{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:00:01Z","timestamp":1760148001523,"version":"build-2065373602"},"reference-count":58,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,3,20]],"date-time":"2023-03-20T00:00:00Z","timestamp":1679270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005416","name":"Research Council of Norway","doi-asserted-by":"publisher","award":["259870"],"award-info":[{"award-number":["259870"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["BDCC"],"abstract":"<jats:p>In this study, the numerical solutions to the Elder problem are analyzed using Big Data technologies and data-driven approaches. The steady-state solutions to the Elder problem are investigated with regard to Rayleigh numbers (Ra), grid sizes, perturbations, and other parameters of the system studied. The complexity analysis is carried out for the datasets containing different solutions to the Elder problem, and the time of the highest complexity of numerical solutions is estimated. An approach to the identification of transient fingers and the visualization of large ensembles of solutions is proposed. Predictive models are developed to forecast steady states based on early-time observations. These models are classified into three possible types depending on the features (predictors) used in a model. The numerical results of the prediction accuracy are given, including the estimated confidence intervals for the accuracy, and the estimated time of 95% predictability. Different solutions, their averages, principal components, and other parameters are visualized.<\/jats:p>","DOI":"10.3390\/bdcc7010052","type":"journal-article","created":{"date-parts":[[2023,3,20]],"date-time":"2023-03-20T05:46:42Z","timestamp":1679291202000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analysis of the Numerical Solutions of the Elder Problem Using Big Data and Machine Learning"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2849-6525","authenticated-orcid":false,"given":"Roman","family":"Khotyachuk","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Natural Sciences, University of Bergen, 5020 Bergen, Norway"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0600-2715","authenticated-orcid":false,"given":"Klaus","family":"Johannsen","sequence":"additional","affiliation":[{"name":"NORCE Norwegian Research Center AS, 5008 Bergen, Norway"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1017\/S0022112067000576","article-title":"Transient convection in a porous medium","volume":"27","author":"Elder","year":"1967","journal-title":"J. Fluid Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1016\/S0309-1708(02)00063-5","article-title":"Variable-density flow and transport in porous media: Approaches and challenges","volume":"25","author":"Diersch","year":"2002","journal-title":"Adv. Water Resour."},{"key":"ref_3","first-page":"2097","article-title":"Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone","volume":"26","author":"Voss","year":"1987","journal-title":"Water Resour."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Nield, D.A., and Bejan, A. (2013). Convection in Porous Media, Spinger. [4th ed.].","DOI":"10.1007\/978-1-4614-5541-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1023\/A:1025515229807","article-title":"On the validity of the Boussinesq approximation for the Elder problem","volume":"7","author":"Johannsen","year":"2003","journal-title":"Comput. Geosci."},{"key":"ref_6","first-page":"63","article-title":"Numerical modelling of convection dominated transport coupled with density driven flow in porous media","volume":"24","year":"2001","journal-title":"Adv. Water Resour."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Elder, J., Simmons, C.T., Diersch, H.-J., Frolkovic, P., Holzbecher, E., and Johannsen, K. (2017). The Elder Problem. Fluids, 2.","DOI":"10.3390\/fluids2010011"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3607","DOI":"10.1029\/1999WR900254","article-title":"On a test case for density-dependent groundwater flow and solute transport models: The salt lake problem","volume":"35","author":"Simmons","year":"1999","journal-title":"Water Resour. Res."},{"key":"ref_9","first-page":"1","article-title":"Insights from a pseudospectral approach to the Elder problem","volume":"45","author":"Mathias","year":"2009","journal-title":"Water Resour. Res."},{"key":"ref_10","unstructured":"(2022, December 23). SUTRA: A Model for 2D or 3D Saturated-Unsaturated, Variable-Density Ground-Water Flow with Solute or Energy Transport, Available online: https:\/\/www.usgs.gov\/software\/sutra-model-2d-or-3d-saturated-unsaturated-variable-density-ground-water-flow-solute-or."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1029\/94WR02272","article-title":"Dispersive transport dynamics in a strongly coupled groundwater\u2013brine flow system","volume":"31","author":"Oldenburg","year":"1995","journal-title":"Water Resour. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/S0309-1708(96)00034-6","article-title":"Coupled groundwater flow and transport: 1. Verification of variable density flow and transport models","volume":"21","author":"Kolditz","year":"1998","journal-title":"Adv. Water Resour."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4-1","DOI":"10.1029\/2002WR001290","article-title":"Unstable density-driven flow in heterogeneous porous media: A stochastic study of the Elder \u201cshort heater\u201d problem","volume":"39","author":"Prasad","year":"2003","journal-title":"Water Resour. Res."},{"key":"ref_14","first-page":"485","article-title":"The Elder problem\u2014bifurcations and steady state solutions","volume":"47","author":"Johannsen","year":"2002","journal-title":"Dev. Water Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1029\/2001WR000586","article-title":"Numerical error in groundwater flow and solute transport simulation","volume":"39","author":"Woods","year":"2003","journal-title":"Water Resour. Res."},{"key":"ref_16","first-page":"1549","article-title":"Lattice Boltzmann model for the elder problem","volume":"55","author":"Thornea","year":"2004","journal-title":"Dev. Water Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.advwatres.2010.11.008","article-title":"The effect of dispersion on the stability of density-driven flows in saturated homogeneous porous media","volume":"34","author":"Musuuza","year":"2011","journal-title":"Adv. Water Resour."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1111\/gwat.12067","article-title":"Influence of Boundary Condition Types on Unstable Density-Dependent Flow","volume":"52","author":"Simmons","year":"2014","journal-title":"Groundwater"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1111\/gwat.12593","article-title":"The Elder Problem","volume":"55","author":"Simmons","year":"2017","journal-title":"Groundwater"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"8609682","DOI":"10.1155\/2019\/8609682","article-title":"Impact of Low- or High-Permeability Inclusion on Free Convection in a Porous Medium","volume":"2019","author":"Yan","year":"2019","journal-title":"Geofluids"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Shafabakhsh, P., Fahs, M., Ataie-Ashtiani, B., and Simmons, C.T. (2019). Unstable Density-Driven Flow in Fractured Porous Media: The Fractured Elder Problem. Fluids, 4.","DOI":"10.3390\/fluids4030168"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"e2022WR032307","DOI":"10.1029\/2022WR032307","article-title":"Efficient numerical simulation of density-driven flows: Application to the 2- and 3-D Elder problem","volume":"58","author":"Bahlali","year":"2022","journal-title":"Water Resour. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1944","DOI":"10.1029\/2011WR011346","article-title":"Prediction and uncertainty of free convection phenomena in porous media","volume":"48","author":"Xie","year":"2012","journal-title":"Water Resour. Res."},{"key":"ref_24","unstructured":"Kutz, J.N. (2013). Data-Driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data, Oxford University Press."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Brunton, S., and Kutz, J.N. (2022). Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Oxford University Press.","DOI":"10.1017\/9781009089517"},{"key":"ref_26","unstructured":"(2022, December 23). Apache Hadoop. Available online: https:\/\/hadoop.apache.org\/."},{"key":"ref_27","unstructured":"(2022, December 23). Apache Spark. Available online: https:\/\/spark.apache.org\/."},{"key":"ref_28","unstructured":"Fein, E. (1998). d3f\u2014Ein Programmpaket zur Modellierung von Dichtegetriebenen Str\u00f6mungen, GRS."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s007910050003","article-title":"UG\u2014A Flexible Software Toolbox for Solving Partial Differential Equations","volume":"1","author":"Bastian","year":"1997","journal-title":"Comput. Vis. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ferziger, J., Peri\u0107, M., and Street, R. (2020). Computational Methods for Fluid Dynamics, Springer. [4th ed.].","DOI":"10.1007\/978-3-319-99693-6"},{"key":"ref_31","unstructured":"(2022, December 23). ISO Random (The GNU C Library). Available online: https:\/\/www.gnu.org\/software\/libc\/manual\/html_node\/ISO-Random.html#index-rand."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Ajibola, J., Adam, A., and Ann Muggeridge, A. (2016, January 25\u201327). Gravity Driven Fingering and Mixing During CO2 Sequestration. Proceedings of the the SPE Asia Pacific Oil & Gas Conference and Exhibition, Perth, Australia.","DOI":"10.2118\/182317-MS"},{"key":"ref_33","unstructured":"Aggarwal, C. (2014). Data Classification: Algorithms and Applications, Chapman & Hall\/CRC."},{"key":"ref_34","unstructured":"Bishop, C. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_35","unstructured":"Kuhn, M., and Johnson, K. (2018). Applied Predictive Modeling, Springer. [2nd ed.]."},{"key":"ref_36","first-page":"189","article-title":"Bootstrap confidence intervals","volume":"11","author":"Efron","year":"1996","journal-title":"Stat. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1109\/34.990132","article-title":"Complexity measures of supervised classification problems","volume":"24","author":"Ho","year":"2002","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1016\/j.patrec.2006.01.006","article-title":"Data complexity assessment in undersampled classification of high-dimensional biomedical data","volume":"27","author":"Baumgartner","year":"2006","journal-title":"Pattern Recognit. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Eld\u00e9n, L. (2007). Matrix Methods in Data Mining and Pattern Recognition, Society for Industrial & Applied Mathematics.","DOI":"10.1137\/1.9780898718867"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Dulhare, U., Ahmad, K., and Bin Ahmad, K.A. (2020). Machine Learning and Big Data: Concepts, Algorithms, Tools and Applications., John Wiley & Sons, Inc.","DOI":"10.1002\/9781119654834"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Pulliam, T.H., and Zingg, D.W. (2014). Fundamentals Algorithms in Computational Fluid Dynamics, Scientific Computation; Springer.","DOI":"10.1007\/978-3-319-05053-9"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Chakraverty, S., Mahato, N.R., Karunakar, P., and Rao, T.D. (2019). Advanced Numerical and Semi-Analytical Methods for Differential Equations, John Wiley & Sons, Inc.","DOI":"10.1002\/9781119423461"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Rapp, B. (2017). Microfluidics: Modeling, Mechanics and Mathematics, Elsevier Inc.","DOI":"10.1016\/B978-1-4557-3141-1.50009-5"},{"key":"ref_44","unstructured":"(2022, December 23). HDFS Architecture Guide. Available online: https:\/\/hadoop.apache.org\/docs\/r1.2.1\/hdfs_design.html."},{"key":"ref_45","unstructured":"(2022, December 23). Apache ORC\u2014High-Performance Columnar Storage for Hadoop. Available online: https:\/\/orc.apache.org\/."},{"key":"ref_46","unstructured":"Brownlee, J. (2022, December 23). What Is the Difference Between Test and Validation Datasets?. Available online: https:\/\/machinelearningmastery.com\/difference-test-validation-datasets\/."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Calvetti, D., and Somersalo, E. (2020). Mathematics of Data Science: A Computational Approach to Clustering and Classification, Society for Industrial & Applied Mathematics.","DOI":"10.1137\/1.9781611976373"},{"key":"ref_48","unstructured":"(2022, December 23). Scikit-Learn\u2014Machine Learning in Python. Available online: https:\/\/scikit-learn.org\/."},{"key":"ref_49","unstructured":"(2022, December 23). Apache Spark MLlib. Available online: https:\/\/spark.apache.org\/mllib\/."},{"key":"ref_50","unstructured":"(2022, December 23). Project Jupyter. Available online: https:\/\/jupyter.org\/."},{"key":"ref_51","unstructured":"(2022, December 23). Matplotlib: Visualization with Python. Available online: https:\/\/matplotlib.org\/."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Duboue, P. (2020). The Art of Feature Engineering: Essentials for Machine Learning, Cambridge University Press.","DOI":"10.1017\/9781108671682"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer. [2nd ed.].","DOI":"10.1007\/978-0-387-84858-7"},{"key":"ref_54","unstructured":"(2022, December 23). Univariate Feature Selection\u2014Scikit-Learn 1.2.0 Documentation. Available online: https:\/\/scikit-learn.org\/stable\/modules\/feature_selection.html#univariate-feature-selection."},{"key":"ref_55","unstructured":"(2022, December 23). Scipy.Signal.Find_PEAKS\u2014SciPy v1.9.1 Manual. Available online: https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.signal.find_peaks.html."},{"key":"ref_56","unstructured":"Tingle, M. (2023, February 26). Preventing Data Leakage in Your Machine Learning Model. Available online: https:\/\/towardsdatascience.com\/preventing-data-leakage-in-your-machine-learning-model-9ae54b3cd1fb."},{"key":"ref_57","unstructured":"(2022, December 23). Random Forest Classifier\u2014Scikit-Learn 1.2.0 Documentation. Available online: https:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier.feature_importances_."},{"key":"ref_58","unstructured":"(2022, December 23). Gradient Boosting Classifier\u2014Scikit-Learn 1.2.0 Documentation. Available online: https:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier.feature_importances_."}],"container-title":["Big Data and Cognitive Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2504-2289\/7\/1\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:59:16Z","timestamp":1760122756000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2504-2289\/7\/1\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,20]]},"references-count":58,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["bdcc7010052"],"URL":"https:\/\/doi.org\/10.3390\/bdcc7010052","relation":{},"ISSN":["2504-2289"],"issn-type":[{"type":"electronic","value":"2504-2289"}],"subject":[],"published":{"date-parts":[[2023,3,20]]}}}