high-resolution acoustic imaging tool and an ultra-high-resolution handheld metrology-grade laser scanner from CREAFORM for measurement comparison and validation. Fig. 25 shows the test sample with each of the five perfs labeled for reference.

Fig. 26 visualizes the slices measured with the laser scanner through the thickness of the casing sample that were used to calculate perf tunnel volumes. For the reasons discussed in the section detailing perf tunnel volume calculation methods, slices were taken at the entry hole along the casing’s ID (0%), 10%, 20%, 50%, and at the exit hole along the casing’s OD (100%). Again, this corresponds to the C5 method.

After collecting and analyzing the data, the measurements of this sample obtained through high-resolution acoustic imaging technology and a laser scanner produced the results shown in Table 2. The average absolute variation in acoustic area measurements compared to the laser scan was found to be 0.02 in². This demonstrates the exceptional precision of acoustic measurement techniques applicable in the downhole environment of a stimulated horizontal well, a setting unsuitable for laser scanners. It also provides confidence in perf tunnel volumes calculated using the C5 method, which can be compared against the more precise ML volumetric model.

Two of the most commonly used diagnostic tools to assess SDE in unconventional horizontal wells are permanent fiber optics and high-resolution acoustic imaging. Both are designed to provide operators with the same type of data to better inform completion design strategies. Pros and cons of these tools are discussed by Wien et al. (2024), but a monumental difference between them is the cost. Permanent fiber is substantially more expensive than acoustic imaging, but it is capable of providing real-time feedback with a high degree of spatial and temporal accuracy. What if operators could obtain analytical results that rival fiber in terms of decision-making quality at a fraction of the cost?

In 2023, Oxy completed a horizontal well in the Niobrara formation of the Powder River Basin that was equipped with permanent fiber optics for DAS and Distributed Temperature Sensing (DTS) data collection during stimulation. High-resolution acoustic imaging data, comprised of perf entry and exit hole measurements, was also acquired after the post-frac cleanout was complete. The specifics of the completion Design of Experiment (DoE) are discussed at length by Jones et al. 2025, but a simplified table of the key parameters for each design tested in the well are provided below in Table 3. Treatment rate, cluster spacing, proppant per cluster, and fluid per cluster were kept consistent for all designs.

Before comparisons are made between fiber optics and perf volumetrics for the horizontal Niobrara well, it is important to discuss the initial uneroded perf measurements used in the analysis. As mentioned earlier, the well was equipped with permanent fiber on the outside of the production casing. Perfs were shot in-line with 0 degree phasing, and orientations around the borehole were contingent on the position of the fiber throughout the lateral for each stage. Any unstimulated reference perfs shot in the heel of the well would have been subjected to the same constraints, and they would not constitute an accurate baseline that adequately represented all of the stimulated stages. Therefore, reference perfs were intentionally omitted for this well.

The initial perf entry and exit hole measurements used in this analysis were determined using the empirical approach demonstrated in Figs. 3-7, and the initial perf volumes were represented as conical frustums as shown in Fig. 9 and Fig. 17. Two different types of perforating charges, a standard deep penetrator used in the multi-perf cluster (MPC) stages and "big hole" charge used in the single-perf cluster (SPC) stages, were analyzed. The initial MPC perf measurements were determined for each borehole region, but the initial SPC perf measurements were determined for all borehole regions collectively due to sample size limitations. Table 4 shown below lists the initial entry and exit hole measurements as well as the initial perf volumes used to compute growth in the analysis. Fig. 27, also shown below, provides radar plots of these measurements for the MPC designs. The production casing for this well was 5-1/2 in. (20#) casing, which has a nominal wall thickness of 0.361 in.

After initial volumes were determined, volumetric growth was calculated by subtracting the appropriate baseline value listed in Table 4 from the final volume of each stimulated perf throughout the lateral that had all entry and exit hole measurements available for analysis. The results were then directly compared to those obtained from the fiber optic analysis detailed by Jones et al. (2025). Fig. 28 shown below compares PUI values from fiber and VUI values from perf erosion analysis, and Fig. 29 compares HFPs between fiber and perf erosion analysis from three select stages.

It is clear from Figs. 28 and 29 that volumetric perf erosion analysis using entry and exit hole measurements generates analytical results that are strikingly similar to those of permanent fiber optics. In general, the relative ranking of design performance is the same between both analyses, with Design I as the only outlier in this regard. Of particular interest is the directional alignment between the results where Designs C/D and G/H are concerned, which are labeled as "Pair #1" and "Pair #2", respectively (Fig. 28). These pairs have the exact same parameters between designs with the exception of perf strategy. Designs C and G are MPC stages that used a standard deep penetrator while Designs D and H are SPC stages that used a "big hole" charge. Both diagnostic tools indicate an uplift in SDE when this change was made.

The comparison of HFPs is also quite interesting as both analyses indicate similar trends with respect to treatment bias. Notice, however, that the locality of a dominant cluster as determined by fiber may differ slightly from the locality of a dominant cluster as determined by perf erosion. For example, in Stage 5, fiber indicates that Cluster 5-4 is dominant while perf erosion indicates that Cluster 5-3 is dominant. While this may initially seem disconcerting, it is important to consider each diagnostic tool’s sensitivities and how a small discrepancy such as this can be explained by physical phenomena. Although fiber is used to estimate proppant placement, the allocation is based on acoustics, and it does not measure proppant throughput directly. It is perhaps a better predicter of total slurry placement rather than proppant outright. Perf volumetrics, on the other hand, is a direct reflection of erosion, which is a combination of proppant concentration, particle mass, and particle velocities throughout the stage. It is reasonable that a dominant cluster as determined by fiber may appear downstream according to perf erosion analysis, accounting for differences between fluid and proppant velocities and vectors.

The following points summarize the key learnings and conclusions obtained from this study:

• Traditional perf erosion analyses, which use 2D exit hole measurements explicitly, fail to capture the intricate details of erosional phenomena caused by proppant placement in hydraulically stimulated horizontal wells. On the other hand, volumetric perf erosion analyses with 3D measurements obtained from high-resolution acoustic imaging technology can yield results comparable to permanent fiber optics but at a significantly lower cost to operators.

• Entry hole measurements on the casing’s ID and exit hole measurements on the casing’s OD can be used collectively to fundamentally explain perf erosion behavior that is predicated on theoretical growth cases. Moreover, a novel diagnostic plot of entry and exit hole area ratios versus differences can be used to empirically determine uneroded reference perf measurements that serve as the foundation for growth calculations in a volumetric perf erosion analysis. This reduces the reliance on a dedicated set of unstimulated reference perfs.

• Two new metrics for volumetric perf erosion analyses are VUI and PDE. VUI measures the relationship between standard deviation and mean of perf tunnel growth volumes in a stage due to erosion, without indicating deviations from the initial proppant placement design. PDE compares planned DV with actual DV at the cluster level for a stage, quantitatively reflecting how the stimulation aligns with the original design. Evaluating both metrics is essential for fully assessing stimulation performance.

• Without the use of ML models to ascertain highly accurate perf tunnel volumes, measuring five cross-sectional areas at specific contours along the tunnel provides an accurate volume representation. Comprehensive analysis indicated that this technique had only a minor deviation from the ML-based method. The discrepancy in absolute volume difference between the conical frustum approximation using entry and exit hole measurements and the ML-based method is equally minimal.

Going forward, further ML model validation and training are necessary to ensure robust performance across all scenarios, including those involving debris. Debris in perfs is commonly observed, as perf imaging usually occurs post-stimulation and after drilling out frac plugs. Most debris issues can be managed when deploying high-resolution acoustic imaging tools, though some residual debris may still remain despite an operator’s best efforts. Currently, high-resolution acoustic analyses use the conical approximation method with entry and exit hole measurements only (C2). Additional steps and processes are being developed to reliably implement the five cross-sectional area volumetric analysis method (C5).

The authors wish to express their heartfelt thanks to the leadership teams of Oxy and DarkVision for their approval and support of this significant work. The extensive effort required to analyze countless datasets and formulate new ideas is always taxing, but the hope is that the insights presented in this study will advance the oil and gas industry’s comprehension of perf erosion behavior and the knowledge obtained from submillimetric perf measurements. Considerable appreciation also goes to those who have laid the foundation for perf erosion analyses over the past 65 years leading up to this work.