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 Carbon capture and storage is currently the only technology that may allow significant reductions in CO2 emissions from large point sources. Seismic images of geological CO2 storage show the rise of CO2 is influenced by horizontal shales. The buoyant CO2 spreads beneath impermeable barriers until a gap allows its upward migration. The large number and small scale of these barriers makes the prediction of the CO2 migration path and hence the magnitude of CO2 trapping very challenging. We show that steady buoyancy dominated flows in complex geometries can be modeled as a cascade of flux partitioning events. This approach allows the analysis of two-dimensional plume dispersal from a horizontal injection well. We show that the plume spreads laterally with height y above the source according to (y/h)1/2L, where L is the width of the shales and h is their vertical separation. The fluid volume below successive shale layers, and therefore the magnitude of trapped CO2, increase as (y/h)5/4 above the source, so that every additional layer of barriers traps more CO2 than the one below. Upscaling small scale flow barriers by reducing the vertical permeability, common in numerical simulations of CO2 storage, does not capture the dispersion and trapping of the CO2 plume by the flow barriers.
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 Carbon capture and storage (CCS) is currently the only technology that may allow significant reductions in CO2 emissions from large point sources, in particular coal-fired power stations (for an introduction, see Metz et al. ). The technology calls for the capture of CO2 from flue gases and the injection into underground geological formations for permanent storage. Given the drastic projected increase in coal-fired power generation and other CO2 emissions in the next decades, CCS is considered to be “the only way forward” by the previous chief scientific advisor of the UK (see D. King, Scientist hopes for CO2 storage, BBC News, 6 December 2005, http://news.bbc.co.uk/2/hi/uk_news/4501964.stm), and a “grand engineering challenge of the 21st century” by the US National Academy of Engineering (http://www.engineeringchallenges.org/). The buoyancy of the injected CO2 provides a driving force for leakage back into the atmosphere, which may create a dangerous legacy of CO2 emissions for future generations. The leakage of CO2 back into the atmosphere may be prevented by the formation of disconnected immobile residual CO2 in the wake of a migrating CO2 plume [Kumar et al., 2005; Juanes et al., 2006] and dissolution into the brine [Ennis-King et al., 2005; Riaz et al., 2006]. The special report on CCS by the Intergovernmental Panel on Climate Change (IPCC) has identified such trapping processes as the key to long-term storage security.
Figure 1 illustrates that storage aquifers generally include thin, interbedded, and laterally extensive layers of impermeable shale [Chadwick et al., 2006] or other low porosity materials which may form barriers to vertical CO2 migration due to high capillary entry pressures [Saadatpoor et al., 2009; Woods and Farcas, 2009]. These barriers lead to substantial lateral spreading and capillary trapping of the CO2 as it rises through the formation (Figure 1). Numerical simulations [Johnson et al., 2004; Audigane et al., 2007] and theoretical models, reviewed in section 2, developed to describe the CO2 migration in layered rocks show that these flow barriers can have a dominant effect and present a key challenge in determining the capacity and security of potential storage aquifers.
 While Johnson et al.  and others have shown that it is possible to explicitly model flow barriers and their effect on CO2 migration in numerical simulations, very high numerical resolution is required. Given the large dimensions of storage aquifers and the long time scales of CO2 migration the permeability field is typically upscaled to a very coarse grid. On the coarse grid the layering is represented by an anisotropic permeability tensor [Durlofsky, 1991]. Using an idealized geometry we show that this simplification is not appropriate, if the flow is dominated by the buoyancy of the injected fluid.
2. Previous Work
 Two types of analytical models for buoyant CO2 migration in layered media have been proposed: vertical one-dimensional models of gravity-segregation of CO2 and water (for introduction, see Barenblatt et al. ) and gravity current models of CO2 migration (for an introduction, see Woods ). Silin et al.  and Hayek et al.  have presented analytical and semi-analytical solutions for the Riemann problems arising from vertical two-phase flow in a porous medium. They show that even a moderate reduction of permeability leads to the formation of a layer of high CO2 saturation beneath the flow barrier. The flow barriers are often laterally extensive, and flow along the barriers cannot be captured by one-dimensional models. In these cases the lateral migration of the CO2 is much longer than the vertical migration, and the flow can be modeled as a series of gravity currents. Pritchard  and Neufeld and Huppert  have considered lateral spreading of a gravity current beneath continuous flow barrier with some seepage through the barrier. However, in two-phase flow the seepage of CO2 through flow barriers is prevented by the capillary entry pressure [Woods and Farcas, 2009; Saadatpoor et al., 2009], and the buoyant rise of CO2 depends on lateral discontinuities in the flow barriers.
3. Model Problem
 We explore the steady shape of an idealized two-dimensional CO2 plume which develops during the injection of a constant volume flux per unit length, Q, from a long horizontal well into a reservoir which includes a set of horizontal flow barriers of characteristic length, L, much greater than h, the vertical spacing of the layers containing the barriers. Within a layer the barriers are separated by regularly spaced gaps of width, G ≪ L, which allows the fluid to drain upwards (Figure 1). In this flow geometry, the offset in the position of the gaps between layers is key. As the CO2 rises, it spreads laterally under the barriers; this generates an increasing number of smaller plumes that disperse the CO2 laterally as it ascends. Each barrier divides the incident flux per unit length Qi into Qa and Qb as shown in Figures 2a–2d, which can be estimated using the Dupuit approximation [Bear, 1972; Huppert and Woods, 1995], given by
where k is the effective permeability of the rock and α ∈ [a, b], Δρ is the density difference between water and CO2, g is the acceleration of gravity, the CO2 viscosity, η is the current depth, and η(0) is the current depth at the point of impact of the plume on the barrier. Equation (1) implies that the fraction fb of the incident flux per unit length Qi which spills over the right end of the layer at x = Lb is given by
Figure 2e illustrates that equation (2) describes the experimentally measured partitioning of the flux in a glycerol current spreading over a barrier in a Hele-Shaw analogue of buoyancy driven flow in a porous medium [Bear, 1972].
4. Buoyant Dispersal
Figure 3 shows experiments in which a constant flux of glycerol was supplied at the top of a Hele-Shaw cell containing N = 8 horizontal layers of barriers with the gaps off-set so that (1) fb = 0.5 (Figures 3a–3c) and (2) fb = 0.25 (Figures 3d–3f). Case 1 is similar to the conceptual scenario shown in Figure 1a, and case 2is similar to Figure 1b. At the nth layer the plume intersects n barriers and from Pascal's triangle, it follows that the flux per unit length incident on the kth barrier (with k = 1, …, n corresponding to the barriers which interact with the plume) is given by
Our experiments are therefore a deterministic analogue to the classic probability experiment known as the bean machine, Galton Box, or quincunx [Bulmer, 2003]. Figures 3a and 3d show that the distribution of the experimental fluxes draining off the last layer matches the binomial distribution in both cases. Curiously, the center of mass of the vertical flux always remains directly above the source even if fb ≠ 0.5; this is because the fraction dispersed to the right side of each barrier, fb, migrates a distance (1 − fb) L laterally, while the fraction dispersed to the left side, 1 − fb, only migrates a distance fbL. The standard deviation of the incident fluxes increases with layer number n according to L.
 If the number of shale layers n is sufficiently large, the binomial distribution for the flux may be approximated by the normal distribution, leading to the expression for the flux density (i.e. flux per unit area), q(x, y), in terms of the position (x, y) in the plume, as
where h is the spacing between the layers, and L the lateral extent of the barriers. Equation (4) demonstrates that the plume continues to spread laterally as it rises, with characteristic width
This macroscopic dispersion associated with the presence of barriers is different from the capillary dispersion in immiscible two-phase flow in porous media and may exceed it by several orders of magnitude depending on the geometry of the barriers.
5. Plume Area and Trapping
Hesse et al.  have shown that the amount of residual trapping is the product of the vertical sweep of the plume and the magnitude of the average residual saturation. The residual saturation that forms after the end of injection and subsequent upward drainage of the CO2 is therefore proportional to the volume per unit length of the steady plume. The agreement between observed and theoretical current shape (Figure 2) suggests the volume per unit length of the steady current which spreads out beneath the kth barrier in the nth shale layer can be found from equation (1) and has the form
Figure 4 shows that the theoretical prediction of the total fluid volume per unit length in an N layer system, VN = Σn ΣkVnk, is in accord with our experimental observations. Taking the limit of a large number of layers, and combining equation (6) and equation (4), we find that the total volume of the plume increases with height above the source as (y/h)5/4. This scaling is observed in both the discrete model and the experimental data shown in Figure 4.
 In many geological systems, the flow may persist for considerably longer than the transient time to establish the flow. The steady flow pattern shows that the barriers disperse the CO2, in the sense that the flux density q is determined by a diffusive rather than an advective process. This distinction is important because small scale flow barriers are often represented by a reduced effective vertical permeability in reservoir simulation [Durlofsky, 1991]. This may be appropriate for pressure driven flows, but not for buoyancy dominated flow. In this case the upscaled model should lead to the introduction of a cross-flow diffusive term rather than a reduction of the vertical advection.
 Residual trapping is an important trapping mechanism and most residual trapping occurs during the buoyant rise of CO2 from the injection point to the top of the aquifer. It is therefore important that the trapping induced by small flow barriers is upscaled appropriately to capture the scaling demonstrated above. The practice of reducing vertical permeability to account for horizontal flow barriers is unlikely to accomplish this.
Hesse et al.  have argued that low viscosity of the CO2 relative to the ambient brine requires that the displacement is modeled as confined even if the CO2 plume is not in contact with the bottom boundary of the aquifer. An unconfined model was chosen here to simplify the experimental work and the basic theory. To extend the ideas presented here to confined geometries, capillary dispersion, loss due to dissolution, or sloping barriers, the model of the flux partitioning, given by equation (2), has to be changed. Large viscosities of the ambient fluid may lead to some coupling between the plumes.
 We have introduced a new modeling paradigm for steady buoyant fluid rise, and verified it against experiments of two-dimensional plume dispersal from a horizontal injection well. We show that steady buoyant fluid rise can be modeled as a cascade of independent flux partitioning events. This paradigm provides a flexible framework to investigate the interplay of complex geometries and physical processes. The model of two-dimensional plume dispersal from a horizontal injection well highlights two important principles of buoyant rise of CO2, that can be quantified in the new modeling paradigm. We show that flow barriers, of lateral extent, L, and vertical separation h, lead to dispersion of the flux according to (y/h)1/2L and the volume of the steady plume increases upwards as (y/h)5/4, where y is the height above the injection point. Numerical simulations of CO2 storage, which represent the flow barriers by a reduced effective vertical permeability do not capture these effects and may therefore miss-represent both the plume dispersal and the amount of CO2 trapped during buoyant rise.
 A.W.W has funding through the BP Institute and BP plc. M.A.H. was supported by a David Crighton Fellowship from the Department of Applied Mathematics and Theoretical Physics at Cambridge University.