Analytical Predictions and Lattice Boltzmann Simulations of Intrinsic Permeability for Mass Fractal Porous Media

AU›o®ƒa®EAÝ : LBM, lattice Boltzmann method; MPA, Marshall’s probabilistic approach; PCC, probabilistic capillary connectivity. SO›‘®ƒ½ S›‘a®EA : FUƒ‘aƒ½Ý We derived two new expressions for the intrinsic permeability (k) of fractal porous media. The fi rst approach, the probabilisƟ c capillary connecƟ vity (PCC) model, is based on evaluaƟ ng the expected value of the cross-secƟ onal area of pores connected along various fl ow paths in the direcƟ on in which the permeability is sought. The other model is a modifi ed version of Marshall’s probabilisƟ c approach (MPA) applied to random cross matching of pores present on two parallel slices through a fractal porous medium. The Menger sponge is a three-dimensional mass fractal that represents the complicated pore space geometry of soil and rock. PredicƟ ons based on the analyƟ cal models were compared with esƟ mates of k derived from laƫ ce Boltzmann method (LBM) simulaƟ ons of saturated fl ow in virtual representaƟ ons of Menger sponges. Overall, the analyƟ cally predicted k values matched the k values from the LBM simulaƟ ons with <14% error for the determinisƟ c sponges simulated. While the PCC model can represent variaƟ on in permeability due to the randomizaƟ on process for each realizaƟ on of the sponge, the MPA approach can capture only the average permeability resulƟ ng from all possible random realizaƟ ons. TheoreƟ cal and empirical analyses of the surface fractal dimension (D 2 ) for successive slices through a random Menger sponge show that the mean D 2 value 〈D 2 〉 = D 3 − 1, where D 3 is the threedimensional mass fractal dimension. IncorporaƟ ng 〈D 2 〉 into the MPA approach resulted in a k that compared favorably with the modal value of k from LBM simulaƟ ons performed on 100 random realizaƟ ons of a random Menger sponge.

[1]  Haibo Huang,et al.  Proposed approximation for contact angles in Shan-and-Chen-type multicomponent multiphase lattice Boltzmann models. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Fractal Porous Media , 1993 .

[3]  J. L. McCauley,et al.  Implication of fractal geometry for fluid flow properties of sedimentary rocks , 1992 .

[4]  Tortuosity factor for permeant flow through a fractal solid , 2000 .

[5]  K. Tsujii,et al.  Menger sponge-like fractal body created by a novel template method. , 2006, The Journal of chemical physics.

[6]  D. Wolf-Gladrow.  Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction , 2000 .

[7]  J. Quirk,et al.  Permeability of porous solids , 1961 .

[8]  M. Takeda,et al.  Strong localization of microwave in photonic fractals with Menger-sponge structure , 2006 .

[9]  T. J. Marshall.  A RELATION BETWEEN PERMEABILITY AND SIZE DISTRIBUTION OF PORES , 1958 .

[10]  M. Borkovec,et al.  ON PARTICLE-SIZE DISTRIBUTIONS IN SOILS , 1993 .

[11]  Cass T. Miller,et al.  An evaluation of lattice Boltzmann schemes for porous medium flow simulation , 2006 .

[12]  Henry Lin,et al.  A comparison of fractal analytical methods on 2- and 3-dimensional computed tomographic scans of soil aggregates , 2006 .

[13]  W. Rawls,et al.  Comment on''Fractal processes in soil water retention''by Scott W , 1992 .

[14]  Walter J. Rawls,et al.  Fractal models for predicting soil hydraulic properties: a review , 1997 .

[15]  C. Jacquin,et al.  Fractal porous media II: Geometry of porous geological structures , 1987 .

[16]  E. Perfect,et al.  Water retention of prefractal porous media generated with the homogeneous and heterogeneous algorithms , 2001 .

[17]  Boming Yu,et al.  Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry , 2008 .

[18]  Michael C. Sukop,et al.  Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers , 2005 .

[19]  D. Hillel.  Environmental soil physics , 1998 .

[20]  Wei Liu,et al.  Fractal Analysis of Permeabilities for Porous Media , 2004 .

[21]  R. R. Filgueira,et al.  Particle-size distribution in soils: A critical study of the fractal model validation , 2006 .

[22]  Boming Yu,et al.  Permeability of the fractal disk-shaped branched network with tortuosity effect , 2006 .

[23]  Yanjun Liu,et al.  Analysis of permeability for the fractal-like tree network by parallel and series models , 2006 .

[24]  John S. Tyner,et al.  Water Retention Models for Scale‐Variant and Scale‐Invariant Drainage of Mass Prefractal Porous Media , 2007 .

[25]  Allen G. Hunt,et al.  Applications of percolation theory to porous media with distributed local conductances , 2001 .

[26]  W. Rawls,et al.  Predicting Saturated Hydraulic Conductivity Utilizing Fractal Principles , 1993 .