1.1. The Importance of Predicting Bottomhole Pressure
While it can be challenging to extract conventional oil and gas efficiently, fracturing operations play a crucial role in enhancing their productive development [
1,
2]. To achieve a successful hydraulic fracturing treatment, accurate real-time bottomhole pressure (BHP) prediction is necessary [
3]. BHP data are vital for real-time fracture treatment diagnostics, facilitating engineer decisions concerning pumping schedules, proppant placement, and the ability to detect problems such as screenouts [
4,
5]. More importantly, accurate BHP measurements are an integral part of interpreting the fracture geometry and evaluating the effectiveness of the treatment once the job is complete [
6,
7]. However, it is difficult to accurately predict BHP because the rheological behavior of fracturing fluids is complex, especially with proppant, and pressure losses along the wellbore are numerous [
8,
9]. The fracturing fluids exhibit non-Newtonian, shear-thinning behavior, which makes predicting friction pressures challenging [
10]. Discrepancies measured to predicted BHP have been attributed to fluid rheology, proppant properties, tubular geometry, and temperature [
11].
1.2. Traditional Prediction Methods
The BHP inside the wellbore at perforation depth can be measured directly or calculated during fracturing. In the first scenario, the bottom hole pressure over time must be monitored using a downhole pressure gauge attached to the fracturing string [
12]. In contrast, the second method implies the calculation of BHP, which comprises two main parts. The first part is a hydrostatic head that includes the effect of proppant weight [
13]. The second part is the calculation of pipe friction during pumping. Pipe friction losses can be estimated using several models and correlations. For example, traditional fluid mechanics equations can be used by physics-based models to calculate pipe friction losses [
14]. Additionally, empirical correlation models are built based on observed relationships between pressure loss and several parameters, such as the flow rate, fluid properties, and tubular dimensions [
15]. Nevertheless, these correlations typically need calibration to field data for specific fluid types, proppant types, and wellbore geometries to improve their accuracy. However, hybrid models have been developed to take advantage of the complementary strengths of purely physics-based and empirical models, which provide richer interpretability. A more comprehensive pressure loss model is developed by integrating laboratory flow loop test results with field data. This approach exploits the controlled environment of laboratory experiments to understand fluid behavior but also incorporates some of the real-world fluid complexity of the field [
16].
Using estimates of the power law fluid rheology constants n′ and k′, Barree et al. (2009) developed a model to calculate pipe friction in a hydraulic fracturing process [
17]. Pressure drop in the laminar flow regime is first estimated for pipe friction. The laminar pressure drop (P
l) is given in psi per length of pipe using Equation (1).
where (d) is the pipe inner diameter in inches, (ρF) is the fluid density in g/cm
3, (k′) is the fluid consistency index in conventional oilfield units (lb-sec
n′/ft
2), (n′) is the power law exponent, and the factor (Xs) is an increase in friction with a volume fraction of solids (proppant) in suspension in the injected fluid. The volume-fraction solids injection (Cv) is related to the sand friction factor by Equation (2).
Equation (2) is adequate for linear gels or delayed cross-link fluid systems. To model the friction increase with the solids addition increase when using rapid cross-linked fluids and foams, the friction increase is corrected with an empirical correction factor (FS).
The Reynolds number required to transition to turbulent flow must be estimated once the laminar flow friction pressure has been estimated. (R
et) is estimated by Equation (3).
The final part of the friction model is the estimation of the pressure drop during the turbulent flow regime. The turbulent flow factor (T) is given by Equation (4). The transition to turbulent flow is a function of pipe inner diameter (d), which is input in inches. There is another empirical correction factor (F
D) included in Equation (4). This factor modifies the fluid k′ to handle the impact of delay in the onset of cross-link in the pipe.
Finally, the pipe friction pressure drop is calculated in Equation (5). The total wellbore pump rate (Q) is input in barrels per minute (bpm), and the pipe length increment (L) is in feet.
Much research has been conducted on the key drivers and constraints of the conventional BHP calculation methods [
18,
19]. While conventional BHP calculation methods are typically used in hydraulic fracturing, they also have several limitations and are inaccurate, leading to incorrect predictions during treatment optimization [
20,
21]. The reliance on simplified friction pressure correlations may not accurately describe the complex behavior of fracturing fluids, particularly those with rapid cross-linking and/or time-dependent rheology, which represents at least one primary limiting factor [
22,
23]. For example, borate cross-linked fluids are known to rapidly cross-link and generally show higher friction pressures than are predicted by conventional models, resulting in an underestimation of BHP [
24]. This is because standard correlations cannot describe the special flow behavior of these fluids, which can depart significantly from the assumptions of Newtonian or simple non-Newtonian models [
25,
26]. Accurately accounting for the impact of proppant on friction pressure is also another challenge [
27,
28]. While some try to rescale the base fluid friction to include correction factors to put into the proppant effects, correcting base fluid friction may not be enough to capture the complexity of two-phase flow dynamics [
29]. The presence of proppant changes the density, viscosity, and flow regime of the fluid, which generates higher friction losses, which can have a major impact on BHP calculations [
30,
31]. Additionally, the complexity introduced by the heterogeneous flow patterns that can occur when pumping proppant-laden slurries—especially in long vertical wellbore locations—is not well understood and makes it difficult to predict friction pressure [
32]. Severe limitations on the accuracy of conventional methods include the inability to seamlessly integrate real-time data into BHP calculations [
33]. Most approaches use known fluid properties and operational parameters that may not be representative of the dynamic conditions in a treatment. Fluctuating rates of flow, changes in fluid rheology due to temperature or shear effect, and changes in proppant concentration are important factors affecting BHP [
34,
35]. Not taking into account these real-time variations can cause predictions to differ from actual downhole pressures, thereby creating operational inefficiencies and poor treatment outcomes. As an example, Keck et al. (2000) analyzed measured BHP data for 45 fracture treatments and found significant deviations from predicted values using existing correlations, particularly in the proppant stages [
36]. However, they stressed that such errors can cause the misinterpretation of a net pressure trend, leading to premature treatment termination or continuation of the treatment beyond the point of imminent screenout.
Researchers have developed a number of ways to improve BHP prediction accuracy. They describe one approach in which friction pressure correlations are calibrated against field data for well condition and fluid property conditions [
37]. Real BHP data can be collected during treatments by using downhole pressure gauges, which can then be used to adjust friction factors to further represent observed pressure behavior better. Fragachan et al. (1993) used measured BHP data from ten wells in northern Mexico to calibrate a pseudo three-dimensional fracture model, which was successfully matched to observed pressure behavior [
38]. Another good approach also involves incorporating real-time rheological data. The impact of pressure losses from the fracturing fluid properties is particularly affected by viscosity. Monitoring of the fluid rheology in real time, with appropriate specialized equipment, provides the ability to make continuous updates to BHP calculations based on actual fluid behavior during the treatment [
39]. To properly calculate the BHP, it must be taken into account that proppant effects exist. An increase in friction pressure losses occurs when proppant is added to the fracturing fluid, changing its density and flow characteristics. Proppant friction pressure correction techniques can be used in conjunction with the adjustment of friction factors based on proppant concentration to improve BHP prediction during proppant stages [
40].
1.3. Machine Learning Models for Predicting BHP
The acquisition of the intricate interplay of factors influencing BHP during fracturing by machine learning models, given the large datasets readily available, makes it a promising tool [
41,
42]. Machine learning algorithms excel at analyzing large and complex datasets to detect correlations that can be applied to forecasting, improvement, and control [
43]. The application of data-driven models to all aspects of the oil and gas industry, including petrophysics [
44], drilling [
45,
46], reservoir engineering [
47,
48], production engineering [
49], geology [
50] and completion engineering [
51] has benefited from much advancement within the last several years, but little published work has focused on real-time BHP prediction during fracturing. New work has demonstrated the use of ML to predict wellhead pressure during hydraulic fracturing. By training neural networks on the first few minutes of each fracturing stage, Ben et al. (2020) developed a real-time wellhead pressure prediction model, achieving accurate forecasts while continuously updating the model with new data as it became available [
52]. An approach of continuous learning allowed for the capture of the evolving system dynamics during the fracturing process, resulting in increased achievable prediction accuracy when compared with static models. Their work highlights ML’s capability to predict pressure trends in complex processes like hydraulic fracturing, suggesting its potential applicability to BHP prediction.
The second promising avenue for BHP prediction is integrating ML with physics-based models. An example of the use of combined experimental data from flow loop tests and fracturing stage field data to develop a hybrid physics-augmented ML model for predicting friction pressure loss during hydraulic fracturing is given by Abdulwarith and Ammar et al. (2024) [
16]. The model includes critical parameters such as fluid properties, proppant concentration, and perforation parameters that account for both wellbore and near-wellbore friction loss. The resulting hybrid approach adopts strengths from both the physics-based models and data-driven algorithms with the possibility of improved prediction accuracy and BHP robustness. This is evidence that integrating ML with domain knowledge could yield better BHP forecasting, as the results of their model predicting wellhead pressure had an average absolute relative error of less than 5%.
Although research on direct BHP prediction with ML is still growing, several studies demonstrate the application of ML in related domains (fracture diagnostics and treatment optimization), which can be used to indirectly inform the BHP prediction models. For example, Reznikov et al. (2024) built a computationally efficient approach to interpret the fluid injection in layered formations, given BHP measurements [
53]. Solving such an inverse problem enables their approach to determine parameters such as skin factor and fluid partitioning across zones. This technique significantly enhances the analysis of matrix treatments by coupling hydrodynamic models with real-time pressure and injection rate data to aid in identifying the efficiency of fluid placement and potential pressure effects.
More research is required to gain a full understanding of the capabilities of ML in the prediction of BHP in hydraulic fracturing. This research intends to fill the gap in developing an advanced model by developing and contrasting 11 advanced ML models using readily available field data. The goal is to predict real-time BHP in the wellbore at perforation depth during hydraulic fracturing. In fact, these ML models can be seamlessly embedded within fracturing dashboards and hydraulic fracturing simulation software for real-time BHP prediction during fracturing. These machine learning models are an effective alternative to expensive and time-consuming deployed bottomhole pressure gauges, as well as inaccurate physics-based models, empirical correlations, and hybrid models due to high accuracy and low price. Therefore, hydraulic fracturing engineers can benefit from employing it as a capable tool for rapid diagnostic assessments and real-time optimization of operational variables such as rate of cross-linked gel injection, proppant concentration, and size to produce an optimal fracture geometry and keep ahead of premature proppant screen out [
54]. Additionally, these ML models perform better than a single correlation or model at capturing a broad range of cross-linked fracture fluid systems, proppant types, and tubing configurations, which is difficult for any single conventional correlation or model.