Modeling hydraulic fracture fluid efficiency in tight gas reservoirs using non-linear regression and a back-propagation neural network

ABSTRACT This study introduces a back-propagation (BP) neural network model and a regression model for estimating the fracture fluid efficiency based on a data set consisting of 1261 staged and ramped simulation runs of tight gas reservoirs subjected to hydraulic fracturing treatment. Data were generated using a 3-D commercial simulator which is a versatile software portfolio that models many well configurations, proppant placement, and fracture geometries. The BP network inputs consist of shear rate/fracture conductivity ratio, the injection rate, reservoir permeability, formation closure stress, reservoir thickness, effective viscosity, and fracture height. The neural network model was able to generate satisfactory estimates of the fracture fluid efficiency for the training dataset, and for the blind testing data. An average error of approximately 2.5% was obtained for the training set, and an average error of 3% was obtained for the testing set. An empirical non-linear regression model has been constructed based on dimensionless groups derived by applying dimensional analysis to a set of variables consisting of the maximum fracture width, fracture length, fracture height, effective viscosity, shear rate/fracture conductivity ratio, reservoir thickness, injection rate, reservoir permeability, and formation closure stress. The average error for estimating the fluid efficiency using the non-linear regression empirical model was approximately 6.17%. Since the non-linear regression model has an explicit formulation, it is easier to apply than the neural network model. The empirical regression model estimates of the fluid efficiency appeared to be unbiased and were more precise than those estimates obtained using either the KGD or the PKN 2-D models. The introduced BP model and the non-linear regression model offer fast and inexpensive alternatives to the application of three-dimensional simulators for estimating the fluid efficiency.


Hydraulic fracturing background
Hydraulic fracturing treatment is applied for the purpose of increasing the productivity index of hydrocarbon formations (Eom et al., 2014;Fan et al., 2015;Gao et al., 2018;Lin et al., 2018;Rayudu et al., 2019). Hydraulic fracturing is a means of stimulating primarily low permeability reservoirs. It can also be applied to stimulate moderate permeability reservoirs via bypassing formation damage that may not be efficiently removed by matrix acidizing. The process, however, is complicated by design constraints, and operational difficulties.
The treatment method consists of increasing the surface area of the formation open to flow by creating a fracture and filling it with a proppant. The hydraulic fracturing process consists of three stages: the pad stage, the slurry stage, and the flush stage. The pad stage consists of injecting a pad fluid at a pressure greater than the fracture pressure of the reservoir rock, causing a fracture to initiate and to propagate in the formation (H. Wang & Sharma, 2018). A slurry of pad fluid and proppant at various increasing concentrations is then pushed into the fracture so that the fracture gains width and stays open once the applied pressure is released. Finally, the slurry stage is followed by a flush stage that consists of injecting one wellbore volume of fluid (Cipolla et al., 2008). The flush stage is meant to sweep the wellbore clean of proppant. The well is then shut for some period to allow pad fluid to leak off from the fracture. Consequently, the in-situ stresses compress the proppant pack, and the fracture establishes its initial producing geometry (Sesetty & Ghassemi, 2018). In retrospect, the final design parameters must honor the actual well-operational constraints dictated by the wellcompletion type and the pump horsepower requirements. When the fracturing job is 'successfully' completed, well productivity can be increased by several folds (Mohaghegh et al., 1999;Zhao et al., 2018).
Although hydraulic fracturing is a rather old method of well stimulation, it is often applied with mixed results. The key to hydraulic fracturing success is the ability to optimize conditions for its application. The goal of a treatment design is to optimize four basic variables that consist of the maximum fracture width, the fracture length, the fracture height, and the fracture fluid efficiency. To achieve these goals, engineers are allowed to control a few parameters like the pad and slurry volume, the proppant final concentration, the fluid additives, the pumping schedule pattern (ramped or staged), and the pumping rate (Detournay, 2016;Popa, 2004). The next section discusses common simple models, available in the literature, for estimating the fracture fluid efficiency.

Literature review
Recently, Al-Zaabi (2019) applied dimensional analysis and neural networks for the development of a model for estimating the fracture fluid efficiency. Even though the obtained dimensionless groups claimed to give good correlations for discrete pad volumes and discrete dimensionless fracture conductivity, the final estimated fracture fluid efficiency from Al-Zaabi (2019) models suffered from considerable uncertainty.
Explicit models for the fracture fluid efficiency are rare in the literature. The PKN model (Perkins & Kern, 1961) gives the following implicit expression for the fracture fluid flow efficiency as a function of fracture geometry and operational parameters: Equation 1 has been re-arranged to give an explicit expression for the fluid efficiency as follows: η ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi w max 0:8573t 0:2 In the above notation, υ is the Poisson's ratio, q i is the injection rate in bpm, μ eff is the effective pad fluid viscosity in cp, H F is the fracture height in ft, t is the pumping time in minutes, w max is the maximum fracture width in inches, η is the hydraulic fracture fluid efficiency expressed as a fraction, and G is given as a function of the Young's modulus (E) in psi by the following expression: Likewise, the KGD model (Geertsma & De Klerk, 1969) gives an implicit expression for the fracture fluid efficiency as follows: (4) Equation 4 has been re-arranged to give an explicit expression for the fracture fluid efficiency as follows: In both PKN and KGD models, the fracture height is assumed to be constant. Only length and width are allowed to vary (Economides & Nolte, 2000). Fracture width is determined from a form of Sneddon's equation for a plane-strain crack (Ham & Kwon, 2019), whereas fracture length is determined by material balance considering injected fluid volume, leak-off, and fracture volume (Ham & Kwon, 2019).

Scope of study
Since the publication of the two-dimensional PKN model (Perkins & Kern, 1961) and the KGD model (Geertsma & De Klerk, 1969), there has been a shortage of effort in attempting to predict fracture geometry parameters, using simple models. The usual hydraulic fracturing optimization process involves the application of three-dimensional simulators. These simulators require large data sets of petrophysical and rock mechanical properties (Gao et al., 2018;Zhang et al., 2012). Hence, in general they are difficult to run, and are computationally expensive. The objective of this paper is to apply dimensional analysis in conjunction with neural networks to develop general models that can predict the hydraulic fracture fluid efficiency, as a function of key reservoir and operational variables. These anticipated models are meant to provide an alternative to three-dimensional simulators and to the simple PKN and KGD models. The performance of the models developed in this study for the fracture fluid efficiency will be compared later in the text to the performances of the PKN and the KGD models, given by Equations 2 and 5, respectively. The data used, in this study, are described next.

Data description
This research study uses simulation data of hydraulic fracturing treatments conducted on tight gas sand reservoirs (Popa, 2004). Tight gas sand reservoirs constitute a natural extension of gas-bearing conventional sandstone reservoirs with permeability less or equal to 0.1 mD. The simulation data were generated using the three-dimensional simulator FRACPRO (Popa, 2004) which was the simulator of choice because of the software's widespread use in the oil and gas industry, and because of its powerful 3-D capabilities. To generalize the study as much as possible, input parameters were selected with ample variations (Table 1). Fracture length, maximum width, height, conductivity, and fracture fluid efficiency were generated by inputting the following parameters: 1. Total fluid volume (V T ) -ranging from 10,544 gallons to approximately 200,000 gallons.
2. Pad volume (V p ) -typically ranging between 39% to approximately 44% of the total fluid volume.
3. Final proppant concentration (C F ) -typically considered between 6 and 16 ppg for staged treatments. For ramped treatment jobs, the final proppant concentration varied between 1 and 10 ppg.
4. Average injection rate (q i ) -ranging from 15 bbl/ min to 40 bbl/min. 5. Depth (D) to the top of the 'productive formation'varied between 5000 ft to 15,000 ft.
6. Reservoir pore pressure (P R ) -It was calculated as depth multiplied by the pore pressure gradient.
10. Closure stress gradient for the pay zone 13. Porosity (ϕ) was taken as a fixed value of 9%. 14. The leak-off fluid permeability ratio (K p /K l ) varied between 25 and 30. K p /K l stands for the ratio of formation pore-fluid permeability to formation leakoff fluid permeability.
A pad fluid from the class Boragel was used in the tight gas sand simulations. Fluids from this class are typically used in the stimulation of tight gas reservoirs (Popa, 2004). The proppant Brady 20/40 sand was used for generating the simulation data since 20/40 mesh size is one of the most popular sands used in fracturing tight gas sands.
A total of 1261 ramped and staged treatments were simulated. The simulation runs consisted of 630 ramped treatment cases and 630 five-staged treatment cases. In general, the staged schedule treatment consisted of pumping different volumes of slurry at a specific proppant concentration for fixed amounts of time. In the staged treatment, the number of slurry stages was usually less or equal to 10. For instance, the five-stage treatment consisted of injecting the following fluids in increments: 1. Pad fluid 2. First slurry volume at an initial proppant concentration (C 1 ).
The proppant concentrations were increased from C 1 to C 4 , as a step-function. Usually, one wellbore volume of slick water was pumped at the end of the treatment. This was referred to as the post-flush stage. Its purpose was to clean all the sand from the tubing, or casing. Conversely, in the ramped schedule treatment, state of the art blenders allowed the entire slurry stages to be combined into one single stage. Therefore, the proppant concentration was ramped up continuously from its initial value to the final desired value. Likewise, the slurry volume was followed by a post-flush stage that consisted of one well-bore volume of slick water. The Table 1. Summary of simulation data from Popa (2004) used in this paper. required input parameters for the simulation run for both the ramped treatment and the staged treatment were identical. Ten input parameters, discussed earlier, were required. The next section illustrates how the data mining methodology was applied for formulating the hydraulic fracture fluid efficiency models.

Methodology and discussion
The methodology applied in this research consists of using dimensional analysis in conjunction with artificial neural network to develop models for the fracture fluid efficiency (η). Dimensional analysis is described next, followed by non-linear regression analysis and artificial neural network development.

Dimensional analysis
The advantage of dimensional analysis consists of its ability to study similar systems for which the defining equations are not completely articulated (Garrouch, 2018;Garrouch & Al-Sultan, 2019). Studies that apply dimensional analysis for estimating hydraulic fracturing design parameters are scarce in the literature. He et al. (2018) have recently applied dimensional analysis in hydraulic fracturing optimization. He et al. (2018) used dimensional analysis tools to come up with transforms that affect the creation of controllable hydraulic fractures in cave mining. In fact, by applying dimensional analysis, He et al. (2018) reduced the number of variables influencing hydraulic fracturing from 19 to 10. A set of dimensionless groups governing mining hydraulic fracturing has been derived, consequently. An FLT dimensional analysis for the dependence of the fracture fluid efficiency (η) on the hydraulic fracturing design parameters was performed. The objective of this dimensional analysis was to ultimately use the derived dimensionless groups for developing an empirical model for η as a function of reservoir and hydraulic fracturing operational parameters. FLT stands for the fundamental dimensions of force (F), length (L), and time (T). It was postulated that η was a function of the maximum fracture width (wmax), the tip-to-tip fracture length (LF), the fracture height (HF), the effective viscosity (µ eff ), the shear rate (γ), the dimensionless fracture conductivity (FCD), the reservoir thickness (HR), the injection rate (qi), the reservoir permeability (k), and the formation closure stress (σ c ). The postulated relationship is given as follows: The parameters, w max , HF, and HR, were grouped since they have the same fundamental dimension. The new composite variable (wmax HF/HR) has a fundamental dimension of length. Since the dimensionless fracture conductivity FCD is dimensionless, it allows some freedom for the parameters they are grouped with. This parameter is grouped arbitrarily into a transformed is, therefore, rearranged as follows: A unique requirement of dimensional analysis dictates that the variables to the right-hand side (RHS) of the above equation must have low correlations among each other. The correlation coefficients between variables used in Equation 7 are illustrated in Table 2. The highest correlation coefficient among the variables shown at the RHS of Equation 7 is approximately 0.59 (Table 2). It is evident that parameters of the RHS of Equation 7 are not well correlated and may be considered independent. For a dimensionally homogeneous equation, with the three fundamental dimensions (FLT) used in the analysis, and the total six variables expressed in Equation 7, the Buckingham pi theorem (Munson et al., 2010) depicts three dimensionless groups (π 1 ,π 2 , and π 3 ). The formulation of these dimensionless groups is constrained by rules that relate to the selection of the repeating and the non-repeating variables of these groups (Garrouch, 2018;Garrouch & Al-Sultan, 2019;Munson et al., 2010). Taking η, k, and σ c as the non- ,and q i as the repeating variables allowed to formulate the following three dimensionless groups: The variables of Equation 8 were replaced by their FLT fundamental dimensions shown in Table 3. The exponent powers of each fundamental dimension were summed up and were equated to zero. The unique solution of exponents a, b, and c depicts the following expression for the dimensionless group π 1 : Likewise, the variables of Equation 9 were replaced by their FLT fundamental dimensions given in Table 3. The exponent powers of each fundamental dimension were summed up and were equated to zero. The unique solution of exponents a 1 ,b 1 , and c 1 depicts the following expression for the dimensionless group π 2 : The dimensionless number (π 2 ) of Equation 12 appears to normalize the rock permeability to variables that reflect the fracture geometry. The latter variables consist of the fracture maximum width (w max ), the fracture height (H F ), and the fracture height, which are same as the pay-zone thickness (H R ). Therefore, π 2 is designated as the fracture geometry transform. Finally, the variables of Equation 10 were replaced by their FLT basic dimensions given in Table 3. The exponent powers of each fundamental dimension were summed up and were equated to zero. The unique solution of exponents a 2 , b 2 , and c 2 depicts the following expression for the dimensionless group π 3 : The dimensionless number (π 3 ) in Equation 13 may be considered as a fracture flow property transform. The dimensionless number (π 3 ) appears to normalize the fracture width (w max ) with flow design parameters like the shear rate, the effective pad fluid viscosity, and the dimensionless fracture conductivity. Equations 11, 12, and 13 reduced the physical relationship between the fracture fluid efficiency (ƞ) and the remaining variables of Equation 7 into a concise functional form as follows: The dimensionless groups in the RHS of Equation 14 were used to develop a non-linear regression model for ƞ.

Non-linear regression
The contraction of the physical relationship between η and the remaining variables, given by Equation 7, into a more succinct functional form given by Equation 14 illustrates the main utility of dimensional analysis. The simulation data, generated by Popa (2004) for tight gas reservoirs, have been used to validate the derived dimensionless groups (Khouli, 2020). The n-dimensional data fitting software (ndCruveMaster) was applied for establishing a relationship between the fracture fluid efficiency (π 1 ) and the dimensionless groups (π 2 ) and (π 3 ). Searching through all possible variants manually is timeconsuming. Hence, an automated approach has been implemented in ndCurveMaster software to solve this problem by selecting the nonlinear functions and variable combinations using the 'Auto-fit' method. The 'AutoFit' method utilizes an algorithm where variables are randomized, and many exponents corresponding to these variables are iterated. This algorithm is fast and efficient. The final relationship between dependent and independent variables with the least amount of error between actual and predicted target variables is reported. This algorithm finishes searching when the coefficient of determination value reaches its maximum value (Khouli, 2020). By applying the 'Auto-fit' method in the 'ndCurveMaster' software, the non-linear relationship between the estimated fracture fluid efficiency and the dimensionless groups was suggested, as follows: In the above notation, d 0 ,d 1 , and d 2 are constants. Equation 15 affirms a high degree of non-linearity between ƞ and the dimensionless groups π 2 and π 3 . The hydraulic fracture efficiency appears to increase as the fracture geometry number (π 2 ) decreases. This is an intuitively expected result. As the rock permeability increases, the fluid losses increase and the efficiency decreases. By substituting Equations 12 and 13 in Equation 15, the following expression was obtained for the fracture fluid efficiency: (16) Figure 1 shows estimated fracture fluid efficiency using Equation 16 versus the simulated fracture fluid efficiency. A coefficient of determination of 0.85 is obtained. The average relative error obtained, using the model expressed by Equation 16, is approximately 6.17%.

Neural network model
Nowadays, the application of artificial neural network (ANN) is increasing for petroleum engineering fields. Machine learning potential in solving petroleum industry problems has attracted researchers to enhance process performance. Ebrahimi et al. (2013) used artificial neural networks to extract the pore network from micro-computed tomography images. Ramah et al. (2022) constructed an ANN to predict the fold-ofincrease in production rate post a hydraulic fracturing job. Jamshidian et al. (2017) constructed an ANN for estimating the minimum horizontal stress using well log data. Al-Dousari et al. (2022) have developed an ANN model to predict the flow zone indicator using open hole log data for carbonate reservoirs. In several studies, results from the ANN models were as precise as those obtained from large-scale simulations. The objective of this part of the research was to develop a back-propagation neural network model to predict the fracture fluid efficiency (η). The backpropagation neural network paradigm consisted of a ten-neuron input layer, three hidden layers of twelve neurons each, and a one-neuron output layer as shown in Figure 2. For a single neuron with x p inputs, the output y i of a neuron i is given as follows: Where, x 1 ,x 2 , . . ., x p are inputs to neuron i, w i1 ,w i2 , . . ., w ip are synaptic weights of neuron i, u i is the linear combiner output, b i is the bias, λ is the activation function, and y i is the neuron i output. The output of the neuron is described by the activation function. The linear transfer function takes linear values between 0 and 1 (Al-Dousari & Garrouch, 2013). The linear transfer function is given by: Also, the log-sigmoid and tan-sigmoid transfer functions are given, respectively, by the following expressions: The Gaussian transfer function is given by: In the above notation, µ o is the sample mean, and σ o is the standard deviation. As summarized in Table 4, a logistic activation transfer function was used for layer 1. A Gaussian transfer function was used for layer 2 (first hidden slab). A tan-sigmoid transfer function was used for layer 3 (second hidden slab). A logistic transfer function was used for layer 4 (third hidden slab), and a Gaussian correlation transfer function was used in layer 5 (output slab). The learning rate was 0.1, and the momentum rate was also 0.1. The weight matrix and the bias vector are shown in Table 5. A database of 1261 hydraulic fracturing jobs (Popa, 2004), giving the hydraulic fluid efficiency as a function of the ten operational and tight gas reservoir parameters, was used for training and testing a backpropagation neural network. Since the input variables were different by several orders of magnitude, they were normalized so that they would have the same order of magnitude variation ranging from −1 to +1. The input variables were transformed into standardized normalized values using the following equation:  where, � Y is the normalized variable value, Y is the original variable value, δ is the minimum variable value, and ς is the maximum variable value. The performance of the neural network is assessed by (i) calculating the coefficient of determination (R 2 ) and by (ii) calculating the average relative error for the target output value.
The input file consisting of 1261 hydraulic fracturing jobs database for tight gas reservoirs (Popa, 2004) has been uploaded into NeuroShell software (Vonk et al., 1995, May). The input file consisted of ten input vectors and one output vector. The data were randomly divided into a training data set that includes 80% of the data and a blind testing data set that consists of 20% of the data. Ten essential input parameters of the neural network model consisted of the shear rate/fracture conductivity ratio (γ/F CD ), the injection rate (q i ), the reservoir permeability (k), the formation closure stress (σ c ), the reservoir thickness (H R ), the effective viscosity (μ eff ), the fracture permeability (k f ), the fracture length (L f ), the fracture height (H F ), and the maximum fracture width (w max ). The network output consisted of fracture fluid efficiency (ƞ). For the effective viscosity, the pad fluids used are non-Newtonian with viscosity being shear dependent. Therefore, the study useed the in-situ effective viscosity in the fracture. This was calculated by using Equations A-1 and A-2 shown in the appendix.
The back-propagation neural network paradigm design process was iterative as shown in Figure 3 (Al-Dousari & Garrouch, 2013). It started by selecting a suitable architecture. Then, the ANN was trained by using 80% of the original available data. If the training results were satisfactory, i.e., they gave an average relative error of approximately less than 5% between estimated target output and actual output, then the network performance was verified using the remaining 20% of the data not seen by the network. If the precision of the blind test results were weak, the ANN structure and parameters were altered, and the ANN training and testing were repeated until the precision of the training and blind test sets were satisfactory (Al-Dousari & Garrouch, 2013).
Using both training and testing data sets, the neural network model was able to estimate the fracture fluid efficiency (ƞ), with satisfactory precision. The data for the estimated fracture fluid efficiency (ƞ) have been fit against the actual fracture fluid efficiency for the training data set (Figure 4), yielding a coefficient of determination of 0.967. This corresponded to an average absolute relative error of 1.8%, implying an excellent degree of learning by the network model. The data for  the estimated fracture fluid efficiency (η) have been also fit against the actual fracture fluid efficiency for the blind test data set (Figure 5), yielding a coefficient of determination of 0.9274. The average absolute relative error for the testing data set of the neural network model was 2.7% for the fracture fluid efficiency (ƞ). These ANN training and testing results showed that the ANN model was able to predict the fracture fluid efficiency without memorizing and with a remarkable precision.

Sensitivity analysis
A sensitivity study has been performed by making individual runs while removing a single input variable at a time. Table 6 gives a comparison of the coefficient of determination (R 2 ) for all sensitivity study runs. The fracture height (H F ) appeared to have the most significant effect on the fracture fluid efficiency since the coefficient of determination (R 2 ) was reduced from 0.967 to 0.9298 for the training set and from 0.93 to 0.88 for the testing test. The network architecture appeared to be equally sensitive to the fracture thickness (H F ), as well. The testing coefficient of determination was also reduced from approximately 0.93 to 0.89 when H F was removed. The removal of the effective viscosity from the input variables set appeared not to have a significant effect on the prediction capability of the network. Similarly, removal of the fracture permeability (k f ), fracture length  (L F ), and maximum fracture width (w max ) did not appear to significantly alter the prediction capability of the network. This was not a surprising result since these variables k f ,L F , and w max were included in the dimensionless conductivity F CD . Parameter F CD is given as follows: In summary, the neural network may be able to run with only seven input nodes instead of ten input nodes. The final network architecture is given in Figure 6. Its paradigm consisted of a seven-neuron input layer, three hidden layers of twelve neurons each, and a one-neuron output layer. A final run was performed with the following input nodes γ/F CD ,q i , k, σ c ,H R ,µ eff ,H F for estimating the fracture fluid efficiency. Figure 7 illustrates the estimated efficiency versus simulated efficiency for the training data set using the seven-input neuron paradigm. The ANN yielded a coefficient of determination of approximately 0.94, and a relative error of 2.5% for the training set. Figure 8 illustrates the estimated efficiency versus simulated efficiency for the testing data set using the seven-input neuron paradigm. The ANN yielded a coefficient of determination of approximately 0.91, and a relative error of 3% for the testing set. These results appear to suggest that the ANN with the seven-input neuron paradigm was adequate for predicting the fracture fluid efficiency with a reasonable precision.

Comparing results with PKN and KGD models
The PKN and KGD models, expressed by Equations 2 and 5, respectively, have been applied to estimate the fracture fluid efficiency using Popa (2004) simulation data for comparison. Popa (2004) data did not include values for the in-situ effective pad fluid viscosity (μ eff ) given in Equation 2 and 5. Appendix A gives details how the in-situ effective pad fluid viscosity has been estimated in this study. As shown in Figures 9 and 10 both the KGD and PKN models generated biased estimates of the fracture fluid efficiency. The average relative error obtained using the PKN model was approximately 39%. The average relative error obtained using the KGD model was about 30% (Table 7). In the PKN model, the fracture width is allowed to increase due to an increase in the viscous pressure drop, along the increasing fracture length. This might be the reason why the PKN model did not match the simulated η with reasonable precision. In the KGD model, the pressure needed to maintain a constant width decreases as the fracture length grows. It predicts a smaller pressure gradient along the fracture length than the simulated values (Economides & Nolte, 2000). Perhaps, this was the reason the KGD lost precision in estimating values of ƞ. As shown in Figure 11, the estimates of the fracture fluid efficiency using the non-linear regression model appeared to be less biased than the estimates obtained using either the PKN or the KGD models. On average, the non-linear regression model estimates of η appeared to be closer to the simulation estimates than either KGD or PKN model estimates (Table 7). The back-propagation neural network model outperformed the non-linear regression model as      well as the PKN and the KGD models in predicting the hydraulic fracture fluid efficiency as shown in Figure 11.

Conclusions
A unique data mining approach has been implemented, using dimensional analysis in conjunction with regression analysis and a back-propagation neural network, for formulating empirical models that predict the hydraulic fracture fluid efficiency (η) in tight gas reservoirs. Dimensional analysis has proven to be useful for unravelling distinct transforms controlling the fracture fluid efficiency. The back-propagation neural network model gave an average absolute error for both the training and testing data sets of 2.5% and 3%, respectively. These results indicate that the back-propagation model was able to generalize solutions to unseen data sets during training. The non-linear regression model prediction capability has improved by the inclusion of dimensionless groups, yielding an average relative error of approximately 6.17%. The non-linear regression model estimates of η were unbiased and were more precise than those estimates obtained using the KGD, or the PKN 2-D models. The non-linear regression model featured an essential advantage of being explicit, though. The empirical models introduced in this study for the fracture fluid efficiency offer a fast and cost-effective alternative for simulations that is costly, timeconsuming, and requires input of many variables.

Disclosure statement
No potential conflict of interest was reported by the author(s). Figure 1. Calculated effective viscosity versus shear rate at the wellbore cross section using Popa (2004) data.