Analytical solutions for co- and counter-current imbibition of sorbing, dispersive solutes in immiscible two-phase flow

We derive a set of analytical solutions for the transport of adsorbing solutes in an immiscible, incompressible two-phase system. This work extends recent results for the analytical description for the movement of inert tracers due to capillary and viscous forces and dispersion to the case of adsorbing solutes. We thereby obtain the first known analytical expression for the description of the effect of adsorption, dispersion, capillary forces and viscous forces on solute movement in two-phase flow. For the purely advective transport, we combine a known exact solution for the description of flow with the method of characteristics for the advective transport equations to obtain solutions that describe both co- and spontaneous counter-current imbibition and advective transport in one dimension. We show that for both cases, the solute front can be located graphically by a modified Welge tangent. For the dispersion, we derive approximate analytical solutions by the method of singular perturbation expansion. The solutions reveal that the amount of spreading depends on the flow regime and that adsorption diminishes the spreading behavior of the solute. We give some illustrative examples and compare the analytical solutions with numerical results.

[1]  S. E. Buckley,et al.  Mechanism of Fluid Displacement in Sands , 1942 .

[2]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[3]  Morris Muskat,et al.  Physical principles of oil production , 1949 .

[4]  H. J. Welge,et al.  A Simplified Method for Computing Oil Recovery by Gas or Water Drive , 1952 .

[5]  John Crank,et al.  The Mathematics Of Diffusion , 1956 .

[6]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[7]  M. A. Collins,et al.  General Analysis of Longitudinal Dispersion in Nonuniform Flow , 1971 .

[8]  G. Dagan.  Perturbation Solutions of the Dispersion Equation in Porous Mediums , 1971 .

[9]  G. Dagan,et al.  Solutions of hydrodynamic dispersion in porous media , 1972 .

[10]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

[11]  J. Philip.  ON SOLVING THE UNSATURATED FLOW EQUATION: 1. THE FLUX‐CONCENTRATION RELATION , 1973 .

[12]  John L. Wilson.  Dispersive mixing in a partially saturated porous medium. , 1974 .

[13]  M. V. Dyke.  Perturbation methods in fluid mechanics /Annotated edition/ , 1975 .

[14]  L. Lake,et al.  CATION EXCHANGE IN CHEMICAL FLOODING - 2. THE EFFECT OF DISPERSION, CATION EXCHANGE, AND POLYMER/SURFACTANT ADSORPTION ON CHEMICAL FLOOD ENVIRONMENT. , 1978 .

[15]  G. Pope,et al.  CATION EXCHANGE IN CHEMICAL FLOODING - 1. BASIC THEORY WITHOUT DISPERSION. , 1978 .

[16]  Gary A. Pope,et al.  The Application of Fractional Flow Theory to Enhanced Oil Recovery , 1980 .

[17]  L. Gelhar,et al.  Analysis of longitudinal dispersion in unsaturated flow: 1. The analytical method , 1981 .

[18]  Eva Farkas,et al.  General Purpose Compositional Model , 1985 .

[19]  A. F. Zazovskii.  Structure of discontinuities in problems of oil displacement by reactants that influence the phase equilibrium , 1985 .

[20]  Ragnar Winther,et al.  The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding , 1988 .

[21]  G. I. Barenblatt,et al.  Theory of Fluid Flows Through Natural Rocks , 1990 .

[22]  David B. McWhorter,et al.  Exact integral solutions for two‐phase flow , 1990 .

[23]  D. McWhorter,et al.  Reply [to Comment on Exact integral solutions for two-phase flow by David B. McWhorter and Daniel , 1992 .

[24]  G. Bodvarsson,et al.  Comment on “Exact integral solutions for two-phase flow” by David B. McWhorter and Daniel K. Sunada , 1992 .

[25]  Pavel Bedrikovetsky,et al.  Mathematical Theory of Oil and Gas Recovery , 1993 .

[26]  Magne S. Espedal,et al.  Macrodispersion for two-phase, immiscible flow in porous media , 1994 .

[27]  Tissa H. Illangasekare,et al.  Analytical Solutions for Water Flow and Solute Transport in the Unsaturated Zone , 1995 .

[28]  M. Blunt,et al.  A generalized streamline method to predict reservoir flow , 1995, Petroleum Geoscience.

[29]  Gedeon Dagan,et al.  Reactive transport and immiscible flow in geological media. II. Applications , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[30]  Akhil Datta-Gupta,et al.  A semianalytic approach to tracer flow modeling in heterogeneous permeable media , 1996 .

[31]  Gedeon Dagan,et al.  Reactive transport and immiscible flow in geological media. I. General theory , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[32]  R. Helmig.  Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems , 2011 .

[33]  Rainer Helmig,et al.  Multiphase flow in heterogeneous porous media: A classical finite element method versus an implicit pressure–explicit saturation‐based mixed finite element–finite volume approach , 1999 .

[34]  Eric James Mackay,et al.  Mixing of injected, connate and aquifer brines in waterflooding and its relevance to oilfield scaling , 2000 .

[35]  Norman R. Morrow,et al.  Recovery of oil by spontaneous imbibition , 2001 .

[36]  Eric James Mackay,et al.  Predicting in Situ Sulphate Scale Deposition and the Impact on Produced Ion Concentrations , 2002 .

[37]  O. Cirpka.  Choice of dispersion coefficients in reactive transport calculations on smoothed fields. , 2002, Journal of contaminant hydrology.

[38]  S. Attinger,et al.  Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media , 2003 .

[39]  Dag Chun Standnes,et al.  Spontaneous imbibition of water into oil-wet carbonates , 2003 .

[40]  M. Panfilow,et al.  Nonlinear Two-Phase Mixing in Heterogeneous Porous Media , 2004 .

[41]  S. Geiger,et al.  Combining finite element and finite volume methods for efficient multiphase flow simulations in highly heterogeneous and structurally complex geologic media , 2004 .

[42]  Brian Berkowitz,et al.  Mixing-induced precipitation and porosity evolution in porous media , 2005 .

[43]  Knut-Andreas Lie,et al.  A front-tracking method for the simulation of three-phase flow in porous media , 2005 .

[44]  R. Johns,et al.  Analytical solutions for surfactant‐enhanced remediation of nonaqueous phase liquids , 2005 .

[45]  Margot Gerritsen,et al.  MODELING FLUID FLOW IN OIL RESERVOIRS , 2005 .

[46]  Alberto Guadagnini,et al.  A procedure for the solution of multicomponent reactive transport problems , 2005 .

[47]  Hussein Hoteit,et al.  Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media , 2005 .

[48]  S. Geiger,et al.  Multiphase Thermohaline Convection in the Earth’s Crust: II. Benchmarking and Application of a Finite Element – Finite Volume Solution Technique with a NaCl–H2O Equation of State , 2006 .

[49]  Karsten Pruess,et al.  TOUGHREACT - A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration , 2006, Comput. Geosci..

[50]  Patrick Jenny,et al.  Multiscale finite-volume method for compressible multiphase flow in porous media , 2006, J. Comput. Phys..

[51]  S. Geiger,et al.  Multiphase Thermohaline Convection in the Earth’s Crust: I. A New Finite Element – Finite Volume Solution Technique Combined With a New Equation of State for NaCl–H2O , 2006 .

[52]  M. Blunt,et al.  Simulation of counter-current imbibition in water-wet fractured reservoirs , 2006 .

[53]  Ruben Juanes,et al.  Analytical Solutions to Multiphase First-Contact Miscible Models with Viscous Fingering , 2006 .

[54]  R. Helmig,et al.  A mixed-dimensional finite volume method for two-phase flow in fractured porous media , 2006 .

[55]  S. G. Roberts,et al.  Numerical simulation of multi-phase fluid flow in structurally complex reservoirs , 2007 .

[56]  Jesús Carrera,et al.  A mixing ratios‐based formulation for multicomponent reactive transport , 2007 .

[57]  R. H. Brooks,et al.  Hydraulic properties of porous media , 1963 .

[58]  David B. McWhorter.  Infiltration affected by flow of air , 2007 .

[59]  R. Helmig,et al.  Black-Oil Simulations for Three-Component -- Three-Phase Flow in Fractured Porous Media , 2007 .

[60]  N. Morrow,et al.  Waterflood Performance By Injection Of Brine With Different Salinity For Reservoir Cores , 2007 .

[61]  C. J. Seto,et al.  Analytical Solutions for Multicomponent, Two-Phase Flow in Porous Media with Double Contact Discontinuities , 2009 .

[62]  Christopher C. Pain,et al.  Simulation of Solute Transport Through Fractured Rock: A Higher-Order Accurate Finite-Element Finite-Volume Method Permitting Large Time Steps , 2010 .

[63]  L. Cathles,et al.  The Impact of Pore Water Chemistry on Carbonate Surface Charge and Oil Wettability , 2010 .

[64]  S. Geiger,et al.  Non‐Fourier thermal transport in fractured geological media , 2010 .

[65]  Kenneth Stuart Sorbie,et al.  Semianalytical solutions for cocurrent and countercurrent imbibition and dispersion of solutes in immiscible two‐phase flow , 2011 .

[66]  Martin Frey,et al.  Using discharge data to reduce structural deficits in a hydrological model with a Bayesian inference approach and the implications for the prediction of critical source areas , 2011 .

[67]  Andreas Englert,et al.  Mixing, spreading and reaction in heterogeneous media: a brief review. , 2011, Journal of contaminant hydrology.