Modeling Apparent Viscosity, Plastic Viscosity and Yield Point in Water-Based Drilling Fluids: Comparison of Various Soft Computing Approaches, Developed Correlations and a Committee Machine Intelligent System

Abstract

Soft computing (SC) techniques are being used by drilling experts to have proper control on drilling processes and decrease the overall expenditure of these operations by providing the possibility of pre-estimating the drilling incidents. In this research study, several methods including generalized reduced gradient (GRG) method and SC techniques such as multilayer perceptron (MLP) and radial basis function (RBF) are employed to model apparent viscosity (AV), plastic viscosity (PV) and yield point (YP) values of water-based drilling fluids (WBDFs). Furthermore, to ameliorate the estimation capability of the SC models, the performance of different MLP learning algorithms such as Levenberg–Marquardt (LM), Bayesian regularization (BR), scaled conjugate gradient (SCG) and resilient backpropagation (RB) were investigated and finally the four top models were unified into a paradigm using a committee machine intelligent system (CMIS). Moreover, to facilitate the on-site estimation of AV, PV and YP parameters, GRG method was adopted and three novel correlations were proposed. Finally, the performance analysis of all the constructed models is done and with respect to the obtained results, the CMIS method is found to be the most successful method for the tested dataset. The average absolute percent relative errors (AAPRE) of this model was found to be 6.51%, 8.01% and 11.72% for AV, PV and YP respectively, indicating that this model outperforms almost all the other predictive approaches existing so far. Although other models in the literature produced a better accuracy, dataset used in this study is more complicated and generalized. Additionally, in comparison with the existing correlations, more accurate results for the tested dataset were provided by the correlations developed in this study.

Appendix

1.1 Estimation Techniques

1.1.1 Generalized Reduced Gradient

GRG can be used as a method to solve multivariable problems. To be more specific, this method is capable of solving linear and nonlinear problems and is capable of providing a significant prediction accuracy by choosing the most appropriate variables for target equations [47]. Indeed, GRG generates a linear estimation for the gradient at a specified point (e.g., x). The gradient and restrictions are concurrently explained and the objective function could be defined as the gradients of restrictions. Thereafter, it is possible to move the search space in a possible direction and the search area size can be reduced, consequently. For the objective function of f(x) which is subjected to h(x), we have:

$${\text{Minimum}}:f(x) = x$$

(A.1)

Subjected to:

$$h_{k} (x) = 0$$

(A.2)

This method can also be described as shown in Eq. (A.3):

$$\frac{df}{{dx_{k} }} = \nabla x_{k}^{t} f - \nabla x_{i}^{t} f\left( {\frac{\partial h}{{\partial x_{i} }}} \right)^{ - 1} \frac{\partial h}{{\partial x_{k} }}$$

(A.3)

It is noteworthy that a necessary situation for minimizing f(x) is that df(x) = 0 or the same condition for an unlimited least is that $\frac{df}{{dx_{k} }} = 0$ (Ameli et al., 2016). To get more informed about this concept, reviewing prior research studies on this subject [48,49,50,51,52] is suggested.

1.1.2 Multilayer Perceptron

There are different tools and methods for explaining intricate issues and nonlinear relations between input and output parameters, including typical ANNs [53,54,55,56,57,58]. ANNs can be described as mathematical systems in which data is processed in a similar way to that of a biological neural system with similar intricacy and functionality [43, 59, 60]. Neurons are considered as primary units of ANN models, therefore adjusting the optimum number of neurons along with other key parameters (e.g., weights and biases) facilitates the control of network’s performance [61,62,63,64].

The most frequently used ANNs are known to be MLP and RBF [65, 66]. There are several layers in an MLP neural network, in which the first and the last ones represent the inputs and outputs of the model and the middle layers are known as hidden layers [67, 68]. In order to specify the number of hidden layers, the most common method is to evaluate different ANN structures based on trial and error. For ordinary problems, an MLP model with one hidden layer may be proper and enough, while using two hidden layers or more would be considered in more complicated problems [69]. There is an interconnection between the neurons of one determined hidden layer and the neurons of prior and next layers, in a particular network with Purlin function for its output layer, and Logsig and Tansig activation functions for the hidden layers. The model output can be calculated from Eq. (A.4):

$$Output = purelin~(w_{3} \times [logsig~\left( \begin{gathered} w_{1} \times \left( {Tansig~\left( x \right) + b_{1} } \right) \hfill \\ + {\text{logsig~}}\left( {w_{2} \times Tansig~\left( x \right) + b_{2} } \right) \hfill \\ \end{gathered} \right] + b_{3} )$$

(A.4)

where b₁ and b₂ stand for the bias vectors of the first and the second hidden layers, respectively, and b₃ represents the bias of the output layer. In addition, w₁ and w₂ indicate the weight matrix of the first and the second hidden layers, in order, and w₃ represents the weight matrix of the output layer.

Furthermore, four optimization algorithms are applied in current research, including LM, BR, SCG and RB. For more information and details about the mentioned algorithms, checking the study conducted by Hemmati-Sarapardeh et al. [64] is recommended.

1.1.3 Radial Basis Function

RBF network is among the most famous kinds of ANNs that is typically applied for regression purposes. RBF neural networks contain three feed-forward layers, an input layer, a hidden layer and an output layer [70]. The input layer is formed of input nodes, where the input nodes and input factors of the model are equal in number [65]. The hidden layer can be regarded as the most important part of an RBF network, as it transfers the information from the input space to the hidden space [71].

RBF is considered to be a widely used method in mathematics and physics problems to estimate different types of properties. This approach has several features, including the generalization of data to different dimensional spaces, the possibility to handle scattered data and the resolution of spectral precision [68, 72, 73]. Primary models are feed-forward neural networks which have two layers [74, 75]. A training process with three steps is also used to specify all of the parameters for RBF units. First of all, K-means grouping algorithm are used to specify the units’ centers. In the second step, an optimization procedure like gradient descend is applied to specify the weights. Finally, a superposition rule is used to determine the weights connecting the RBF elements to the output. Some localized basis functions are also applied in the RBF approximation technique, e.g., ϕ (x_i) for the regression of y (x). As it can be seen, the linear combination produces the following output functions:

$$f(x_{i} ) = w^{T} \varphi (x_{i} ) + b$$

(A.5)

where φ (x_i) indicates the transport function, b indicates the bias parameter and w^T is the vector of the transferred output layer.

In the initial step, centers are determined utilizing random methods. The hidden layer constitutes the most crucial part of the algorithm. Gaussian function is considered to be among the most practical types of transfer functions which are commonly used [76]:

$$\varphi (r) = \exp \left( {0.5 \times \left( {\frac{r}{\delta }} \right)^{2} } \right),\,\,\,\,With \, spread \, coefficient \, \delta > 0$$

(A.6)

Using the transport function, a correlation is attained as shown in Eq. (A.7):

$$\varphi_{ki} (x) = \exp \left[ { - \frac{{\left\| {x_{k} - c_{i} } \right\|}}{{2\delta^{2} }}} \right],\,\,\,i = 1 \ldots .N, \, K \, = 1, \ldots ,M$$

(A.7)

In Eq. (A.7), c_i indicates the centers, δ represents the spread coefficient, φ_ki (x) represents the Gaussian transport function, and M and N signify the numbers of data points and kernels, respectively.

If a linear superposition method is used, the model output is expressed as shown in Eq. (A.8):

$$y_{k} = \sum\limits_{i = 1}^{N} {w_{i} \varphi {}_{{}}\left( {\left\| {x_{k} - c_{i} } \right\|} \right)} + w_{0,i} = 1,...N,K = 1,...,M$$

(A.8)

where N represents the clusters number, w₀ indicates the bias factor, M stands for the number of inputs and outputs and finally y_k represents the output of the model.

The main purpose of this equation is to underrate mean square error (MSE) by using optimization design like gradient descent. The following equation calculates the optimized weight:

$$w_{i} (t + 1) = w_{j} (t) - \eta_{i} \left. {\frac{{\delta_{MSE} }}{{\delta_{wi} }}} \right|_{wi(t)}$$

(A.9)

Using the above scheme, it is possible to characterize the centers and optimize the weights, which is called “the approximated outputs.” This algorithm has two main factors, including the spread coefficient and the neurons number, which are optimized in the developed neural network to improve the estimation of mud rheological properties. These optimum values could be obtained by trial-and-error approach. To meet this purpose, the amounts of spread coefficient and neurons number were changed and various Gaussian RBF networks were designed. Finally, the optimum amounts are chosen by reducing the mean square error.

1.2 Optimization Techniques

1.2.1 Levenberg–Marquardt Algorithm

Several techniques are available to optimize the biases and weights in an MLP model among which, the LM algorithm is regarded as one of the best and most common ones. This technique is utilized to solve the nonlinear least square issues. Furthermore, this technique does not cause a universal minimization, yet it can find the ultimate solution even based on an unsuitable initial surmise. Another point to mention about this method is that it does not calculate the Hessian matrix. To put it in other way, since the performance function is represented as sum of the squares, the approximation of the Hessian matrix and the gradient are expressed as follows [68, 76, 78]:

$$g = J^{T} e$$

(A.10)

$$H = J^{T} J$$

(A.11)

where J stands for the Jacobian matrix that consists of the first system formative of the network error in relation to biases and weights, and e represents the network errors vector.

The order can be updated as follows:

$$x_{k + 1} = x_{k} - (J^{T} J - \eta I)^{ - 1} J^{T} e$$

(A.12)

where x denotes the connection weight, η represents a constant which is altered to less amounts when the steps are taken successfully and is changed to greater amounts as a hesitant stage expands the performance function.

As it is seen, after each iteration the performance function value is reduced, which has a detailed explanation provided and stated in the literature [68, 78].

1.2.2 Bayesian Regularization Algorithm

BR technique is considered as a training algorithm that is able to update biases and weights based on LM optimization [79, 80]. This algorithm decreases a combination of squared errors and weights and afterward specifies the appropriate composition to develop a network [81]. The weights of the networks for the objective function are specified as shown in Eq. (A.13) [68, 82]:

$$F(\omega ) = \beta \times E_{D} + \alpha \times E_{\omega }$$

(A.13)

where E_D, E_ω and F(ω) represent the total of network errors, the total of squared network weights and the objective function, respectively. Moreover, α and β are factors of the objective function.

Weight of network can be introduced as the random parameter of the BR neural network. Additionally, the training sets and network weight can be created according to the Gaussian distribution. Then, the best amounts for the determined variables are computed and after that, learning phase of the algorithm is performed by LM algorithm to minimize the objective function and update the weight space. If a divergence occurs, the parameters should be updated and the procedure should be repeated [68, 82].

1.2.3 Scaled Conjugate Gradient Algorithm

The weights used in this technique are adopted from the most negative gradient; however, it is not considered as the most rapid algorithm [68, 77]. Using a search algorithm like the conjugate gradient leads in a faster convergence. In this algorithm, assuming P_k as the search direction (or the conjugate direction) and to optimize the current search direction, the following equations are used [68, 77, 83]:

$$x_{k + 1} = g_{k} \times \alpha_{k} + x_{k}$$

(A.14)

$$P_{k} = - g_{k} + \beta_{k} P_{k - 1}$$

(A.15)

The procedure of computing β specifies the degree of conjugate gradient. Line search method, which is regarded as a prediction method for the stage size, is not cost-effective in terms of computation expenses. In another method, known as SCG, the CG algorithm is combined with trust area method, but with a less cost [68, 84].

1.2.4 Resilient Backpropagation Algorithm

Different transfer functions, such as Sigmoid or Tansig, can be used in an MLP algorithm. The task of these functions is to reduce the infinite input domain to the finite output domain. In an activation function like Tansig, the line slope becomes closer to zero by entering a big input, which might lead to some issues while applying steepest descent to train the network. This is due to the fact that the gradient has a minor amount and hence, small variations are made in weights and biases. In order to solve these problems, the RB algorithm has been suggested to remove the undesirable effects of the partial derivatives [85].