Eulerian Moment Equations for 2-D Stochastic Immiscible Flow

We solve statistical moment differential equations (MDEs) for immiscible flow in porous media in the limit of zero capillary pressure, with application to secondary oil recovery. Closure is achieved by Taylor expansion of the fractional flow function and a perturbation argument. Previous results in one dimension are extended to two dimensions. Comparison to Monte Carlo simulation (MCS) shows that the MDE approach gives a good approximation to total oil production. For such spatially integrated or averaged quantities MDEs may be substantially more efficient than MCS.

[1]  S. P. Neuman,et al.  Nonasymptotic Behavior of a Common Eulerian Approximation for Transport in Random Velocity Fields , 1991 .

[2]  T. F. Russell,et al.  Moment Equations for Stochastic Immiscible Flow , 2001 .

[3]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[4]  J. P. Delhomme,et al.  Spatial variability and uncertainty in groundwater flow parameters: A geostatistical approach , 1979 .

[5]  R. Freeze.  A stochastic‐conceptual analysis of one‐dimensional groundwater flow in nonuniform homogeneous media , 1975 .

[6]  Sean Hatch,et al.  General joint conditional simulations using a fast fourier transform method , 1997 .

[7]  Hamdi A. Tchelepi,et al.  Stochastic Analysis of Immiscible Two-Phase Flow in Heterogeneous Media , 1999 .

[8]  Shlomo P. Neuman,et al.  Gaussian Closure of One-Dimensional Unsaturated Flow in Randomly Heterogeneous Soils , 2001 .

[9]  S. P. Neuman,et al.  Eulerian‐Lagrangian Theory of transport in space‐time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximation , 1993 .

[10]  C. S. Simmons.  A stochastic‐convective transport representation of dispersion in one‐dimensional porous media systems , 1982 .

[11]  Zhangxin Chen,et al.  Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment , 2002 .

[12]  Gedeon Dagan,et al.  A solute flux approach to transport in heterogeneous formations: 2. Uncertainty analysis , 1992 .

[13]  Randall J. LeVeque,et al.  CLAWPACK Version 4.3 User's Guide , 2002 .

[14]  G. Dagan,et al.  A solute flux approach to transport in heterogeneous formations: 1. The general framework , 1992 .

[15]  E. Paleologos,et al.  Stochastic Methods for Flow in Porous Media, Coping With Uncertainties , 2003 .

[16]  Timothy R. Ginn,et al.  Stochastic‐Convective Transport with Nonlinear Reaction: Mathematical Framework , 1995 .

[17]  Daniel M. Tartakovsky,et al.  Erratum: Transient flow in bounded randomly heterogeneous domains, 1, Exact conditional moment equations and recursive approximations (Water Resources Research (1998) 34:1 (1-12)) , 1999 .

[18]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[19]  G. Marsily.  Quantitative Hydrogeology: Groundwater Hydrology for Engineers , 1986 .

[20]  L. Gelhar.  Stochastic Subsurface Hydrology , 1992 .

[21]  Felipe Pereira,et al.  Crossover from Nonlinearity Controlled to Heterogeneity Controlled Mixing in Two-Phase Porous Media Flows , 2003 .

[22]  S. P. Neuman,et al.  Stochastic theory of field‐scale fickian dispersion in anisotropic porous media , 1987 .

[23]  Dennis McLaughlin,et al.  Stochastic analysis of nonstationary subsurface solute transport: 1. Unconditional moments , 1989 .

[24]  Dongxiao Zhang,et al.  Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media , 1999 .

[25]  Benoît Nœtinger,et al.  Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 2 – Isotropic Media , 2004 .

[26]  B. Welch.  The structure , 1992 .

[27]  C.A.J. Appelo,et al.  FLOW AND TRANSPORT , 2004 .

[28]  Dongxiao Zhang.  Quantification of uncertainty for fluid flow in heterogeneous petroleum reservoirs , 1999 .

[29]  L. Dake.  Fundamentals of Reservoir Engineering , 1983 .

[30]  Benoît Nœtinger,et al.  Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 1 – Stratified Media , 2004 .

[31]  Hamdi A. Tchelepi,et al.  Stochastic Formulation for Uncertainty Assessment of Two-Phase Flow in Heterogeneous Reservoirs , 1999 .

[32]  Andrew F. B. Tompson,et al.  Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media , 1990 .

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

[34]  G. Dagan.  Flow and transport in porous formations , 1989 .

[35]  D. A. Barry,et al.  On the Dagan Model of solute transport in groundwater: Foundational aspects , 1987 .

[36]  Daniel M. Tartakovsky,et al.  Extension of “Transient flow in bounded randomly heterogeneous domains: 1, Exact conditional moment equations and recursive approximations” , 1999 .

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

[38]  Dongxiao Zhang.  Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media , 1998 .

[39]  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.

[40]  A. S. Monin,et al.  The Structure of Atmospheric Turbulence , 1958 .

[41]  Dexi Zhang,et al.  Moment-equation approach to single phase fluid flow in heterogeneous reservoirs , 1999 .