Abstract
When CO_{2} is injected in saline aquifers, dissolution causes a local increase in brine density that can cause RayleighTaylortype gravitational instabilities. Depending on the Rayleigh number, densitydriven flow may mix dissolved CO_{2} throughout the aquifer at fast advective timescales through convective mixing. Heterogeneity can impact densitydriven flow to different degrees. Zones with low effective vertical permeability may suppress fingering and reduce vertical spreading, while potentially increasing transverse mixing. In more complex heterogeneity, arising from the spatial organization of sedimentary facies, finger propagation is reduced in low permeability facies, but may be enhanced through more permeable facies. The connectivity of facies is critical in determining the largescale transport of CO_{2}rich brine. We perform highresolution finite element simulations of advectiondiffusion transport of CO_{2} with a focus on faciesbased bimodal heterogeneity. Permeability fields are generated by a Markov Chain approach, which represent facies architecture by commonly observed characteristics such as volume fractions. CO_{2} dissolution and phase behavior are modeled with the cubicplusassociation equationofstate. Our results show that the organization of highpermeability facies and their connectivity control the dynamics of gravitationally unstable flow. We discover new flow regimes in both homogeneous and heterogeneous media and present quantitative scaling relations for their temporal evolution.
Introduction
Geological sequestration of carbon dioxide (CO_{2}) has been proposed as a technology to reduce the amount of CO_{2} emitted into the atmosphere^{1,2,3,4,5,6}. The main processes controlling the trapping of CO_{2} during geological sequestration are storage of supercritical CO_{2} in a gas cap, CO_{2} dissolution in brine, known as solubility trapping^{7,8}, residual trapping due to hysteresis^{9,10}, and mineral trapping by CO_{2} precipitation as secondary carbonates^{11}. In this work, we focus on solubility trapping, which can be enhanced when gravitational instabilities (fingering) are triggered by a local increase in brine density as CO_{2} dissolves into brine in the top of an aquifer. This RayleighTaylortype instability^{12} is sometimes referred to as gravitoconvective mixing and involves both diffusive and advective motion of dissolved CO_{2}, with advection being the dominant driving force. Whether the interface between CO_{2}bearing brine and fresh brine becomes gravitationally unstable depends on the ratio of advection to molecular diffusion (Rayleigh or Péclet numbers). When a fingering instability is triggered, dissolved CO_{2} is mixed throughout the aquifer at advective timescales, which can be much faster than diffusive transport^{12,13,14}. This improves the storage capacity of a given aquifer and decreases the leakage risk in case of cap rock failure.
There is extensive literature on gravitational instabilities^{14,15,16,17,18}. Stability analyses and numerical simulations predict a critical wavelength for the instability that depends on fluid and reservoir properties as^{13,14,15,16,17,18,19}:
with brine viscosity, μ, porosity, ϕ, CO_{2} diffusion coefficient, D, permeability, k, maximum density increase of the aqueous phase upon CO_{2} dissolution, Δρ, and gravitational acceleration, g. A critical onset time is predicted by:
Different proportionality constants c_{1} and c_{2} have been obtained by the different authors cited above.
It is evident from Eq. (1) and Eq. (2) that the critical wavelength and onset time depend on porosity and permeability. Since the spatial variability of porosity is much smaller than that of permeability, the latter is often used to represent the heterogeneity of porous media^{20,21,22}.
Various studies have characterized the transport of CO_{2}rich brine through heterogeneous systems. Using numerical simulations, Farajzadeh et al.^{23} studied the influence of spatial heterogeneity in permeability on gravitational fingering, and explained three different flow regimes for stable trapping of dissolved CO_{2}: fingering, dispersion, and channeling. Different models of permeability heterogeneity have been tested by previous authors^{24,25}. These studies concluded that permeability heterogeneity could have a significant impact on the onset, rate, and characteristic behavior of densitydriven flow of CO_{2}. On the other hand, theoretical^{26}, numerical^{27}, and experimental^{6} results suggest that densitydriven flow may not always lead to significant convective mixing especially in layered systems containing lowpermeability zones; CO_{2} may actually become immobilized in these zones. In order to quantify the effects of vertical heterogeneity, Green and EnnisKing^{28} developed a simple model consisting of a random and uncorrelated distribution of horizontal, impermeable barriers with a given overall volume fraction (0.04% and 0.15%). They found that in a homogeneous medium with equivalent effective vertical permeability, compared to heterogeneous medium, convection begins at later onset times.
In reality, geological heterogeneity is controlled by the spatial organization of sedimentary facies with different physical attributes, such as permeability^{29}. Facies distributions therefore affect species transport through a formation^{9}. Architectures of sedimentary facies can exhibit sharp discontinuities across boundaries between depositional features, for instance across the sandstoneshale contacts that are common in fluvial deposits. The complex patterns in such fluvial architectures^{30} exist in many candidate aquifers for CO_{2} sequestration^{9,31,32} and result in variable connectivity and tortuous flow pathways^{33}. Representing such discontinuous, correlated heterogeneity in reservoir simulations is nontrivial^{34}, and its influence on the characteristics of densitydriven flow has not been studied in detail. In this work, we generate many volumepreserving realizations of bimodal facies architectures using a Markov Chain approach combined with an indicator simulation with quenching. For comparison, homogeneous domains are defined by the equivalent geometric mean permeability for each realization. We then perform highresolution twodimensional simulations of gravitational fingering in both homogeneous and heterogeneous media to investigate the influence of faciesbased heterogeneity and connectivity on the transport of dissolved CO_{2}.
For accurate numerical simulations of gravitational fingering, the physics of fluid flow and thermodynamics have to be represented rigorously. Highresolution numerical discretizations are required to minimize numerical dispersion which can obscure the onset of smallscale fingers. We adopt a combination of the mixed hybrid finite element (MHFE) method –to simultaneously solve for continuous pressure and velocity fields– and a higherorder discontinuous Galerkin (DG) method for the species transport. Phase behavior is described by the cubicplusassociation (CPA) equation of state (EOS). CPAEOS has been shown to accurately reproduce measured densities of CO_{2}brine mixtures and the volume swelling due to CO_{2} dissolution^{1,35,36}. Volume swelling and the associated movement of the interface between free CO_{2} in a gas cap and the underlying brine, cannot be modeled by considering only the aqueous phase^{15} and replacing the gas cap with a constantCO_{2}composition Dirichlet boundary condition^{37}. In this work, we inject CO_{2} from the upper boundary at a sufficiently low rate that the simulated domain remains a single aqueous phase (i.e., without a capillary transition zone^{22}). We do not adopt the Boussinesq approximation in order to fully study the effects of fluid compressibility and the pressure response due to swelling effect when increasing volumes of CO_{2} are injected in a confined domain (impermeable boundaries) and subsequently dissolve in the brine^{1,18}.
Numerical Experiments
We consider CO_{2} sequestration in saline aquifers by injecting CO_{2} uniformly from the top without production of brine (alternatively, this can be interpreted as a CO_{2} dissolution rate from an overlying CO_{2}gas cap). The rate is kept sufficiently low (0.1% pore volume (PV) per year) to insure that brine is nearly CO_{2}saturated in the top but without forming a gas phase in the computational domain. Because saline aquifers are large, injection of 1–2% PV in 10–20 years is sufficient, and higher injection volumes result in excessive pressure increases. We model a 12 m × 30 m subdomain, discretized by a fine 240 × 300 grid and with a uniform (or heterogeneous average) porosity of 10%. The constant temperature is 77 °C and the initial pressure is 100 bar at the bottom. At these conditions, the pure and CO_{2}saturated brine densities are ρ_{w} = 978 kg/m^{3} and ρ = 978 kg/m^{3}, with Δρ = (ρ−ρ_{w})/ρ_{w} = 0.9%, for a CO_{2} solubility of 1.7 mol%. The diffusion coefficient is D = 1.33 × 10^{−8} m^{2} s^{−1}.
Whether spatial heterogeneity of an aquifer enhances densitydriven flow of CO_{2}rich brine depends on the volume fraction of highpermeability facies and their connectivity. Our longterm goal is to perform large two and threedimensional simulations with multiple facies types. In this preliminary study we show the fundamental importance of heterogeneities arising from facies architecture. We choose a parsimonious set of two facies types representing a bimodal architecture^{29,38}. Different structures are created using a Markov Chain model and indicator simulation with quenching in TPROGS^{39}. Rather than focusing on any single site or data set, we represent general permeability distributions consisting of high and lowpermeability facies. The highpermeability facies volume fraction is increased from 10% to 90% in increments of 10% (Fig. 1), which also increases the facies connectivity^{29,40}. In all cases, the mean length of the less abundant facies are 2 m in both horizontal and vertical directions representative of meterscale sedimentary features. For each volume fraction, we generate five realizations of facies structures and for each realization we superimpose a lognormal permeability distribution with a variance of 0.1 within each facies. The geometric mean of high and lowpermeability facies are 8.5 (5,000 md) and 4.6 (100 md), respectively^{29}. For comparison, we create homogeneous domains with the same mean permeabilities (Table 1).
Results and Discussions
To characterize densitydriven flow of CO_{2}, we use the method from a seminal paper by Sudicky^{41}. The spatial moments of the CO_{2} distribution provide an insightful measure for the transport of CO_{2} at various timescales. The first three central moments of the distribution are employed to define the spatial variance, , of the CO_{2} molar density as an indicator of vertical spreading (dispersion) and a reasonable proxy to mixing. The square root of spatial variance, σ_{z}, is defined as the dispersionwidth (scale) to determine the extent to which the descending plume of CO_{2} has stretched (i.e., the volume of CO_{2}enriched brine). These definitions are in line with models developed for flow and transport in systems having complex spatial variability in physical and chemical properties^{42,43}, but also take into account the density change due to dissolution and compressibility. In our analyses, we consider the dynamics from the first contact of CO_{2} with brine until or σ_{z} reach their maximum values –determined by domain dimensions– above which they plateau (see Methods). Next, we study the temporal evolution of and note that faster vertical motion implies reaching the maximum variance in a shorter amount of time (Eq. (7)). Fast mixing and spreading towards higher depths improve the longterm storage of dissolved CO_{2} throughout the aquifer^{6}, because CO_{2} is likely to remain in aquifer due to residual and mineral trapping even if the pressure were to drop due to a failure of the cap rock.
We summarize our main findings in Figs 2 and 3, which show the molar density of dissolved CO_{2} at 2, 6, 12, 15, and 25 years for heterogeneous media and their homogeneous equivalents. Densitydriven flow in homogeneous domains has been extensively studied before, though with simplifying assumptions in terms of boundary conditions and phase behavior. Phase behavior is critical because the driving force for gravitational instabilities is the density contrast, Δρ, upon CO_{2} dissolution. From our scaling analyses (Fig. 4) for fully compositional EOSbased simulations of compressible flow, we observe new flow regimes even for homogeneous domains due to the nontrivial evolution of maximum Δρ over time (Fig. 5). In the following, we first discuss these results for homogeneous media and then for different bimodal heterogeneous facies architectures.
Homogeneous media
We observe four distinct flow regimes in all simulations for homogeneous media, but most clearly identifiable at low mean permeabilities (Fig. 4a,b). Dependencies on the mean formation permeability can be related to the Rayleigh number, which is the ratio of buoyancy to diffusive flux: Ra = kgΔρH/μϕD, with H the depth of the domain^{44}. In this study, the Rayleigh number is equivalent to the Péclet number because the advective Darcy flux is predominantly gravitydriven.
Traditionally, a constant pressure and CO_{2}composition are assumed at the top boundary of a brinesaturated subdomain. Consequently, the density contrast between pure and CO_{2}saturated brine in the top remains at a constant, maximum value Δρ_{max}, which has been used to estimate t_{c}, λ_{c}, and Ra^{15}. For a nearly constant brine viscosity and CO_{2} diffusion coefficient, the critical wavelength and time only depend on rock properties k and ϕ. In our work, we continuously inject CO_{2} and the composition in the top depends both on the injection rate, and on the downward transport rate of dissolved CO_{2}. The latter also depends on permeability: for low permeabilities, buoyancydriven Darcy fluxes are slower and more injected CO_{2} accumulates in the top. This gradually increases the brine density and makes the dissolution front increasingly unstable (positive feedback). Conversely, for high permeabilities, downward transport of dissolved CO_{2} is faster, which decreases the CO_{2} composition and thus the unstable densitycontrast in the top (negative feedback). Swelling, and the associated pressure response, due to different rates of CO_{2} dissolution further complicate the evolution of Δρ_{max}. These different feedbacks between transport dynamics and the buoyant driving force result in the four flow regimes (Figs 4 and 5) that we discuss in the next paragraphs.
Initially, CO_{2} transport is controlled by diffusion as long as the CO_{2}front remains stable and advective flow is less than the diffusive flux (with small Δρ_{max} resulting in low Rayleigh/Péclet numbers). Molecular diffusion involves low dissolution and mixing rates. The front travels to a ‘diffusion penetration depth’ which obeys a ~(Dt)^{0.5} Einsteintype scaling^{45}. Likewise, the dispersionwidth shows a classical Fickian scaling of σ_{z} ∝ t^{0.5} in Fig. 4b. We also keep track of the vertical propagation speed of the most advanced finger tip, which is shown in Fig. 4d for one case. The tip velocities are consistent with our measure of the dispersion coefficient (). This first flow regime is observed in all homogeneous cases and as long as Rayleigh numbers are small, diffusiondominated transport is insensitive to permeability (and for nearly constant μ and D and small Δρ_{max}, also independent of Ra). Interestingly, while the total amount of introduced CO_{2} increases linearly with time (due to constant injection rate), the driver for advective buoyant flow, Δρ_{max}, also follows Fickian ~t^{0.5} scaling until it has increased sufficiently to trigger gravitational instabilities (Fig. 5). The onset time, t_{c}, of this next flow regime does depend on permeability (and Ra). The instability onset for the 60% volume fraction can also be observed in Fig. 3 after 2 years.
The onset of fingering can be recognized as the transition to a convectiondominated regime^{15} (high Rayleigh numbers), which manifests itself as a sharp increase in and σ_{z} (Fig. 4). This regime occurs after ~6 years in Fig. 3 for the 40% volume fraction. Table 1 summarizes the critical times from stability analysis^{13,14,19,46} as well as the critical times derived from the deviation of σ_{z} from Fickian scaling (inset of Fig. 4b). It has been recognized that the initial exponential growth of a dominant harmonic perturbation, as predicted by linear stability analyses, may not necessarily be apparent in numerical simulations or even in experiments. Observed t_{c} are generally higher by at least an order of magnitude^{1,14}. As discussed above, Eq. (2) also does not fully take into account the nonlinear evolution of Δρ_{max}. The dependence of critical times and wavelengths on permeability (Eqs (1)–(2), and Table 1) imply that higher permeabilities result in an earlier onset time with a larger number of smallscale fingers, illustrated in Fig. 3. However, Fig. 4b shows a new scaling relation of t_{c} ∝ k^{−1}, rather than the t_{c} ∝ k^{−2} from stability analyses in Eq. (2). This is because Δρ_{max} itself evolves. Since Δρ_{max}(t) ∝ t^{0.5} in the first regime (Fig. 5), we find that Δρ_{max}(t) ∝ k^{−0.5}. Using this scaling of Δρ_{max} in Eq. (2) is consistent with our independent finding for the scaling of t_{c} with k.
A second interesting new finding is that the initial scaling of the dispersionwidth in the fingering regime is σ_{z} ∝ t^{3.5} for all permeabilities. This is comparable to the Rayleighindependent convective mixing rate found in Hidalgo et al.^{12}. We argue that this independence from Rayleigh number is because , and the critical time t_{c} in Eq. (2) is related to a Rayleigh number in which H is the thickness of the propagating diffusive layer before the onset of instabilities^{47}, rather than the domain height. Because the initial diffusive layer grows as t^{0.5}, Δρ_{max}(t_{c}) × H(t_{c}) ∝ t_{c} ~ k^{−5} and the Rayleigh number at the onset of fingering (Ra(t_{c}) ∝ kΔρ_{max}H) becomes insensitive to permeability.
We observe a third flow regime that has not been clearly recognized in the literature before. Figure 4 shows an inflection point in the spatial variance curves, followed by another regime of diffusiontype scaling. This regime can be seen in Fig. 3 for a volume fraction of 10% after 25 years. The cause of this diffusive regime is threefold. First, persistent coalescence and merging of growing fingers involve lateral diffusion, which result in coarsened fingers. Second, as the fingers stretch, the compositional gradients across the finger surface become steeper due to compression, which enhances diffusive fluxes across the finger boundaries^{48}. Third, diffusion is further promoted by the nature of RayleighTaylor fluid dynamics: downward motion of (growing) dense fingers and subsequent upwards displacement of fresh lighter brine increase the surface area of the CO_{2} dissolution front. The evolution of mixing rates is controlled by the competition between the advective stretching mechanism and such lateral compositional diffusion^{12}. Following the onset of the previous regime, fast advective downward transport of fingers exceeds the diffusive rate of CO_{2} dissolution at the top. This depletion of CO_{2} concentrations in the fingers causes Δρ_{max} to drop soon after the fingering onset (Fig. 5). As the driving force for advection/buoyancy (Δρ_{max}) weakens, an inflection point is reached where diffusion exceeds advection and stretching stops. The inflection point corresponds to the local minimum of Δρ_{max} and to the onset of a diffusiondominated restoration regime (Figs 4d and 5), after which the dispersionwidth returns to diffusive scaling as well (Fig. 4b). This third regime is comparable to a stagnation point in the advective flow where the rate of compression balances the rate of diffusive expansion^{49}. During this transition period, CO_{2} transport is slower, concentrations are replenished (owing to the constant injection rate), and Δρ_{max} increases, similar to the first regime, until the dissolution front becomes gravitationally unstable once again.
The fourth regime is again convectiondominated and shows a sharp increase in the growth rates of the CO_{2} spatial variance and dispersionwidths. The duration of the first convective regime is shorter for highpermeability cases and the reduction in Δρ_{max} is smaller (Fig. 5). This regime occurs after 2 years for the 90% volume fraction in Fig. 3. This is in line with experimental results^{17} that show a reduced Δρ scaling as ~Ra^{−0.2} rather than a Rayleighindependent constant dissolution flux^{14,50}. Likewise, the third transition periods are shorter in highpermeability domains with less replenishment with CO_{2}. Counterintuitively, this results in finger growth rates that are slower for the highest permeabilities (σ_{z} ∝ t^{1.5}) than for the lowest permeability cases (σ_{z} ∝ t^{4}) in this regime. This suggests that while Rayleigh and Péclet numbers are in the convectiondominated range for all permeabilities, they are lower for the highpermeability than for the lowpermeability domains, due to the complex fingerbrine interface at the end of the previous regime and the different depletionreplenishment histories of CO_{2} concentrations (and densities) inside the fingers.
Heterogeneous media
In the previous section, we considered densitydriven flow in homogeneous aquifers and found four alternating diffusion and convectiondominated flow regimes, the extent of which depends on Rayleigh numbers. Next, we investigate whether similar behavior persists in realistic heterogeneous media that have regions of different facies with tortuous and potentially connected pathways. We expect that our global measures of CO_{2} spreading will be a mixture of the flow regimes observed for a range of homogeneous mean permeabilities. How the gravitoconvective mixing of dissolved CO_{2} is controlled by realistic permeability fields may dramatically affect the longterm CO_{2} storage efficiency.
Comparing Figs 2 and 3 shows that fingering is more pronounced in the heterogeneous media. This is different from flow in unimodal heterogeneous media, characterized by (pressuredriven) permeabilitychanneling that dominates (gravitydriven) hydrodynamic fingering. The reason for this is the spatial organization of facies: for a given mean permeability, the corresponding bimodal facies distribution has both regions with much higher and with much lower permeabilities. In large connected regions of the highpermeability facies, the critical time and wavelength are shorter than those for the mean permeability, and fingering is more pronounced. In addition, smallscale fingers may survive longer without merging because they are shielded from each other by lowpermeability facies.
In terms of our quantitative measures, Fig. 4a exhibits a higher degree of CO_{2} spreading throughout the domain at early times in heterogeneous compared to homogeneous cases. At later times (during the fourth regime in homogeneous domains), we observe a crossover after which CO_{2} spreading in heterogeneous media is no longer higher than that in homogeneous domains. The reason is that for all heterogeneous cases the onset of convective fingering (first deviation from Fickian scaling: σ_{z} ∝ t^{β>0.5}) is almost instant, leading to more initial spreading but with a lower growth rate than for the homogeneous cases (Fig. 4c). Higher initial spreading ends in an earlier convectiveshutdown regime in bimodal heterogeneities once the fingers reach the lower boundary and the domain starts to be saturated with CO_{2}. This phenomenon has been studied in both numerical simulations and experiments as a latetime reduction of mixing rates^{12,51}.
At late times, there is a delay in reaching the maximum spreading and a slower growth of σ_{z} for heterogeneous domains. This subconvective behavior is caused partly by slow diffusion of CO_{2} into isolated lowpermeability islands that locally impede the vertical propagation of descending fingers. More tortuous flowpaths can also be a factor, resulting in either a longer transit time of CO_{2}rich brine or favorable channeling of flow depending on the facies structure. Blockage of flow may also happen in bimodal structures, especially in aquifers with lower volume fractions of highpermeability facies.
The global measure of CO_{2} spreading is a superposition of fast convective (albeit tortuous) flow of CO_{2}rich fingers through highpermeability pathways and slower diffusive transport of CO_{2} through the lowerpermeability facies. The former explains the higher initial mixing, while the latter causes the delay in reaching the maximum spreading. This phenomenon can be clearly seen in the 30% case (Figs 2 and 4a). For larger highpermeability volume fractions (above 50%), the delay in the spatial variance of CO_{2} over time is reduced. This is because we exceed the percolation threshold above which highpermeability clusters form large and connected preferential pathways that span the full extent of the domain^{52}.
Scaling of the dispersionwidth with time in Fig. 4c provides further insight into the complex flow dynamics of bimodal architectures. The four distinct flow regimes observed in homogeneous domains are smoothed out in heterogeneous formations. We find that CO_{2} transport in architectures below the percolation threshold scales from a diffusiondominated Fickian regime (~t^{0.5}) at early time, associated with flow through lowpermeability facies, to a subsequent ballistic regime (~t) within highpermeability facies, and eventually to a subdiffusive nonFickian regime (~t^{β<0.5}). The latter is due to a contribution from diffusive transport through lowpermeability facies in a subset of the domain at late times. The ballistic regime is due to advective flow through connected highpermeability pathways with a linear growth rate of fingers^{13}. Similar regimes of dispersionwidths have been observed in viscously unstable solute transport through heterogeneous porous media, in which flow starts ballistically and reaches a Fickian scaling in the latetime asymptotic stage, but showing no subdiffusive spreading^{48,53}.
We note that for facies architectures below the percolation threshold, the chronological order of Fickian and ballistic flow regimes are realization dependent: when highpermeability facies are concentrated at top of the domain, a ballistic period is likely to precede a Fickian diffusive regime and vice versa. Above the percolation threshold, we see predominantly ballistic flow through fully connected clusters, followed by subdiffusive scaling in a final delay period that is shorter than that below the percolation threshold. For the highpermeability heterogeneous simulations, σ_{z} reaches the maximum spreading at comparable times as the equivalent homogeneous cases, because its temporal scaling is similar (t^{1} versus t^{1.5}).
We also model gravitational fingering in heterogeneous media at infinite Rayleigh numbers (D = 0) to isolate the predominantly advective flow (channeling) through highpermeability facies (for 30% and 90% volume fractions). Diffusive effects are generally more pronounced in heterogeneous (particularly, layered or fractured) aquifers in which compositional gradients tend to occur across interfaces between different permeabilities^{54}. Expectedly, in the absence of diffusion as a restoring force, we see more dramatic fingering throughout the highpermeability facies (Fig. 6a). Our quantitative spreading measures show correspondingly higher values of σ_{z} and in Fig. 6b. Comparing the evolution of the spatial variance and dispersionwidths for simulations with and without diffusion, we see that diffusion has little impact on the global mixing rates in the 90% case because the flow is advectiondominated from the start. For the 30% case, diffusion causes more stagnation at early times and mainly delays the onset of ballistic regime. This suggests that while bimodal formations with low proportions of highpermeability facies may show less vertical spreading of CO_{2}rich brine, they can promote more lateral mixing due to diffusion into lowpermeability zones at early times compared to those with higher volume fractions of such facies. For both the 30% and 90% facies distributions, maximum spreading is achieved at similar times with or without diffusion, confirming that the transport of CO_{2} is ultimately driven by advectiondominated, unstable, gravitoconvective mixing.
Finally we study the impact of heterogeneous porosity distributions on densitydriven CO_{2} mixing (Fig. 7). We select one realization each of the 10%, 40%, and 90% volume fractions. A ratio of 10 is assigned between the porosities of the two facies while keeping the average porosity at 10% to allow comparison to homogeneous media and to keep the injection rates the same. This means that the bimodal porosities for the low and highpermeability facies are 5.3% and 53% for the 10% case, 2.7% and 27% for the 40% case, and 1% and 10% for the 90% case. Lower porosities formally reduce λ_{c}, t_{c}, and the diffusive flux (see Eq. (4)) and also affect CO_{2} transport (Eq. (3)) and its spatial variance (Eq. (7)). Lower permeabilityporosity facies become saturated with CO_{2} earlier on, which increases the density contrast and triggers instabilities. As a result, while the advective flux is lower in lowpermeability facies, the finger growth rates can be comparable to that in the higher permeability regions. This effect becomes more pronounced as the volume fraction of lowpermeability facies increases. Figure 7b,c compare the temporal evolution of and σ_{z} for these cases. We observe a diffusiondominated initial regime, followed by a ballistic regime for all bimodal porosity distributions, with flow in the low porosity facies more unstable than in the corresponding uniform porosities of 10%.
Conclusions
We perform physically robust simulations of gravitoconvective mixing of CO_{2} in saline aquifers without the limiting assumptions of prior studies. CO_{2} is injected into a confined domain and phase behavior is described by the cubicplusassociation EOS. CPA provides accurate densities, and accounts for both volume swelling due to CO_{2}dissolution, and compressibility due to injectioninduced pressure increases. Heterogeneity, typical of fluvial deposits, is represented by bimodal facies architectures (e.g., shale and sandstone). To quantify the spreading of dissolved CO_{2} during the complex gravitoconvective mixing, we evaluate global measures for the spatial variance and dispersionwidth of CO_{2} as well as the evolution of a dispersion coefficient, the maximum density contrast Δρ_{max}, and the velocity of the fastest growing unstable finger. From these measures, new flow regimes emerge at all Rayleigh numbers, caused by changes in Δρ_{max}, the driving force for fast advective mixing. Due to competing diffusive and advective processes, we find in homogeneous media that for a constant dissolution rate in the top, Δρ_{max} initially scales as k^{−0.5}. Moreover, the critical onset time of fingering scales as k^{−1}, rather than the k^{−2} found from stability analyses that assume a constant Δρ_{max}. Topological structures of sedimentary facies and their connectivity are responsible for additional complexities. Fingering is enhanced in connected highpermeability facies (especially above the percolation threshold). Lowpermeability regions can delay downward advective transport, but diffusion through such regions results in a longer mixing tail. Solubility trapping is an important process in predicting the fate of injected CO_{2} in feasibility studies of carbon storage, and the subsurface distribution of CO_{2} is critical when evaluating, e.g., the risk of leakage through nearby abandoned wells. A promising result is that in bimodal facies architectures, convective mixing of dissolved CO_{2} starts much earlier leading to higher degrees of mixing than predictions for equivalent homogeneous media. Our detailed analyses of the scaling relations of various global measures of the flow dynamics can be used to upscale the computationally expensive finegrid simulations to the large dimensions of saline aquifers.
Methods
Governing equations: species transport and pressure equations
We consider a fluid mixture consisting of 2 components, labeled by index i with i = 1 for brine and i = 2 for CO_{2}. The advectiondiffusion transport equation is represented in terms of each component’s molar density (cz_{i}) as follows:
where c is the molar density, z_{i} the mole fraction of component i with z_{1} + z_{2} = 1, F_{i} [mol/m^{3}s] a source term of component i, ϕ the porosity, t the time, the advective flux and the diffusive flux, given by^{55}:
Velocities follow from Darcy’s law as:
with ρ[kg/m^{3}] the brine mass density, which is related to the molar density through the molecular weight of each component, M_{i}, as ρ = c(M_{1}z_{1} + M_{2}z_{2}).
Most studies of carbon sequestration assume an incompressible aqueous phase, while we allow for both fluid compression due to the pressure buildup during CO_{2} injection into an aquifer with impermeable boundaries, and volume swelling of brine due to CO_{2} dissolution. The pressure response is computed from^{56}:
where C_{f} is the brine compressibility and is the total partial molar volume of each component in the mixture.
Numerical framework: higherorder finite element methods for flow and transport
We use numerical methods presented in earlier works^{1,18,55,57}. A secondorder discontinuous Galerkin (DG) method is used to update the transport equations. This approach has been demonstrated to reliably capture the smallscale onset of viscous and gravitational fingers on relatively coarse grids by reducing numerical dispersion^{54}. Shahraeeni et al.^{18} showed that the higherorder mass transport update can resolve the critical wavelength with only a few elements. A Mixed Hybrid Finite Element (MHFE) method is used to solve the Darcy and pressure equations simultaneously, which provides accurate velocity and pressure fields. MHFE is particularly useful in the simulations presented in this article due to the sharp contrasts in the heterogeneous permeability fields.
Phase behavior is based on the cubicplusassociation (CPA) EOS. The CPAEOS is suitable for watercontaining mixtures and takes into account the hydrogen bonding of water molecules by thermodynamic perturbation theory. The physical interactions follow from the PengRobinson EOS. In addition, the CPAEOS allows for crossassociation of CO_{2} with water, which is induced by polarpolar interactions^{1,35,36}. This approach is an improvement over previous studies that relied on Henry’s law for CO_{2} solubilities and empirical correlations for the brine density.
Characterization of CO_{2} spreading: spatial variance and dispersionwidth of CO_{2} molar density
In order to quantify the degree of mixing due to densitydriven flow of CO_{2}, we compute the spatial variance of the molar density of CO_{2} and the dispersionwidth σ_{z}^{41}. The choice of CO_{2} molar density, rather than composition, accounts for both the density change due to CO_{2} dissolution and the compressibility of the aqueous phase.
The first three central moments of the distribution define the spatial variance of the molar density of CO_{2} ():
where N_{01}/N_{00} expresses the coordinate location of the center of mole in the vertical (z) direction. The N_{ik} is the ikth moment of the CO_{2} molar density distribution in space, analogous to the moment of concentration distribution^{58}, defined here as:
with L_{x} and L_{z} the domain size in x and zdirections, respectively. The zeroth moment, N_{00}, physically represents the total number of moles of CO_{2} in the domain. The N_{00} does not remain constant over time because CO_{2} is continuously injected into the domain, unlike previous studies involving a pulse input of a solute in solution^{59}. However, by normalizing the first and second moments by N_{00}, the spatial variance becomes insensitive to increasing N_{00}. The variance in the xdirection is assumed to be negligible due to the dominant downward migration of CO_{2}rich brine. As a result, i equals to 0 in the ikth moments.
While spreading and dispersion can fundamentally differ from mixing and dilution, especially in highly heterogeneous media^{48}, it is a reasonable proxy to mixing, especially when there is a variable total volume of the dissolving solute in the system, because more spreading usually leads to more dilution and mixing^{60}. The or σ_{z} reach their maximum values, above which they plateau, shortly after entering a convectionshutdown regime. During this period, it can be concluded that spatial fluctuations of CO_{2} concentration about its mean value will gradually vanish as CO_{2} begins to saturate the entire domain. This implies a mixing degree that will finally increase to unity at the perfect mixing state, though with dissipative rates. To obtain an estimate of the maximum value of , we can assume a constant and spatially invariant nonzero C(x, z, t) throughout the domain and analytically solve the integral in Eq. (8). This results in m^{2}, i.e. close to the values observed in Fig. 4a (higher values are partly due to compressibility as the pressure increases).
Additional Information
How to cite this article: Soltanian, M. R. et al. Critical Dynamics of GravitoConvective Mixing in Geological Carbon Sequestration. Sci. Rep. 6, 35921; doi: 10.1038/srep35921 (2016).
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Acknowledgements
The first author was supported by the U.S. Department of Energy’s (DOE) Office of Fossil Energy funding to Oak Ridge National Laboratory (ORNL) under project FEAA045. ORNL is managed by UTBattelle for the U.S. DOE under Contract DEAC0500OR22725.
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School of Earth Sciences, the Ohio State University, Columbus, Ohio, USA
 Mohamad Reza Soltanian
 , Mohammad Amin Amooie
 , David Cole
 & Joachim Moortgat
Los Alamos National Laboratory, Los Alamos, New Mexico, United States
 Zhenxue Dai
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Contributions
M.R.S. designed and carried out the simulations, performed data analysis. M.R.S. and M.A.A. wrote the draft manuscript. M.A.A. and J.M. helped on model development and parameter characterization. D.C. and Z.D. helped in manuscript preparation. All authors reviewed the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Joachim Moortgat.
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