Evaluation of permeability and non-Darcy flow in vuggy macroporous limestone aquifer samples with lattice Boltzmann methods

Authors


Corresponding author: M. C. Sukop, Florida International University, Miami, FL 33199, USA. (sukopm@fiu.edu)

Abstract

[1] Lattice Boltzmann flow simulations provide a physics-based means of estimating intrinsic permeability from pore structure and accounting for inertial flow that leads to departures from Darcy's law. Simulations were used to compute intrinsic permeability where standard measurement methods may fail and to provide better understanding of departures from Darcy's law under field conditions. Simulations also investigated resolution issues. Computed tomography (CT) images were acquired at 0.8 mm interscan spacing for seven samples characterized by centimeter-scale biogenic vuggy macroporosity from the extremely transmissive sole-source carbonate karst Biscayne aquifer in southeastern Florida. Samples were as large as 0.3 m in length; 7–9 cm-scale-length subsamples were used for lattice Boltzmann computations. Macroporosity of the subsamples was as high as 81%. Matrix porosity was ignored in the simulations. Non-Darcy behavior led to a twofold reduction in apparent hydraulic conductivity as an applied hydraulic gradient increased to levels observed at regional scale within the Biscayne aquifer; larger reductions are expected under higher gradients near wells and canals. Thus, inertial flows and departures from Darcy's law may occur under field conditions. Changes in apparent hydraulic conductivity with changes in head gradient computed with the lattice Boltzmann model closely fit the Darcy-Forchheimer equation allowing estimation of the Forchheimer parameter. CT-scan resolution appeared adequate to capture intrinsic permeability; however, departures from Darcy behavior were less detectable as resolution coarsened.

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 m2. 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 ( math formula). The values of k calculated using LBM on simulated random sphere packs agreed well with those calculated using the semiempirical Kozeny-Carman equation, math formula, 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 105 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 × 107 and 108 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.

2. Darcy's Law With Respect to Reynolds Number

[10] The hydraulic conductivity of a sample is defined in terms of Darcy's law, which states that the volume discharge rate of flow per unit area, q (specific discharge vector), is proportional to the head gradient and can be written as

display math(1)

where math formula is the hydraulic conductivity tensor, and the minus sign indicates that flow is in the direction opposite to the head gradient. In a system where there are no elevation head differences (e.g., a horizontal column), the head (h) can be expressed in terms of the pressure alone via h = P/ρg, and if the flow is driven by pressure difference (ΔP = PinPout) across a sample of length L, it will be macroscopically one dimensional, and the specific discharge (q) (=|q|) can be written as follows:

display math(2)

where g is gravity, ρ is the density of water, and K is the scalar horizontal hydraulic conductivity. Finally, we can replace the hydraulic conductivity with the more fundamental, porous medium-specific intrinsic permeability (k) using the following equation:

display math(3)

where μ is the dynamic viscosity of water, leading to

display math(4)

[11] The validity of Darcy's law has a limited range, because when the Re, based on the specific discharge and average grain diameter, increases beyond approximately 1, the linear relation assumed by Darcy's law begins to fail [Todd, 1959; Bear, 1972; Freeze and Cherry, 1979; Ford and Williams, 1989; Barree and Conway, 2004]. The Reynolds number denotes the relative importance of inertia compared with viscous drag in a flowing fluid and is a measure of a flow's propensity to form eddies and turbulence. For very low Re, creeping flow is present. As the Re increases, inertial forces begin to play a role and eventually dominate. If Re increases even further, the flow transitions into turbulent flow. Intrinsic permeability and hydraulic conductivity are presumed to represent fixed properties of porous media that accurately fit Darcy's law, i.e., where the flow is linearly proportional to the head or pressure gradient, as in equation (1). This no longer holds true where inertial effects become evident. Here, we compute absolute intrinsic permeabilities at very low Re (≪ 1) and refer to intrinsic permeability computed at higher Re as “apparent” intrinsic permeability and corresponding “apparent” hydraulic conductivity.

[12] The Re is dimensionless, and hence whether computed in real or consistent LBM units (defined below), it will have the same value for hydrodynamically similar flows. Based on the specific discharge (q), and using the mean grain diameter (d) as the characteristic length, one possible macroscopic Re is given by [Bear, 1972, p. 125]

display math(5)

where math formula is the kinematic viscosity, which is equal to the dynamic viscosity divided by the fluid density. Identifying the maximum pore-scale Re is more involved, although it would be more appropriate for purposes of identifying non-Darcy flow in porous media that are not granular. The standard “pipe” Re is given as follows:

display math(6)

where u is the flow velocity and Lp is the pipe diameter. It turns out that for monodisperse porous media, the standard porous media Re based on grain size and Darcy flux (5) is approximately equivalent to the pore-scale pipe Re (6). Given that the pore size is approximately half of the grain size [Adkins and Davis, 1986], we have Lpd/2. For an average approximate porosity of n = 0.5, the mean pore water velocity is math formula = q/n ≈ 2q. Combining these observations, Re = u × Lp/ν ≈ (2q d/2)/ν = q × d/ν. Thus, under the assumptions of monodisperse grain size distribution and 50% porosity, equations (5) and (6) are equivalent, which suggests that the pore-scale pipe Re is still actually the relevant one for granular media and that the 1-to-10 Re range proposed by Bear [1972] for departures from Darcy's law therefore remains relevant to nongranular media.

[13] With the availability of a complete CT-based virtual model of the macroporous limestones considered in this study and the complete knowledge of the simulated flow fields from the lattice Boltzmann model, the pore sizes and velocities are known. However, such data are not routinely available, and determining the proper length and velocity to apply in calculating the Re for such porous materials remains problematic, although many authors recommend using math formula as the length scale. Fortunately, for purposes of scaling lattice Boltzmann simulations via the Re, we can use any length (e.g., the CT voxel size) to ensure that we have hydrodynamic similarity between the experiment and the model, although the absolute magnitude of the Re would not be comparable with other Re such as Bear's [1972] 1-to-10 rule in that case. Instead of Re, Zeng and Grigg [2006] argued for the use of a “Forchheimer number” involving the Forchheimer coefficient discussed below.

3. Methods

3.1. Samples and CT Scanning

[14] Samples representative of biogenic vuggy macroporosity in the Biscayne aquifer were collected from the Fort Thompson Formation, the Miami Limestone, and the Holocene limestone by the U.S. Geological Survey (Figure 1). Centimeter-scale biogenic vuggy macroporosity (readily visible without magnification) in the limestone of the Biscayne aquifer typically manifests as one or more of the following: interburrow and intraburrow macroporosity, interroot and intraroot macroporosity, and fossil-moldic macroporosity (Cunningham et al., 2009, 2010). Figures 2 and 3 show the seven samples selected to represent the most characteristic biogenic vuggy macropore types found in the karstic limestone of the Biscayne aquifer.

Figure 1.

Sample location maps in Florida. (a) Location of sample area in southeastern Florida. (b) Samples submitted for CT analysis and subsequently simulated with LBM for intrinsic permeability estimation are indicated.

Figure 2.

(a–c) Limestone samples and macropore types for which high-resolution X-ray CT scans were performed. (d–f) Computer renderings of high-resolution X-ray CT scans that were conducted on samples in Figures 2a–2c. (g–i) Computer renderings of subsamples for which lattice Boltzmann-computed intrinsic permeability were performed. Subsamples in Figures 2g and 2i are 9.01 cm on a side, whereas subsample in Figure 2h is 9.12 cm. Macroporosity and hydraulic conductivity of subsamples in Figures 2g–2i are shown. In situ orientation of ML-01 is unknown. Figure modified from Cunningham et al. [2009].

Figure 3.

(a–d) Limestone samples and macropore types for which high-resolution X-ray CT scans were performed. (e–h) Computer renderings of high-resolution X-ray CT scans that were conducted on samples in Figures 3a–3d. (i–l) Computer renderings of subsamples for which lattice Boltzmann-computed intrinsic permeability were performed. Subsamples in Figures 3i and 3j are 7.95 cm on a side, whereas subsamples in Figures 3k and 3l are 6.83 cm. Macroporosity and hydraulic conductivity of subsamples in Figures 3i–3l are shown. Figure modified from Cunningham et al. [2009].

[15] Macroporous limestone samples were imaged at the University of Texas High-Resolution X-ray CT facility, and individual 8-bit slice images were combined to obtain a 3-D binary file, which is used as the LBM input file. Cunningham et al. [2009, 2010] presented details on the nature of the samples and the imaging.

3.2. CT Data Processing

[16] Here, we discuss the preparation of the data and calculations for macroporous limestone sample ML-1, which is an example of the burrow porosity type (Figure 2b), in some detail. The other samples were treated similarly but were generally of different overall size.

[17] Sample ML-1 was scanned into 401 slices, each being 1024 × 1024 pixels. The images produced from the scan had a resolution of 0.271 mm in length per pixel in the x and y directions. We used a direct correspondence between pixels and lattice length units. Therefore, this 0.271 mm/pixel ratio is used as the scaling relation for determining physical values of permeability from permeabilities produced by the model in lattice units, as described below. As described in Cunningham et al. [2009], the z direction spacing of the slices was 0.8 mm, and therefore, interpolation between the slices was necessary to obtain proportional scaling in all three space dimensions. A simple approach was adopted wherein each slice was repeated three times to create the LBM input file; the z spacing was then 0.267 mm/pixel, which is close to the 0.271 mm/pixel in the x and y directions. Such an interpolation is appropriate because CT data do not represent a zero-thickness planar slice but rather are an integration over a thickness of rock (1 mm in this case); therefore, it is appropriate to assign the single slice to the entire 0.8 mm interslice thickness. Moreover, to alleviate concerns about this interpolation, we obtained a new set of uniformly spaced slices from the High-Resolution X-ray Computed Tomography Facility at the University of Texas at Austin, which collected the CT data, and reran some of the simulations. No difference in the average flow velocities, which are used to compute the permeabilities, was observed between simulation domains based on the original interpolation scheme or the new uniform slices.

[18] In our LBM simulations, a volume of 336 × 336 × 336 voxels (0.09 m × 0.09 m × 0.09 m) that unambiguously lies inside the irregular boundaries of the scanned rock is created and used for intrinsic permeability calculations. The CT images were in jpeg format with grayscale values ranging from 0 to 255. Figure 4 shows the histogram of voxel grayscale values for the entire volume. A threshold grayscale value of 75 was chosen that would assign the color black to the areas that were macropores and white to rock matrix. More details on thresholding can be found in Cunningham et al. [2009]; however, we point out here that there was little ambiguity in the assignment of the large macropore and rock matrix voxels at the CT data resolution for these rocks. The inset of Figure 4 shows a well-defined minimum, and calculations show that the selection of thresholds between grayscale 64 and 80 lead to only small changes in the porosity from 61.5% to 64%. Even considering a much broader (and less likely) range of potential thresholds from 48 to 96 only corresponds to a possible porosity change from 59% to 68%. Kaestner et al. [2008] pointed out that simple thresholding as we apply here truncates the overlapping tails of the pore and solid class distributions resulting in some incorrectly classified voxels. The steep dropoff of the pore and solid voxel counts shown on the inset of Figure 4 and the small number of voxels that lie between the peaks suggest that the number of incorrectly classified voxels should be small in this case.

Figure 4.

Histogram of voxel grayscale levels for sample ML-1. Vertical lines at grayscale 75 indicate selected threshold value. Inset shows expanded vertical axis scale.

3.3. Lattice Boltzmann Computations

[19] Detailed introductions to LBM can be found in Sukop and Thorne [2006], Succi [2001], and Wolf-Gladrow [2000]. A brief summary of 3-D LBM that uses a single relaxation time (also referred to as the “BGK” model after Bhatnagar et al. [1954]) is provided in Sukop et al. [2008] and many other sources. Single-relaxation-time LBMs are subject to limitations [Pan et al., 2006a, 2006b; Chukwudozie, 2011], but are adequate for the determination of intrinsic permeability of the macroporous media considered here, as we demonstrate. Similarly, the simplest no-slip wall boundary condition—the “bounceback” we use here—also has limitations. More complex boundary conditions, such as multireflection [Pan et al., 2006b; Ginzburg and d'Humières, 2003] or the link-averaged one-point approach [Junk and Yang, 2005; Maier and Bernard, 2010], generally improve grid convergence. We include a grid convergence evaluation of one sample that indicates an approach to a constant intrinsic permeability value. We also include a simulation that demonstrates excellent agreement between our simulations and experimental data for flow at higher Re in a sinusoidal pipe.

[20] Our parallel LBM code and computational resources available when this work was completed made it possible to run a sample as large as 8733 voxels. This code was run on the MAIDROC cluster “TESLA-128” located at the Florida International University's Engineering Center. The code is scalable, and thus, there are no inherent limits to the size of domain in which flows can be simulated, although the scalability has not been tested in detail. In general, we use a set of internally consistent LBM units we designate as lattice length units (lu), lattice time steps (ts), and lattice mass units (mu) in our simulations. Conversion to and from physical units is discussed below.

[21] Several analytical solutions for simple 3-D flow systems are available and allow partial verification of our code. Verifications for flow in an axisymmetric pipe and for flow in a square duct can be found in Alvarez [2007]. These verifications of the LBM code applied here show that it is adequate for the purpose of intrinsic permeability determination. However, more stringent tests are needed to demonstrate the code's effectiveness at higher Re when eddies are present. Recently, Llewellin [2010] presented solutions for flows in a rotationally-symmetric sinusoidal pipe over a range of Re. This is a problem that has an analytical solution (like the straight pipes) at low Re and has experimental measurement results for model comparison at higher Re. Figure 5 shows the low-Re analytical solution, the experimental data, the recent 3-D lattice Boltzmann simulations of Llewellin [2010], and our current 3-D simulations. Llewellin overpredicted the product of the Reynolds number and the friction factor over the entire range of Re. Our own simulations show good agreement with the analytical and experimental results for this domain. This confirms the capabilities of the current code for higher Re flows in domains more complex than straight pipes and ducts.

Figure 5.

Experimental data of Deiber et al. [1992], low-Re analytical solution of Sisavath et al. [2001], simulations of Llewellin [2010], and current simulations for sinusoidal pipe (inset) “Dieber1” of Llewellin [2010]. Figure modified from Llewellin [2010]. Inset shows half of 3-D rotationally symmetric sinusoidal pipe.

[22] In Cunningham et al. [2009], we give a detailed example that illustrates the use of LBM to obtain the intrinsic permeability of the macroporous limestone samples. In brief, the flux through a sample under a small, known pressure gradient is determined from the simulation in internally consistent LBM units and used to determine the intrinsic permeability k values in LBM units from a rearrangement of equation (2). This can be converted to physical units via

display math(7)

where the Lphysical and LLBM are the lengths of any identical feature in the physical sample and the LBM domain in their respective units, for example, 0.271 mm = 1 lu based on the physical size of one voxel. Finally, the hydraulic conductivity can be obtained from equation (3).

[23] Darcy's law typically applies to any noninertial flow with a Reynolds number less than approximately 1. When Re increases, some deviation from Darcy's law can be observed. At higher Re values, when inertial bending of streamlines or eddies are present, apparent intrinsic permeability is expected to be smaller than the true intrinsic permeability because head is being dissipated through inertial processes, which results in flow rates less than the product of the true intrinsic permeability and the head gradient.

[24] To compare an easier-to-compute macroscopic Re in the form of equation (6), with u as the mean pore water velocity and L as a characteristic burrow diameter, with an explicit pore-scale Re, we identified the locations of both the largest pore and of the highest velocity in the simulation domain and computed the local Re for each. The largest velocity in the case of sample ML-1 was u = 0.0016 lu ts−1, which occurred in a pore diameter of approximately 32 lu. The LBM kinematic viscosity is given by math formula, where math formula is the sound speed squared (1/3 lu2 ts−2), τ is the relaxation time, and δt is the time step length, which is always 1 here. Hence, the pore-scale Re = u L/ ν = 0.0016 lu ts−1 × 32 lu/[(1/3 lu ts−1) (1 – 1/2) ts] = 0.31 as calculated using the greatest velocity. The largest pore in sample ML-1 is on slice z = 285 and its diameter is about 120 lu. The average velocity in the pore is 3.5 × 10−4 lu ts−1. Hence, the Re = 0.00035 lu ts−1 × 120 lu/[(1/3 lu ts−1) (1 – 1/2) ts] = 0.25 as calculated using the largest pore size.

[25] Although these estimates of Re correspond to the traditional pipe Reynolds number, they could not be computed without detailed knowledge of the pore geometry and intrapore velocities. However, an Re can be calculated using the mean pore water velocity math formula and an average burrow diameter (chosen as 60 lu for ML-1) that would be available from macroscopic observations. In this case, it is given as follows:

display math(8)

[26] The two detailed pore-scale Re estimates are larger than the macroscopic estimate, but agree adequately considering that we are most interested in order-of-magnitude changes in Re, and all three estimates suggest that non-Darcy inertial effects should be negligible under these circumstances. We apply the macroscopic Re in the remainder of the paper.

3.4. Non-Darcy Flow

[27] To investigate the non-Darcy flow behavior in the macroporous Biscayne aquifer limestone subsamples, we performed 10 simulations of flow at different gradients and viscosities in the ML-1 3363 lu3 cube subsample. Simulation parameters and calculated quantities are listed in Table 1. In Table 1, the first column values, Δh/L, are the imposed physical head gradients. The second column values, math formula, are the density differences corresponding—through the LBM equation of state, P = ρ/3—to the pressure difference we applied across the simulation domains. The conversion of the physical head gradient to LBM pressure difference is as follows: because math formula is a nondimensional variable, it is expected that this variable is the same in the real physical world and the LBM model. In addition, we can rearrange the dimensionless Re as math formula. In the LBM simulation, we assume a horizontal flow system without elevation potential differences where the fluid is driven by pressure gradients only so that the head difference across the domain that is responsible for the gradient can be expressed purely as a pressure difference via ΔP = ρg × Δh. Then, math formula. We use the same Re in both the physical and LBM systems, and therefore, the Re values are equal and cancel. Replacing L2 with L3/L on both sides of the equation gives

display math(9)

where the physical hydraulic gradient appears on the left-hand side and the LBM pressure gradient on the right-hand side. Incorporating the LBM equation of state and the relationship between the relaxation time and the kinematic viscosity, we obtain the following equation:

display math(10)

which can readily be solved for the LBM simulation density gradient math formula required for the LBM simulations to recover hydrodynamic similarity with the physical system:

display math(11)
Table 1. Pressure Gradient Simulation Parameters and Results for ML-1 3363 lu3 sample
Simulation Numberh/L)physicalΔρ (mu lu−3)P/L)LBM (mu lu−2 ts−2)τ (ts)ū (lu/ts)Rek (lu2)K (m/s)
11.62E−098.858E−068.76E−0912.53E−060.000947.9238.70
21.00E−085.46E−055.42E−0811.56E−050.00647.9238.70
31.00E−075.46E−045.42E−0711.56E−040.0647.9338.70
41.00E−060.005465.42E−0611.56E−030.647.9138.05
51.00E−050.05465.42E−0510.0143544.0435.57
62.52E−050.005465.45E−060.60.00651040.1232.40
78.25E−050.01801.78E−050.60.01513028.1422.73
81.25E−040.027262.70E−050.60.01853022.8018.41
91.00E−078.72E−058.65E−080.76.13E−050.0647.1838.10
101.00E−072.18E−052.16E−080.62.97E−050.0545.6936.90

[28] Alternatively, one can solve for the physical head gradient Δh/Lphys

display math(12)

[29] Hilpert [2011] offered a related approach for dimensionalizing LBM simulations.

[30] The LBM pressure gradient in Table 1 can be obtained from the density difference through math formula. The fifth column math formula is the relaxation time parameter used in the various simulations. LBM simulation results produced math formula.

[31] The purpose of the high-Reynolds-number cases, 6–8 in Table 1, is to investigate potential non-Darcy behavior. It is important to consider the work of Pan et al. [2006b] in interpreting these model results. A principal conclusion of that work was that the single-relaxation time, single bounce-back model we apply here is unreliable for relaxation times τ ≠ 1 ts. This poses a challenge for the simulation of higher Reynolds number flows as we explain presently. Considering case 5 of Table 1 for example, the physical head gradient across the sample is only 10−5 and yet we find that the average velocity is somewhat large for the LBM technique as the average Mach number (the speed relative to the sound speed in the fluid) is about math formula. The highest velocity in the model has an even higher Mach number. To simulate the larger pressure gradient cases by keeping the τ value at 1.0 and increasing the math formula, the Mach number would be expected to be much larger, which would violate the incompressible flow condition (usually M < 0.3). In addition to that, the numerical simulations generally become unstable. To overcome these difficulties, it is possible and desirable to achieve higher Reynolds number flows by reducing the kinematic viscosity of the simulated fluid by changing τ: as math formula, we can make τ closer to 1/2 ts to achieve arbitrarily low kinematic viscosity. When the fluid kinematic viscosity is reduced (e.g., going from simulations 5 to 6 where τ changes from 1 to 0.6 ts), the velocity is actually lower under a higher physical head gradient and a higher Re can be reached. However, as reported by Pan et al. [2006b], results for a hypothetical glass bead medium [Pan et al., 2004] simulated with 323 lattice nodes showed that the apparent intrinsic permeability was reduced to less than 90% of its true value going from τ = 1 to 0.6 ts. Here, we explore how much τ values are likely to affect the intrinsic permeability computation in our LBM simulations.

[32] Cases 3, 9, and 10 in Table 1 test the dependence of intrinsic permeability on τ for our sample ML-1. A similar test was made for a Menger Sponge fractal [e.g., Cihan et al. 2009]. In all cases, the bulk Reynolds numbers are about 0.3, which satisfies Darcy's law applicability. Figure 6 demonstrates that the τ value has a minimal effect on intrinsic permeability when our model is applied to the ML-1 (3363) sample and an 813 lu3 Menger Sponge; at τ = 0.6, the maximum reduction in intrinsic permeability is 5%. This reduction is less severe than demonstrated in the work of Pan et al. [2006b], in which a glass bead medium with uniform, smaller pores was simulated. τ can be viewed as affecting the exact position of the solid/pore boundaries “sensed” by the fluid. It is likely that media where the permeability is dominated by a few large well-connected macropores simulated by many lattice nodes, such as our simulations of Biscayne aquifer samples, would be less affected by τ than simulations in poorly resolved glass bead media. In Pan et al. [2006b], for example, pore throats were resolved with 2.8 lattice nodes. Our excellent match with experimental data from the sinusoidal pipe at τ = 0.55 ts (Figure 5) is also consistent with this hypothesis. The test results for different τ applied to the ML-1 sample in Figure 6 are presented along with the non-Darcy flow results in section 4.3 below to further support the appropriateness of the smaller τ in the current work.

Figure 6.

Viscosity dependence of computed relative intrinsic permeability using single relaxation time model. Relative intrinsic permeability k/k0 given as a function of relaxation time τ (ts) and kinematic viscosity ν (lu2 ts−1). Black squares show results obtained from flow through 3363 lu3 cubic subsample from macroporous limestone sample ML-1. Red squares show results obtained from flow through 813 lu3 Menger sponge. In all simulations, Re ∼ 0.30 to ensure validity of Darcy's law.

4. Results and Discussion

4.1. Intrinsic Permeability of Seven Macroporous Limestone Samples

[33] The methods described above were applied to each of the seven samples for which CT scan data are available. Largest sample diameter was 0.3 m and macroporosity was as high as 81%. Individual macropores for whole samples can be up to 0.1 m in diameter. In Figures 2 and 3, the images in the top row are photographs of the aquifer samples, images in the middle and bottom rows are computer renderings of whole samples and subsamples, respectively. LBM computations were completed on the subsample renderings. The volume of the subsample renderings are 3363 lu3 (759 cm3) for sample ML-01 and 3183 lu3 (from 319 to 731 cm3) for the other six samples. The macroporosities, which were determined by voxel counting after thresholding, and the LBM-calculated hydraulic conductivities of all seven samples considering only macropores are also shown in Figures 2 and 3.

[34] The intrinsic permeabilities of six of the macroporous limestone samples as a function of macroporosity are illustrated in Figure 7. Matrix porosity is expected to make only a very small contribution to intrinsic permeability and was not considered in the permeability calculations. The intrinsic permeabilities agree relatively well with a fitted Kozeny-Carman curve (following Bear [1972, p. 166], where we lump the shape factors into a single empirical constant) math formula (with R2 = 0.98), indicating the overriding importance of macroporosity in determining a sample's permeability.

Figure 7.

Intrinsic permeability versus macroporosity for six of the seven macroporous limestone aquifer samples with fitted Kozeny-Carman curve. With a poorly interconnected macroporosity of 0.16, sample FPL-Q3a-1 did not percolate and hence had zero intrinsic permeability.

4.2. Anisotropy

[35] The in situ field orientations of all macroporous limestone samples other than ML-1 were known. The hydraulic conductivities of the subsamples were computed in the vertical and horizontal directions and are listed in Table 2. The results indicate relatively minor hydraulic conductivity differences with direction (always less than a factor of 2), and there was no clear dependence of the magnitude of the conductivity on direction. There was also no clear trend in anisotropy with sample macroporosity. Outcrop-scale samples would exhibit vertical, strata-related variations in lithology and would be expected to show higher hydraulic conductivity in the horizontal directions. Given that these subsamples display no strong anisotropy attests to the random spatial distribution of each subsample's macropore network at the current LBM simulation scale; this is consistent with expectations from visual inspection of the samples.

Table 2. Horizontal and Vertical Hydraulic Conductivitiesa
Sample NumberKBM-1C100-Q5e-1G-3837-22G-3837-18FPL-Q3a-2FPL-Q3a-1
  1. a

    Macroporosity decreases from left to right.

Horizontal hydraulic conductivities (m/s)140132.20.150.390
Vertical hydraulic conductivities (m/s)170201.60.300.300
Horizontal/vertical hydraulic conductivity ratio0.820.581.30.501.3Na

4.3. Non-Darcy Flow Results

[36] Figure 8 shows the apparent hydraulic conductivity results for a range of applied hydraulic gradients and corresponding Reynolds numbers. Figure 8 also shows the small effect of τ on K at small hydraulic gradient (10−7) and low Re in comparison with the much larger effect of higher gradient and Re. These results suggest that it is appropriate to conclude that the use of smaller τ to achieve higher Re flows (section 3.4; cases 6–8 in Table 1) is reasonable in our media and that apparent hydraulic conductivity reductions on the order of 50% with increasing gradient (Figure 8) are mainly an effect of non-Darcy behavior and not the τ value.

Figure 8.

Effect of increasing gradient on apparent hydraulic conductivity for 336 lu cube (ML-1 sample) and fitted Darcy-Forchheimer equation. τ values for different simulations are indicated. Three simulations at dh/dx = 10−7 and different τ show small effect of τ relative to large effect of gradient between dh/dx = 10−5 and 10−4. The smallest K value shown on the extrapolated Darcy-Forchheimer equation curve at a gradient of 10−1 is approximately 1 m/s, which is still a high value by most standards.

[37] Based on the analysis of maps in Parker et al. [1955], we compute regional hydraulic gradients for the Biscayne aquifer on the order of 10−5 and 10−4 depending on location, season, and stage. Higher gradients are likely near canals, water control structures, and wells. The results in Figure 8 indicate a reduction in the apparent hydraulic conductivity as the applied gradient increased to realistic field gradients. This suggests that under field conditions in the Biscayne aquifer, inertial flows and departures from the linear gradient-flow relation assumed in Darcy's law are likely to be common and important. It is well known that slug and aquifer tests in the limestone of the Biscayne aquifer commonly exhibit inertial oscillations [DiFrenna, 2005; Renken et al., 2005]. This indicates that hydraulic conductivity and transmissivity measurements may often be reflective of inertial flow conditions and consequently may return smaller estimates of the hydraulic conductivity and transmissivity than they would if they represented the true intrinsic permeabilities. The observation that non-Darcy flow is likely to be occurring under normal field conditions argues for the use of models capable of incorporating the relevant physics in critical aquifer simulation efforts. See Shoemaker et al. [2008] for one proposed approach that captures the apparent intrinsic permeability reduction even though it does not simulate inertial flow.

[38] It is possible to use the results in Figure 8 to estimate the Darcy-Forchheimer parameter for this particular porous medium. Darcy's law can be written as follows:

display math(13)

[39] The Darcy-Forchheimer equation includes a nonlinear drag term as follows:

display math(14)

[40] A least squares fitting procedure was used to estimate the value of the β parameter as 100 m−1; the fitted equation is plotted along with the simulation results in Figure 8. The Darcy-Forchheimer equation can then be used for the prediction of the flow or apparent k within its range of applicability. The simulation results are well fitted by the Forchheimer equation considering that, because k is known from the low-gradient simulations and μ is considered fixed, the Forchheimer fit has only one free parameter. The value of the β parameter estimated from the simulation results can be considered in the context of an estimate obtained from the correlation proposed by Geertsma [1974],

display math(15)

where math formula is the porosity (0.64) and, for k = 3.51 × 10−6 m2, the β parameter estimated from the correlation is 31 m−1. This close agreement is despite the fact that the correlation was developed with just a few samples approaching 50% porosity and with no sample having math formula as low as obtained from the simulation. β parameter values for typically studied porous media seem to be greater than 5000 m−1 [Balhoff and Wheeler, 2009]. See Newman and Yin [2011] for evidence that large contrasts between the size of pore bodies and pore necks are responsible for the early onset of non-Darcy behavior and for an alternate correlation that incorporates a pore surface area-to-volume ratio.

[41] Here, we present simulation results at hydraulic gradients up to slightly more than 10−4. Sukop et al. [2008] included streamlines that illustrate differences in the nature of the flow at different gradients. Although gradients associated with the low topographic relief of the south Florida region are often considerably smaller than this, higher gradients are likely near canal water level control structures and close to production wells. It is likely that the decrease in the apparent conductivity would not continue indefinitely at the steep rate suggested by the simulation results between gradients of 10−5 and 10−4. Extrapolation of the fitted Darcy-Forchheimer equation predicts that as gradients and the Reynolds number increase further and flows within the aquifer assuming a more inertial character (fully turbulent), there will be a leveling off of the apparent hydraulic conductivity, at least on the linear hydraulic conductivity scale of Figure 8. Double logarithmic plots (not shown here) of the extrapolation show that the apparent K computed from the Darcy-Forchheimer equation continues to decline indefinitely by a factor of 10 for each factor of 100 increase in gradient. In addition, Barree and Conway [2004] showed that the Forchheimer β is itself a function of the Reynolds number and that the Darcy-Forchheimer equation is also inadequate to capture the full range of possible flows. Consequently, an alternative equation might be necessary for more reliable extrapolation [Barree and Conway, 2004]. These additional complexities have been attributed [Barree and Conway, 2004; Miskimins et al., 2005] to the phenomema of “streamlining,” where flow moves rapidly through larger pore spaces in a coherent jet rather than slowing and spreading across the pore, and a “participation factor” in which more of the flow moves through smaller pores as the overall flow rises.

4.4. Effect of Simulation Resolution on Apparent Hydraulic Conductivity

[42] Not surprisingly, the resolution of the virtual porous media used to create the LBM simulation domain can affect the calculated apparent hydraulic conductivity results. As the resolution degrades, fewer voxels define the details of the flow channel walls, and simulated flows will begin to deviate significantly from those that actually occur in the macropore system. It seems clear that the required resolution will be sample specific or at least rock fabric specific; porous media with highly regular channels may not require resolution as high as media with complex channels. Thus, it may not be possible to determine the necessary resolution a priori. To investigate the effects of resolution, we computed the apparent hydraulic conductivity at different resolutions and looked for convergence to a fixed value as the resolution increased. Figure 9 shows the ML-1 sample at different resolutions.

Figure 9.

Resampling of ML-1 at reduced resolution. Resolution decreases left to right from 0.271 to 1.084 mm lu−1. The 336 cube represents the highest available resolution of the macroporous limestone subsample.

[43] We conducted simulations at all three resolutions for different head gradients, and the results are plotted in Figure 10. The smaller change in the apparent hydraulic conductivity going from the 1683 to the 3363 lu3 simulation domains when compared with the change going from the 843 to 1683 lu3 domains indicates that the results tend to approach a true K (at least at low gradient values) as the resolution approaches that of the original CT-scan data. The other important conclusion from these results is that non-Darcy effects at higher gradients are more difficult to capture at lower resolution.

Figure 10.

Apparent hydraulic conductivities at three different resolutions as a function of gradient.

4.5. LBM as a High-Permeability Extension to Standard Petrophysical Laboratory Methods

[44] The intrinsic permeabilities computed with LBM for the Biscayne aquifer subsamples range from 10−8 to 10−5 m2 (107 to 1010 md), with the exception of FPL-Q3a-1, which did not have sufficient well-connected macroporosity to percolate. These intrinsic permeability values are 3 to 6 orders of magnitude higher than values a typical petrophysical core laboratory using air permeability measurement techniques routinely measures as a maximum and appear to be well beyond the capability of many standard petrophysical laboratory instruments used to measure intrinsic permeability. Laboratory and equipment manufacturer websites report maximum capabilities up to 3 × 104 md [Core Laboratories, 2011; Temco, 2011; Coretest Systems Inc., 2011, http://www.coretest.com/KA-210-Gas-Permeameter.html], which corresponds to SI unit permeability of 3 × 10−11 m2 and hydraulic conductivity of about 3 × 10−4 m s−1. The U.S. Army Corps of Engineers [2011] described the method and apparatus in general use and indicates a 104 md maximum. Personal communications with a laboratory manager suggest 3 × 103 md as a practical limit.

[45] We expect that these limits are primarily the result of air flow restrictions in the tubing and holes in the Hassler-type core holders that deliver air flow to the sample. The relevant ASTM standard (ASTM, 2001) indicates that these should be large enough “…to prevent pressure loss at maximum flow rate.” Every Hassler-type core holder has the same fundamental design: air delivered through narrow tubes passes through a narrow fitting and then enters the sample. For most rocks of interest, the size of the tubing and fittings are significantly larger than the pore size. The laboratory that conducted the sample measurements indicated that the diameter of these holes is approximately 7 mm, which is much smaller than pores in our samples. As the permeability varies as the radius of the size of a pipe-like channel squared, 2 order of magnitude differences in measured versus real permeabilities can occur if the tubing and fittings are one-tenth the size of the connected pores. This type of size difference is easily the case for some of our samples. Laboratories attempt to reduce the effect of the small diameter tubes and fittings by reducing the flow rate; however, this reduces the pressure difference and in turn the accuracy with which it can be measured, subsequently affecting the permeability estimation. Furthermore, we maintain that if the connected pore path size is larger than the tubes and fittings, then reducing the flow rate is effective at reducing turbulence inside the pores; however, because the porous medium has little or no restrictive effect on the flow, the permeability cannot be accurately measured.

[46] Petrophysical laboratories use other strategies that can affect the permeability measurements. Hassler-type core holders incorporate a rubber sleeve that is used to exert confining pressure on rock samples during permeability measurement to replicate in situ conditions, which can be important for deeply buried rock. Normally, when fluid pressure is applied to the outside of the sleeve, the sleeve pushes snuggly against the sample. However, for rocks like the macroporous Biscayne aquifer samples, the size of the pores raises the possibility that the sleeve will deform sharply, enter the large pores, and potentially rupture. Laboratory personnel circumvented this by stuffing large open pores on the sample surface with cotton. This is expected to reduce the measured permeability by an unknown amount.

[47] The application of confining pressure has another potential effect on low strength rocks like the highly macroporous Biscayne aquifer samples we focus on, that is, they can break. The permeabilities of two samples were measured at both 400 and 100 pounds per square inch (psi) confining pressure, and two samples that were considered especially fragile were measured at 25 psi. The final condition of the samples is unknown; breakage could reduce or increase the permeability.

[48] Cunningham et al. [2009, 2010] reported petrophysical core laboratory intrinsic permeability measurements for samples C100-Q5-e-1, FPL-Q3a-2, G-3837-18, and G3837-22 (Figures 2 and 3) collected from the Biscayne aquifer. The laboratory samples were larger than the subsamples used in the LBM analysis. The maximum of vertical and horizontal intrinsic permeability laboratory values were approximately 125,000, 66,000, 11,000, and 3000 md (∼1.25 × 10−10 m2, ∼6.6 × 10−11 m2, ∼1.1 × 10−11 m2, and ∼3 × 10−12 m2), respectively. These laboratory values are consistently about 3 to 4 orders of magnitude smaller than the corresponding LBM simulation results. Some of the difference between the laboratory and simulation estimates of permeability might be due to the smaller sample sizes used in the LBM simulations.

[49] Repeat laboratory measurements on core G3837-18 (Figure 3) at different confining stresses had reported permeabilities 2 to 8 times larger under higher (400 psi) confining stress, whereas core G3837-22 (Figure 3) had reported permeabilities 1.5 to 3 times smaller under higher confining stress.

[50] We demonstrated that the LBM is an accurate tool for flow simulation through comparison with flow in a sinusoidal pipe that has an exact analytical solution at low Re and experimental data at higher Re (Figure 5). However, this is not a conclusive proof that it will work as well in the rock samples. Nevertheless, for the highly porous, large-macropore rocks we work with, the analytical Poiseuille model for the permeability of a pipe shows that a 2 cm pipe has a 12,700 md permeability (hydraulic conductivity of 123 m s−1), which is consistent with the magnitude of our results. Thus, we expect permeabilities of the magnitude we estimate with LBM for these rocks. Recognition that standard core laboratory measurements are limited to a maximum permeability of 3 × 104 md and that the agreement of Poiseuille equation-based permeability calculations with the simulation results strongly suggests that the gross lack of agreement between the LBM results and standard petrophysical core laboratory methods arises because the laboratory methods are not appropriate for the highly macroporous rocks considered here. LBMs appear to provide a versatile tool for the estimation of intrinsic permeability in these media. Experimental verification of Biscayne aquifer permeabilities using specialized techniques and equipment is underway in our laboratory and should ultimately confirm or refute the value of LBM for permeability estimation in highly macroporous rocks.

5. Conclusions

[51] In this paper, the LBM was successfully applied to simulate 3-D flows in biogenic vuggy macropore networks of seven subsamples that were collected from limestone of the karstic Biscayne aquifer in southeastern Florida. Intrinsic permeability values of seven subsamples were obtained from LBM simulations at very small hydraulic gradients. Reductions in the apparent hydraulic conductivity were observed as the applied gradient increased to levels representative of those observed in the field and the results follow the Darcy-Forchheimer equation. Standard petrophysical core laboratory measurements are limited to a maximum permeability of 3 × 104 md (and sometimes 10 or more times less). Poiseuille equation-based permeability calculations using sample pore sizes give permeabilities with the same orders of magnitude as LBM simulation results. These facts suggest that the lack of agreement between the LBM results and standard petrophysical core laboratory methods arises because the laboratory methods are not appropriate for the highly macroporous rocks considered here. Newly specialized laboratory measurements are in progress.

Acknowledgments

[52] The Priority Ecosystems Science (U.S. Geological Survey) and Critical Ecosystems Studies Initiative (Everglades National Park) programs provided major project funding. Kateryna Ananyeva assisted with the simulations in Figure 5. Early reviews by Joe Hughes, Barclay Shoemaker, and Dorothy Payne and subsequent reviews by three anonymous reviewers improved the manuscript.

Notation

c

constant in Kozeny-Carman equation, L2.

cs

sound speed on the lattice, LT−1.

d

grain diameter for Reynolds number, L.

g

gravitational acceleration, LT−2.

h

hydraulic head, L.

k

intrinsic permeability, L2.

k0

true intrinsic permeability, L2.

K

hydraulic conductivity, LT−1.

K

hydraulic conductivity tensor, LT−1.

L

sample length, L.

Lp

pipe or characteristic pore diameter, L.

P

pressure, ML−1 T−2.

q

specific discharge, LT−1.

q

specific discharge vector, LT−1.

Re

Reynolds number.

t

time, T.

math formula

mean pore water velocity, LT−1.

u

fluid velocity magnitude, LT−1.

β

Darcy-Forchheimer parameter.

µ

dynamic viscosity, ML−2 T−1.

math formula

kinematic viscosity, L2 T−1.

math formula

porosity.

math formula

fluid density, ML−3.

math formula

average fluid density, ML−3.

τ

relaxation time, T.

Ancillary