C.C. Ezeuko* S.R. McDougall

Transcription

1 Published August, 2010 Special Section: Pore-Scale Processes C.C. Ezeuko* S.R. McDougall Modeling Flow Transi ons during Buoyancy-Driven Gas Migra on in Liquid-Saturated Porous Media We conducted a pore network modeling study of the influence of different fluid and rock proper es on gas migra on under the influence of gravita onal forces. Transi ons from con nuous, capillary-dominated flow, through gravita onally biased fingering and discon- nuous braided migra on, to discon nuous dispersive (bubbly) flow were inves gated using both small-scale (1-cm) and macro-scale (100-cm) networks. Our simula on results were found to closely match a wide range of published experimental observa ons and the model is the first to comprehensively explain the influence of several pore-scale proper es on buoyancy-driven migratory pa erns. We found a transi on from con nuous, compact, capillary-dominated flow to discon nuous dispersive flow as the rela ve strength of buoyancy forces increased. We found that the governing regime is strongly affected by the mean capillary radius, pore-size distribu on variance, gas liquid interfacial tension, and the underlying connec vity of the porous medium. We also inves gated the implica ons of these system proper es and associated boundary condi ons for air sparging efficacy. Abbrevia ons: PSD, pore-size distribu on. A dynamic pore network model was u lized for extensive parametric studies of the dependency of gas flow pa erns on the buoyancy or capillary forces balance, quan fied as the local Bond number. Classifica ons of flow are provided and implica ons of several fascina ng observa ons for the performance of in situ air sparging are systema cally explored. S.R. McDougall and C.C. Ezeuko, Ins tute of Petroleum Engineering, Heriot-Wa Univ., Edinburgh, EH14 4AS, UK. *Corresponding author (cosmas.ezeuko@pet.hw.ac.uk; steve. mcdougall@pet.hw.ac.uk). Vadose Zone J. 9: doi: /vzj Received 1 April Published online 3 Aug Soil Science Society of America 5585 Guilford Rd., Madison, WI USA. All rights reserved. No part of this periodical may be reproduced or transmi ed in any form or by any means, electronic or mechanical, including photocopying, recording, or any informa on storage and retrieval system, without permission in wri ng from the publisher. Buoyancy-driven, immiscible gas liquid flow in porous media plays a key role in a number of engineering applications, such as biodegradation stimulation via gas injection, volatilization of organic contaminants in saturated aquifers, and oil recovery by solution gas drive or gas flooding. For example, during in situ air sparging, compressed gas is directly injected into the formation beneath the water table and the contaminated vapor reaching the vadose zone after migration is extracted with the aid of vapor extraction wells. In addition to information on contaminant characteristics, the pore-scale properties of the associated porous medium and its influence on gas-phase distribution patterns must be understood to design an optimized gas injection protocol. A key factor that significantly controls the effectiveness of air sparging and the contaminant removal rate is the spatial distribution of the gas. During the sparging process, it is desirable for the injected gas to contact the maximum possible area of the saturated zone as it migrates upward under buoyancy. Several experimental investigations (Brook et al., 1999; Ji et al., 1993; Dumoré, 1970) showed a transition from channelized to dispersive (bubbly) flow during gas injection in liquid-saturated porous media. This transition is commonly described in terms of the grain size distribution and injection rate. Experimental investigations of gas flow in systems characterized by different grain-size diameters (Brook et al., 1999; Peterson et al., 1999; Ji et al., 1993) have shown that the more favorable dispersive (bubbly) flow generally occurs in porous media characterized by large grain-size diameters (generally >1 mm, although this may be dependent on the gas liquid interfacial tension, σ). A number of injection experiments have shown that the overall pattern of migrating gas during dispersive (bubbly) flow generally takes the form of a symmetrical cone around the injection point (Stöhr and Khalili, 2006; Geistlinger et al., 2006; Peterson et al., 1999; Ji et al., 1993). For example, Geistlinger et al. (2006) observed conically shaped gas structures during gas injection into water-saturated tanks of glass beads with 1- and 2-mm diameters (so-called glass bead sediments). Dumoré (1970) also observed a conical-shaped gas pattern during dispersive (bubbly) flow in a coarse grain model in the context of solution-gas drive. Accurate predictions of the expected flow pattern that emerges during air sparging in the field requires a model that is not only capable of capturing the full range of migratory patterns observed at the smaller (?centimeter) experimental scale, but one that is also readily 2010, Vol

2 scaled up to facilitate prediction at the larger (?meter) scale. Although the pore network modeling approach has previously been adapted to model gas migration in porous media (Bondino et al., 2007; Tsimpanogiannis and Yortsos, 2004; Meakin et al., 1992, 1999; McDougall and Mackay, 1998; Birovljev et al., 1995), it appears that no systematic modeling work has been published to date that describes the buoyancy-driven transitions from capillary-dominated growth to gravity-biased growth to nondispersive braided migration to conical-shaped, discontinuous dispersive flow. We believe that the model described here is the first pore network model to successfully highlight the physical mechanisms underpinning the full range of gas evolution behavior observed in the laboratory, including the important transition from nondispersive or slug flow to dispersive flow. The objectives of this work therefore were (i) to present a novel pore network simulator that captures the full range of experimentally observed migratory regimes, (ii) to offer insightful discussion of the pore-scale mechanisms responsible for these transitions, and (iii) to highlight how such information as may be derived from our network simulations can broaden our understanding of the migratory regimes associated with air sparging and related engineering applications. Model Descrip on Model Background The pore network simulator described here is an extension of a model originally developed in the context of reservoir depressurization by McDougall and Sorbie (1999), McDougall and Mackay (1998), and Bondino et al. (2005). The porous medium is modeled using a three-dimensional regular network of interconnected pore elements. Pore size distributions derived from either Hg intrusion experiments or void space image reconstruction (aimed at producing geometrically and topologically equivalent networks) are implemented by systematically assigning capillary entry radii, pore lengths, and shape factors. Pore-level visual experiments (typically from micromodels) and theoretical formulations representing the physical principles provide the foundation for multiphase mass transfer calculations and the governing displacement rules. The fundamentals of the pore network simulator described here are represented pictorially in Fig. 1. An internally seeded modified invasion percolation approach is adopted for modeling the advance of the gas liquid interface and the consequent displacement of liquid-filled pores by the evolving gas. Injection, depressurization, and repressurization processes can be modeled with the simulator (including nonequilibrium effects). A brief description of the key algorithms used follows additional details can be found in Bondino et al. (2005) and McDougall and Sorbie (1999). (which may be either live or dead). If the fluid is live (i.e., contains dissolved gas), then a free gas phase can nucleate from the liquid phase during pressure depletion once a certain level of supersaturation has been attained. Both live and dead fluids may be considered when modeling external gas injection. To better describe the range of pore-scale processes that occur during sparging operations, it is perhaps beneficial to first consider the physical mechanisms underpinning gas evolution during pressure depletion. The impact of these mechanisms on injection will be made clear later. The first depletion-related process to consider is bubble nucleation. Bubble nucleation in a liquid phase can occur progressively or instantaneously, depending on a range of system parameters (σ, the presence of impurities, the presence of solid surfaces, inter alia). Both mechanisms are accounted for in the model and the predominant mechanism will depend on the rate at which supersaturation builds up within the system, instantaneous nucleation essentially being a very rapid manifestation of progressive behavior. Nucleation is implemented numerically by distributing sites of increased nucleation potential, which can be thought of as microscopic gas-filled crevices. Bubble nucleation (or crevice activation) is assumed to occur when the local supersaturation exceeds the capillary threshold of the crevice (see Yortsos and Parlar, 1989). A crevice of radius ω is therefore activated as a result of a reduction in liquid pressure, when the following inequality is satisfied: 2σcosθ KC P [1] ω l where C is the local dissolved gas concentration, P l is the liquid pressure, K is the gas equilibrium constant, σ is the gas liquid interfacial tension, and θ is the contact angle. The number of activated cavities at any stage of depletion depends on both the crevice size distribution (predefined) and the degree of supersaturation (a Forma on of the Free Gas Phase At the start of a simulation, the network model is a single-phase system that is completely saturated with liquid Fig. 1. Integration of key information for the development of our process-based pore network simulator; MICP = Hg injection capillary pressure experiment

3 function of other factors including pore connectivity, depletion rate, and diffusivity). Mass Transfer, Bubble Growth, and Expansion Before the appearance of a free gas phase (either by injection or nucleation), the system is at equilibrium and all liquid pores have a constant dissolved gas concentration. Following nucleation, the concentration of dissolved gas at the gas liquid interface is assumed to immediately reach an equilibrium value determined by the current system pressure. This creates a concentration gradient in the system (as liquid-filled pores at a distance from the gaseous phase will be effectively supersaturated) and initiates the diffusion of dissolved gas toward gas liquid interfaces. The diffusion of gas through the liquid phase toward gas bubbles means that the dissolved gas concentration should be higher in regions away from the gas liquid interface. This is corroborated by the graphics in Fig. 2, showing dissolved gas concentration contours at different pressures during a depletion simulation. Mass diffusion is implemented in the model following Fick s laws. Consider two neighboring liquid-filled pores i and j (with dissolved gas concentrations C i and C j, respectively), separated by the distance L. The gas flux J ij (mass/time) across pores i and j is modeled according to Fick s first law: j ij ( j Ci) DC = [2] L Depending on the orientation of the local concentration gradient (toward or away from the pore of interest), positive or negative gas fluxes may exist. Typically, pores have different sizes (and hence different cross-sectional areas in contact with neighbors) and diffusion occurs across the minimum cross-sectional area between adjacent pores. In addition, the interconnectivity of pore network models means that the dissolved gas concentration in a given pore is a result of the simultaneous mass transfer across all of the neighbors. Knowledge of the mass flux, J, allows the computation of the instantaneous mass of solute, Δm, that can diffuse into a pore w after time Δt and is given by where V i is the volume of the liquid-filled pore. Early bubble growth periods involve the expansion of subpore bubbles and this stage is primarily controlled by diffusivity, resulting in an increase in the moles of gas in each nucleated bubble, n bub. Eventually, the host pore is filled by gas and further expansion is now controlled by both diffusivity and the entry thresholds of the neighboring liquidfi l l e dp o r e s.th e expansion of a gas bubble beyond its host pore requires that the gas liquid capillary pressure (P g P l ) exceeds the minimum entry threshold of the neighboring liquid-filled pores; otherwise, the gas bubble is treated as being constrained. A continued depletion of liquid pressure leads to additional diffusive mass transfer into gas bubbles, resulting in an increase in gas pressure hence, a constrained bubble will eventually become unconstrained as the depletion proceeds. Interface advancement into liquid-filled neighbor pore(s) is permitted once the local capillary pressure (P g P l ) exceeds the minimum entry threshold associated with the perimeter liquid-filled pores, i.e., when min g l c P P > P [5] In the absence of gravity or viscous forces, gas expansion is modeled as a pure invasion percolation process with one interface advancing at a time and P c min calculated from the Young Laplace equation: min c P σcos = min η θ i r i where σ is the gas liquid interfacial tension, η = 2 for circular cylindrical pore elements, and θ is the gas liquid contact angle (assumed = 0 in this study). In multiple bubble systems, bubble coalescence is permitted (Hoshen and Kopelman, 1976), with the gas volume and moles of gas treated as additive for updating the pressure of the new gas cluster. In the presence of gravity or viscous forces, the minimum capillary entry threshold is perturbed and other important phenomena such as bubble mobilization, fragmentation, and liquid reimbibition become essential elements of the model. [6] 6 Δm( Δ t) = Δt J min ( A, A ) [3] n= 1 n n w where A is the cross-sectional area, n represents an index running through all perimeter liquidfi l l e dn e i g h b o r so fp o r ew, and J n is the gas flux between pores n and w. The gas concentration in pore i is subsequently updated using t+δt t Δm( Δt) Ci = Ci + [4] V i Fig. 2. Dissolved gas concentration contours at different pressures during pressure depletion; the black color is the gas cluster (80 by 80 by 1 nodes); S g is gas saturation, R s is the initial gas concentration

4 Buoyancy Effect on Bubble Growth In the presence of gravitational forces, the minimum capillary entry threshold can be perturbed via the relation NET c P 2σ = min ΔρgH i i r i where Δρ is the density difference between liquid and gas, r is the capillary entry radius of a perimeter pore, g is gravitational constant, i is an index running across all liquid-filled perimeter pores surrounding the gas cluster under consideration, and H is the height of a liquid-filled pore above the bottom of the network (Tsimpanogiannis and Yortsos, 2004; Birovljev et al., 1995) (the bottom of the network is assumed to be at the current system reference pressure and liquid density decreases toward the top of network according to the hydrostatic gradient), and σ is the gas liquid interfacial tension. All other parameters being constant, gas bubbles nucleated at higher vertical elevations may be expected to expand more readily into liquid-filled pores due to the decreasing hydrostatic liquid pressure (increasing capillary pressure) toward the top of the system. Unlike capillary-dominated growth (invasion percolation without buoyancy or viscous forces [Wilkinson, 1984]), a gas liquid interface does not simply invade the perimeter pore with minimum entry radius it is possible for a smaller pore to be invaded before a larger pore or for several pores to be invaded simultaneously when gravity is operating. The latter case can occur during bubble migration and is discussed next. Buoyancy-Driven Migra on and Mul ple Pore Filling Visual experimental studies in the literature (Wagner et al., 1997; Birovljev et al., 1995; Mumford et al., 2009) have shown that buoyancy-induced mobilization of gas bubbles in porous media involves a series of drainage events, imbibition events, and the possible fragmentation of gas clusters. All three mechanisms are included in the pore network simulator described here. A gravitationally stable gas cluster grows by invading the perimeter liquid pore characterized by the minimum entry threshold defined in Eq. [7]. The onset of gravitational migration (mobilization), however, occurs when NET c P Δρghj 0 andbo= > 1 2σ r gl i where Bo is the local Bond number that quantifies the relative importance of gravity (buoyancy) and capillarity (note that Bo increases toward the top of a gas cluster). The net driving pressure for meniscus movement in the presence of gravity, P MIG, is calculated for each (untrapped) liquid-filled pore that has a mobile gas-filled pore as a neighbor according to MIG 2 ij ri P j [7] [8] σ = Δρ gh [9] where h j is the height of the liquid-filled pore above the bottom of the gas cluster being considered and other variables are as defined previously. Note that the height of a pore above the base of the model affects the net capillary entry pressure, whereas the buoyancy force acting on a bubble once it is mobile depends on the local bubble height. The algorithm described here extends the model described in McDougall and Mackay (1998), which restricted the number of migration events during a time step. The current model now allows the simultaneous mobilization of all gas liquid interfaces that satisfy the mobilization condition. Several migration events may be possible during a depletion step. The first liquidfilled pore to fill is identified based on the minimum filling time, t min. Migration events are accompanied by simultaneous liquid reimbibition and the possible fragmentation of gas clusters. A stepwise description of the dynamic migration algorithm following each pressure simulation time step is presented below. Step 1 Calculate the buoyancy pressure of each gas pore ( j); Δρgh j. The value of h j depends on the height of the gas pore relative to the bottom of its host cluster. Step 2 Identify all liquid-filled pores that are not trapped (i.e., surrounded by gas) and have a gas-pore neighbor and a P c NET < 0. Step 3 For each liquid pore identified in Step 2, assign a net driving pressure, calculated as MIG 2 ij ri P σ = Δρ gh [10] j where j is the gas pore neighbor of the liquid-filled pore, i. Step 4 Find the first pore to fill based on the minimum filling time, t min. The filling time is the time it takes for a gas bubble advancing at a pore-level velocity, v, to complete the filling of a pore of total length L i containing a gas liquid meniscus at a normalized position f i along the pore (initially, all pores are liquid filled, and so f i = 0 for all pores). That is to say, find L i tmin = min ti = min ( 1 fi) v i where MIG i ij [11a] gp vi = [11b] A where g i is the pore conductance and A is the pore body crosssectional area. Note that a volume-weighted effective viscosity, μ eff 600

5 = f i μ g + (1 f i )μ l, is used for the calculation of local pore conductance here and that the meniscus velocity depends on the buoyancy pressure gradient across a gas cluster of a given height. Step 5 Invade the liquid pore identified in Step 4 with gas and set its meniscus position, f i = 1.0 Step 6 Following the invasion of the relevant liquid pore in Step 4, all other perimeter gas liquid interfaces are advanced and the meniscus position updated via t f = f + f [12] ( ) new old min old i i 1 i ti Step 7 Reimbibe gas-filled pores with liquid. The liquid reimbibition process is modeled here following Lenormand and Zarcone (1984) and is governed by local cluster topology and local capillary pressure, with gas preferentially retreating from pores characterized by the highest local capillary entry pressure. Piston-like retraction of the nonwetting gas phase is assumed to occur in dangling gas-filled pores (i.e., gas pore throats that are completely surrounded at one end by oil-filled throats) where the capillary pressure is given by P ce pistonlike = 2P ce snap-off ; thus dangling gas-filled pores are more likely to be reimbibed by piston-like displacement before snap-off can occur (although this actually depends on the variance of the pore size distribution, as is shown below). In addition, a gravity term is included in the threshold condition for liquid reimbibition because capillary pressure (P g P l ) is lowest at the bottom of a gas cluster and so liquid is more likely to reimbibe there, all other things being equal this is the corollary of the discussion above for drainage (relating to gas invasion). Network Simula ons Core-Scale Simula on of Deple on As a preface to the studies of gravity-destabilized gas flow in disparate porous media (covering deeply buried, consolidated formations common in petroleum engineering and shallow, unconsolidated sands more relevant to groundwater studies), we conducted a corescale simulation of gas evolution during depletion in a consolidated network. Factors such as bubble density, depletion rate, diffusion coefficient, connectivity, and boundary conditions all affect the temporal evolution pattern of nucleated bubbles (Bondino et al., 2005). For the illustrative case presented here, 600 bubbles were nucleated at saturation pressure in a network completed filled with live liquid. Capillary radii were distributed using a uniform probability function, and a mean radius characteristic of Berea sandstone was selected (r min = 1 μm; r max = 30 μm). Temporal occupancy graphics following a two-dimensional, core-scale (100 by 10 cm) simulation of 72.5 Pa/d depletion are presented in Fig. 3. Network dimensions were 1000 by 100 nodes. The subpore bubbles grow initially by diffusive mass transfer from the live liquid and eventually coalesce as the bubble size increases. With further depletion, these small clusters are seen to coalesce and eventually form a continuous gas phase that spans the system from top to bottom (not shown). Step 8 Redo fluid clustering and update the total migration time thus far. Repeat Steps 1 to 8 until the total migration time (T) equals the simulation time step or until all P NET 0. Step 9 At this stage, migration events stop and the next simulation time step is taken. The algorithm described above thus allows for time-dependent interface advance and multiple pore-filling events during gravitational migration. Network simulations have been performed using the new algorithm for both depletion with mass diffusion and injection of gas into a dead-liquid-saturated network and the results are presented and discussed together with a comparison with experimental observations. Fig. 3. Simulated fluid occupancies at different stages of depletion for a gas liquid interfacial tension σ = 30 mn/m, 100- by 10-cm network model (two-dimensional coordination number z = 3.3) with mean capillary radius R m = 15.5 μm, slow depletion at 72.5 Pa/d, gravity 0, gas is white and liquid is black

6 Table 1. Summary of network simulation fluid parameters. Parameter Value Liquid saturation pressure (P s ), Pa 10,933 Gas liquid interfacial tension (σ gl ), mn/m 3 Liquid density (surface conditions), kg/m Gas density (surface conditions), kg/m Initial gas concentration (R s ), m 3 /m Fig. 4. Schematic description of the two-dimensional network setup designed to maximize gas migration observation window, gravity (g) is oriented downward. Model Design for Gas Migra on Studies Pore network simulations of both single-point gas injections and pressure depletions were used in this work to develop an understanding of regime transitions observed during unstable gas migration. Simulations utilizing two-dimensional, homogeneous network models have been set up in a manner schematized in Fig. 4, with the boundary effect being minimized by nucleating or injecting close to the bottom of the network and away from the sides. In all cases, gas is treated as a nonwetting phase and, to facilitate extensive parametric studies within a reasonable time frame, simulations were predominantly performed using smaller scale (1-cm) networks. Larger scale results are presented below. Gas Produc on Boundary Condi on The capillary characteristics of the water table (i.e., the boundary between the saturated and vadose zones) play a key role in determining the temporal evolution of vapor saturation in the vadose zone. In addition, temporal evolution of vapor saturation in the vadose zone may also be influenced by the production rate of the associated vapor extraction wells. To investigate this assertion, two types of gas production boundary conditions have been implemented in the pore network simulator: (i) boundary condition (BC) Type A, where gas is produced via a 20-node buffer; and (ii) BC Type B, where gas is produced via a single-node buffer. A summary of the base fluid properties utilized for the simulations is presented in Table 1 and highlights the large difference in density between gas and liquid. Gas saturation (defined as the macroscopic average across entire network, excluding the buffer) plots against time for the different production boundary conditions are shown in Fig. 5 following a depletion at 72.5 Pa/d. Gas saturation histories for both boundary types are similar up to 40 d of depletion (i.e., before the gas reached the outlet pores). Beyond this point, the effect of the production boundary condition is quite marked, with BC Type A showing a dramatic change in slope due Fig. 5. Gas saturation (S g ) vs. time (72.5 Pa/d depletion) for different gas production boundary conditions (BC) (gravity 0, 1- by 1-cm network, mean capillary radius R m = 321 μm, gas liquid interfacial tension σ = 3 mn/m). to the continued production of gas (and reduced back-filling of the network) from the 20-node buffer. In contrast, the slope of gas saturation is relatively constant for BC Type B, as a secondary gas cap formed in this case due to the rate of gas production significantly lagging the rate of gas migration (see Fig. 6). In relation to air sparging, although the area of influence of the migrating gas clusters in the saturated zone may increase for BC Type B, it can be envisaged that this may also result in the spreading of potentially Fig. 6. Occupancy graphics after 6, 9, and 49 d of depletion for the different gas production boundary condition (BC) types showing that boundary type only starts to affect the gas migration pattern after breakthrough; gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm), gas is white and liquid is black

7 dangerous vapors in the vadose zone. In light of these observations, it is evident that the rate of fluid withdrawal has major implications for the redistribution of the injected gas. Parametric Studies with Boundary Condi on Type B (Single-Node Buffer) Mean Capillary Radius Effect As a preface to investigating gravity-unstable gas migratory regimes, a control simulation was performed ignoring gravity forces to highlight the base-case gas evolution pattern under full capillary domination. A single bubble was nucleated as shown in Fig. 4 and the liquid pressure was dropped below the saturation pressure at steps of 72.5Pa/d. Figure 7 shows the dendritic pattern of gas growth during quasi-static, capillary-dominated growth. To evaluate the effect of mean capillary radius on gas evolution, a uniform pore-size distribution function was used. Starting from a base distribution, the mean capillary radius was sequentially increased while keeping the distribution variance constant. This first sensitivity is summarized in Table 2. Following the nucleation of a single embryonic bubble (near the bottom of the network), gas is supplied to the developing bubble by neighboring liquid pores at a rate controlled by the diffusion coefficient, the gas liquid interface area, and the diffusion length. Since gravity is active here, local Bond numbers can be measured for each gas structure throughout the course of a depletion. To ascertain the relative strengths of the gravity and capillary forces during depletion, the maximum local Bond number (BoMax), taken across all gas structures in the system, can be calculated. At the scale under consideration here (1 cm), the BoMax remained less than unity throughout the base case simulation (mean capillary radius R m = 21 μm). As the mean capillary radius was increased further, however, the average local Bond number in the system also increased and gravitational migration became more apparent. Figure 8 shows the pressure history of the maximum local Bond number from simulations characterized by different mean capillary radii. These figures give a direct visual measure of periods of spontaneous gas migration. It should be noted that the nonmonotonic behavior evident in Fig. 8 for the base 25 case (R m = 321 μm) relates to reductions in the local Bond number due to temporary reductions in cluster height when fragmentation occurs during migration (see Eq. [8]). Hence, migration during a depletion step will cease when all values of the local Bond number fall below unity. Figure 8 also clearly demonstrates that, for this particular system, the migration condition is satisfied for the base 3 (R m = 64.2 μm) simulation with less fragmentation. To extend this discussion, observations will be classified by means of the maximum local Bond number observed during a simulation. Bond Number 1.0. For the base case with gravity activated, cluster growth becomes gravitationally biased, resulting in the formation of thin gas fingers, i.e., a fingering regime emerges. Clearly, for network systems with a mean capillary radius much less than Fig. 7. Gas and liquid occupancy at pressure P = 2958 Pa for a fully capillary controlled simulation, gravity = 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm), gas is white and liquid is black. Table 2. Sensitivity parameters for mean capillary radius ( R ) effect study. Case R R min R max Pore-size distribution variance μm Base Base Base Base the base case, the effect of capillarity on growth of the gas cluster would be even more significant and growth should approach the capillary regime. Bond Number > 1.0. As the mean capillary radius is increased (base 3), the Bond number exceeds unity after 1363-Pa depletion (i.e., P g P l = 1363 Pa). It was observed that although gas migration occurred in this case, no dispersive pattern was observed, only braided migration. For a still larger mean capillary radius (base 25), Bond numbers exceeded unity far sooner and a clear dispersive pattern of gas topology emerged. A closer look at the gas liquid occupancy graphics (Fig. 9) also reveals that the average size of migrating clusters decreases with an increase in the mean capillary radius. This can also be interpreted as the size of migrating clusters decreasing with increasing buoyancy influence (Bond number). Such an interpretation is validated when Fig. 8. Pressure (liquid pressure, P) history of the maximum Bond number (BoMax) for network models with varying mean capillary radius, gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm)

8 Fig. 9. Gas and liquid occupancy at different pressures (P) showing the evolution of flow patterns with varying mean capillary radius (R m ) and maximum Bond number (BoMax), gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm), gas is white and liquid is black. mean radii. For migration to occur in a system with a smaller mean capillary radius, the gas cluster must grow to a larger height. As a result, there is an increased number of dangling gas-filled pores associated with this type of gas structure the structure is stable for a longer time and so is able to explore more of the surrounding pore space. Preferential reimbibition into the dangling pores therefore results in less fragmentation of the initial structure and this, in turn, reduces dispersion. The converse argument can be adopted for a system with a large mean capillary radius, where migration thresholds are exceeded earlier and fragmentation is likely to occur to a larger extent in such systems (remember that the inclusion of multiple pore-filling capabilities in our buoyancy model allows several pores that satisfy the migration condition to be invaded simultaneously within a depletion step and this is accompanied by an increased number of liquid reimbibition events). Hence, for two systems with the same pore-size distribution (PSD) variance but different mean capillary radii, dispersion is more likely to occur in the system characterized by the larger mean radius. compared with the experimental results of Birovljev et al. (1995), where similar behavior was observed. The foregoing results point to the fact that four distinct evolution regimes can be identified when gas displaces a liquid in a porous medium: (i) the capillary regime; (ii) the fingering regime; (iii) the braided migration regime; and (iv) the dispersive regime. An important question to address at this stage is, what are the porescale explanations for the transition from static to dynamic growth and from braided to dispersive flow in two systems? The first aspect is clearly answered with recourse to Eq. [7], which simply shows that a gravitational perturbation applies throughout any gas liquid system, with this perturbation increasing with increasing cluster height. The second, however, requires additional discussion. Figure 8 shows that, as expected, the migration condition was first exceeded at a higher pressure for systems characterized by larger Dependence of Gas Satura on on Regimes The impact of migration regimes on the average gas saturation was also evaluated. Figure 10 shows plots of the average (S g ) and final network gas saturation (S gf ) vs. pressure for models with different mean capillary entry radii. An important observation is that the final gas saturation decreases initially with increasing mean capillary radius and then increases for a larger mean radius. It is important to recall that gas production is allowed through the top of the system via a single-node buffer. If the mean capillary radius is very small and capillarity dominates gas evolution, then gas clusters are more compact and large regions of the network are efficiently invaded by the gas, resulting in a relatively large S gf. As the mean capillary radius increases, however, the relative strength of gravitational forces also increases and this initially results in vertically biased gas growth and fingering, leading to decreased sparging Fig. 10. (a) Average (S g ) and (b) final network gas saturation (S gf ) vs. pressure (P) history showing the effect of the mean capillary radius (R m ) on S g during depletion at 72.5 Pa/d; gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm). Shaded region in (b) shows trends in the S gf during buoyancydriven gas migration occurring when the Bond number (Bo) is >

9 efficacy. The minimum in the S gf plot corresponds to a mean entry radius where gas growth is highly unstable, resulting in braided migration without dispersion. Fragmentation occurs to a lesser degree at this radius due to preferential reimbibition into dangling pores. For still larger mean radius values, significant fragmentation and dispersion occurs. The fragmented bubbles (essentially constituting an increased bubble density) eventually grow and may coalesce, resulting in further migration, reimbibition, and fragmentation. Repeated cycles of this phenomenon during pressure depletion results in a migratory pattern with increased gas saturation, increased area of influence, and, consequently, increased sparging efficacy. In summary, the observed increase in final gas saturation for a very large mean capillary radius coincides with the onset of dispersion in the network. These results are in agreement with the experimental observations of Dumoré (1970), which showed that gas saturation is significantly higher with dispersion compared with braided migration. events following each depletion step also decreased. Hence, for systems with a large mean radius and similar variance, the extent of dispersive flow decreases with decreasing connectivity, as observed in Fig. 12. As the coordination number is further reduced to z = 2.33, we observe a transition from dispersive flow to braided migration. Again, this transition can be attributed to the reduction in the number of liquid-filled neighbors adjacent to migrating gas clusters. A nondispersive migratory pattern is therefore more likely to occur in a poorly connected system simply because of the limited availability of potential migratory routes. These results clearly show that the mean capillary radius is not the only factor that determines the transition from nondispersive to dispersive migration additional properties of the system also play important and subtle roles. Having previously established the effect of dispersion on the average gas saturation, it is not surprising therefore that the final gas saturation decreased with decreasing coordination number; this figure is not shown for brevity. Impact of Pore Connec vity To evaluate the effect of the average network connectivity on the migratory regime, we compared simulations using networks with two-dimensional coordination numbers of 2.33, 2.67, 3.33, and 4.0. The average coordination number was reduced by randomly removing bonds from the network. It is well known that a reduction in pore connectivity can lead to a slowing of gas evolution. Here, however, we were particularly interested in how connectivity impacts the onset of dispersive, gravity-driven flow. A fully connected network model with a large mean capillary radius (R m = 321 mm) was selected as the new base case simulation to ensure that the sensitivity study began in the dispersive regime. Figure 11 shows the maximum local Bond number at different pressures from networks with different average coordination numbers. The first point to note is that all simulations satisfied the condition for spontaneous migration (Bo > 1.0) at some point during the depletion, with the onset of migration delayed as connectivity decreased. When connectivity was reduced from 4 to 3.33 to 2.67, the number of liquid-filled neighbors of a gas cluster subject to migration was steadily reduced. Consequently, the number of migration The Effect of Pore Size Distribu on Variance We explored the effect of PSD variance by keeping the mean capillary radius (R m = 107 mm) and connectivity (z = 4) constant for this sensitivity study. The effect of increasing and decreasing the variance by a factor of 10 from a base value of 114 was investigated. It should be noted that the PSD variance also directly represents the variance of the local capillary pressure thresholds. A plot of the maximum local Bond number vs. pressure for each case (Fig. 13) once again shows that all simulations exhibited some degree of migration. Figure 14 shows that for a constant mean capillary radius, a transition from nondispersive to dispersive migration Fig. 11. Pressure (P) history of the Bond number (Bo) for network models with varying average coordination number z; gravity 0 (two-dimensional, 1 by 1 cm). Fig. 12. Gas and liquid occupancy at the end of the simulation, showing the evolution of flow patterns for different average coordination numbers z; gravity 0 (two-dimensional, 1 by 1 cm), gas is white and liquid is black

10 Fig. 15. Pressure (P) history of gas saturation (S g ) for simulations with different gas liquid interfacial tensions showing non-monotonic final gas saturation; gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm). Fig. 13. Pressure (P) history of the Bond number (Bo) for network models with constant mean capillary radius and different variances; gravity 0, two-dimensional coordination number z = 4 [(twodimensional, 1 by 1 cm). Fig. 14. Gas and liquid occupancy after 5800-Pa depletion showing the effect of pore-size distribution variance (σ 2 ) on the development of gas migration patterns; gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm), gas is white and liquid is black. Fig. 16. Gas and liquid occupancy after 1740-Pa depletion showing the effect of the gas liquid interfacial tension (σ gl ) on the development of gas migration patterns; gravity 0, two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm), gas is white and liquid is black. takes place as the PSD variance is decreased. When the PSD variance is small, it is probable that many pores will exceed the migration threshold at the same time, leading to many more simultaneous meniscus movements and increased gas fragmentation. Effect of Gas Liquid Interfacial Tension For the sensitivities investigated here, we assumed pressureindependent gas liquid interfacial tension. A fully connected network was considered with R m = 42 μm, and the gas liquid interfacial tension was varied from 0.1 to 300 mn/m. Intuitively, in the absence of gravitational migration, decreasing gas liquid interfacial tension (σ) should result in increased gravitationally biased gas fingering and, consequently, should be expected to lead to a decrease in sparging efficacy. Simulation results show that, for the scale under consideration, the maximum local Bond number remained less than unity for high gas liquid interfacial tensions between 10 and 300 mn/m. Consequently, we do not expect any gravitational migration to occur under these conditions (σ 10 mn/m) at the 1-cm scale. This is consistent with an initial decrease in S gf as σ decreased (Fig. 15). Local Bond numbers did exceed unity at lower σ (σ = 1 and 0.1 mn/m), however, and the associated dispersive gas evolution led to an increase in S gf (Fig. 15). Figure 16 shows snapshots of gas buildup for different interfacial tensions after 7540 Pa of depletion. The value of S g again clearly increases when dispersion occurs. It may be 606

11 concluded, therefore, that different gas evolution regimes emerge depending on the gas liquid interfacial tension characterizing the fluid system. The impact of pressure-dependent interfacial tension on evolution would be very intriguing, with the regime possibly varying as the depletion proceeds. This will be examined in more detail in future work. Impact of Gas Produc on Boundary Condi on on Final Gas Satura on A comparison of S gf using BC Type A (gas production via a 20-node buffer) exhibits some differences when compared with observations above using the single-node buffer, as the dispersive regime is entered. With all other parameters remaining fixed (and interfacial tension = 3 mn/m), increasing mean capillary radius beyond 100 μm with increased production capacity leads to a reduction in S gf (Fig. 17a). This contrasts with the single-node buffer results and is due to migrating bubbles now being able to completely leave the network instead of forming a gas cap because of reduced production capacity. Indeed, the production capacity of a core experiment (which is tied to the pump withdrawal rate) could be an important factor in promoting gas cap formation, leading to inappropriate sparging efficacy predictions. In addition, when moving from low to high coordination number (mean radius fixed at 321 μm), the final gas saturation initially increases as the gas evolution moves toward a more dispersive regime. Further increase in the coordination number for BC Type A (increased production capacity via the 20-node buffer) shows, however, that S gf begins to decrease again. This is because all the bubbles reaching the outlet are effectively produced and are therefore unable to promote gas cap formation (Fig. 17b). A similar trend was observed by varying σ for a fixed mean capillary radius. Within the dispersive regime, decreasing σ essentially increases the frequency of migration of small bubbles, leading to a decrease in S gf at low σ values bubbles have little time to expand before they are again mobilized and produced from the outlet buffer. This provides a possible explanation for the observations of the gas saturation profiles presented by Kim et al. (2006, 2004), who investigated the σ effect for different outlet boundary conditions. Modeling a depletion process involves a finite mass of solute, with nucleated gas bubbles simply growing via diffusive mass transport. Experiments involving gas injection into dead liquid represent a rather different process because any given gas cluster can only grow by means of either the external source or bubble coalescence during migration (assuming that diffusion across gas liquid interfaces remains insignificant). We now present a comparative simulation by scaling the average network Bond number to that characterizing the experiment of Dumoré (1970) and by modeling single-point gas injection (this is achieved by setting the gas liquid diffusion coefficient in the modified depletion simulator to m 2 /d and using a single bubble nucleus as an injection port through which gas is injected). An appropriately scaled 1-cm-high simulation model (see Eq. [13]) of the experiments was run by injecting a fixed mass of gas ( kg) per day at a centrally positioned pore close to the network inlet (as per the Dumoré experiment): Δρgh R Δρgh R = 2σ 2σ sim avg exp avg sim exp [13] At the start of gas injection, the gas cluster is initially pinned in the host pore by capillary forces. Following continued injection, the gas pressure eventually exceeds capillary resistance and the initial pattern of expanding gas cluster is dendritic. Buoyancy forces eventually exceed capillary forces, however, as the height of the injected gas column increases (see Eq. [7]). This is accompanied by mobilization of the gas cluster and subsequent fragmentation. We clearly observe that, following buoyancy-driven gas mobilization, a dispersive pattern emerges for an absolute permeability of K = m 2 (350Darcy in the Dumoré experiment) and a conical shaped gas envelope is seen to extend toward the top of the system, while a discontinuous slug flow occurs in the lower permeability case, with K = m 2 (15Darcy in the Dumoré experiment) (Fig. 18). This compares very favorably with the corresponding experiments and provides a reasonable validation of our model. These results highlight the strong influence of system permeability on gas migratory regime. Single-Point Gas Injec on Simula ons Thus far, the results presented have been associated with pressure depletion in porous media. While many of the same mechanisms could be expected to operate during gas migration in the context of injection protocols, some additional injection simulations are nevertheless presented for completeness. Fig. 17. Comparison of variation of final gas saturation for boundary condition (BC) Type A and Type B vs. (a) mean capillary radius (R m ), and (b) average coordination number z; gravity 0 (two-dimensional, 1 by 1 cm)

12 Core-Scale Simula ons of Regime Transi on Before proceeding, it is important to state that several realizations of each simulation were evaluated to confirm that no observation was pathological. Before this point, we utilized relatively small networks (1 cm) and small gas liquid interfacial tensions to capture the full range of migratory regimes. While this may be suitable for examining many near-critical fluids, we must expand our length scale of investigation to 100 cm to investigate migratory regimes associated with gas water systems. To this end, pore lengths were fixed at 1 mm and a two-dimensional, 100- by 10-cm network was constructed. We utilized a mean capillary radius of 210 μm calculated from a 1.0-mm mean grain size bead pack, as reported in Geistlinger et al. (2006). In reality, porous media are unlikely to be fully connected, prompting our use of a system with a two-dimensional average coordination number of 3.3. Interfacial tension values were varied between 50 and 70 mn/m to capture the upper range of σ where discontinuous flow has been reported in similar PSD systems (Glass et al., 2000; Birovljev et al., 1995). Simulations were designed to mimic lowrate depletion experiments, with vertically oriented core samples and fluid withdrawal from the top of the system (via a 20-node buffer). Phase occupancy graphics (Fig. 19a) show that for these practical parameter combinations, a dispersive regime occurred for a gas liquid interfacial tension of 50 mn/m. Interestingly, increasing the interfacial tension to 70 mn/m resulted in a dramatic change in the migration pattern (Fig. 19b), which may be described as a transition to discontinuous braided migration. These results are highly encouraging when compared with the experimental observations of Glass et al. (2000), who utilized a bead pack with a 1.1-mm mean grain size. In this study, discontinuous channelized flow was observed for a CO 2 water system with σ = 71 mn/m, and more dispersed migration was seen for a trichloroethylene water system with σ = 27 mn/m. The implication of the regime on gas saturation is obvious. Fig. 18. Network model simulation of constant-mass, single-point gas injection for different absolute permeabilities (K) at time t = 2, 3, and 7 d and the end of the experiment (Dumoré, 1970); gravity 0, (two-dimensional coordination number z = 4 (two-dimensional, 1 by 1 cm), gas is white and liquid is black. The exact parameter combinations that characterize regime transition have not been exhaustively resolved experimentally in the literature (and indeed may be very difficult to resolve satisfactorily due to the uniqueness of different experiments). It is highly encouraging, however, that our core scale network model, parameterized within realistic experimental ranges, predicts realistic flow regimes. Conclusions In this study, we have developed and applied a pore-scale network simulator capable of reproducing a wide range of migratory behavior observed during buoyancy-driven gas evolution in liquid-saturated porous media. Our simulation results provide Fig. 19. Simulated fluid occupancies at different stages of depletion in terms of pressure (P) for a gas liquid interfacial tension σ = 50 mn/m (a) and (b) σ = 70 mn/m; twodimensional, 100- by 10-cm network model (two-dimensional coordination number z = 3.3) with mean capillary radius R m = 210 μm, slow depletion at 72.5 Pa/d, gravity 0, gas is white and liquid is black