Mathematical modelling of drilling mud plastic viscosity at downhole conditions using multivariate adaptive regression splines

Abstract

Viscosity is an essential mud property which enables it to transport cuttings out of a well. Determining the real time value of mud viscosity at downhole conditions is complex and till date lacks a precise model. Literature is filled with models that are only applicable at surface conditions with most of the models being artificial neural network (ANN) based. Neural networks have been criticized for their long training time because they rely on a trial-and-error approach to determine their optimal architecture. The few models related to mud viscosity at downhole conditions are based on non-linear regression which are not flexible in nature. However, Multivariate Adaptive Regression Splines (MARS) has at no time been used notwithstanding some of its known benefits over ANN and other regression methods. The focus of this study is to model mud plastic viscosity at downhole conditions with the MARS method using parameters such as temperature, pressure and initial mud viscosity. MARS is a flexible non-parametric splines-based technique that extends linear regression models by including nonlinearities and interactions between predictors. An experimental database comprising 88 and 149 data points culled from literature for oil-based mud (OBM) and water-based mud (WBM) respectively were utilized for developing the models. For each mud type, the most suitable model was determined by choosing the model with the maximum coefficient of determination (R2) value and the least values of mean square error (MSE), root mean square error (RMSE), and generalized cross-validation criterion (GCV). Using these criteria, the R2, MSE and RMSE of the MARS models according to the mud type were found to be 0.64, 104.35, 10.21 and 0.71, 12.8, 3.57 for OBM and WBM respectively while the GCV values were 129.3 and 14.38 for OBM and WBM respectively. Due to lack of mud rheology data at downhole conditions, it was difficult to compare the performance of the developed models with existing models. Parametric importance analysis indicate that temperature has the highest effect on mud viscosity estimation at downhole conditions. The developed models are valid for 20 °C ≤ Temperature ≤315 °C and 0 psi ≤ Pressure ≤40,000 psi. This pioneer work on using MARS in drilling fluid property prediction points out the implementation protocol, the dataset utilized, the relative importance of input variables and the explicit model for estimating mud viscosity at downhole conditions.