Modeling Temporal Dependence of Average Surface Treating Pressure in the Williston Basin Using Dynamic Multivariate Regression

Abstract

The oil and gas industry has shifted paradigms after seeing the drastic decrease in oil prices since 2015. Companies are now focused as much on cost reduction as much as production maximization to drive profitable operations. This aspect is more prevalent in unconventional plays with the need for long horizontal drilling and hydraulic fracturing (HF) operations to develop and produce from the tight reservoirs. There exists an optimum point between the costs of HF treatment and the expected production. Because of the paradigm shift, many operators are now focused on re-developing existing assets at much lower costs instead of developing newer, more costly assets. Re-fracturing existing wells provides an opportunity for companies to add economical wells to their portfolio. Re-fracturing consists of pumping HF treatments in wells that were previously drilled and completed. Although it may seem that the HF process on a well would be easier the second time around, this is not always the case. There are often numerous operational and engineering parameters that may cause screen outs due to excessively high surface treating pressure (STP) that can drastically affect the economics of a re-fractured well. Being able to isolate the effects of these parameters and estimate their marginal effect on treatment will help engineers design to better HF treatments and surface equipment to effectively implement treatments in the field. This novel study uses field treatment data from re-fractured wells to create dynamic multivariate regression models to characterize the effects of treatment parameters on the average STP. The model allows for engineers to isolate the effects of other treatment parameters and estimate their marginal effects on average STP by holding other variables of interest constant. The model also attempts to account for the temporal dependence of stress shadow effects from the previous zones by using the average STP as a good approximation. It was found that the distance between zones (perforation standoff) was statistically significant at the 90% level, average pump rate, acid volume displaced, and the presence of a 3.5” liner were all statistically significant predictors of average STP at the 95% level and average surface treating pressure from the previous stage at 99% significance. The model was used to predict the STP for another re-fractured well, which showed reasonable results.

1. Introduction

The combined technological development of horizontal drilling and HF has not only made the U.S. energy independent in terms of oil and gas (O&G), but has also made the country an exporter of these resources [1]. However, the downturns and economic conditions in the O&G industry in 2015 and 2020 have created a paradigm shift. Many companies are now focused on minimizing costs as much as maximizing production [2]. Due to high degrees of leveraging and persistent lower prices, many US operators have been forced to focus on developing current wells as opposed to investing in new drills [1]. However, low recovery factors for both shale gas and oil and quick declines during the first couple years of production highlight the need for solutions to combat the rapid production declines [1]. Re-fracturing existing wells is one such solution that is shown to be cost effective [1,3].

Re-fracturing has seen an increase in implementation due to inadequate initial stimulation [1]. It is also appealing in times of low oil prices when comparing costs of drilling and completing new wells [1]. Production from re-fracturing treatment can be as high as 92% of initial production after the initial HF treatment in the Bakken [1].

There are numerous sources using data mining to investigate HF and re-fracturing viability and production. Maldonado and Aoun (2019) were able to successfully structure data from 50 producing wells in the Messaoud Field in Algeria and select the best candidates for re-fracturing treatment [4]. There have been recent attempts to simplify these models. Peirce and Bunger (2014) predicted fracture growth and propagation using a coupled mathematical model that included the effects of fluid flow, rock breakage, and pressure loss. The authors argued that this simpler model was necessary to: (1) create models that can make broad, but useful, predictions that may not be leveraged by a few individuals with niche expertise, and (2) provide more complex models with a starting point and provide them with guidance and valuable insights [5].

Mohaghegh et al. (2017) developed “Shale Analytics” as an all-encompassing workflow using machine learning to optimize production from shale resources in the Utica, Marcellus, Niobrara, and Eagle Ford plays [6]. Shale Analytics uses machine learning to perform multiple tasks, including optimizing well spacing, predict candidates for re-fracturing, and perform Decline Curve Analysis (DCA).

Mohaghegh (2016) also developed a re-fracturing candidate selection machine learning algorithm that was able to change parameters for re-fracturing design [7]. Mohaghegh (2016) found that the optimal HF job was defined by creating the largest fracture network possible that maximizes hydrocarbon production. While this was certainly a primary goal, the authors of this study would add that the optimum HF treatment creates the largest possible fracture network given a certain cost. Optimizing between predicted production and cost is vital to re-fracturing economics.

Due to cost constraints, the best possible data (fiber optic, microseismic) are usually not collected regularly for initial completions and are most likely not economical for re-fracturing treatments. What is collected regularly, however, is actual treatment data from HF treatments. This resource is often overlooked, presumably due to the complexity of the interactions between fluid, proppant, wellbore, formation, and equipment that occur simultaneously during HF treatments. This makes it hard to isolate effects and to attempt to draw any conclusions. Figure 1 shows the problem posed by Mohaghegh (2019) in trying to draw conclusions from correlations between well productivity and field treatment data [8].

Mohaghegh (2019) approaches this problem using an innovative version of soft cluster analysis in the Shale Analytics workflow to estimate how these parameters will affect well productivity and predictions.

While previous work has been done to optimize field operations for production, little has been done for optimized field implementation. This study was undertaken in an attempt to understand the operational parameters that affect treatment after a re-fracturing attempt of an offset well from the wells used in this study. The offset well encountered high treating pressures and eventually had to be abandoned with 4 stages remaining before completion. One hypothesis for this is that there are significant stress shadow effects at work due to things like depletion. Understanding the factors that affect STP in the field is important because the STP is often the binding constraint for operations, especially on re-fracture treatments. This is because STP will often dictate how much proppant may be placed in formation, the maximum concentration of proppant, total fluid volume, and the maximum pump rate. Treatment fluid volume, proppant mass, and compartmentalization are primary drivers for production, and all affect the cost of treatment [9]. Pump time and Hydraulic Horsepower (HHP) will also have an impact on treatment cost.

This study will attempt to build off of the work of Pierce and Bunger (2014) and Mohaghegh (2019) by using many of the same parameters from field treatment data but focus on simpler linear multivariate regression models to aid treatment implementation in the field and provide new tools for engineers to work within the new paradigm of cost minimization, which depends on successful treatment implementation.

2. Materials and Methods

Re-fracturing success in terms of production depends largely on the well selection [1]. There is no universal method for candidate screening, but it should be based on well potential, production performance, and the success of the initial treatment [1]. Candidate selection is beyond the scope of this study and, here, it is assumed that the best candidates were chosen for re-fracturing treatment. Some of the wells selected for re-fracturing treatment were also completed during zipper frac operations on pads with new drills. Any effects of these operations are assumed to be negligible compared to the effect from treatment of previous stages.

Although well selection may play the primary role in production, effective implementation of HF treatments in the field are dependent on many other factors. These include, as examples, pump rate, fluid design, wellbore construction, and mechanical and chemical interactions that occur downhole. The models in this study attempt to characterize the marginal effects of some independent variables that might affect the average STP. This is important because the pressure limitations of the wellbore and HF equipment are usually the limiting factors in executing an effective HF treatment. High treating pressures reduce the pump rate that can be achieved and will often restrict the amount of sand that is able to be placed for fear of screen-outs. HF treatments may also be cut short as a precaution to avoid screen-outs. Here, some of the important parameters that affect the re-fracturing operations are briefly discussed.

2.1. Perforation Standoff

Net pressure and stress contrast both increase substantially with the number of sequential fractures and decreased fracture spacing [10]. Induced stress shadow from previously fractured stages also has effects on the subsequent zone as the effects extend beyond the top perforation and may thus change the fracture initiation at the next bottom perforation for the next stage [2]. Therefore, the standoff between the top perforation for one stage and the bottom perforation for the subsequent stage may affect treatment through stress shadow effects and will be included in the regression models. Figure 2 is a schematic of the perforation standoff in the wellbore. The hypothesis is that increasing the perforation standoff will decrease stress shadow effects from previous stages and thus decrease average STP.

2.2. Stage Proppant Weight

The amount of proppant placed during a stage affects bottom hole treating pressure by increasing the hydrostatic pressure. This in turn affects the STP. The amount of proppant pumped may also affect STP through perforation erosion as treatment progresses through hole erosion and rounding [11]. The hypothesis is that the effects of increasing stage proppant weight will be negative as more sand increases in hydrostatic pressure and increases perforation erosion, thus decreasing the average STP.

2.3. Total Clean Volume

It is reasonable to assume that treatment size will affect the STP. Total clean volume pumped should be a good indication of stage size. The hypothesis is that an increase in total clean volume will increase average STP.

2.4. Previous Stage Average Surface Treating Pressure

This variable will attempt to capture the stress shadow effects from the previous stage treatment. HF treatment with multiple stages and clusters leads to complex interactions resulting from different propagating fractures [12]. When multiple fractures are growing near each other, the stress fields created around each fracture must be considered, as these will affect fracture growth and treatment pressures [2].

This may be even more significant in re-fracturing treatments, since decreases in pore pressure can lead to redistributing stresses or even reversal of principle stress in regions around initial fractures [13]. Stress interference from reorientation increases with the number of fractures created and depends on the sequence of fracturing [10].

Fractures on each end of the stage will behave as individual fractures, regardless of the number of fractures between the two [2]. These end fracture will tend to dominate and become the primary fractures [2]. Poro-elastic effects change the net vertical stress along the wellbore [2]. Although it is not possible to know the rock properties and poro-elastic effects for each stage, average STP may capture these effects during and after treatment. This is also related to the perforation stand-off. Considering the dominance of the end fractures and the associated stress and poro-elastic effects that fractures create, the hypothesis is that a higher STP from previous stages will increase the average STP for the subsequent stage.

2.5. Number of Perforations

The number of perforations is important to any treatment, as these are the conduit from the wellbore that will ultimately deliver the treatment slurry into formation. Although only a limited number of perforations may take fluid and preferentially propagate major fractures, usually the toe and heel perforations [10], it is still necessary to have a certain number of perforations above the number of fractures for pressure relief during treatment. The hypothesis is that an increase in the number of perforations will decrease average STP.

2.6. Presence of a 3.5” Liner

The presence of a 3.5” liner will increase friction due to the smaller inner diameter relative to treatments pumped through a 4.5” liner. Installation of a new liner in the horizontal section is usually necessary. If the well was initially completed with sliding sleeves and packers, these will need to be removed or milled and a new liner installed to provide a means for zonal isolation. If the well was initially completed using a cemented liner in the horizontal, a new liner may be required to avoid the issues that will arise with wireline running into previous perforations and possible casing issues from previous treatment and production. Previous open hole completions also require the installation of a new 4.5” liner. The hypothesis is that the presence of a 3.5” liner will increase the average STP. This is a binary variable, with “1” indicating the presence of a 3.5” liner and “0” indicating a 4.5” liner.

2.7. Average Pump Rate

Increased pump rate creates more friction along the wellbore as well as the fracture face. The average pump rate will most likely have an effect on treatment costs as service companies generally charge for pump time. Therefore, longer pump times will increase treatment costs. The assumption is that an increase in average pump rate will increase the average STP.

2.8. Acid Volume

Acid is important in hydraulic fracturing designs to clean up any near wellbore damage caused by the perforation and cement. It can also have drastic effects on treating pressure if the formation is carbonate. The assumption is that an increase in acid volume will decrease the average STP.

2.9. Formation Type

In an attempt to capture any geologic differences between the formations that may affect treatment, a binary variable will be used in Model 2 to see if there are any statistical differences between the models and if accounting for geological differences has any statistical impact on the models and coefficient estimates for the other independent variables. These estimates will be relative to the Three Forks formation.

3. Multivariate Regression

Equation (1) shows the bivariate model that is the basis for the models used in this study [14]. Here, $Y_i$ is the dependent variable of interest, $X_i$ represents the independent variable used that is thought to affect the dependent variable, ${\beta }_0$ is the expected value when the independent variable is 0, and ${\beta }_1$ represents the marginal effect that the independent variable will have on the dependent variable. Therefore, if the independent variable $X_i$ is increased by one unit, the dependent variable is expected to change by the coefficient ${\beta }_1$, all else being equal. Bivariate regression can yield insight into the correlation between a dependent variable and an independent variable, but in complex systems, these models yield little insight as to the other factors that may be affecting the independent variable. When using bivariate models for purposes of drawing conclusions about the system, there is an implicit assumption that there are no other factors at work.

$$Y_i={\beta }_0+{\beta }_1{X}_i$$

(1)

In HF, because of factors like reservoir heterogeneity and wellbore dynamics, there is a need to account for more independent variables in treatment analysis. This study will use multivariate regression models to not only construct predictive models, but to estimate the marginal effects of multiple independent variables of interest. Just as the coefficient estimate for ${\beta }_1$ in the bivariate model gives an estimate of the marginal effects $X_i$ will have on $Y_i$, so too do the ${\beta }_n$ coefficient estimates provide marginal effect estimates on the dependent variable. Multivariate regression models are an extension of bivariate regression models. Equation (2) shows the basic construction of multi-variate regression models [14].

$$Y_i={\beta }_0+{\beta }_1{X}_1+\cdots +{\beta }_n{X}_n+\epsilon ,$$

(2)

One important addition to the multivariate model is the error term $\epsilon$. The error term is meant to capture all other factors not included in the regression model that may have a causal relationship with the dependent variable. Leaving variables in the error term that have a causal relationship with the dependent variable will lead to what is known as endogeneity. Broadly speaking, endogeneity is defined as explanatory variables being correlated with the error term [15]. Endogeneity leads to the deduction of causal relationships that may be spurious. One example may be a potential relationship between ice cream sales and an increase in drowning rates in swimming pools [16]. To regress one variable on the other and then conclude that an increase in ice cream sales causes people to drown would be spurious. In fact, it would most likely be the case that heat was driving people to buy more ice cream and drive more people to pools and thus increasing the likelihood of drownings [16].

The same type of problem exists for re-fracturing applications, and complex problems in the O&G industry in general. For example, when constructing a model to predict the 180-day cumulative production from Mogaghegh (2019) for a re-fracturing treatment, a model that only contains total clean volume may be a good place to start. The model may be represented by Equation (3).

$$180 day cumulative produciton={\beta }_0+{\beta }_1\ast Total Clean Volume$$

(3)

While it is reasonable to think that total clean volume accounts for a certain amount of the 180-day cumulative production, it is unreasonable to think that it is the only variable affecting production, which is what Equation (3) does. There are other variables like sand, pump rate, geologic properties, perforation erosion, etc. that almost certainly affect the production as well. As Mohaghegh (2019) notes, it is hard to determine the extent to which these affect production by looking at each in isolation. These would be the other variables that are contained in the error term and should be pulled out and accounted for. This is what multivariate regression attempts to do and allows engineers to account for more than one independent variable in complex systems to try and figure out what is going on and what is actually affecting the dependent variable.

The same logic applies to the field treatment data in this study. The intersection of stimulation equipment, wellbore design, formation properties, stress shadows, fluid design, amount of proppant pumped, and perforation designs make utilization of bivariate regression to draw any useful conclusion ineffective. Therefore, more complex methods must be employed. Here is where we introduce dynamic multivariate regression to account for the temporal dependence of stress shadow effects. Temporal dependence is simply an assumption that the current state depends on a previous state. So, a lagged dependent variable (average STP) must be included to try and account for this temporal relationship. Equation (4) shows the basic form for a dynamic multivariate regression model for well, i, in a given stage, t. The $\gamma$ coefficient represents the effect of the lagged dependent variable, $\beta$ coefficients represent the effects of the independent variables, and the error term, ${ϵ}_{it}$, is assumed to be uncorrelated with the independent variables [14].

$$Y_{it}=\gamma Y_{i,t-1}+{\beta }_0+{\beta }_1{X}_{1it}+\cdots +{ϵ}_{it}$$

(4)

This final dynamic model constructed in this study will use the independent variables described above in an attempt to see which variables have statistically significant effects on average STP.

4. Input Data

The summary and definitions for the data used for this study are shown in

Table 1. Summary statistics for re-fracturing datasets used in this study.

Variable	N	Mean	Std. Dev.	Min	Pctl. 25	Pctl. 75	Max
avg_pump_rate	121	50.331	9.966	10.5	42.6	57.8	75.4
avg_stp	121	8234.355	465.448	6411	7903	8726	9083
acid_volume	121	20.711	30.92	0	11	23	227
total_clean_volume	121	3950.039	839.746	2920.091	3606.364	4024.941	8940.621
stage_prop_weight	121	220,207.884	37,671.035	25,649	212,565	218,373	405,515
perfs	121	24.165	1.562	22	24	24	36
liner_3.5	121	0.686	0.466	0	0	1	1
formation	121	0.752	0.434	0	1	1	1
perf_standoff	117	25.573	4.415	18	22	26	49
avg_stp_prev	117	8235.778	467.02	6411	7907	8726	9083

and

Table 2. Variable definitions for data set used in this study.

Variable	Variable Definition
avg_pump_rate	Average pump rate, bpm
avg_stp	Average surface treating pressure, psi
acid_volume	Acid volume, bbls
total_clean_volume	Total clean volume for stage, bbls
stage_prop_weight	Proppant weight pumped for stage, lbs.
perfs	Number of perforations over treatment interval
liner_3.5	Presence of 3.5” lateral liner (1 = yes, 0 = no)
formation	Binary variable indicating formation (1 = middle Bakken, 0 = Three Forks)
perf_standoff	Distance between top perforation of one stage and bottom perforation of subsequent stage, ft
avg_stp_prev	Average surface treating pressure for previous stage, psi

. Data was collected from four recently re-fractured wells in the Hector field in Dunn County North Dakota. The location of the field and the operator’s acreage are shown in Figure 3 [17]. Specific well locations and names were not allowed to be included. Figure 4 shows the location of the field in the larger context of North Dakota. The data only include 117 observations for perforation standoff and previous stage average STP, as the first stage of each well were excluded as they were not applicable. These were excluded from the regression models.

Data for this study were not normalized as it is unnecessary because of the uniqueness of multivariate regression models. These models attempt to measure marginal (per unit) changes in the independent variable and how these will affect the dependent variable. Therefore, since the units of each of the independent variables used in the regression are used in the field, they will be kept in these units to make coefficient interpretation easier. If the models attempted to account for something like Poisson’s ratio, which has a range from 0 to 0.5, then a marginal unit increase does not make sense in this context. In other words, the units used for the variables in the regression models are in context with the units used in the field so there is no need for standardization. In fact, statistical significance and model fit would be the same for standardized and unstandardized results [14].

All data, R code, and other materials will be provided during the review.

5. Results

Table 3. Regression results showing two different models.

	Dependent Variable
	Average Surface Treating Pressure (psi)
	Model (1)	Model (2)
Perforation Standoff (ft)	14.752 *	14.615 *
	(7.723)	(7.796)
Stage Proppant Weight (lb)	−0.001	−0.001
	(0.002)	(0.002)
Total Clean Volume (bbl)	−0.043	−0.044
	(0.090)	(0.090)
Number of Perforations	−22.859	−23.170
	(15.884)	(16.050)
3.5-inch Liner	228.412 **	221.913 **
	(91.925)	(99.258)
Average Pump Rate (bpm)	9.561 **	9.471 **
	(3.692)	(3.742)
Acid Volume Pumped (bbl)	8.075 ***	7.785 **
	(2.628)	(3.098)
Previous Stage Average Surface Treating Pressure (psi)	0.714 ***	0.713 ***
	(0.074)	(0.075)
Formation (1 = middle Bakken, 0 = Three Forks)		−14.937
		(83.683)
Constant	2217.077 ***	2260.782 ***
	(821.258)	(860.533)
Observations	117	117
R2	0.712	0.712
Adjusted R2	0.691	0.688
Residual Std. Error	259.541 (df = 108)	260.712 (df = 107)
F Statistic	33.423 *** (df = 8; 108)	29.446 *** (df = 9; 107)

Note: * p < 0.1; ** p < 0.05; *** p < 0.01.

summarizes the regression results from R, including the independent variables discussed above. The model investigates whether these have a statistically significant effect on the dependent variable (average STP) from the given data set. Two models were run, with model 2 including a binary variable for which formation the well was drilled (1 for middle Bakken and 0 for Three Forks). The coefficient estimate for formation is then relative to the Three Forks formation. This was done to see if there were any statistical difference between models accounting for geologic properties and those that do not. It is important to note that whenever coefficient estimates and results are discussed, they are only expectations as the true coefficient estimates, and therefore marginal effects, are not known [14]. This extends from the fact that statistical models do not yield facts, only expectations. To test the marginal effects, experiments would have to be conducted in the field by altering only one parameter by one unit while holding all others constant.

Model 1 and model 2 yielded similar results, with statistical significance for acid volume being the only change. Coefficient estimates were also largely unchanged. The coefficient estimate for the formation type was also not statistically significant, indicating that any heterogeneity between the formations had no statistical impact on the average STP.

The significance codes’ corresponding significance levels are shown in the results as well. The number for each independent variables not in parenthesis is the coefficient estimate for the corresponding independent variable. For example, in model 1, the coefficient estimate for perforation standoff is 14.752. Although not all independent variables were statistically significant, all of the coefficient estimates for model 2 will be discussed.

5.1. Perforation Standoff

Perforation standoff was found to have a statistically significant effect at the 90% level. This means that the default null hypothesis that the coefficient estimate is 0 can be rejected with 90% confidence. However, the coefficient estimate is (+), which runs counter to the initial hypothesis. The positive coefficient estimate indicates that, in expectation, a 1 ft. increase in perforation standoff between the top perforation from one zone and the bottom perforation for the subsequent zone would yield a 14.615 psi increase in average surface treating pressure.

5.2. Stage Proppant Weight

The initial hypothesis was that increasing stage proppant weight would decrease average surface treating pressure due to increased hydrostatic pressure and perforation erosion effects. The coefficient estimate of −0.001 indicates the initial hypothesis was correct based on the data set. The coefficient estimate means that a 1 lb. increase in stage proppant weight yields a decrease of 0.001 psi in average surface treating pressure. However, stage proppant weight was not a statistically significant predictor of average surface treating pressure.

5.3. Total Clean Volume

The initial hypothesis was that an increase in total clean volume would tend to increase the average surface treating pressure simply based on the fact that there is only a finite fracture volume. The coefficient estimate of −0.044 counters the initial hypothesis and estimates that a 1 bbl. increase in total clean volume will decrease the average surface treating pressure by 0.044 psi. Total clean volume was not a statistically significant predictor of average surface treating pressure.

5.4. Number of Perforations

The initial hypothesis that an increase in the number of perforations will tend to decrease average surface treating pressure was backed by the model. The coefficient estimate of −23.17 indicates that the addition of one perforation will decrease the average surface treating pressure by 23.17 psi, although it was not statistically significant.

5.5. Presence of a 3.5” Liner

The initial hypothesis that a 3.5” liner will increase average surface pressure was backed by the model. The coefficient estimate of 221.913 indicates that a 3.5” liner will increase the average surface treating pressure by 221.913 psi, which agrees with the initial hypothesis. The presence of a 3.5” liner is also a statistically significant predictor of average surface treating pressure at the 95% level, meaning the null hypothesis that the coefficient estimate is 0 can be rejected with 95% confidence.

5.6. Average Pump Rate

The initial hypothesis that an increase in average pump rate will tend to increase the average surface treating pressure is backed by the model. The coefficient estimate of 9.471 indicates that a 1 bpm increase in average pump rate will increase the average surface treating pressure by 9.471 psi. The coefficient is statistically significant at the 95% level, indicating that the null hypothesis that the coefficient estimate is 0 can be rejected with 95% confidence.

5.7. Acid Volume

The initial hypothesis was that an increase in acid volume would decrease the average surface treating pressure. This hypothesis was not backed by the model. It is also interesting to note that the coefficient estimate decreased from model 1 to model 2 and significance level dropped from 99% to 95%. This may be because of endogeneity issues and may be resolved by including omitted variables or gathering more data. It is also possible that the variable is endogenous because the pressure relief seen from acid allows for greater treatment rate which will also increase the STP. The coefficient estimate of 7.785 indicates that a 1 bbl. increase in acid volume will increase average surface treating pressure by 7.785 psi. The estimate is statistically significant at the 99% level, indicating that the null hypothesis that the coefficient estimate is 0 can be rejected with 99% confidence.

5.8. Previous Stage Average Surface Treating Pressure

This variable was included to try and capture stress shadow effects on treating pressure from one stage to the next. The (+) coefficient estimate confirms the hypothesis that higher treating pressures from the previous stage will tend to increase the surface treating pressure for the subsequent stage. The coefficient indicates that a 1 psi increase in average surface treating pressure from the previous stage will increase the average surface treating pressure of the next stage by 0.713 psi in expectation.

Previous stage average surface treating pressure was statistically significant at the 99% level, meaning the null hypothesis that the coefficient estimate is 0 can be rejected with 99% confidence. This provides statistical evidence that stress shadow effects may tend to dominate treatment from an operational standpoint.

5.9. Formation Type

Accounting for the two different formations (middle Bakken and Three Forks) from model 1 to model 2, they did not have a significant effect overall on the model. Although the coefficient estimate for acid decreased in magnitude and significance, there was no clear evidence that the different formations, and therefore differing geological and mechanical properties between the two, had any statistically significant effect on average STP. The coefficient estimate of −14.937 indicates that if the well were drilled in the middle Bakken, the average STP is expected to be 14.937 psi lower relative to the Three Forks formation.

5.10. Model Visual and Verification

Figure 5 shows the added variable (AV) plots for model 2 from R. Since creating 2-D visuals for multiple regression is impossible, the best way to view individual effects is through AV plots. The plots are constructed so that the dependent variable (average STP) is on the y-axis and an independent variable is on the x-axis. These plots differ from the bivariate regression constructed in Mohaghegh (2019) in that these are accounting for other independent variables that are being held constant. Therefore, each plot shows the individual effects of one independent variable, with all other independent variables being held constant on the dependent variable.

The trendline shows the direction of effect with a positive slope indicating a positive marginal effect and a negative slope indicating negative marginal effect. For a visual comparison between AV plots and simple bivariate regression plots, Figure 6 shows the bivariate regression model and associated equation for the effect of average pump rate on the average STP taken from the dataset. Notice the positive relationship between average pump rate and average STP in the AV plot (which we would expect due to friction) and the negative relationship in the bivariate regression plot. In the bivariate model, we would conclude that the average STP will decrease as the average pump rate increases which does not make sense intuitively. Therefore, without accounting for other factors, we might draw an incorrect causal inference about the relationship between pump rate and STP. This further highlights the usefulness of multivariate regression models as opposed to bivariate models.

Figure 7 is a plot of the residuals vs. the fitted values for model 2. A residual is what is not explained by the model and is simply the fitted value minus the actual value at each data point. The most important thing to note is that there is not a clear trend in the residual values, which would indicate a different regression model may work better. With no clear and present trend in the residuals, this is the first piece of evidence that the linear model constructed in this study may be appropriate.

One other concern is that there is the possibility of induced bias in the model by including a lagged dependent variable as an independent variable. This makes sense intuitively because average STP is essentially being used as a control variable for its’ future self. This may create a positive feedback loop with the other independent variables since they will affect average STP in a given period, which will then be used as an independent variable in the next period, thus propagating the effects. This feedback would create patterns or serial correlation in the errors of the model and induce bias [14]. Therefore, we need to make sure that the errors are not serially correlated using the following model shown in Equation (5) [14]:

$${ϵ}_t=\rho {ϵ}_{t-1}+{\upsilon }_t$$

(5)

where ${ϵ}_t$ and ${ϵ}_{t-1}$ are the residuals and lagged residuals, respectively, from the model in question. Here, we are simply looking for a statistically significant value for $\rho$, which would indicate a correlation between the errors for each period.

Table 4. Regressions results from error model.

	Errors
	Err1
Lagged Errors	0.067
	(0.093)
Constant	−1.836
	(23.327)
Observations	116
R2	0.005
Adjusted R2	−0.004
Residual Std. Error	251.236 (df = 114)
F Statistic	0.523 (df = 1; 114)

Note: * p < 0.1; ** p < 0.05; *** p < 0.01.

shows the results from the regression.

From the model, we see a coefficient estimate of 0.067 for the lagged error that is not statistically significant, indicating that there is no correlation of the errors in the model from one period to the next. This further indicates that the model constructed in this study is appropriate for characterizing the factors that affect the average STP and the results obtained are robust.

6. Model Prediction

The models need to be able to make predictions about future events to be useful [19].

Table 5. Predicted average surface treating pressure for offset well using the model from this study.

Stage	Avg. Pump Rate (bpm)	Acid Volume (bbls)	Total Clean Volume (bbl)	Perforations	Perforation Standoff	Stage Prop Weight (lb)	Average STP (psi)	Predicted Average STP (psi)
1	21.6	35.7	4230.2	NA	26	61,748	8719	NA
2	29.3	31.5	8316.8	27	26	225,181	8903	7990
3	28	31.5	6065.2	24	26	218,337	8749	8285
4	29.7	23.8	5688.9	24	26	224,202	8968	8142
5	29.9	23.2	5224.8	24	26	227,257	8799	8313
6	28.2	23.8	5007.3	24	26	226,392	8731	8191
7	29.1	23.8	4959.8	24	26	226,608	8848	8153
8	29.9	11.1	5077.1	24	26	226,521	8730	8140
9	33.6	11.1	4665.6	24	26	227,421	8626	8108
10	35.3	11.3	4395.7	24	26	226,288	8859	8065
11	37.6	11.3	4300.1	24	26	227,810	8806	8255
12	39.1	11.1	4185.1	24	26	228,025	8732	8235
13	42.6	11.1	4365	24	26	228,048	8582	8207
14	46.5	11.1	4171.8	24	26	228,459	8780	8145
15	44.6	11.1	4186.9	24	26	230,832	8881	8266
16	43.9	11.1	3933	24	26	213,499	8608	8359
17	42	11.1	3903.2	24	26	227,737	8631	8134
18	44.6	11.1	3782	24	26	217,414	8650	8191
19	47.8	11.1	3770.9	24	26	217,405	8728	8235
20	41.2	0	4018.4	24	26	217,358	8258	8131
21	49.1	11.1	3741.7	24	26	217,387	8813	7969
22	48.3	11.1	4441.2	24	26	220,905	8301	8323
23	51.5	11.1	3812.6	24	26	216,973	8339	8020
24	49	11.1	3796.2	24	26	217,031	8347	8024
25	50.6	11.1	3876.7	24	26	217,689	8277	8040

shows the results using the regression model 1 to predict the STP for a re-fracture treatment in the middle Bakken. Table 3 shows that the multi-variate models produce reasonable predictions. While these models can be used for predictions, the primary benefit is estimating marginal effects of each independent variable on the dependent variable. This allows engineers to alter parameters in treatment design (average pump rate for example) while holding all other variables constant and predict how the average surface treating pressure, and therefore treatment in the field, will react. The predictions were included to give the reader some context as to how altering the treatment parameters might alter the average STP.

7. Further Analysis

There are additional considerations regarding the models constructed in this study. First, although there were numerous independent variables included in the regression, there still may be variables in the error term that are causing endogeneity. This may be why the acid volume seems to increase average STP and the perforation stand-off coefficient does not match the intuitive understanding of stress shadow effects from previous zones. One way to combat this is to simply add more data from future re-fracturing treatments. The assumption of negligible interaction between wells on the same pad may also need to be addressed.

The proppant concentration of the treatment should also be considered, although this data was not available for this study. Since the proppant travels at a different speed than the slurry and falls out due to gravity [20], this may influence the surface treating pressure near wellbore and in the fracture. Cleary et al. (1993) implemented a system using proppant slugs to estimate pressure responses and clean up near wellbore issues that affect treating pressure [21]. This indicates that sand concentration may be statistically significant predictor of surface treating pressure.

It is also important to note that the dynamic model did not predict an exceedingly high pressure for stage 25 that would have translated to stage 26, where the well had to be abandoned. There was also no indication from the actual average STP data of significant pressure build up prior to stage 26. This may indicate that there are unobservable geologic and wellbore effects that need to be accounted for and may be a source of endogeneity. These geologic and wellbore effects may be specific to each individual well. Further analysis will need to be conducted to attempt to isolate individual well characteristics. Things such as percentage of wellbore drilled out of zone, natural fracture networks, variations in geomechanical properties along the wellbore, perforation characteristics, etc. may also have a causal role in determining average STP.

8. Discussion

The model constructed in this study may also have benefits in combination with simultaneous HF treatments. Over 200 wells have been completed using simultaneous HF treatments in the Bakken and Permian [9]. Simultaneous treatment allows for separate treatment designs to be pumped on different well concurrently [9]. Having the ability to predict how STP of re-fracturing treatments will react to altering variables will help engineers better understand the wellbore integrity and surface equipment limitations and allow for any necessary changes to treatment design. This may become especially important in the case of simultaneously fracturing one re-fracture treatment and one treatment on a newly drilled and completed well.

9. Conclusions

• Dynamic multi-variate regression models can be constructed using field treatment data to provide understanding of causal factors on STP and provide engineers a tool to isolate effects of different independent variables of interest to see how they may affect treatment implementation in the field;
• Stress shadow effects from the previous stage on a subsequent stage have a statistically significant effect on the average STP. According to the dynamic model developed in this study, based off of the data set used, a 1 psi increase in the average STP from the previous zone will create a 0.713 psi increase in average STP in the subsequent zone. This type of pressure translation can be thought of as temporal dependence of pressure and may be the cause of the issues encountered in the offset well that lead to abandonment of the final 4 stages which was the motivation for this study;
• The models constructed in this study may be used in the field during simultaneous HF treatments that allow different treatments to be pumped on different wells simultaneously. Models to estimate the expected average STP of re-fracture treatments while simultaneously pumping treatments on newly drilled wells will provide a tool to better select surface equipment and alter treatment designs;
• Further research needs to be done to isolate other affects such as mineralogy and geological factors, and other geologic and wellbore factors that may affect treatment.