*3.2. Optimization Process Findings*

Consequently, the optimized network would use certain parameters, summarized as follows:


Figure 8 shows a schematic diagram of the architecture of the developed ANN model used to estimate the *PRstatic* values for sandstone formations. The results obtained from the developed ANN model show a significant match between the measured and predicted *PRstatic* values from the ANN model. This is indicated by high values of *R* of 0.96 and MAPE of 2.39% between the measured and predicted *PRstatic* values for the training process as shown in Figures 9 and 10. Also, the results showed *R* of 0.95 and MAPE of 2.10% between the measured and predicted *PRstatic* values for the testing process as depicted in Figures 11 and 12. These findings are only guaranteed if the new input data are within the same range of the dataset used to train the network in order to get accurate predictions of *PRstatic*, otherwise a large error may be encountered. If the new data are out of that range, the network should be re-trained from the beginning to be updated and to get the new optimized parameters for the new dataset.

**Figure 8.** Schematic diagram of the architecture of the developed ANN model showing the input and output parameters with the optimized number of neurons (13 neurons) and assigned weights and biases between the model layers.

**Figure 9.** Comparison profile between predicted *PRstatic* vs. measured *PRstatic* during the training process showing a high match between the predicted and measured values with a correlation coefficient (*R*) of 0.96 and mean absolute percentage error (MAPE) of 2.39%.

**Figure 10.** Cross plot between predicted *PRstatic* vs. measured *PRstatic* for the training process with an *R*<sup>2</sup> of 0.9 between the predicted and measured values.

**Figure 11.** Comparison profile between predicted *PRstatic* vs. measured *PRstatic* during the testing process, showing a significant match between the predicted and measured values with *R* of 0.95 and MAPE of 2.10%.

**Figure 12.** Cross plot between predicted *PRstatic* vs. measured *PRstatic* for the testing process with *R*<sup>2</sup> of 0.90 between the predicted and measured values.

### *3.3. Development of an ANN-Based Mathematical Model*

The developed ANN model can be mathematically expressed by Equation (4), which includes the linking weights and biases of the aforementioned three layers of the ANN model (input layer, hidden layer, and output layer).

$$PR\_{\text{static},n} = \sum\_{i=1}^{N} w\_{2\_i} \frac{w\_{1\_{i1}} R H OB\_{\text{n}} + w\_{1\_{i2}} \Delta t\_{\text{comp},n} + w\_{1\_{i3}} \Delta t\_{\text{shear},n} + b\_{1\_i}}{1 + \left| w\_{1\_{i1}} R H OB\_{\text{n}} + w\_{1\_{i2}} \Delta t\_{\text{comp},n} + w\_{1\_{i3}} \Delta t\_{\text{shear},n} + b\_{1\_i} \right|} \tag{4}$$

where *PRstatic,n* is the normalized value, *N* is the optimized number of neurons in the hidden layer (*n* = 13), *i* is the index of each neuron in the hidden layer, *w*<sup>1</sup> is a matrix of weights linking the input and hidden layers, *b*<sup>1</sup> is a vector of biases linking the input and hidden layers, *w*<sup>2</sup> is a matrix of the weight linking the hidden and output layers, *b*<sup>2</sup> is a bias (scalar) between the hidden and output layers (*b*<sup>2</sup> = 0.544), *w*1*i*,1 represents the weight (associated with neuron of index (*i*) in the hidden layer) which will be multiplied by the normalized value of the first input (*RHOBn*), *w*1*i*,2 represents the weight (associated with neuron of index (*i*) in the hidden layer) which will be multiplied by the normalized value of the second input (Δ*tcomp*,*n*), and *w*1*i*,3 represents the weight (associated with neuron of index (*i*) in the hidden layer) which will be multiplied by the normalized value of the third input (Δ*tshear*,*n*).

The development of this empirical equation converts the developed ANN model from a black-box mode into a white-box mode. This provides the ability to predict *PRstatic* values for sandstone formations using Equation (4) by only substituting the required input parameters (*RHOB*, Δ*tcomp* , Δ*tshear*) and the optimized weights and biases listed in Table 3, without the need to run the ANN model. Hence, the feasibility of practical implementation of the developed ANN model is high.


**Table 3.** The optimized weights and biases for the developed ANN model.

#### *3.4. Procedure to Use the Developed Empirical Equation to Predict PRstatic Values*

The developed empirical equation can be used to estimate *PRstatic* values for sandstone formations according to the steps described below.

First, the input parameters (*RHOB*, Δ*tcomp* and Δ*tshear*) should be normalized using Equations (5)–(7). The normalized values (*RHOBn*, Δ*tcomp*,*<sup>n</sup>* and Δ*tshear*,*n*) are substituted in Equation (4) to calculate the normalized value of the static Poisson's ratio (*PRstatic,n*) with the optimized weights and biases listed in Table 3.

$$RHOB\_n = 2.994(RHOB - 2.312) - 1\tag{5}$$

$$
\Delta t\_{comp,n} = 0.0578 \left( \Delta t\_{comp} - 44.341 \right) - 1 \tag{6}
$$

$$
\Delta t\_{\text{shear},n} = 0.0318(\Delta t\_{\text{shear}} - 73.18\mathcal{T}) - 1\tag{7}
$$

Then, the actual value of the static Poisson's ratio (*PRstatic*) can be obtained by denormalizing its normalized value (*PRstatic,n*) using Equation (8). Figure 13 shows a summary of the procedure needed for applying the developed correlation.

$$PR\_{static} = \frac{PR\_{static,n} + 1}{7.1174} + 0.2\tag{8}$$

**Figure 13.** Procedure to apply the developed ANN-based correlation.

*3.5. Validation of the Developed ANN Model and the Extracted Equation*

The validation process of the developed ANN model is conducted in two main phases:

**Phase 1:** includes using unseen data from other drilled wells within the same area to predict *PRstatic* and comparing the results with the actual values.

**Phase 2:** validates the developed model vs. common previous approaches.

#### 3.5.1. Phase: Validation Using Field Data

For validating the developed ANN model, actual field data from two other wells are used. These data are not included in building the ANN model (training and testing).

#### **Case Number 1**

The data collected from well number 1 comprise a continuous profile of petrophysical log data including *RHOB*, Δ*tcomp*, and Δ*tshear* measurements of an interval of 550 ft of the sandstone formation, in addition to five core data points representing core samples of the formation within this interval. The log data of these three parameters were used as the inputs to estimate *PRstatic* using the ANN-based empirical equation expressed in Equation (4). Then, the results obtained from the ANN model are compared with the laboratory measured *PRstatic* core data. Figure 14 shows that the model estimates *PRstatic* values within this 550 ft-interval with good match, indicated by *R* of 0.93 and an MAPE of 4.2% between the predicted and the actual values.

**Figure 14.** Comparison of the predicted *PRstatic* values by the ANN model with the measured values for cores from well number 1 (*R* = 0.93, MAPE = 4.2%).
