## 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]):

where *ξ*_{max} is the maximal dimensionless pore diameter, *ξ*_{min} is the minimal dimensionless pore diameter, = *ξ*_{max}/*ξ*_{min} 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.

[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.

[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.