{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T04:04:14Z","timestamp":1774929854794,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,6]],"date-time":"2019-06-06T00:00:00Z","timestamp":1559779200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In the current paper, an iterative algorithm is developed to simulate the problem of two-phase flow with heat transfer in porous media. The convective body force caused by heat transfer is described by Boussinesq approximation throughout with the governing equations, namely, pressure, saturation, and energy. The two coupled equations of pressure and saturation are solved using the implicit pressure-explicit saturation (IMPES) scheme, while the energy equation is treated implicitly, and the scheme is called iterative implicit pressure, explicit saturation, implicit temperature (I-IMPES-IMT). In order to calculate the pressure implicitly, the equations of pressure and saturation are coupled by linearizing the capillary pressure which is a function of saturation. After that, the equation of saturation is solved explicitly. Then, the velocity is computed which is used in the energy equation to calculate the temperature implicitly. The cell-centered finite difference (CCFD) method is utilized for spatial discretization. Furthermore, a relaxation factor along is used with the Courant\u2013Friedrichs\u2013Lewy (CFL) condition. Finally, in order to illustrate the efficiency of the developed algorithm, error estimates for saturation and temperature for different values of time steps and number of iterations are presented. Moreover, numerical examples of different physical scenarios of heterogamous media are presented.<\/jats:p>","DOI":"10.3390\/a12060117","type":"journal-article","created":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T03:56:31Z","timestamp":1559879791000},"page":"117","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2513-6791","authenticated-orcid":false,"given":"Mohamed F.","family":"El-Amin","sequence":"first","affiliation":[{"name":"College of Engineering, Effat University, Jeddah 21478, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Aswan University, Aswan 81528, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,6]]},"reference":[{"key":"ref_1","unstructured":"Aziz, K., and Settari, A. (1979). Petroleum Reservoir Simulation, Applied Science Publishers."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"259","DOI":"10.2118\/15133-PA","article-title":"An efficient approach to adaptive implicit compositional simulation with an equation of state","volume":"7","author":"Collins","year":"1992","journal-title":"SPE Res. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1023\/A:1011521413158","article-title":"A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton- Krylov solver","volume":"1","author":"Dawson","year":"1997","journal-title":"Comput. Geosci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1137\/0732037","article-title":"Implicit-Explicit methods for time-dependent partial differential equations","volume":"32","author":"Ascher","year":"1995","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1600","DOI":"10.1137\/060656929","article-title":"Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems","volume":"45","author":"Boscarino","year":"2007","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"181","DOI":"10.2118\/84924-PA","article-title":"IMPES stability: Selection of stable time steps","volume":"8","author":"Coats","year":"2003","journal-title":"SPE J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1108\/HFF-01-2015-0008","article-title":"A novel discontinuous Galerkin model for two-phase flow in porous media using an improved IMPES method","volume":"26","author":"Jamei","year":"2016","journal-title":"Int. J. Num. Meth. Heat Fluid Flow"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chen, Z., Huan, G., and Ma, Y. (2006). Computational Methods for Multiphase Flows in Porous Media, SIAM Computational Science and Engineering.","DOI":"10.1137\/1.9780898718942"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"243","DOI":"10.2118\/12244-PA","article-title":"A compositional formulation of the pressure and saturation equations","volume":"1","author":"Watts","year":"1986","journal-title":"SPE Reserv. Eng."},{"key":"ref_10","first-page":"727","article-title":"A generalized compositional approach for reservoir simulation","volume":"23","author":"Young","year":"1983","journal-title":"SPE J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1137\/S106482750240443X","article-title":"Iterative solution methods for modeling multiphase flow in porous media fully implicitly","volume":"25","author":"Lacroix","year":"2003","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s12182-009-0008-x","article-title":"Iterative coupling reservoir simulation on high performance computers","volume":"6","author":"Lu","year":"2009","journal-title":"Pet. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1923","DOI":"10.1016\/j.compfluid.2010.06.022","article-title":"A new treatment of capillarity to improve the stability of IMPES two-phase flow formulation","volume":"39","author":"Kou","year":"2010","journal-title":"Comput. Fluids"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1344","DOI":"10.1016\/j.procs.2016.05.423","article-title":"An iterative implicit scheme for nanoparticles transport with two-Phase flow in porous media","volume":"80","author":"Kou","year":"2016","journal-title":"Procedia Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2289","DOI":"10.1108\/HFF-05-2016-0210","article-title":"Convergence analysis of the nonlinear iterative method for two-phase flow in porous media associated with nanoparticle injection","volume":"27","author":"Kou","year":"2017","journal-title":"Int. J. Num. Meth. Heat Fluid Flow"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1029\/91WR02527","article-title":"Numerical Modelling of Steam Injection for the Removal of Nonaqueous Phase Liquids from the Subsurface. 2. Code Validation and Application","volume":"28","author":"Falta","year":"1992","journal-title":"Adv. Water Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0309-1708(93)90014-7","article-title":"A positivity preserving method for simulation of steam injection for NAPL site remediation","volume":"16","author":"Forsyth","year":"1993","journal-title":"Adv. Water Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1108\/HFF-02-2017-0080","article-title":"Two-phase explicit CBS procedure for compressible viscous flow transport in porous materials","volume":"28","author":"Cortellessa","year":"2018","journal-title":"Int. J. Num. Meth. Heat Fluid Flow"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1306\/10240808059","article-title":"Pore-throat sizes in sandstones, tight sandstones, and shales","volume":"93","author":"Nelson","year":"2009","journal-title":"AAPG Bull."},{"key":"ref_20","unstructured":"Bear, J. (1972). Dynamics of Fluids in Porous Media, American Elsevier."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/0309-1708(93)90029-F","article-title":"Toward an improved description of the physics of two-phase flow","volume":"16","author":"Hassanizadeh","year":"1993","journal-title":"Adv. Water Resour."},{"key":"ref_22","unstructured":"Helmig, R. (1997). Multiphase Flow and Transport Processes in the Subsurface, Springer. [1st ed.]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1023\/A:1006749505937","article-title":"Vapor transport in low permeability unsaturated soil with capillary effects","volume":"40","author":"Olivella","year":"2000","journal-title":"Trans. Porous Med."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (2000). Iterative Solution of Nonlinear Equations in Several Variables, SIAM.","DOI":"10.1137\/1.9780898719468"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2082","DOI":"10.1137\/050638473","article-title":"A Multipoint Flux Mixed Finite Element Method","volume":"44","author":"Wheeler","year":"2006","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"828","DOI":"10.1137\/S0036142994262585","article-title":"Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences","volume":"34","author":"Arbogast","year":"1997","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.petrol.2006.04.019","article-title":"Characteristics of high-permeability zones using core analysis, and production logging data","volume":"55","year":"2007","journal-title":"J. Pet. Sci. 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