Development of optimal models of porous media by combining static and dynamic data: the permeability and porosity distributions.

We describe a method for the development of the optimal spatial distributions of the porosity phi and permeability k of a large-scale porous medium. The optimal distributions are constrained by static and dynamic data. The static data that we utilize are limited data for phi and k, which the method honors in the optimal model and utilizes their correlation functions in the optimization process. The dynamic data include the first-arrival (FA) times, at a number of receivers, of seismic waves that have propagated in the porous medium, and the time-dependent production rates of a fluid that flows in the medium. The method combines the simulated-annealing method with a simulator that solves numerically the three-dimensional (3D) acoustic wave equation and computes the FA times, and a second simulator that solves the 3D governing equation for the fluid's pressure as a function of time. To our knowledge, this is the first time that an optimization method has been developed to determine simultaneously the global minima of two distinct total energy functions. As a stringent test of the method's accuracy, we solve for flow of two immiscible fluids in the same porous medium, without using any data for the two-phase flow problem in the optimization process. We show that the optimal model, in addition to honoring the data, also yields accurate spatial distributions of phi and k, as well as providing accurate quantitative predictions for the single- and two-phase flow problems. The efficiency of the computations is discussed in detail.

[1]  J. Quiblier.  A new three-dimensional modeling technique for studying porous media , 1984 .

[2]  S. Patz,et al.  Probing porous media with gas diffusion NMR. , 1999, Physical review letters.

[3]  M. Sahimi.  Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing , 1993 .

[4]  S. Torquato,et al.  Reconstructing random media. II. Three-dimensional media from two-dimensional cuts , 1998 .

[5]  J. Thovert,et al.  Grain reconstruction of porous media: application to a low-porosity Fontainebleau sandstone. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  W. V. Pinczewski,et al.  Velocity‐porosity relationships, 1: Accurate velocity model for clean consolidated sandstones , 2003 .

[7]  John W. Cahn,et al.  Phase Separation by Spinodal Decomposition in Isotropic Systems , 1965 .

[9]  Hilfer,et al.  Stochastic reconstruction of sandstones , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  S. Torquato,et al.  Reconstructing random media , 1998 .

[11]  Thomas Bohlen,et al.  Paralel 3-D viscoelastic finite difference seismic modelling , 2002 .

[12]  Roberts,et al.  Structure-property correlations in model composite materials. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Berk Nf,et al.  Scattering properties of a model bicontinuous structure with a well defined length scale. , 1987 .

[14]  J. Garat,et al.  A Petrophysical Interpretation Using the Velocities of P and S Waves (Full-Waveform Sonic) , 1990 .

[15]  Berk Nf,et al.  Scattering properties of the leveled-wave model of random morphologies. , 1991 .

[16]  K. Binder,et al.  A Guide to Monte Carlo Simulations in Statistical Physics: Preface , 2005 .

[17]  M. Sahimi,et al.  Self-affine fractal distributions of the bulk density, elastic moduli, and seismic wave velocities of rock. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[19]  M. Sahimi,et al.  Renormalization group analysis and numerical simulation of propagation and localization of acoustic waves in heterogeneous media , 2007 .

[20]  Jack K. Cohen,et al.  Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion , 2001 .

[21]  M. Sahimi,et al.  Generation of long-range correlations in large systems as an optimization problem. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Clayton V. Deutsch,et al.  Practical considerations in the application of simulated annealing to stochastic simulation , 1994 .

[23]  M. Sahimi,et al.  Shape of a wave front in a heterogenous medium. , 2006, Physical review letters.

[24]  Muhammad Sahimi,et al.  Development of optimal models of porous media by combining static and dynamic data: the porosity distribution. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.