Water Resources Research

Pore-scale and continuum modeling of gas flow pattern obtained by high-resolution optical bench-scale experiments

Authors


Abstract

[1] High-resolution optical bench-scale experiments were conducted in order to investigate local gas flow pattern and integral flow properties caused by point-like gas injection into water-saturated glass beads. The main goal of this study was to test the validity of the continuum approach for two-fluid flow in macroscopic homogeneous media. Analyzing the steady state experimental gas flow pattern that satisfies the necessary coherence condition by image processing and calibrating the optical gas distribution by the gravimetrical gas saturation, it was found that a pulse-like function yields the best fit for the lateral gas saturation profile. This strange behavior of a relatively sharp saturation transition is in contradiction to the widely anticipated picture of a smooth Gaussian-like transition, which is obtained by the continuum approach. This transition is caused by the channelized flow structure, and it turns out that only a narrow range of capillary pressure is realized by the system, whereas the continuum approach assumes that within the representative elementary volume the whole spectrum of capillary pressures can be realized. It was found that the stochastical hypothesis proposed by Selker et al. (2007) that bridges pore scale and continuum scale is supported by the experiments. In order to study channelized gas flow on the pore scale, a variational treatment, which minimizes the free energy of an undulating capillary, was carried out. On the basis of thermodynamical arguments the geometric form of a microcapillary, macrochannel formation and a length-scale-dependent transition in gas flow pattern from coherent to incoherent flow are discussed.

1. Introduction

[2] There exists a comprehensive literature on modeling of gas transport in porous media that is based on concepts of multiphase flow theory [Ho and Webb, 2006]. Review articles on modeling of air sparging by McCray [2000] and by Thomson and Johnson [2000] discuss analytical and numerical approaches. All research papers discussed in these review articles are based on the continuum hypothesis and the representative elementary volume (REV) assumption assuming that the porous medium consists of two overlapping fluid continua (one air and the other water) and that these fluids occupy their own pathways or network within the REV. Both fluid networks build a coherent cluster and the corresponding flow is driven by the applied pressure. These continuum models have been applied, successfully, both on the bench scale and on the field scale [McCray and Falta, 1997; Hein et al., 1997] using homogeneous and heterogeneous permeability fields. However, as Thomson and Johnson [2000] criticized, continuum models produce a continuous range of air saturations throughout the sparge zone, while observed flow pattern consist of a series of stochastic and discrete continuous gas flow channels [Clayton, 1998]. Glass et al. [2000] conclude from optical bench-scale experiments that this characteristic channelized flow behavior can only be explained by pore-scale phenomena. Selker et al. [2007] proposed a stochastic picture that bridges pore scale and continuum scale. The authors assumed that the porous medium possesses many equiprobable channel realizations due to pore-scale heterogeneity and that only few of them are used at low flow rates. The ensemble average over all possible channel realizations is described by the continuum model. From this stochastic hypothesis one anticipates that at high flow rates most of the channels are occupied and the averaged gas flow can be described by a continuum model. However, at low flow rates the continuum model is not able to describe the asymmetric flow pattern produced by few occupied flow channels.

[3] An inherent assumption of all continuums models is the stability and coherence of the fluid network or fluid channels. In a recent paper [Geistlinger et al., 2006] (hereinafter referred to as Gei06) we investigated both experimentally and theoretically the stability of gas flow pattern in a typical range of flow rates from 10 to 1000 mL/min using glass beads (GBS) and natural sands [Ji et al., 1993; Brooks et al., 1999; Elder and Benson, 1999; Glass et al., 2000; Selker et al., 2007]. We found both a rate-dependent and grain size– (or pore size) dependent transition from incoherent to coherent flow. In order to classify our experimental results, we first applied a standard instability criterion (quasi-static criterion [see Dullien, 1992]):

equation image

where ξmax is the maximal dimensionless pore diameter, ξmin is the minimal dimensionless pore diameter, equation image = ξmax/ξmin is the aspect ratio, and Bo is the Bond number. As can be seen from Figure 1 the standard criterion gives small positive values for all glass beads, and hence unstable flow (I > 0). As our experimental results clearly indicate the 0.5 mm GBS show for all flow rates, especially for the lowest flow rate of 10 mL/min, stable flow. For the sediment with the largest grain diameter, the 2 mm GBS, we found unstable flow already for the lowest flow rate and a transition in flow pattern from unstable to stable flow at a flow rate of about 100 mL/min. As discussed by Gei06 this has strong consequences for gas transport, the gas flow pattern, and the gas volume that will be stored or trapped by the sediment.

Figure 1.

Quasi-static criterion: instability function versus averaged grain radius. The black dots indicate the 0.5, 1, and 2 mm GBS (Rk = 0.19, 0.44, and 1.05 mm). The thick solid lines frame the physical region between loosest (simple cubic (sc)) and closest (face centered cubic (fcc)) packing. The dashed line corresponds to close random packing with a porosity of 0.36.

[4] Using a dynamic stability criterion [Geistlinger et al., 2006] we were able to classify the experimental results. The proposed criterion was derived at the pore scale, approximates core-annular two-fluid flow through a variable diameter pore channel by a mean straight capillary, and accounts for the stabilizing viscous forces in contrast to the standard quasi-static Bond number models in literature [Joseph and Renardy, 1993]. Figure 2 shows the critical flow rate (thick solid line) at which the coherent channel or coherent network breaks down. The pairs of dashed lines indicate the realistic range of capillary radii for the 0.5 mm GBS, 1 mm GBS, and 2 mm GBS, respectively. At the lowest flow rate of 10 mL/min (indicated by the thin solid line) only the 0.5mm GBS lies well above the critical flow rate. For the 1mm GBS the thin solid lines crosses the critical curve; therefore one expects some transitional behavior. This was indeed observed. The gas flow of the 2mm GBS is described by an incoherent gas flow pattern, e.g., bubbly flow.

Figure 2.

Dynamic stability criterion: critical flow rate versus capillary radius. Each pair of dashed lines marks the interval of the capillary radii for 0.5, 1, and 2 mm GBS. The thin solid line indicates the smallest experimental flow rate of 10 mL/min.

[5] Gei06 argue that continuum models are not able to describe the essential features of gas flow pattern, when the coherence condition for stable flow is not satisfied. For coherent flow McCray and Falta [1997] concluded that the continuum approach describes the main characteristics of air sparging bench-scale experiments performed by Ji et al. [1993]. Because of the channelized flow structure and according to the stochastic hypothesis one expects that the continuum model will describe gas flow caused by high flow rates and will fail for flow rates smaller than a critical one. To verify this statement, i.e., to test the validity of the continuum approach for two-fluid flow, is the main objective of this paper. In order to satisfy the necessary coherence condition, we have to consider our experimental results for the 0.5mm GBS for flow rates between 10 to 844 mL/min. For modeling we use the TOUGH2 program [Pruess et al., 1999]. For the comparison between theory and experiment image processing was conducted to the gray scale images obtained by high-resolution optical bench-scale experiments. We note that the visualization experiments for the 0.5 mm GBS are published in previous papers; that is, Lazik et al. [2008] presented a k space image analysis for injection rates Q = 10, 59, 233, and 844 mL/min investigating the spectrum and identifying different maxima that characterize mobile and trapped gas phases. Here, we conduct a real-space image analysis for injection rates Q = 10, 59, 146, 233, and 844 mL/min analyzing the upscaled saturation profiles and comparing these with the theoretical ones. Furthermore, new experimental results are presented; that is, the steady state integral gas volumes of a second experimental series are compared to those of the first experimental series.

2. Experiment

[6] We designed point-like gas injection experiments within a flow cell using quantitative light reflection techniques to record gas flow patterns in time.

2.1. Experimental Setup

[7] A Plexiglas flow cell with inner dimensions, Lx × Ly × Lz = 40 × 1.2 × 45 cm, was placed vertically in front of the optical system at a distance of about 50 cm as shown in Figure 3. The flow cell was sealed with 1 mm silicon foil placed between the sediment chamber frame and the front/back walls. The airflow rate was controlled by a mass flow controller (MFC 8712, Qg = 0.01 – 5 L/min ± 0.5% from measure; Bürkert GmbH and Company, KG). Air entry pressure was measured at the injection point by a difference pressure transmitter (HCXM350D6V, 0 … 350 mbar ± 0.1% from measure; Sensortechnics GmbH). For gas injection, a cylindrical diffuser (inner diameter: 5 mm) with openings of about 1.5 mm was inserted orthogonal to the observation plane at a height of 5 cm from bottom of the tank. To prevent clogging at the diffuser, it was embedded in high-porosity, nonwoven material. Before each air injection experiment the flow cell was flooded with CO2 gas and then CO2 was displaced by distilled degassed water from bottom to top as long as the measured water mass within the flow cell became constant. The gas injection experiment started with the lowest gas flow rate of 10 mL/min. When steady state was achieved, the flow rate was stepwise increased. At least two experimental series (in the following denoted as experiment 1 and 2) were conducted for each GBS, where for the repeating experiment smaller flow rate steps were applied. The steady state values of the capillary pressure (gas pressure at the injection point minus hydrostatic pressure) and the total gas volumes are listed in Table 1 for the 0.5 mm GBS (experiment 1).

Figure 3.

Experimental setup.

Table 1. Steady State Values of the Capillary Pressure pc and the Gas Volume for Experiment 1
 Steady State
1234567891011
Qg (mL/min)103159146233321407495582669844
pc (kPa)1.21.92.33.44.96.06.88.19.911.615.8
Vg (mL)17.722.429.150.266.577.488.898.9108.0114.0136.0

2.2. Characterization of Glass Bead Sediments

[8] For the injection experiments three different glass beads (Carl Roth GmbH + Co KG) were used to represent medium to coarse grain sizes, where each glass bead is characterized by a certain size interval, namely, 0.25–0.5 mm, 0.75–1.0 mm, and 2.0–2.2 mm. The GBS had a density of about 2.45 g/cm3, a packing density of 1.51 g/cm3 for the 0.5 mm and 2 mm GBS and 1.57 g/cm3 for the 1 mm GBS, respectively, with corresponding porosities of 0.36 and 0.39.

[9] As mentioned by many authors both the packing procedure and the sediment stability can affect the gas flow geometry. Most of the papers used gradual pouring methods both for glass beads and natural sands [e.g., Ji et al., 1993; Brooks et al., 1999] and had to prevent layering effects by mechanical vibrating or manual layer mixing. Continuous dry sand pouring methods were used by Glass et al. [2000] and Niemet and Selker [2001] using a computer-controlled movable or fixed hopper, respectively. As the authors emphasize, this continuous dry packing method avoids a layered sediment structure. For spherical glass beads the gradual pouring method combined with a mechanical vibration should also produce a macroscopically homogeneous and tight packing as shown by the classical work of Ji et al. [1993]. For future research a comparison between the continuous and gradual pouring method should be conducted in order to estimate the effect on flow geometry that is introduced by the pouring method. In order to achieve sediment stability and to prevent grain rearrangement and sediment fluidization a lithostatic overlayer [e.g., Ji et al., 1993; Brooks et al., 1999], pressure-fixed sediments [Clayton, 1998], and glued beads [Elder and Benson, 1999] are used.

[10] We tested different gradual pouring methods. Starting with gradual pouring without any lithostatic layer, we observed that grain rearrangement influences gas channel geometry already at low flow rates (<100 mL/min). In order to prevent grain rearrangement, we apply a lithostatic pressure using a 5 cm layer of 4 mm glass beads as proposed by Ji et al. [1993]. Without a separating gauze between the two layers we observed channel recombination and sediment fluidization at low flow rates (10−100 mL/min) as discussed by Clayton [1998]. For higher flow rates macroscopic cracks of the glass beads were observed. Finally, the following packing method was used, in order to prevent grain rearrangement, cracks and fluidization in the applied range of flow rates between 10–844 mL/min: The glass beads were cleaned with acetone and water, dried at 105°C, and the wetted glass beads were packed up to a final height of 35 cm into the cleaned flow cell using a gradual pouring method. For each pouring step mechanical pulses were applied in order to reach close packing. The glass beads were covered by a lithostatic layer (thickness 6 cm) of lead beads (dk = 3 mm) to stabilize the packing. Gauze was inserted below the load layer to prevent mixing. Because of the relative large pores of the lead bead layer, we estimated that this layer could store at most 10% of the total gas volume inside the flow cell. The actual value for each injection rate can be estimated using air saturation values of bubbly flow for 3 mm GBS from Brooks et al. [1999] that range from 11 to 16% for corresponding injection rates from 23 to 675 mL/min. For instance, for the highest injection rate we estimated the lateral extension of air bubbly flow of about 30 cm that gives a gas volume of about 13 mL, i.e., 10% of the total gas volume.

[11] In a series of column experiments the hydraulic conductivity (Kf value) of the three different glass beads was determined using the constant-head method. The values are given in Table 2 and compared with Kozeny-Carman values that were obtained from the following equation:

equation image

where dm is the averaged particle diameter.

Table 2. Experimental Kf Values, Derived Permeabilities, and Theoretical Kozeny-Carman Permeabilitiesa
ExperimentKf (m/s)kw (m2)kw,KC (m2)
  • a

    Here kw is derived permeability, and kw,KC is theoretical Kozeny-Carman permeability.

0.5 mm GBS9.05 ± 0.35 × 10−49.23 ± 0.36 × 10−118.90 × 10−11
1 mm GBS5.40 ± 0.14 × 10−35.50 ± 0.14 × 10−104.84 × 10−10
2 mm GBS2.20 ± 0.14 × 10−22.34 ± 0.14 × 10−92.53 × 10−9

2.3. High-Resolution, Optical-Gravimetrical Measurements

[12] The gas flow patterns were investigated using a high-resolution optical-gravimetrical setup. The optical system was designed for multiscale image analysis, consisting of two cameras (1280 × 1024 pixels, size of a pixel is 6.7 μm, 12 bit coding, frame rate 8 Hz). Using the overview camera, the whole flow cell (0.5 × 0.5 m2) was recorded at the bench scale with a resolution of 0.48 mm/pixel, using the detail camera, a subsection of 1.62 × 1.27 cm2 can be investigated with a resolution of 12.7 μm. To study highly dynamic processes, the location of the subregion recorded by the detail camera can be adjusted rapidly using a mirror scanning system. Both cameras are computer operated and communicate via the same software. This enables a direct link between bench-scale observations and the navigation of the detail camera so that the latter can be directed to the most informative locations, either automatically or in an interactive mode. To observe the local multiphase phenomena, oblique incident light (LED arrays are used to avoid heating) was used and the reflected light was analyzed. The reflection is expected to be influenced by the contact angle between the fluid phases and the wall of the container. To minimize this influence, the contact angle of the fluid phases with the observation plane should be 90°. This was approximately achieved by using Plexiglas with a contact angle of about 70°−78° [van Pelt et al., 1985; Mozes et al., 1987; Holländer et al., 2003]. Hence, the visible phase distribution at the 2-D wall of the flow cell is approximately the same as would be expected for three dimensions where the solid matrix is reflected at this plane. Besides the optical measurements the displaced water mass was measured with a weighting system as an equivalent of the injected air volume. Therefore it was possible to calibrate the optical gas saturation by the gravimetrical value. For more details of the experimental setup and multiscale analysis see Gei06, Lazik et al. [2008], and Krauss [2006].

3. Modeling

3.1. Pore-Scale Modeling: Capillary Bundle Model for the Constitutional Relationships

[13] The following physical picture of gas flow is based on the capillary bundle model. Because gas is the nonwetting fluid gas flow is initially induced through the largest capillaries. Hence, we can calculate the gas volume as an integral over all gas-filled capillaries, similar to the proposed picture for trapped gas bubbles [Geistlinger et al., 2005]. For mathematical convenience the total gas saturation is considered:

equation image

where Rmax is the maximal capillary radius, Rgf the smallest gas-filled capillary radius, f(r) the probability density function (pdf) of the capillary radii, and Vcap the volume of a straight capillary. We assume a lognormal pdf as widely accepted in literature [Kosugi, 1996]:

equation image

where Vp is the pore volume, rm is the mean, and σ is the standard deviation. In Figure 4 the normalized pdf is shown for the 0.5 mm GBS with rm = 0.10 mm, σ = 0.02 mm, and where the gas-filled range is indicated by the gray area. Choosing these values one obtains a good fit to the experimental capillary pressure as discussed below. Note that the realistic range of capillary radii between 0.06 and 0.11 mm can be estimated from cubic face–centered and simple cubic coordination (for details see Gei06).

Figure 4.

Normalized pdf of the capillary radii for the 0.5 mm GBS. The gray area indicates the gas-filled capillaries.

[14] Solving the implicit equation for Rgf(3) and using the Laplace equation [Geistlinger et al., 2006] one can calculate the capillary pressure for a given saturation that is denoted as pc(Sg) curve in the following. For the upper integration boundary Rmax a finite capillary radius was used that causes a finite entry pressure. In Figure 5 the derived pc(Sg) curves are shown for three finite radii (thick solid lines), where the radii, 0.11 mm and 0.15 mm, are real cutoffs of the pdf indicated by dashed lines in Figure 4. In contrast to that, the radius of 1 mm is equivalent to an infinite upper integration boundary. The pc(Sg) curve for 0.15 mm (also denoted as 0.15 mm curve) fits to the experimental value observed for the lowest flow rate of 10 mL/min (black dot). Therefore, it was selected for the following calculations. In addition two best fits (van Genuchten and Brooks-Corey parametrization) to the 0.15 mm curve are shown in Figure 5. The corresponding expressions are given in (5a) and (5b), respectively:

equation image
equation image

with the effective water saturation

equation image

where m = 1 − 1/n, Swr is the residual, and Sm the maximal water saturation. As can be seen from Figure 5 the Brooks-Corey fit (dashed curve) deviates significantly; it overestimates the capillary pressure between Sg = 0 and 0.3 and underestimates the capillary pressure for Sg > 0.5. On the other hand, the van Genuchten parametrization (thin solid line; n = 10, α = 7 m−1), which is nearly identical to the 0.15 mm curve, gives a significant better fit and merge to the 0.15 mm curve at a gas saturation of about 0.05. Hence, it underestimates the capillary pressure only in a small range for Sg < 0.05. In this saturation range it overestimates the lateral extension of the gas flow pattern. This has an effect mainly on low injection rates and becomes less important for higher injection rates. The effect of a finite entry pressure on the saturation profiles will be discussed in section 4.3. Since the van Genuchten parametrization gives a better fit and is widely used in multiphase flow literature, especially in air sparging literature [McCray and Falta, 1997], it will be used in the following calculations.

Figure 5.

Capillary pressure for the 0.5 mm GBS using a lognormal radii distribution for Rmax = 0.11, 0.15, and 1 mm (thick solid lines). The thin solid line represents a van Genuchten best fit (n = 10, α = 7 m−1), and the dashed line represents a Brooks-Corey best fit to the 0.15 mm curve. The black dot represents the experimental capillary pressure for the lowest injection rate of 10 mL/min.

[15] Note that the main advantage of the derived relationship from pore size distribution is its physical background. Especially, for homogeneous equal-sized glass beads the mean can be easily estimated and the standard deviation should be small, given by the possible range of coordination. We note that Kosugi [1996] derived an analytical expression, both for the capillary pressure and the permeability using Mualem's [1976] model. Since we used the TOUGH2 program, the calculations were based on the best fit van Genuchten parametrization and Mualem's expression for the relative permeability (for details see Ho and Webb [2006]). To be consistent, the same n both for the capillary pressure and for the permeability was used.

3.2. Pore-Scale Modeling: Scale Dependence of the Flow Channel Geometry

[16] To this point we have assumed cylindrical gas flow channels with a flat gas-water interface. As shown by Gei06 such a flat interface is thermodynamically unstable, and leads to the well-known Rayleigh instability. More realistic, the gas flows through irregular shaped or undulating channels. Since there is no analytical solution for such a curved geometry, we use the Ritz variation principle (for details see Geistlinger and Weller [1985] and Braunschweig and Geistlinger [1993]). Assuming Hagen-Poiseuille flow for the curved flow channel, we obtain the generalized free energy functional:

equation image

where σ is the interface tension, Q is the total flow rate, and Lz is the macroscopic length scale in z direction. For the capillary radius R(z,β) we use two different one-parametric test functions:

equation image
equation image

where β is the variational parameter, dk is the grain diameter, ξmin = 0.33 is the dimensionless minimal capillary diameter, ξmax = 0.64 is the dimensionless maximal capillary diameter, and λ is the length of the elementary cell, which is approximately given by dk and also noted as wavelength. The test function (7a) varies both the pore neck and the pore body, whereas the test function (7b) only varies the pore body. The question, which test function is the more physical one, is decided by the global minimum of the free energy. For the variation of the free energy functional (6) both test functions are used. In order to consider a realistic flow rate, we used the mean radius of the pore size distribution as an upper limit, thus that one needs about 44 capillaries with a single flow rate of 0.23 mL/min for the total gas flow rate of 10 mL/min. The numerical integration was carried out with Mathematica 5.1 and the results for two different length scales, Lz1 = 100λ and Lz2 = 1000λ are shown in Figure 6.

Figure 6.

(top) Free-energy and (bottom) pore-scale channel geometry for two different test functions and for two different macroscopic length scales: (left) Lz1 = 100λ and (right) Lz2 = 1000λ. The thick solid lines represent the absolute free-energy minimum with the corresponding test function: R1 (7a) for Lz1 and R2 (7b) for Lz2. The thin solid lines represent the other test function with the higher free energy. The dashed line shows the free energy of the flat interface.

[17] Figure 6 demonstrates some interesting features of the process dynamics on the pore scale and its correlation to the macroscopic scale. Consider first the free energy in the top row in Figure 6. The dashed line corresponds to the free energy of a cylindrical capillary. Both curved geometries exhibit minimum values of the free energy lower than those value of the flat interface. Hence, the flat interface is not a stable one as discussed earlier. For the length scale Lz1 = 100λ (left plots in Figure 6) the global minimum of the free energy is obtained for the test function R1 (thick solid lines, β = 0.43) that allows a variation both of the pore neck and of the pore body. On the basis of the physics of core-annular flow, one concludes that there is still some thick water film in the pore neck. Consider now a ten times larger length scale (≈ bench scale of our experiments; right plots in Figure 6), then the global energy minimum is obtained for the test function R2 (thick solid lines, β = 0.73). In this case there is only a thin water film in the pore neck and the pore body is still smaller than the mean capillary radius of 0.1 mm (see Figure 4). Note that there is no additivity of the free energy due to the friction term. Because of this length-scale dependence of the free energy the pore-scale dynamics are determined by the macroscopic length scale. This contradicts the usual upscaling theories that search for effective parameters at the larger scale that can be expressed as function of pore-scale parameters.

3.2.1. Length Scale–Dependent Transition

[18] The next question of interest is about stability on different length scales. To investigate this, the dynamic stability criterion for a straight capillary is used as a necessary stability condition (Gei06, equation (12)). In a more stringent manner one has to carry out a linear stability analysis of two-fluid core-annular flow as discussed by Joseph and Renardy [1993]. For the length scale Lz1 one obtains for the mean radius of the undulating capillary R1(z,β): Rmean1 = 0.04 mm, and for the length scale Lz2: Rmean2 = 0.075 mm, respectively; and for the corresponding critical flow rates: Qcrit(Lz1) = 0.027 mL/min and Qcrit(Lz2) = 0.36 mL/min, respectively. In the first case we have stable flow, since Qcrit(Lz1) < 0.23 mL/min. However, for the second case we would anticipate unstable flow. Despite the inconsistent application of the dynamic criterion to an undulating capillary that was derived for a straight capillary, we conjecture that this length-scale-dependent transition from stable coherent to unstable incoherent flow really exists. If so, it has strong implications on upscaling, since the process changes qualitatively with increasing length scale. In other words, upscaling makes only sense, if gas transport is governed by the same process dynamics, i.e., the same partial differential equations.

3.2.2. Formation of Macroscopic Flow Channels

[19] Calculating the free energy of a bundle of 44 capillaries, one obtains about a 12 times larger free energy (again no additivity holds). This might be the thermodynamic reason, why macroscopic channels are formed, because it is more favorable, if the capillaries merge together and form larger flow channels minimizing friction and interface tension. A simple estimate of such a macroscopic flow channel for simple cubic packing gives a capillary bundle that is imbedded by 8 × 8 grains, hence a channel diameter of 8 × dk = 3 mm. This value has the right order of magnitude of the flow channel diameters experimentally observed, which lie in the range of 3 mm [Elder and Benson, 1999] to 8 mm [Selker et al., 2007]. Of course there will be some broadening due to the stochastic nature of porous media.

3.3. Continuum Modeling Using TOUGH2

[20] The TOUGH2 program [Pruess et al., 1999] is a well-established multiphase flow simulation program that was tested against analytical benchmark problems and applied in different subsurface flow problems [McCray, 2000]. The equation of state 3 (EOS3) module describing gas-water flow was used under isothermal conditions. We include the gas phase permeability according to Mualem's model into the source code. The geometry of the experimental flow cell was mapped to a fine-scaled, 2-D grid with 51 × 63 cells, where the single cell dimension was Lx × Ly × Lz = 8 mm × 12 mm × 8 mm. Since the upper 18 layers represent the lead sphere layer, a higher permeability was chosen. The corresponding parameters of the GBS and the lead spheres are listed in Table 3.

Table 3. Parameter for the TOUGH2 Simulation
 Glass BeadsLead Beads
Averaged particle size (mm)0.3753
Permeability (m2)9.2 × 10−119.2 × 10−10
Porosity0.360.37
α (m−1)77
n in capillary pressure equation (6a)1010
n in relative permeability equation105
Swr0.010.01
Sm1.01.0

[21] A special preprocessing program was used for mesh generation, calculation of appropriate inhomogeneous initial conditions, and generation of the input files. The inhomogeneous initial conditions are required, since the water saturation within the flow cell exhibits a transition from complete water saturation at the bottom to complete air saturation at the top. The simulation itself consists of two parts. First, the capillary-gravitational equilibrium was calculated using Dirichlet boundary conditions at the top and at the bottom of the flow cell, i.e., atmospheric pressure at the top and an appropriate hydrostatic pressure at the bottom. Furthermore, we had to use a no-flow boundary condition at the bottom in order to prevent numerically instabilities; for example, gas occurs at the bottom. In the second part, the injection phase, the equilibrium distributions for the pressure and the gas saturation were used as initial conditions. Care has to be taken for choosing the boundary conditions. No-flow boundary condition at the bottom and at the left and right side would lead to an increase of the water table during gas injection. Therefore, at the sides Dirichlet boundary conditions were used. During gas injection the water pressure becomes larger compared to equilibrium pressure and the displaced water flows to the right and left side of the flow cell as long as steady state was achieved. In all simulations the gradients were large enough for keeping the water level constant, thus that the experimental conditions could be mapped. Additionally, a constant pressure condition (atmospheric boundary condition) was applied to the top. Similar to the experiment we started with the lowest flow rate of 10 mL/min. After steady state was achieved, the corresponding results for the primary variables were used as initial conditions for the next flow rate. As standard solver the iterative DLUSTB solver, a stabilized biconjugate gradient solver, and for comparison sometimes the slower direct algebraic solver LUBAND was used [Moridis and Pruess, 1998]. Preconditioning was used as described in Moridis and Pruess [1998], but sometimes it prevented a convergent solution. In such cases the relative convergence criterion has to be smaller than 10−5, in order to obtain convergence within 8 – 16 Newton iterations. Hence, the time steps become rather small and lie approximately between 10−4 to 10−2 s; that is, for a typical relaxation time of 100 s one needs between 104 to 106 time steps. Since the maximal number of time steps for one run is limited by 10000, one usually needs several runs using the last-run results as initial condition for the next run.

[22] Figure 7 shows the transient gas flow for the smallest flow rate of 10 mL/min and the corresponding model parameters are listed in Table 3. The breakthrough occurs at 70 s; that is, the gas phase front has contacted the high permeable lead sphere layer. The steady state has been established at about 130 s. Comparing these results with experimental data shown in Figure 8, it can be seen that the agreement is reasonable. The theoretical steady state value of the dynamical gas volume is 14.4 mL, and the experimental one (see Figure 8, right axis) is 17.7 mL, i.e., a relative error of about 16%. Increasing the flow rate step by step, excellent agreement between TOUGH2 simulation for α = 7 m−1 (open rhombi) and experimental gas volumes (experiment 1, black triangles; experiment 2, black circles) is obtained at higher flow rates (error < 3%), as can be seen in Figure 9. We did not anticipate such good agreement, especially, for flow rates larger than 100 mL/min. It seems that this integral gas flow property has some self-averaging behavior, and therefore the continuum approach can produce the right functional form between injection rate and gas volume.

Figure 7.

TOUGH2 simulation of the gas injection into saturated 0.5 mm GBS for Q = 10 mL/min. The peak represents the injection point. The x and z axes are indicated by the cell indices (51 × 63, Δx = Δz = 8 mm).

Figure 8.

Experimental air entry pressure and gas volume versus time. Gas injection starts at 0 s. The dashed line indicates the hydrostatic pressure at the injection point, and Pc denotes the capillary pressure at steady state.

Figure 9.

Steady state gas volume versus injection rate. The right axis shows the relative error between experiment 1 and the TOUGH2 simulation for α = 7 m−1.

[23] To investigate the influence of the key parameter α on gas volume, a second TOUGH2 simulation is represented in Figure 9 for α = 9.81 m−1 (open circles). As anticipated, a smaller capillary pressure leads to smaller gas volume, too. While the reduction of the capillary pressure is about 30% the gas volume is only reduced by 10%. We emphasize that we had only calculated the gas volume within the glass beads, whereas the experimental values include the gas volume of the lead layer, too. Since TOUGH2 cannot simulate bubbly flow, which occurs in the lead layer, we had to estimate this gas saturation as discussed in section 2.2. If one adds this lead layer gas volume of about 10% to the glass beads value, the second TOUGH2 simulation gives excellent agreement, too. These values are indicated in Figure 9 by crosses. Since the 10% gas saturation of the lead layer is only an estimate, it is difficult to decide, which parameterization is the best fit of the experiments. However, more important than the agreement of the absolute values is the right functional form of the Vg(Q) curve.

[24] From this excellent agreement one can draw an important conclusion concerning coherent or incoherent flow for the GBS with larger radii, i.e., for the 1 mm and 2 mm GBS, respectively. Since the steady state gas flow is mainly buoyancy-driven except within a small region near the injection point, Falta [2000] could derive the following analytical implicit relationship between gas saturation and gas flux:

equation image

where qg is Darcy velocity and kg0 is absolute gas permeability. For simplicity the monotonic krg(Sg) function is expressed by a power law function and resolved for Sg:

equation image

Suppose that the flux is x-independent and integrating (9) over the 2-D flow module yields the required Vg(Q) relationship:

equation image

where V is volume of the flow module and A is the cross section of the flow module. Because of the inverse proportionality between gas volume and absolute permeability kg0, one anticipates that the continuum model yields lower Vg values for the coarser sediments. Since both coarser sediments exhibit experimental steady state Vg values (see Gei06, Figure 8) larger than the 0.5 mm GBS value, one can conclude that a main part of the gas volume is trapped because of unstable flow conditions, i.e., because of incoherent flow. Comparing the different Vg(Q) curves (see Figure 10) one realizes that for the 2 mm GBS almost no Vg increase is possible, since the maximal trapped gas volumes already occurs at the lowest flow rate because of incoherent flow, i.e., because of bubbly flow. The 1 mm GBS exhibits some transitional behavior. Note that the flow behavior of both sediments cannot be described by continuums models like TOUGH2 at the considered flow rates.

Figure 10.

Experimental gas volumes versus injection rate for three different glass beads.

[25] Summarizing the modeling effort at this point, we start with a reasonable lognormal distribution of capillary radii, which yielded a pc(Sg) relationship that was fitted to the experimental steady state capillary pressure at low flow rates. To be conform to standard multiphase-flow literature, the common van Genuchten parametrization of the constitutive relationships was used. The relative permeability was derived by applying Mualem's model, and in a consistent manner the same van Genuchten parameter n was used. The saturated hydraulic conductivity was experimentally determined and no further fit parameters were used. The continuum model (TOUGH2) was able to describe the functional form of the dynamical gas volume, an integral flow property, as a function of the flow rate for the 0.5 mm GBS. The deviation between theory and experiment was most pronounced at low flow rates (<100 mL/min). It was concluded that TOUGH2 is not able to produce the experimental behavior for glass beads with larger spheres (1 mm and 2 mm), because the necessary coherence condition was violated.

4. Discussion

4.1. Image Processing

[26] In the previous section we had considered integral flow properties. In this section we investigate local gas flow pattern using image processing including the following steps: (1) upscaling, (2) background reduction, (3) flow pattern fitting to a pulse-like and Gaussian function, (4) calibration, and (5) 3-D visualization and comparison to TOUGH2 simulation. All matrix operations, numerical and statistical calculations, and visualization were done using Mathematica 5.1. The original gray scale image matrix Aij that had a dimension of 850 × 870 pixels was upscaled by a factor 10. For the upscaled matrix Bnm (dimension: 85 × 87) a background reduction was carried out. The top plots in Figure 11 show the original images, whereas the bottom plots show the upscaled, background reduced images for five different flow rates: 10 mL/min, 59 mL/min, 146 mL/min, 233 mL/min, and 844 mL/min. The typical channelized flow pattern can be seen for all flow rates. Note that there are few horizontal flow paths, especially for Q = 233 mL/min, that are caused by the gradual pouring method. However, these few artifacts have only marginal influence on our main conclusion as will be discussed below.

Figure 11.

Gas flow pattern: (top) original and (bottom) upscaled, background-reduced images for 10, 59, 146, 233, and 844 mL/min. For the Q = 233 mL/min upscaled image the horizontal flow path is indicated by an arrow (row indices = 27, 28).

4.2. Transformation From Local to Integral Scale

[27] Suppose that continuum models, like TOUGH2, describe this channelized flow pattern as an average over many channel realizations according to the stochastical hypothesis, we need a transformation from the local to the integral scale. To find the best transformation, two different fit functions were used, i.e., a pulse-like function

equation image

and a Gaussian function

equation image

where x0 is the center, σ is the width, h is the height, and A0 is the area of the function. These parameters were used as fitting parameters. Testing different values of the stiffness parameter ξ of the pulse-like function between 2 and 10, we found that a significant better fit was achieved for ξ = 3 compared to ξ = 2, and that larger stiffness parameters lead to nearly the same fit goodness. Hence, we use ξ = 3 as a fixed parameter in order to reduce the fitting effort, significantly, i.e., performing a two-parameter fit instead of a three-parameter fit. On the basis of the Marquardt-Levenberg algorithm both the pulse-like and the Gaussian function were fitted row by row to the experimental profiles. Figure 12 shows the fitted curves and the experimental profiles for three different flow rates: 10, 146, and 844 mL/min at 25 cm above the injection point. Again, the deviations increase with decreasing flow rate. For the highly fluctuating structure at the lowest flow rate, the envelope function expressing an averaged flow behavior might be not a meaningful description. However, increasing the flow rate both fit functions give a good average flow behavior. For moderate flow rates, where point-like injection holds, Selker et al. [2007] proposed a single Gaussian function for the channel distribution. It is interesting that not the anticipated Gaussian function, but the pulse-like function shows a smaller standard deviation as shown in Figure 13. Again, the lowest flow rate deviates from this general behavior and yields the same goodness for both fit functions; that is, in this case, where a plateau has not yet established, the Gaussian function can account for the local fluctuating flow pattern.

Figure 12.

Horizontal profiles of channelized gas flow at 25 cm above the injection point for 10, 146, and 844 mL/min. Experimental data, solid circles; best fit to pulse-like function, thick solid line; best fit to Gaussian function, thin solid line.

Figure 13.

Standard deviation of the pulse-like (thick solid line) and Gaussian (thin solid line) function along the centerline in the z direction (row index) for flow pattern shown in Figure 11.

[28] A critical question in our discussion is about the boundary between the range of low and high injection rates, since only in the range of low injection rates a single Gaussian function should be an appropriate description of the gas channel geometry according to Selker et al. [2007]. For higher injection rates the authors proposed a pulse-like function obtained by a superposition of single Gaussian functions assuming line-like injection within the near-source region, whose geometric form is described by an ellipse. The injection rates at which the experimental flow patterns exhibit an elliptic near-source region are: 3.32, 1.56, 1.28, and 0.64 L/min for the 12/20, 20/30, 30/40, and 40/50 sands, respectively [see Selker et al., 2007, Table 1]. These injection rates define a lower boundary for the range of high injection rates. If one considers the mean grain diameter as a suitable criterion for comparison between glass beads and sands, then the 0.5mm GBS would correspond to the 30/40 sand; that is, the range of high injection rates starts at 1.28 L/min. Since our highest injection rate of 0.844 L/min is smaller, it is reasonable to assume that our experimental saturation profiles for injection rates between 0.01 and 0.844 L/min can be described by a single Gaussian function.

[29] As we mentioned above there are some horizontal flow paths in the flow geometry due to the gradual pouring method. In Figure 11 a typical horizontal flow path is indicated in the Q = 233 mL/min upscaled image by an arrow. For the small section of the horizontal path (row index i = 27, 28) the Gaussian function exhibits a smaller standard deviation, as shown in Figure 13 (Q = 233 mL/min, pixel = 27, 28). But most of the rows show that the pulse function yields a smaller standard deviation justifying the conclusion that the pulse function is superior to the Gaussian function.

4.3. Comparison Between Experiment and Theory

[30] In order to compare the experimental gas saturation with the theoretical one, one has to calibrate the gray scale distribution. Assuming linearity that had been proven for the experimental range of flow rates and gas saturations, respectively [Lazik et al., 2008], the calibration constant is obtained by integrating over the gray scale distribution and by constraining it to the gravimetrically measured steady state gas volumes. Having in mind the excellent agreement of experimental and theoretical results for the Vg(Q) dependence, the insufficient agreement of the flow pattern is disappointing as shown in Figure 14. For all flow rates we found a too smooth and too large extension in lateral direction. The discrepancy is as worse as lower the flow rates are, especially, if one compares the pulse-like function fit (Figure 14b) with the TOUGH2 simulation (Figure 14d). The most striking result is that the experimental gas saturation profiles are more likely described by a relatively sharp transition from a region with constant gas saturation to a water-saturated region, than by a smooth Gaussian transition. Obviously, this transition is caused by the channelized flow structure, where the channels have a nearly rate-independent diameter and a constant gas saturation [see Lazik et al., 2008]. It seems to be thermodynamically more favorable to realize a nearly constant channel density, than a widespread Gaussian one; that is, before spreading in lateral direction the system “searches” for equiprobable channels with the same length. Note that this flow characteristic is only elucidated, if one transforms the 2-D parabolic flow pattern into a 3-D graph using image processing.

Figure 14.

Comparison between (a) the experimental gas saturation (z axes), (b) the pulse-like function fit, (c) the Gauss function fit, and (d) the TOUGH2 simulation for three different flow rates (top) 10, (middle) 146, and (bottom) 844 mL/min. The x and y axes are scaled to the same absolute flow cell dimensions, i.e., 0.4 and 0.45 m, respectively.

[31] We note that theoretical profiles are based on a van Genuchten parameterization, which overestimates the lateral extension for Sg < 0.05 as discussed in section 3.1. To obtain an approximate saturation profile that accounts for a finite entry pressure, one has to cut the profile at the Sg = 0.05 isoline. The finite entry pressure reduces the lateral extension, and has an effect mainly on the lowest injection rate. However, the main conclusion about a relatively sharp gas-water interface compared to the too smooth TOUGH2 profile is not questioned by a finite entry pressure.

[32] Since this strange behavior contradicts the widely anticipated picture of a smooth Gaussian transition, we will discuss the underlying physics in more detail. Consider the gas pressure profile in the far source region along an already formed stable gas channel that travels from the source to the upper boundary. Since the steady state is considered, the gas-water interface is at mechanical equilibrium. Therefore, the gas pressure along the vertical gas–water channel interface will be equal to that of the water phase plus the effective capillary pressure, which should by constant for a homogeneous media. Hence, the magnitude of the vertical gradient driving gas movement is identically to the hydrostatic gradient [Selker et al., 2007]. In contrast the pressure distribution obtained from continuum models varies according the pc(Sg) relationship. This physical picture of a macrochannel with nearly constant gas saturation and constant pressure gradient together with the assumption that the system can realize a sufficient number of macrochannels because of the stochastic nature of the porous media, can explain the observed pulse-like gas phase distribution with a relatively sharp transition from gas phase to the water phase.

[33] It is important to realize that only one capillary pressure (or a narrow range of capillary pressures) is realized by the gas flow system in the far source region (see Figure 5). This contradicts the REV concept of the continuum model, which assumed that the whole spectrum of capillary pressures can be realized within the REV. Selker et al. [2007, p. 110] came to a similar conclusion: “…The invasion phenomena of channelized flow are in some sense much simpler than those occurring at higher Bond number, at which connected dendritic networks are formed, where pore size, and pore size distribution, must be included in the analysis….”

[34] To this point it was not necessary to consider the near source region. Selker et al. [2007] parameterized the geometric form both of the near source and of the far source region. In the immediate vicinity of the injection point there may be a region in which pressure gradients due to viscous resistance to the high imposed near-injection flow significantly exceed the energy gradient from buoyancy. In steady state there will be a z-dependent capillary pressure, since near the injection point not enough large capillaries are available. Therefore, the gas flow must displace water from smaller capillaries, too. This additional pressure gradient is used for the higher viscous resistant. The capillary pressures listed in Table 1 can be interpreted like a capillary pressure-saturation relationship that describes a broader pore size distribution. We had tested such a parametrization, too, using a smaller van Genuchten n, without any success. The smaller n yields too large gas volumes and fails therefore in describing of the experimental Vg(Q) dependence. Obviously, a smaller n will lead to a further broadening of the gas flow pattern. Consequently, the near–source region capillary pressures are not used by the flow system in the far source region that determines to a large extent both integral and local flow properties. Note that for the considered flow rates the porous medium provides enough large capillaries merging to macrochannels, and which are only controlled by buoyant and hydrostatic forces.

[35] Concerning the statement of McCray and Falta [1997] that the continuum model describes the bench-scale experiments by Ji et al. [1993] one has to be careful. The comparison is based on a 2-D comparison of the parabolic shape, where the saturation isolines were chosen arbitrary, i.e., without calibration of the experimental saturation profile. Consider the flow rate 844 mL/min in Figure 11, where the parabolic shape is well established, and compare it to the 2-D projection of Figure 14d, bottom plot, one always can define an isoline that describes the experimental flow pattern.

5. Conclusions

[36] Analyzing the steady state experimental gas flow pattern by image processing and calibrating the optical gas distribution by the gravimetrical gas saturation, we found that a pulse-like function yields the best fit for the lateral gas saturation profile. The pulse range of almost constant gas saturation in the fare source region is caused by macrochannels with almost constant pressure gradient, capillary pressure, gas saturation and permeability. Similar conclusions on a more empirical basis were drawn before by Clayton [1998], Thomson and Johnson [2000], and Selker et al. [2007]. This strange behavior of a relatively sharp saturation transition is in contradiction to the smooth and Gaussian-like gas flow pattern obtained by the continuum approach, which assumes that within the REV the whole spectrum of capillary pressures can be realized. The deviations between the experimental and theoretical flow pattern become smaller, if the flow rates increases from 10 to 844 mL/min, however, are still significant at the highest flow rate.

[37] The stochastical hypothesis proposed by Selker et al. [2007] gives a good physical basis for the interpretation of the experimental results and should be the starting point for further model development. We used the proposed Gaussian channel distribution for fitting the experimental data. It turns out that the Gaussian function gives a reasonable fit; however, a pulse-like function with a stiffness factor of three, that means a steep decrease of the gas saturation, yields a smaller standard deviation.

[38] Thus, the main conclusion to be drawn from the presented study is that the continuum model fails in describing the gas flow pattern for the considered glass beads. Statements in literature, which claim this agreement are often based one an arbitrary comparison of uncalibrated 2-D images. Therefore, quantitative evaluation of the optical gray scale images is necessary; both on the bench scale and on the pore scale [see Lazik et al., 2008].

[39] One open question raised by the present study is: Why gives the continuum model excellent agreement between experiment and theory for the integral flow property expressed by the Vg(Q) dependence? One could speculate that the main characteristics of the channel-based physical picture, the constant capillary pressure over a relevant Sg range, is mapped by the continuum model through a flat pc(Sg) parametrization and describes therefore the continuous distributions of channels represented by an effective capillary-saturation relationship. As we have shown a strict parameterization of the experimental capillary pressures listed in Table 1 gives a pc(Sg) parametrization that leads to too broad gas flow pattern with too large gas volumes. Further research is necessary to answer this question.

[40] On the basis of thermodynamical considerations and applying the Ritz variation principle for minimizing the free energy of an undulating gas filled capillary, we found a length-scale-dependent transition from stable coherent to unstable incoherent flow. This must have strong implications on upscaling, since the process dynamics change qualitatively with increasing length scale. However, the conclusion is based on an inconsistent application of the dynamic criterion derived for a straight capillary and then applied to an undulating one. Nevertheless, we believe that the scale-dependent transition is a real physical phenomenon. A general variational treatment has to include the variation of the capillary length.

[41] Concerning the statements of a universal, flow-independent channel diameter [Selker et al., 2007] we give some arguments why it might be favorable to form a macrochannel by a bundle of microchannels and estimate a channel diameter of about 3 mm. However, a strict thermodynamical derivation for channel formation in glass beads or natural sand is still lacking. Attempts to model channel formation by using small-scale heterogeneous permeability fields were unsuccessful (not presented in this paper), confirming the conclusion that channel formation is governed by pore-scale processes. If channel formation is driven by fluid-fluid interaction and not by small-scale heterogeneities, like Saffmann-Taylor fingers, then this phenomenon should be also observed in periodic homogeneous packings. Future work should be focus on the influence of small-scale heterogeneities and fluid-fluid interactions on channel formation and flow pattern generation.

[42] These results, even though some of them are preliminary in nature, are important when applying TOUGH2 simulations of direct gas injection to the field scale. In Figure 15a we show a realistic 2-D permeability profile that exhibits a fine sand/coarse sand/fine sand sandwich structure. Geologically, the high-permeability layer is a quaternary river terrace, and the grain size distribution changes from coarse sand to fine gravel. The presented permeability distribution is based on fine-scale injection logs, EC logs, geological profiles, and sieve analyses. The high-permeability layer is highly contaminated by organic contaminants like BTEX and MTBE. Oxygen gas injection, as a cost-effective bioremediation method for stimulating aerobic microbial activity, is applied at this field site for cleaning up the groundwater. Standard gas injection tries to inject oxygen gas directly into the contaminated layer. Before applying continuum models like TOUGH2, to simulate the corresponding gas flow patterns, one has to check, whether the stability condition is satisfied (Qchannel > Qcrit), and whether the application scale is smaller than the estimated coherence length. Furthermore, the effective pc(Sg) relationship should describe the capillary pressure of the steady state gas channels in the far source region, to obtain reliable steady state gas flow patterns. The simulation will give gas flow patterns similar to those shown in Figure 15b, i.e., low gas saturation within the contaminated layer. This would be the worst case for bioremediation. A more efficient location for gas injection would be the lower fine sand layer (Figure 15c). A smart choice of the injection rate should lead to stable flow (Qchannel > Qcrit) in the fine sand layer, which causes a large lateral extension of the gas flow, and should lead to unstable flow in the coarse sand layer, which causes a high gas saturation due to gas bubble trapping. Again, the TOUGH2 simulation can only describe the gas flow pattern in the fine sand, but would give wrong, low gas saturation in the coarse sand; that is, TOUGH2 is not able to simulate incoherent bubbly flow and bubble trapping. To summarize, a proper application of continuum models to direct gas injection should check, whether stable coherent flow is achieved; estimate the coherence length, and account for the channelized flow pattern by a realistic capillary pressure–saturation relationship. Further research is needed for modeling of direct gas injection to include appropriate stability criteria, the transition from coherent to incoherent flow, and bubble trapping. Whether this is possible within the framework of the continuum approach or within a combination of pore-scale and continuum models is still an open question.

Figure 15.

Schematic representation of gas flow pattern caused by direct gas injection at a realistic field site with dimensions Lx × Lz = 35 m × 12 m. (a) Heterogeneous permeability distribution based on fine-scale injection and EC logs. (b) Gas injection into the high-permeability layer for stable flow conditions. (c) Gas injection into the fine-sand layer below the high-permeability layer. Unstable flow occurs above the coarse sand–fine sand interface and leads to high gas saturation within the high-permeability layer.

Acknowledgments

[43] The authors would like to thank K. Pruess, Earth Sciences Division, E.O. Lawrence Berkeley National Laboratory, for technical support and valuable hints concerning TOUGH2 simulations.

Ancillary