## 1. Introduction

[2] Hydraulic conductivity (*K*) and the more fundamental intrinsic permeability (*k*) are critical hydraulic properties of porous aquifer materials and exhibit an extremely large range of values in natural and man-made materials. *Freeze and Cherry* [1979] suggested a range in *K* values of 13 orders of magnitude between unfractured rock (*K* = 10^{−13} m s^{−1}) and gravel and an uppermost limit of approximately *K* = 10^{−1} m s^{−1} for karst limestone. More common units for intrinsic permeability in the oil industry are millidarcys (md) with 1 md = 10^{−15} m^{2}. Typical petroleum reservoirs have a range of 1–1000 md [*Chen*, 2007]. These parameters are defined in terms of Darcy's law, which assumes a linear increase in flow rate with applied hydraulic gradient. The linear assumption in Darcy's law is usually valid when the Reynolds number (*Re*) based on mean grain size and specific discharge “…does not exceed some value between 1 and 10” [*Bear*, 1972]. However, if the Reynolds number is larger, energy is dissipated through inertial aspects of the flow and increases in flow are no longer exactly proportional to increases in gradient.

[3] Hydraulic conductivity and intrinsic permeability are routinely measured in the laboratory or in the field with aquifer tests; however, in exceptionally permeable materials such as macroporous limestone, these measurements can be complicated to make and challenging to interpret. *Ferreira et al*. [2010] discussed some of the difficulties. As described in section 4.5, conventional petrophysical core laboratory analyses are limited to relatively low permeabilities characteristic of petroleum reservoirs.

[4] Numerical computations of hydraulic conductivity and fluid flow in pore systems in computer renderings generated from X-ray computed tomography (CT) scan data provide a convenient and reliable alternative to laboratory measurements [*Koplik and Lasseter*, 1985; *Rothman* 1988; *Succi et al*., 1989; *Chen et al*., 1991; *Schwartz et al*., 1993; *Chen and Doolen*, 1998; *Lin and Miller*, 2000, 2004; *Li et al*., 2005; *Kutay et al*., 2006; *Pan et al*., 2006a, 2006b; *Cunningham et al*., 2009, 2010]. By computing the detailed flows inside the pores under a particular head gradient, the intrinsic permeability can be determined. However, when conventional computational fluid dynamics methods are used to predict hydraulic conductivity, e.g., finite difference [*Schwartz et al*., 1993] or finite control volume schemes [*Teruel and Rizwan-Uddin*, 2009], they have been often limited to small sample volume, simple physics, and simple geometry [*Chen and Doolen*, 1998].

[5] Unlike traditional computational fluid dynamics methods, the lattice Boltzmann method (LBM) is based on the microscopic kinetic equation for gasses (Boltzmann equation) expressed as a particle distribution function, and the macroscopic pressure and particle velocity can be obtained from the distribution function [*Succi*, 2001; *Wolf-Gladrow*, 2000; *Sukop and Thorne*, 2006]. This kinetic basis provides LBM some valuable advantages, particularly for porous media. LBM is relatively easy to program, and its explicit, noniterative nature makes it easy to parallelize [*Chen and Doolen*, 1998]. Unlike with traditional Darcy's law-based groundwater models, flows with inertia can be simulated. Finally, for porous media, the most attractive feature of the LBM is that the complex spatial geometry of a pore system can be input readily and an easy-to-implement, no-slip bounce-back boundary condition can handle complicated wall geometries.

[6] The LBM has been successfully applied to the study of many complex fluid flow phenomena including flow through porous media [*Chen and Doolen*, 1998; *Chen et al*., 1991; *Pan et al*., 2006b; *Kutay et al*., 2006; *Li et al*., 2005; *Lin and Miller*, 2000, 2004; *Rothman*, 1988; *Succi et al*., 1989]. *Heijs and Lowe* [1995] studied the intrinsic permeabilities (*k*) of three randomly packed arrays of uniform spheres as a function of porosity ( ). The values of *k* calculated using LBM on simulated random sphere packs agreed well with those calculated using the semiempirical Kozeny-Carman equation, , where *c* is a shape factor. In another work, flows through computer renderings of sandstone created from X-ray microtomography scan images were simulated by LBM to determine intrinsic permeability; intrinsic permeability calculated by LBM agreed well with permeameter measurements within experimental uncertainty [*Buckles et al*., 1994; *Soll et al*., 1994].

[7] Previous applications of LBM to porous media have been limited to very small sample sizes due to computer resource and pore geometry data constraints. In addition, most research has focused on validation of Darcy's law, although *Pan et al*. [2006a], *Jeong et al*. [2006], *Chukwudozie* [2011], *Hilpert* [2011], and *Newman and Yin* [2011] are exceptions. *Pan et al*. [2006a] considered non-Darcy flow in two computer-generated nonoverlapping sphere packs that follow a truncated log-normal size distribution with 0.2 mm mean and porosities of 0.33 and 0.45. The simulated domain sizes (128 × 128 × 128 voxels) were just two or three mean sphere diameters on a side and accounted for a cube of physical space approximately 0.5 mm on a side. They arrived at Forchheimer *β* values (described below) of approximately 10^{5} m^{−1}. *Jeong et al*. [2006] studied non-Darcy flow in uniformly sized and distributed circular and square cylinders in two-dimension (2-D) and in uniformly sized and distributed spheres and cubes in two dimensions (3-D). The symmetry of these allowed the use of periodic boundary conditions. They also considered simulated media composed of randomly oriented, overlapping, uniform circular fibers. In addition to simulating non-Darcy flow in computer-generated sphere packs, *Chukwudozie* [2011] used CT data for Castlegate sandstone with 0.15–0.18 mm grains and 18% porosity imaged at 7.57 µm resolution. At 300 × 300 × 300 voxels, a cube representing 2.271 mm on a side was simulated. The Forchheimer *β* was found to be between 3 × 10^{7} and 10^{8} m^{−1}, in close agreement with experiment. *Hilpert* [2011] based a computer-generated nonoverlapping sphere packing at three resolutions (as low as 7.2 µm) on the size distribution and porosity of a real sand. *Hilpert* [2011] used the sphere packing to simulate non-Darcy flow in a 1.445-mm cube at Reynolds number up to approximately 10 and observed a reduction in the apparent permeability that was attributed to inertial effects. Finally, *Newman and Yin* [2011] worked strictly with synthetic media in 2-D. On the basis of simulations, they argue that large contrasts between the size of pore bodies and pore necks are responsible for the early onset of non-Darcy behavior.

[8] Here, the simulated flows in computer renderings of macroporous limestone samples (0.1 m scale) that were collected from the karstic Biscayne aquifer in southeastern Florida [*Cunningham et al*., 2009, 2010] are used to explore potential limitations of Darcy's law. The convergence of intrinsic permeability as a function of sample resolution was also examined to provide confidence in the results for these aquifer samples.

[9] In this paper, first we discuss Darcy's law and its limitations and then briefly present information on the collected macroporous limestone samples and their scanning via X-ray CT. The processing of the scans is described next and is followed by a demonstration of our LBM code's validity for higher Reynolds number flow in a sinusoidal pipe, which has exact analytical solutions at low Reynolds number and experimental measurement results at higher Reynolds number. Then we discuss the use of the 3-D, parallel LBM code we developed to simulate flows in porous medium structures obtained from CT measurements and consider anisotropy, non-Darcy flow, sample resolution, and LBM's role in extending laboratory measurements.