*3.4. New Empirical Correlation for Fishbone Productivity*

An empirical equation was extracted from the optimized neural network model (Equation (2)). This extracted equation can predict the flow rate based the weights and biases of the ANN model. Table 5 lists the values of weights and biases used in Equation (2). The proposed model to predict the fishbone productivity is given by the following equations:

$$\frac{\mathbf{q}}{\mathbf{q}\_{\text{max}}} = \left[ \sum\_{i=1}^{N} \mathbf{w}\_{2i} \text{tan} \, \text{sig} \left( \sum\_{j=1}^{I} \mathbf{w}\_{1i,j} \mathbf{x}\_{j} + \mathbf{b}\_{1i} \right) \right] + \mathbf{b}\_{2} \tag{1}$$

$$\frac{\mathbf{q}}{\mathbf{q}\_{\text{max}}} = \left[ \sum\_{i=1}^{N} \mathbf{w}\_{2i} \left( \frac{2}{1 + \mathbf{e}^{-2(\mathbf{w}\_{\text{Li}}/\mathbf{\tilde{K}\_{\text{h}}})\_{\text{j}} + \mathbf{w}\_{\text{Li}2}\mathbf{I}\_{\text{j}} + \mathbf{w}\_{\text{Li}3}(\mathbf{P}\_{\text{wt}}/\mathbf{P}\_{\text{avg}})\_{\text{j}} + \mathbf{b}\_{\text{li}}}} \right) \right] + \mathbf{b}\_{2} \tag{2}$$

where N is the total number of neurons, w1 is the weights of the hidden layer, w2 is the weights of the output layer, Kh/Kv is the permeability ratio, L is the lateral length, Pwf is the flowing bottomhole pressure, and Pavg is the average reservoir pressure. Note that the ANN model automatically normalizes the input into a range between −1 and 1.

**Figure 8.** Comparison of different artificial intelligence (AI) techniques.



#### *3.5. Model Verification*

Our developed correlation was used to determine the fishbone productivity for the unseen data. More than 70 data sets with different conditions of reservoir parameters and well bore configurations were used. This correlation achieved an average absolute error of 7.23% and a relatively high correlation coefficient of 0.979. Moreover, the developed correlation was compared with different numerical and analytical models proposed in the literature. Ahmed et al. [6] proposed an empirical IPR correlation based on Vogel's productivity model. Ahmed's correlation (Equation (3)) can be used to determine the productivity of a fishbone well drilled into a dry gas reservoir. Based on a literature review, Ahmed's correlation is considered one of the most accurate models that can be used to estimate the gas flow rate for fishbone wells. Therefore, the reliability of the developed ANN-based correlation was compared with that of Ahmed's model. Figure 9 shows a comparison between the gas flow rate predicted using Ahmed's correlation and the ANN-based correlation proposed in this work. It is clear that the flow rates predicted using the developed ANN correlation are a better match with the actual flow rate data than Ahmed's model. Also, at low pressure, the Vogel-form equation overestimates the production rate, while the developed ANN correlation has excellent predictions. The developed empirical equation based on the optimized ANN model predicted the flow rate with an average absolute error of 6.92%.

$$\frac{q}{q\_{\text{max}}} = 1.00756 + 1.154379 \ast \left(\frac{P\_{wf}}{P\_{wf\text{max}}}\right)^{1.35} - 2.15268 \ast \left(\frac{P\_{wf}}{P\_{wf\text{max}}}\right)^{1.7} \tag{3}$$

**Figure 9.** Comparison between the gas flow rate predicted by the ANN model and the Vogel-form equation alongside the actual gas rate.

Overall, this work can increase the confidence of decision makers in deciding to drill more fishbone wells, instead of drilling vertical or horizontal wells. The main issue with drilling fishbone wells is that most of the available production models are inaccurate, which leads to significant deviations between the models' results and the actual production data. As a consequence, most of the petroleum engineering industry prefer to drill vertical or horizontal wells in order to reduce the drilling cost and avoid the risk of fishbone wells, as it is easier to estimate the production rate for a vertical or horizontal well compared to a fishbone well. However, developing accurate models for determining fishbone well productivity can add to the credibility of such complex wells. The AI models presented in this paper can be used to provide an accurate estimation for the hydrocarbon production of multilateral fishbone wells, which will help in designing and optimizing production plans. Ultimately, this work can help in growing the application of complex wells, which can improve the sustainability of hydrocarbon production from underground reservoirs.
