{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T11:10:50Z","timestamp":1788779450880,"version":"build-2803163510"},"reference-count":33,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T00:00:00Z","timestamp":1787875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Research and Development Plan of Shaanxi Province","award":["2024GX-YBXM-506"],"award-info":[{"award-number":["2024GX-YBXM-506"]}]},{"DOI":"10.13039\/501100009103","name":"Education Department of Shaanxi Province","doi-asserted-by":"publisher","award":["23JY066"],"award-info":[{"award-number":["23JY066"]}],"id":[{"id":"10.13039\/501100009103","id-type":"DOI","asserted-by":"publisher"}]},{"award":["23JY066"],"award-info":[{"award-number":["23JY066"]}],"id":[{"id":"https:\/\/ror.org\/00jb7xy46","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Youth Innovation Team of Xi\u2019an Shiyou University","award":["2019QNKYCXTD04"],"award-info":[{"award-number":["2019QNKYCXTD04"]}]},{"award":["2019QNKYCXTD04"],"award-info":[{"award-number":["2019QNKYCXTD04"]}],"id":[{"id":"https:\/\/ror.org\/040c7js64","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Graduate Innovation Fund of Xi\u2019an Shiyou University","award":["YCX2613086"],"award-info":[{"award-number":["YCX2613086"]}]},{"award":["YCX2613086"],"award-info":[{"award-number":["YCX2613086"]}],"id":[{"id":"https:\/\/ror.org\/040c7js64","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Formation pressure labels are often stored as equivalent mud weight (EMW), whereas engineering analysis requires true pressure and validation strategies that account for the strong depth dependence of borehole data. This study develops a dimensionally consistent pressure ratio symbolic regression framework in which vertical overburden stress provides the pressure scale and a compact dimensionless correction is learned from reference-scaled borehole variables. After converting EMW-based labels to MPa and applying transparent quality control criteria, 1877 of 2025 depth-indexed records were retained. Redundant and deterministically derived variables were identified through correlation analysis, variance inflation factors, principal component analysis, and deterministic relation checks and were excluded from the compact symbolic search. Five contiguous depth blocks with a 20 m purge zone were used for leakage-resistant validation, with symbolic-structure selection repeated within each outer training fold. The selected four-coefficient expression, comprising an intercept and three predictor-dependent terms, retained only depth and acoustic slowness. Fixed-structure depth-block validation yielded a coefficient of determination (R2) of 0.3008, a root mean square error (RMSE) of 3.5439 MPa, a mean absolute error (MAE) of 2.6964 MPa, and a mean absolute relative error (MARE) of 10.02%, while the conservative nested symbolic pipeline yielded MARE = 11.60%. External zero-shot evaluation on 974 samples from an external well, 98.36% of which were outside the calibration depth range, yielded R2 = 0.0265, RMSE = 7.0024 MPa, MAE = 5.5723 MPa, and MARE = 11.51% against an independent D-exponent-based engineering pressure reference (PP_DX). Sparse local recalibration using 10 depth-spaced PP_DX engineering reference points, with \u00b120 m neighborhoods excluded from evaluation, improved performance to R2 = 0.8684, RMSE = 2.8431 MPa, MAE = 2.1108 MPa, and MARE = 4.39% on the remaining 564 samples. These results indicate that the proposed framework provides an explicit and independently auditable pressure equation that preserves the overall pressure scale under substantial depth domain shift and can be efficiently recalibrated using sparse local pressure information. Site-specific calibration remains necessary for accurate reproduction of local pressure variations.<\/jats:p>","DOI":"10.3390\/s26175440","type":"journal-article","created":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T12:27:46Z","timestamp":1787920066000},"page":"5440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dimensionally Consistent Formation Pressure Prediction from Borehole Sensing Data Using Pressure Ratio Symbolic Regression"],"prefix":"10.3390","volume":"26","author":[{"given":"Chi","family":"Zhao","sequence":"first","affiliation":[{"name":"College of Petroleum Engineering, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"},{"name":"Key Laboratory of Exploration and Development of Complex and Difficult-to-Produce Oil & Gas Reservoirs (Xi\u2019an Petroleum University), Ministry of Education, Xi\u2019an 710065, China"},{"name":"Shaanxi Key Laboratory of Advanced Stimulation Technology for Oil & Gas Reservoirs, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-2984-7002","authenticated-orcid":false,"given":"Ming","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Petroleum Engineering, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"},{"name":"Key Laboratory of Exploration and Development of Complex and Difficult-to-Produce Oil & Gas Reservoirs (Xi\u2019an Petroleum University), Ministry of Education, Xi\u2019an 710065, China"},{"name":"Shaanxi Key Laboratory of Advanced Stimulation Technology for Oil & Gas Reservoirs, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"},{"role":"project-administration","vocabulary":"credit"},{"role":"funding-acquisition","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Lihao","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Petroleum Engineering, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"},{"name":"Key Laboratory of Exploration and Development of Complex and Difficult-to-Produce Oil & Gas Reservoirs (Xi\u2019an Petroleum University), Ministry of Education, Xi\u2019an 710065, China"},{"name":"Shaanxi Key Laboratory of Advanced Stimulation Technology for Oil & Gas Reservoirs, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"methodology","vocabulary":"credit"},{"role":"software","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Jin","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Petroleum Engineering, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"},{"name":"Key Laboratory of Exploration and Development of Complex and Difficult-to-Produce Oil & Gas Reservoirs (Xi\u2019an Petroleum University), Ministry of Education, Xi\u2019an 710065, China"},{"name":"Shaanxi Key Laboratory of Advanced Stimulation Technology for Oil & Gas Reservoirs, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Wenxuan","family":"Kou","sequence":"additional","affiliation":[{"name":"College of Petroleum Engineering, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"},{"name":"Key Laboratory of Exploration and Development of Complex and Difficult-to-Produce Oil & Gas Reservoirs (Xi\u2019an Petroleum University), Ministry of Education, Xi\u2019an 710065, China"},{"name":"Shaanxi Key Laboratory of Advanced Stimulation Technology for Oil & Gas Reservoirs, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Guojie","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Petroleum Engineering, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"},{"name":"Key Laboratory of Exploration and Development of Complex and Difficult-to-Produce Oil & Gas Reservoirs (Xi\u2019an Petroleum University), Ministry of Education, Xi\u2019an 710065, China"},{"name":"Shaanxi Key Laboratory of Advanced Stimulation Technology for Oil & Gas Reservoirs, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"software","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"929","DOI":"10.2118\/3719-PA","article-title":"The effect of overburden stress on geopressure prediction from well logs","volume":"24","author":"Eaton","year":"1972","journal-title":"J. Pet. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"89","DOI":"10.2118\/27488-PA","article-title":"Pore pressure estimation from velocity data: Accounting for overpressure mechanisms besides undercompaction","volume":"10","author":"Bowers","year":"1995","journal-title":"SPE Drill. Complet."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.earscirev.2011.06.001","article-title":"Pore pressure prediction from well logs: Methods, modifications, and new approaches","volume":"108","author":"Zhang","year":"2011","journal-title":"Earth-Sci. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Law, B.E., Ulmishek, G.F., and Slavin, V.I. (1998). Mechanisms that generate abnormal pressures: An overview. Abnormal Pressures in Hydrocarbon Environments, American Association of Petroleum Geologists. AAPG Memoir 70.","DOI":"10.1306\/M70615"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Krygier, N., Solarin, A., and Orozova-Bekkevold, I. (2020, January 2\u20133). A Drilling Company\u2019s Perspective on Non-Productive Time NPT Due to Well Stability Issues. Proceedings of the SPE Norway Subsurface Conference, Virtual.","DOI":"10.2118\/200732-MS"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Elmgerbi, A., and Thonhauser, G. (2025, January 21\u201323). Assessing the Potential of Advanced Sensor Technologies and IoT in Predicting Downhole Drilling Issues. Proceedings of the GOTECH, Dubai, United Arab Emirates.","DOI":"10.2118\/224796-MS"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1010933404324","article-title":"Random forests","volume":"45","author":"Breiman","year":"2001","journal-title":"Mach. Learn."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chen, T., and Guestrin, C. (2016, January 13\u201317). XGBoost: A scalable tree boosting system. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA.","DOI":"10.1145\/2939672.2939785"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Vapnik, V.N. (1995). The Nature of Statistical Learning Theory, Springer.","DOI":"10.1007\/978-1-4757-2440-0"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1214\/aos\/1013203451","article-title":"Greedy function approximation: A gradient boosting machine","volume":"29","author":"Friedman","year":"2001","journal-title":"Ann. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104548","DOI":"10.1016\/j.cageo.2020.104548","article-title":"A machine learning methodology for multivariate pore-pressure prediction","volume":"143","author":"Yu","year":"2020","journal-title":"Comput. Geosci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"13807","DOI":"10.1021\/acsomega.1c01340","article-title":"Data-driven modeling approach for pore pressure gradient prediction while drilling from drilling parameters","volume":"6","author":"Abdelaal","year":"2021","journal-title":"ACS Omega"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2233","DOI":"10.1016\/j.egyr.2022.01.012","article-title":"A robust approach to pore pressure prediction applying petrophysical log data aided by machine learning techniques","volume":"8","author":"Zhang","year":"2022","journal-title":"Energy Rep."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Li, H., Tan, Q., Deng, J., Dong, B., Li, B., Guo, J., Zhang, S., and Bai, W. (2023). A comprehensive prediction method for pore pressure in abnormally high-pressure blocks based on machine learning. Processes, 11.","DOI":"10.3390\/pr11092603"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"212839","DOI":"10.1016\/j.geoen.2024.212839","article-title":"Enhancing pore pressure prediction accuracy: A knowledge-driven approach with temporal fusion transformer","volume":"238","author":"Cao","year":"2024","journal-title":"Geoenergy Sci. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1016\/j.petsci.2023.11.006","article-title":"An adaptive physics-informed deep learning method for pore pressure prediction using seismic data","volume":"21","author":"Zhang","year":"2024","journal-title":"Pet. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2727","DOI":"10.1016\/j.jrmge.2024.09.049","article-title":"Pore pressure prediction based on conventional well logs and seismic data using an advanced machine learning approach","volume":"17","author":"Ehsan","year":"2025","journal-title":"J. Rock. Mech. Geotech. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2868","DOI":"10.1016\/j.petsci.2025.04.012","article-title":"A novel method for predicting formation pore pressure ahead of the drill bit by embedding petrophysical theory into machine learning based on seismic and logging-while-drilling data","volume":"22","author":"Chen","year":"2025","journal-title":"Pet. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5047","DOI":"10.1016\/j.petsci.2025.09.020","article-title":"A stacking ensemble approach for pore pressure prediction in real time during drilling based on mud log data","volume":"22","author":"Zhang","year":"2025","journal-title":"Pet. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"23357","DOI":"10.1038\/s41598-025-07048-9","article-title":"Machine learning enhanced formation pressure prediction using integrated well logging and mud logging","volume":"15","author":"Liang","year":"2025","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wang, J., Zhang, M., Huang, W., Zhao, C., and Wang, Y. (2026). Pore pressure prediction using DASP-based feature selection and a physics-constrained attention-enhanced CNN. Processes, 14.","DOI":"10.3390\/pr14111779"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1126\/science.1165893","article-title":"Distilling free-form natural laws from experimental data","volume":"324","author":"Schmidt","year":"2009","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"eaay2631","DOI":"10.1126\/sciadv.aay2631","article-title":"AI Feynman: A physics-inspired method for symbolic regression","volume":"6","author":"Udrescu","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","author":"Karniadakis","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2318","DOI":"10.1109\/TKDE.2017.2720168","article-title":"Theory-guided data science: A new paradigm for scientific discovery from data","volume":"29","author":"Karpatne","year":"2017","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1111\/j.1600-0587.2012.07348.x","article-title":"Collinearity: A review of methods to deal with it and a simulation study evaluating their performance","volume":"36","author":"Dormann","year":"2013","journal-title":"Ecography"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"20150202","DOI":"10.1098\/rsta.2015.0202","article-title":"Principal component analysis: A review and recent developments","volume":"374","author":"Jolliffe","year":"2016","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1111\/ecog.02881","article-title":"Cross-validation strategies for data with temporal, spatial, hierarchical, or phylogenetic structure","volume":"40","author":"Roberts","year":"2017","journal-title":"Ecography"},{"key":"ref_30","unstructured":"Molnar, C. (2022). Interpretable Machine Learning, Leanpub. [2nd ed.]."},{"key":"ref_31","first-page":"4765","article-title":"A unified approach to interpreting model predictions","volume":"Volume 30","author":"Lundberg","year":"2017","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_32","unstructured":"Koza, J.R. (1992). Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press."},{"key":"ref_33","unstructured":"Poli, R., Langdon, W.B., and McPhee, N.F. (2008). A Field Guide to Genetic Programming, Lulu Enterprises."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/17\/5440\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T10:33:10Z","timestamp":1788777190000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/17\/5440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,28]]},"references-count":33,"journal-issue":{"issue":"17","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["s26175440"],"URL":"https:\/\/doi.org\/10.3390\/s26175440","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,28]]}}}