Artificial neural network models for reservoir-aquifer dimensionless variables: influx and pressure prediction for water influx calculation

Abstract

Calculation of water influx into petroleum reservoir is a tedious evaluation with significant reservoir engineering applications. The classical approach developed by van Everdingen–Hurst (vEH) based on diffusivity equation solution had been the fulcrum for water influx calculation in both finite and infinite-acting aquifers. The vEH model for edge-water drive reservoirs was modified by Allard and Chen for bottom-water drive reservoirs. Regrettably, these models solution variables: dimensionless influx (WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document}) and dimensionless pressure (PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document}) were presented in tabular form. In most cases, table look-up and interpolation between time entries are necessary to determine these variables, which makes the vEH approach tedious for water influx estimation. In this study, artificial neural network (ANN) models to predict the reservoir-aquifer variables WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document} was developed based on the vEH datasets for the edge- and bottom-water finite and infinite-acting aquifers. The overall performance of the developed ANN models correlation coefficients (R) was 0.99983 and 0.99978 for the edge- and bottom-water finite aquifer, while edge- and bottom-water infinite-acting aquifer was 0.99992 and 0.99997, respectively. With new datasets, the generalization capacities of the developed models were evaluated using statistical tools: coefficient of determination (R2), R, mean square error (MSE), root-mean-square error (RMSE) and absolute average relative error (AARE). Comparing the developed finite aquifer models predicted WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} with Lagrangian interpolation approach resulted in R2, R, MSE, RMSE and AARE of 0.9984, 0.9992, 0.3496, 0.5913 and 0.2414 for edge-water drive and 0.9993, 0.9996, 0.1863, 0.4316 and 0.2215 for bottom-water drive. Also, infinite-acting aquifer models (Model-1) resulted in R2, R, MSE, RMSE and AARE of 0.9999, 0.9999, 0.5447, 0.7380 and 0.2329 for edge-water drive, while bottom-water drive had 0.9999, 0.9999, 0.2299, 0.4795 and 0.1282. Again, the edge-water infinite-acting model predicted WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and Edwardson et al. polynomial estimated WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{eD}$$\\end{document} resulted in the R2 value of 0.9996, R of 0.9998, MSE of 4.740 × 10–4, RMSE of 0.0218 and AARE of 0.0147. Furthermore, the developed ANN models generalization performance was compared with some models for estimating PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document}. The results obtained for finite aquifer model showed the statistical measures: R2, R, MSE, RMSE and AARE of 0.9985, 0.9993, 0.0125, 0.1117 and 0.0678 with Chatas model and 0.9863, 0.9931, 0.1411, 0.3756 and 0.2310 with Fanchi equation. The infinite-acting aquifer model had 0.9999, 0.9999, 0.1750, 0.0133 and 7.333 × 10–3 with Edwardson et al. polynomial, then 0.9865, 09,933, 0.0143, 0.1194 and 0.0831 with Lee model and 0.9991, 0.9996, 1.079 × 10–3, 0.0328 and 0.0282 with Fanchi model. Therefore, the developed ANN models can predict WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document} for the various aquifer sizes provided by vEH datasets for water influx calculation.