{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:24:00Z","timestamp":1787340240691,"version":"3.56.0"},"reference-count":56,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We analyze the problem of uncertainty propagation for nonlinear two-phase transport in heterogeneous porous media. Specifically, we study the evolution of the saturation field associated with nonlinear immiscible two-phase transport (i.e., Buckley--Leverett problem) in the presence of a stochastic velocity field. The uncertainty in the velocity field is due to the limited information that is usually available about the heterogeneous porosity and permeability fields of the particular subsurface formation of interest. The uncertainty in the total-velocity field leads to uncertainty in the saturation of the injected fluid phase, both in space and time. Given information about the spatial statistics of the correlated heterogeneity, we derive the multipoint cumulative distribution functions (CDF) of saturation. The methodology takes full account of the nonlinear hyperbolic nature of the species conservation law. To obtain the multipoint CDF, we first derive the partial differential equation (PDE) of the \u201craw\u201d CDF of saturation at a given point. Then, we describe the development of the PDE that governs the evolution of the multipoint raw CDF of saturation. The resulting equation is linear in the space-time variables and is solved semianalytically for problems in one spatial dimension and numerically for higher spatial dimensions. Ensemble averaging of the raw CDF leads to the multipoint CDF of saturation. We then compute the two-point saturation CDF profiles in one spatial dimension. We also use the two-point CDF to compute the saturation autocovariance function. We demonstrate the accuracy of our new \u201cdistribution method\u201d by comparing the predictions with exhaustive high-resolution Monte Carlo simulations.<\/jats:p>","DOI":"10.1137\/16m1096244","type":"journal-article","created":{"date-parts":[[2017,4,13]],"date-time":"2017-04-13T14:24:59Z","timestamp":1492093499000},"page":"353-377","source":"Crossref","is-referenced-by-count":2,"title":["Multipoint Distribution of Saturation for Stochastic Nonlinear Two-Phase Transport"],"prefix":"10.1137","volume":"5","author":[{"given":"Fayadhoi","family":"Ibrahima","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamdi A.","family":"Tchelepi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,4,13]]},"reference":[{"key":"atypb1","unstructured":"K. Aziz and A. 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Gelhar,\n                      Stochastic Subsurface Hydrology\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1993."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1029\/98WR01573"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1006514109327"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1029\/WR025i002p00215"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1029\/WR025i011p02331"},{"key":"atypb19","unstructured":"W. Hoeffding,\n                      Maszstabinvariante korrelationstheorie\n                      , Schriften des Mathematischen Instituts und des Instituts fu\u0308r Angewandte Mathematik der Universitat Berlin, 5 (1940), pp. 179-233."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s11242-015-0503-z"},{"key":"atypb21","unstructured":"K. D. Jarman and T. F. Russell,\n                      Moment Equations for Stochastic Immiscible Flow\n                      , Technical report 181, Center for Computational Mathematics, University of Colorado at Denver, Denver, CO, 2002."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/S1540345902413176"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1029\/2008GL034495"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1694(88)90111-4"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/0309-1708(94)90033-7"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2001.6889"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.2118\/114802-PA"},{"key":"atypb28","first-page":"280","author":"Li L.","year":"2005","journal-title":"TX"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1002\/2013WR014055"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1002\/2014WR016238"},{"key":"atypb31","unstructured":"P. Likanapaisal,\n                      Statistical Moment Equations for Forward and Inverse Modeling of Mutiplhase Flow in Porous Media\n                      , Ph.D. thesis, Stanford University, Palo Alto, CA, 2010."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1029\/2004WR003389"},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"G. Mariethoz and J. Caers,\n                      Multiple-Point Geostatistics: Stochastic Modeling with Training Images\n                      , Wiley-Blackwell, 2014.","DOI":"10.1002\/9781118662953"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1029\/2010WR009450"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1029\/2009WR008925"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1002\/wrcr.20240"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2013.03.023"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1029\/92WR02062"},{"key":"atypb39","doi-asserted-by":"crossref","unstructured":"P. Pettersson and H. Tchelepi,\n                      Stochastic Galerkin method for the Buckley-Leverett problem in heterogeneous formations\n                      , in Proceedings of ECMOR XIV, Curran, Red Hook, NY, 2014, A33.","DOI":"10.3997\/2214-4609.20141868"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/0360-1285(85)90002-4"},{"key":"atypb41","doi-asserted-by":"crossref","unstructured":"Y. Rubin,\n                      Applied Stochastic Hydrogeology\n                      , Oxford University Press, New York, 2003.","DOI":"10.1093\/oso\/9780195138047.001.0001"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581390"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1016\/j.jconhyd.2010.08.009"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1029\/2008WR007383"},{"key":"atypb45","doi-asserted-by":"crossref","unstructured":"D. M. Tartakovsky and P. A. Gremaud,\n                      Method of distributions for uncertainty quantification\n                      , in Handbook of Uncertainty Quantification, Springer, 2016.","DOI":"10.1007\/978-3-319-11259-6_27-1"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.2118\/27834-PA"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2011.0186"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2013.03.001"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1137\/130940050"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1137\/120865574"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022277418570"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1029\/97WR03607"},{"key":"atypb53","unstructured":"D. Zhang,\n                      Stochastic Methods for Flow in Porous Media: Coping with Uncertainties\n                      , Academic Press, 2002, San Diego, CA."},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.2118\/59802-PA"},{"key":"atypb55","first-page":"59250","volume":"4","author":"Zhang D.","year":"1999","journal-title":"SPE J."},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.2118\/56842-PA"}],"container-title":["SIAM\/ASA Journal on Uncertainty Quantification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1096244","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:26:35Z","timestamp":1787336795000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1096244"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":56,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1096244"],"URL":"https:\/\/doi.org\/10.1137\/16m1096244","relation":{},"ISSN":["2166-2525"],"issn-type":[{"value":"2166-2525","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}