Application of general regression neural network (GRNN) for indirect measuring pressure loss of Herschel–Bulkley drilling fluids in oil drilling

Measurement 85 (2016) 184–191

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Application of general regression neural network (GRNN) for indirect measuring pressure loss of Herschel–Bulkley drilling fluids in oil drilling Reza Rooki ⇑ Birjand University of Technology, Birjand, Iran

a r t i c l e

i n f o

Article history: Received 27 February 2015 Received in revised form 10 February 2016 Accepted 19 February 2016 Available online 27 February 2016 Keywords: Pressure loss GRNN Oil well annulus Herschel–Bulkley fluids

a b s t r a c t Experimental measurements of the pressure losses in a well annulus are costly and time consuming. Pressure loss calculations in annulus is generally conducted based on an extension of empirical correlations developed for Newtonian fluids and extending pipe flow correlations. However, correct estimation of pressure loss of non-Newtonian fluids in oil well drilling operations is very important for optimum design of piping system and minimizing the power consumption. In this paper, a general regression neural network (GRNN) was applied to predict the pressure loss of Herschel–Bulkley drilling fluids in concentric and eccentric annulus. Experimental data from literature were used to train the GRNN for estimating pressure losses in annulus. The predicted values using GRNN closely followed the experimental ones with an average relative absolute error less than 6.24%, and correlation coefficient (R) of 0.99 for pressure loss estimation. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Drilling hydraulic is an essential part of drilling package enabling computation of pressure profiles along the wellbore and particularly in the annulus contributing to well safety and well integrity. Pressure loss occurs in all pipe systems in consequence of elevation changes, turbulence caused by sudden changes in direction, and finally friction within the pipe and fittings. Historically, engineers have had considerable difficulty estimating flowing pressure gradients in a well annulus for drilling or well control operations. The optimum design of a drilling program according to a correct estimation of pressure loss can be lead to economic benefits like decrease in formation damage, increase in the rate of penetration (ROP), and improve of production [1,2]. The calculation of pressure loss in a well annulus is generally based on an extension of empirical correlations ⇑ Correspondence to: Reza Rooki, PhD Assistant Professor of Mining Engineering, Birjand University of Technology, Birjand, Iran. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.measurement.2016.02.037 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.

developed for Newtonian fluids and extending pipe flow correlations. While accurate estimation of the pressure losses in annulus is necessary to determine the required power and selection of suitable mud pump systems. Ineffective estimation of pressure losses may lead to serious drilling problems such as stuck pipe, loss circulation, kicks, and improper rig power selection during drilling operations. Flow of yield-pseudo plastic fluids in annuli is encountered in many situations in a variety of industries particularly in oil-well drilling. Such fluids require at least three rheological parameters for a near-optimum modeling of their rheological behavior showed that the threeparameters model proposed by Herschel and Bulkley [3] describes very well the rheology of the most fluids in oildrilling industry [4,5]. In addition, some analytical and non-analytical studies of laminar flow of Bingham plastic, power-law and Herschel–Bulkley fluids in annuli have been carried out [2,4,6–22]. Regarding pressure loss estimation, the required experimental and analytical tasks are costly, time consuming and complex. Therefore, a simple, reliable and accurate methodology for pressure loss

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Nomenclature AAPE ANN BPNN Di Do di e E f GRNN K n N

p Q R Ri Ro RMS v DP c_

average absolute percent relative error artificial neural network backpropagation neural network diameter of inner tube of annulus (m) diameter of outer tube of annulus (m) Euclidean distance dimensionless eccentricity (–) offset distance between centers friction factor general regression neural network flow consistency index (Pa sn) flow behavior index (–) number of samples (–)

parameter liquid flow rate (gpm) correlation coefficient (–) radius of inner tube of annulus (m) radius of outer tube of annulus (m) average root mean square error (kPa) fluid velocity pressure drop (kPa) shear rate (s1) fluid density (kg/m3) shear stress (Pa) yield stress (Pa) wall shear stress (Pa)

q s sy sw

estimation of Herschel–Bulkley fluids flow in concentric and eccentric annuli without resorting to heavy numerical computations is necessary. It is worth mentioning that the main aim of this paper is to apply a type of artificial neural networks (ANN) in order to estimate pressure loss. ANNs are information processing system, which learn from input/output data to determine the relationships between inputs and outputs. In recent years, ANNs have been used in different fields of engineering because of their capability of extracting complex and non-linear relationships. Back propagation neural networks (BPNN) have been applied in various areas of petroleum industry [23–25], also in friction factor and pressure drop predictions [26– 35]. Most of above mentioned literature do not cover full range concentric and eccentric annuli to calculate the pressure losses. According to benefits of GRNN respect to BPNN such as faster training speed and simple network architecture, the aim of this study is to estimate the pressure losses for flow of Herschel–Bulkley fluids in concentric and eccentric annuli from literature experimental data [17] using GRNN, by knowing the basic parameters of Herschel–Bulkley fluid and annulus. 2. Theory 2.1. Pressure loss of Herschel–Bulkley fluids A rheological model describes the relationship between shear stress and shear rate when a fluid flows through a circular section or an annulus. Various rheological models have been proposed to describe the rheological behavior of drilling fluids. Past investigations [2,4,5] showed that the three-parameter Herschel–Bulkley model describes very well the rheology of the most of the drilling fluids used today. However, the three-parameter Herschel–Bulkley model has not been widely used. This model is defined as follow:

indices while c_ denotes the shear rate. This equation reduces to the more commonly-known rheological models under certain conditions, like the Bingham plastic model when the flow behavior index n ¼ 1:0 and the power law model when sy ¼ 0. It should be mentioned that this model can yield mathematical expressions that are not readily solved analytically but can be solved using non-linear regression, golden section [4] and genetic algorithm [5] methods. Prediction of frictional pressure losses over the entire flow spectrum spanning laminar, transitional and turbulent flows requires knowledge of the transition limits and local power-law assumption for computing friction factor using trial and error procedure [17,20,2]. Then the pressure loss for concentric annuli is computed as:

DP ¼

2f qv 2 4sw ¼ Do  Di Do  Di

ð2Þ

where Di is diameter of inner tube of annulus (m), Do diameter of outer tube of annulus (m), q fluid density (kg/m3), v velocity of fluid (m/s) and sw is wall shear stress (Pa) can be calculated using Eq. (1) as sw ¼ sy þ K c_ nw [17]. In Eq. (2), friction factor (f) depends on rheological parameters in Eq. (1) due to Reynolds number. The determination of f needs a trial and error procedure for different flow regimes due to concentric and eccentric annuli [2,20]. For eccentric annuli, the proposed correlation for power law fluids by Haciislamoglu and Cartalos [36] is used for pressure loss estimation [2,17,20]:






 0:8454   pffiffiffi Di 0:1852 e Di  1:5e2 n n Do Do  0:2527 pffiffiffi Di þ 0:96 n ð3Þ Do

¼ 1  0:072 lami

ð1Þ

 0:8454  0:1852 e Di 2 pffiffiffi Di  e2 n n Do 3 Do  0:2527 pffiffiffi Di þ 0:285 n ð4Þ Do

where s and sy are shear stress and yield stress, respectively, K and n are the fluid consistency and fluid behavior

where e is eccentricity of annulus (e = E/(Ro  Ri)), E is offset distance between the centers of the inner tube and the

s ¼ sy þ K c_ n




Dpecc Dpconc

Dpecc Dpconc



¼ 1  0:048 turb

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outer tube of annulus, and Ri and Ro are the radius of inner tube and the outer tube of annulus respectively. So it is clear that there is no specific equation for pressure loss estimation of Herschel–Bulkley fluids in concentric and eccentric annuli for all flow regimes [2,17,20]. 2.2. General regression neural network (GRNN) Artificial neural networks (ANNs) constitute a class of flexible nonlinear models designed to mimic biological neural systems. ANNs have been widely applied to solve many difficult problems in different areas, including pattern recognition, signal processing, language learning, and etc. Typically, a biological neural system consists of several layers, each of them consists a large number of neural units (neurons) that can process the information in a parallel manner. The models with these features are known as ANN models. A typical neuron structure is shown in Fig. 1. The basic information processing occurs in the following manner; inputs (P) coming from another neuron are multiplied by their individual weights (w1,i), and weighted input connections are combined inside the neuron and the bias term (bi) is added to the summation at the neuron in order to increase or decrease the input (nj) that goes into the activation function (Fig. 2). When these neurons are combined to form a neural network, the system goes through a training stage where input and output pairs are introduced to the network and weights in the neurons are varied so that the network

Fig. 1. A typical neuron of radial basic network [37].

outputs match the desired outputs as close as possible. According to learning algorithms, several types of neural networks such as Backpropagation Neural Network (BPNN), Probabilistic Neural Network (PNN) and general regression neural network (GRNN) have been designed in MATLAB software [37,38]. Since GRNN method is used in this study, it is therefore described below briefly. GRNN which is a type of supervised network has been proposed by Specht [39]. GRNN is able to produce continuous value outputs. GRNN is a three-layer (input, hidden and output layer) network where there is one hidden neuron for each training pattern in hidden layer. The general regression neural network (GRNN) is a memory-based network that provides estimates of continuous variables and converges to the underlying regression surface. GRNNs are based on the estimation of probability density functions, feature fast training times and can model non linear functions. The GRNN is a one-pass learning algorithm with a highly parallel structure. It is that, even with sparse data in a multidimensional measurement space, the algorithm provides smooth transitions from one observed value to another. The algorithmic form can be used for any regression problem in which an assumption of linearity is not justified. GRNN can be thought as a normalized RBF (Radial Basis Functions) network in which there is a hidden unit centered at every training case. These RBF units are usually probability density functions such as the Gaussian. The only weights that need to be learned are the widths of the RBF units. These widths are called ‘‘smoothing parameters (r)”. The main drawback of GRNN is it cannot ignore irrelevant inputs without major modifications to the basic algorithm. So, GRNN is not likely to be the top choice if there are more than 5 or 6 number redundant inputs. The regression of a dependent variable, Y, on an independent variable, X, is the computation of the most probable value of Y for each value of X based on a finite number of possibly noisy measurements of X and the associated values of Y. The variables X and Y are usually vectors. In order to implement system identification, it is usually necessary to assume some functional forms. In the case of linear regression, for instance, the output Y is assumed to be a linear function of the input, and the unknown parameters, ai, are linear coefficients. The procedure does not need to assume a specific functional form. A Euclidean distance

Fig. 2. Three examples of activation functions [37].

R. Rooki / Measurement 85 (2016) 184–191 2

(di ) is estimated between an input vector and the weights, which are then rescaled by the smoothing factor. The radial basis output is then the exponential of the negatively weighted distance. The GRNN equations are as follow: T

2

di ¼ ðX  X i Þ ðX  X i Þ

ð5Þ

  d2 exp  2ri 2 YðXÞ ¼ P   d2i n i¼1 exp  2r2

ð6Þ

Pn

i¼1 Y i

The estimate Y(X) can be visualized as a weighted average of all of the observed values, Yi, where each observed value is weighted exponentially according to its Euclidian distance from X. Y(X) is simply the sum of the Gaussian distributions centered at each training sample. However the sum is not limited to being the Gaussian. In this theory, r denotes the smoothing factor, and optimum smoothing factor can be determined after several runs according to the mean squared error of the estimated values, which must be kept at minimum. This process is referred to as the training of the network. If a number of iterations pass without improvement in the mean squared error, the smoothing factor is determined as the optimum one for that data set. In the production phase, the smoothing factor is applied to data sets that the network has not seen before. While applying the network to a new set of data, an increase in the smoothing factor would result in decreasing the range of output values [39,37]. In GRNN, there are no training parameters such as the learning rate, momentum,

187

optimum number of neurons in hidden layer and learning algorithms as in BPNN. Also, GRNN has high speed in estimation respect to BPNN. In GRNN structure there is a smoothing factor that its optimum value is obtained by a try and error process. The smoothing factor must be greater than 0 and can usually range from 0.1 to 1 with reasonable results. The number of neurons in the input layer is the number of inputs in the problem, and the number of neurons in the output layer corresponds to the number of outputs. The number of hidden layer neurons is training patterns. Because the GRNN networks evaluate each output independently from the other outputs, the GRNN networks may be more accurate than BPNN when there are multiple outputs. The GRNN networks work by measuring how far given samples pattern is from patterns in the training set. The output that is predicted by the network is a proportional amount of all the outputs in the training set. The proportion is based upon how far the new pattern is from the given patterns in the training set.

3. Pressure losses prediction using GRNN In this study 375 data sets collected from flow loop (Fig. 3) from literature [17], were used. Table 1 shows dimensions of annular test sections. Test temperatures and pressures were ranging from 82 °F to 113 °F [27.78–45 °C] and from 20.94 psi to 193.2 psi [144.37–1332 kPa]. Annular geometries and rheological properties of test fluids measured using pipe viscometer are tabulated in Table 2. Sensitivity analyses were conducted using SPSS

Fig. 3. Schematic of the flow loop [17].

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Table 1 Dimensions of annular test sections [17]. Annulus

1 2 3 4

Pipe diameter

Hole diameter

[inch]

[mm]

[inch]

[mm]

0.38 0.5 0.68 0.63

9.53 12.7 17.15 15.88

1.38 1.38 1.38 0.82

35.05 35.05 35.05 20.93

Di/Do

0.27 0.36 0.49 0.76

Table 2 Annular geometries and rheological properties of test fluids [17]. Test fluid and annulus

e

Di/Do

sy [Pa]

K [Pa sn]

n

XCD1 (Annulus #3) XCD2 (Annulus #3) XCD3 (Annulus #3) XCDPAC5 (Annulus #1) XCDPAC5 (Annulus #2) XCDPAC5 (Annulus #4) XCD10 (Annulus #1) XCD10 (Annulus #2) XCD10 (Annulus #4) XCPAC8 (Annulus #1) XCPAC8 (Annulus #2) XCPAC8 (Annulus #4) XCD9 (Annulus #1) XCD9 (Annulus #2) XCD9 (Annulus #4) XCD5 (Annulus #1) XCD5 (Annulus #2) XCD5 (Annulus #4)

1 1 1 0.75 0.6 0.8 0.65 0 0.85 0.65 0 0.85 0.75 0.6 0.8 1 1 1

0.49 0.49 0.49 0.27 0.36 0.76 0.27 0.36 0.76 0.27 0.36 0.76 0.27 0.36 0.76 0.27 0.36 0.76

4.3 11.3 14.4 7.31 7.31 7.31 9 9 9 3.8 3.8 3.8 8.1 8.1 8.1 6.5 6.5 6.5

1.29 1.85 1.56 6.82 6.82 6.82 1.01 1.01 1.01 2.98 2.98 2.98 0.9 0.9 0.9 0.64 0.64 0.64

0.38 0.35 0.39 0.39 0.39 0.39 0.48 0.48 0.48 0.4 0.4 0.4 0.45 0.45 0.45 0.48 0.48 0.48

software. According to the correlation matrix (Table 3), Q and Di/Do have most depending with pressure loss. Input parameters of GRNN as affecting parameters on pressure loss include diameter ratio (Di/Do), eccentricity of annulus (e = E/(Ro  Ri)) where E is offset distance between the centers of the inner tube and the outer tube of annulus, and properties of the non-Newtonian liquid, i.e., flow behavior index (n), consistency index (K), yield stress (sy ), liquid flow rate (Q). The pressure loss (DP) was used as output of network. In view of the requirements of the neural computation algorithm, both the inputs and output data were normalized to an interval by transformation process. In this study, normalization of data (inputs and outputs) was done for the range of [1, 1] using Eq. (7),

Table 3 Correlation matrix between pressure loss and independent variables.

sy

K n DP

e

Di/Do

sy

1 0.163 0.013 0.138 0.170 0.046 0.250

1 0.306 0.177 0.168 0.020 0.174

1 0.022 0.006 0.028 0.739

1 0.154 0.001 0.114

K

n

DP

Fig. 5. (a) General scheme of network and its layers, (b) structure of hidden layer (Layer 1). 1 .621 0.084

1 0.045

1 Data Fit Y=T

350

300

Measured pressure loss (KPa)

Q e Di/Do

Q

250

200

150

100

R=0.99992

50

50

100

150

200

250

300

350

Predicted pressure loss (KPa)

Fig. 4. GRNN structure.

Fig. 6. The predicted pressure losses versus experimental values for the training data.

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R. Rooki / Measurement 85 (2016) 184–191

300

Measured pressure loss (KPa)

(R). The root mean square error (RMS) indicates the discrepancy between the measured and predicted values. It can be calculated as follows:

Data Fit Y=T

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ 2 i¼1 ðyi  yi Þ RMS ¼ N

250

The AAPE gives an idea of absolute relative deviation of estimated from the measured data. It can be calculated as follows:

200

150

AAPE ¼ 100

50

100

150

200

250

300

Predicted pressure loss (KPa) Fig. 7. The predicted pressure losses versus experimental values for the test data.

p  pmin 1 pmax  pmin

ð7Þ

where pn is the normalized parameter, p denotes the actual parameter, pmin represents a minimum of the actual parameters and pmax stands for a maximum of the actual parameters. During the training of the GRNN for pressure loss data, out of 375 of the data sets from literature [17], 260 were randomly selected for training the network and 115 were used for testing purposes. Several varied smooth factors (r) were tried to achieve the optimum r to predict pressure loss using GRNN. Three criteria were used in order to evaluate the effectiveness of each network and its ability to make accurate estimations; the root mean square error (RMS), average absolute percent relative error (AAPE) and the correlation coefficient

400

N ^ 1X ðyi  yi Þ N i¼1 yi

ð9Þ

^i denotes the predicted where yi is the measured value, y value, and N stands for the number of samples. The lowest RMS and AAPE present the more accurate estimation. Furthermore, the efficiency criterion, R2 represents the percentage of the initial uncertainty explained by the model, is given as:

R=0.99487

50

pn ¼ 2

ð8Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u PN u ^ 2 i¼1 ðyi  yi Þ R¼u PN 2 t1  P ^y N 2 i¼1 i i¼1 yi  N

ð10Þ

The best fitting between measured and predicted values, which is unlikely to occur, would have RMS = 0 and R = 1. The final network with optimal r of 0.015, has one input layer with six inputs (K, n, e, Q, sy , Di/Do), one hidden layer with 260 neurons with radbas transfer function and on output layer with one neuron (DP) with linear transfer function (Fig. 4). In Fig. 5a, Layer 1 is hidden layer and Layer 2 is output layer. Fig. 5b is the structure of hidden layer. The pressure loss for laminar flow of Herschel–Bulkley drilling fluids in concentric and eccentric annular was predicted using GRNN as described above. In Fig. 6, the predicted pressure losses are compared with the experimental values for the training set of 260 data. The correlation coefficient (R) to the linear fit (y = ax) is 0.999 with an RMS and AAPE values of 1.037 kPa and 1.18%

Predicted pressure loss(KPa) Measured pressure loss(KPa)

350

Pressure loss (KPa)

300 250 200 150 100 50 0

0

50

100

150

200

250

300

sampels Fig. 8. Comparison of the predicted pressure losses and the measured pressure losses in the training data.

190

R. Rooki / Measurement 85 (2016) 184–191 350 Predicted pressure loss(KPa) Measured pressure loss(KPa)

Pressure loss (KPa)

300

250

200

150

100

50

0

0

20

40

60

80

100

120

sampels Fig. 9. Comparison of the predicted pressure losses and the measured pressure losses in the testing data.

giving an almost perfect fit, something of course expected since it was this data set used for the training of the network. The very good fitting values indicate that the training was done very well. For testing and evaluation the network, the test data group, which was not seen by the GRNN during training, was used. The comparison of the estimations of the network with the measured values for the test data set (population of 115) is shown in Fig. 7. The correlation coefficient (R) is 0.999 with an RMS and AAPE of 8.45 kPa and 6.24%, indicates a very satisfactory model performance. These results show the average relative error for all data is less than 3.7%. These results verified the success of neural networks to recognize the implicit relationships between input and output variables. These results show that GRNN is able to predict pressure losses of nonNewtonian Herschel–Bulkley fluids flow through inclined annulus with high accuracy without complex procedure for estimating of pressure losses of these fluids in previous rare works [17]. Comparison of the predicted pressure loss and measured pressure loss in the training and testing data is shown in Figs. 8 and 9. 4. Conclusion GRNN was applied to predict pressure losses of nonNewtonian Herschel–Bulkley fluids flowing through horizontal annular geometries. The GRNN has three layers; input layer, Hidden layer and Output layer. Input layer has six neurons including, diameter ratio (Di/Do), eccentricity of annulus (e), and properties of the non-Newtonian liquid (n, K and sy ) and liquid flow rate (Q). Hidden layer has 260 neurons with radbas activation function in all neurons. Output layer has one neuron, pressure loss (DP) with purelin activation function. The neural network predicted pressure loss closely followed the experimental values. The correlation coefficient of train and test data is 0.999 and 0.995 respectively and the AAPE of train data and test data are 1.18% and 6.24% respectively. These results showed that GRNN is able to predict pressure losses of

non-Newtonian Herschel–Bulkley fluids flow through drilling wellbore annulus with high accuracy and high speed without complex procedure.

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