# American Institute of Mathematical Sciences

June  2010, 5(2): 217-256. doi: 10.3934/nhm.2010.5.217

## A mathematical model for dynamic wettability alteration controlled by water-rock chemistry

 1 University of Stavanger (UiS), 4036 Stavanger, Norway 2 International Research Institute of Stavanger (IRIS), Prof. Olav Hanssensvei 15, NO-4068 Stavanger

Received  November 2009 Revised  March 2010 Published  May 2010

Previous experimental studies of spontaneous imbibition on chalk core plugs have shown that seawater may change the wettability in the direction of more water-wet conditions in chalk reservoirs. One possible explanation for this wettability alteration is that various ions in the water phase (sulphate, calcium, magnesium, etc.) enter the formation water due to molecular diffusion. This creates a non-equilibrium state in the pore space that results in chemical reactions in the aqueous phase as well as possible water-rock interaction in terms of dissolution/precipitation of minerals and/or changes in surface charge. In turn, this paves the way for changes in the wetting state of the porous media in question. The purpose of this paper is to put together a novel mathematical model that allows for systematic investigations, relevant for laboratory experiments, of the interplay between (i) two-phase water-oil flow (pressure driven and/or capillary driven); (ii) aqueous chemistry and water-rock interaction; (iii) dynamic wettability alteration due to water-rock interaction.
In particular, we explore in detail a 1D version of the model relevant for spontaneous imbibition experiments where wettability alteration has been linked to dissolution of calcite. Dynamic wettability alteration is built into the model by defining relative permeability and capillary pressure curves as an interpolation of two sets of end point curves corresponding to mixed-wet and water-wet conditions. This interpolation depends on the dissolution of calcite in such a way that when no dissolution has taken place, mixed-wet conditions prevail. However, gradually there is a shift towards more water-wet conditions at the places in the core where dissolution of calcite takes place. A striking feature reflected by the experimental data found in the literature is that the steady state level of oil recovery, for a fixed temperature, depends directly on the brine composition. We demonstrate that the proposed model naturally can explain this behavior by relating the wettability change to changes in the mineral composition due to dissolution/precipitation. Special attention is paid to the effect of varying, respectively, the concentration of $\text{SO}_4^{2-}$ ions and $\text{Mg}^{2+}$ ions in seawater like brines. The effect of changing the temperature is also demonstrated and evaluated in view of observed experimental behavior.
Citation: Steinar Evje, Aksel Hiorth. A mathematical model for dynamic wettability alteration controlled by water-rock chemistry. Networks & Heterogeneous Media, 2010, 5 (2) : 217-256. doi: 10.3934/nhm.2010.5.217
 [1] Liping Yu, Hans Kleppe, Terje Kaarstad, Svein M. Skjaeveland, Steinar Evje, Ingebret Fjelde. Modelling of wettability alteration processes in carbonate oil reservoirs. Networks & Heterogeneous Media, 2008, 3 (1) : 149-183. doi: 10.3934/nhm.2008.3.149 [2] T. L. van Noorden, I. S. Pop, M. Röger. Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness. Conference Publications, 2007, 2007 (Special) : 1013-1020. doi: 10.3934/proc.2007.2007.1013 [3] Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks & Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669 [4] Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267 [5] Shannon Dixon, Nancy Huntly, Priscilla E. Greenwood, Luis F. Gordillo. A stochastic model for water-vegetation systems and the effect of decreasing precipitation on semi-arid environments. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1155-1164. doi: 10.3934/mbe.2018052 [6] Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307 [7] Xiaoli Wang, Junping Shi, Guohong Zhang. Interaction between water and plants: Rich dynamics in a simple model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2971-3006. doi: 10.3934/dcdsb.2017159 [8] Igor Chueshov, Tamara Fastovska. On interaction of circular cylindrical shells with a Poiseuille type flow. Evolution Equations & Control Theory, 2016, 5 (4) : 605-629. doi: 10.3934/eect.2016021 [9] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217 [10] Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69 [11] Bogdan-Vasile Matioc. A characterization of the symmetric steady water waves in terms of the underlying flow. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3125-3133. doi: 10.3934/dcds.2014.34.3125 [12] Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial & Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041 [13] Oliver Penrose, John W. Cahn. On the mathematical modelling of cellular (discontinuous) precipitation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 963-982. doi: 10.3934/dcds.2017040 [14] Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012 [15] Urszula Foryś. Some remarks on the Gottman-Murray model of marital dissolution and time delays. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 181-191. doi: 10.3934/dcdsb.2018012 [16] Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks & Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211 [17] Marie Levakova. Effect of spontaneous activity on stimulus detection in a simple neuronal model. Mathematical Biosciences & Engineering, 2016, 13 (3) : 551-568. doi: 10.3934/mbe.2016007 [18] Peter Bednarik, Josef Hofbauer. Discretized best-response dynamics for the Rock-Paper-Scissors game. Journal of Dynamics & Games, 2017, 4 (1) : 75-86. doi: 10.3934/jdg.2017005 [19] Włodzimierz Bąk, Tadeusz Nadzieja, Mateusz Wróbel. Models of the population playing the rock-paper-scissors game. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 1-11. doi: 10.3934/dcdsb.2018001 [20] Qiang Du, Chun Liu, R. Ryham, X. Wang. Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation. Communications on Pure & Applied Analysis, 2005, 4 (3) : 537-548. doi: 10.3934/cpaa.2005.4.537

2018 Impact Factor: 0.871