J. Fluid Mech. (2011), vol. 678, pp. 248–270. doi:10.1017/jfm.2011.110 c Cambridge University Press 2011 Two-phase gravity currents in porous media M A D E L E I N E J. G O L D I N G†, J E R O M E A. N E U F E L D, M A R C A. H E S S E A N D H E R B E R T E. H U P P E R T Department of Applied Mathematics and Theoretical Physics, Institute of Theoretical Geophysics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK (Received 15 October 2010; revised 20 January 2011; accepted 2 March 2011; ﬁrst published online 26 April 2011) We develop a model describing the buoyancy-driven propagation of two-phase gravity currents, motivated by problems in groundwater hydrology and geological storage of carbon dioxide (CO2 ). In these settings, ﬂuid invades a porous medium saturated with an immiscible second ﬂuid of diﬀerent density and viscosity. The action of capillary forces in the porous medium results in spatial variations of the saturation of the two ﬂuids. Here, we consider the propagation of ﬂuid in a semi-inﬁnite porous medium across a horizontal, impermeable boundary. In such systems, once the aspect ratio is large, ﬂuid ﬂow is mainly horizontal and the local saturation is determined by the vertical balance between capillary and gravitational forces. Gradients in the hydrostatic pressure along the current drive ﬂuid ﬂow in proportion to the saturationdependent relative permeabilities, thus determining the shape and dynamics of twophase currents. The resulting two-phase gravity current model is attractive because the formalism captures the essential macroscopic physics of multiphase ﬂow in porous media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that can permanently immobilize CO2 in the societally important example of geological CO2 sequestration. The magnitude of residual trapping is set by the areal extent and saturation distribution within the current, both of which are predicted by the two-phase gravity current model. Hence the magnitude of residual trapping during the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that residual trapping increases in the presence of a capillary fringe, despite the decrease in average saturation. Key words: gravity currents, multiphase ﬂow, porous media 1. Introduction The buoyancy-driven ﬂow of immiscible ﬂuids within a porous medium is a fundamental process that has applications to a host of engineering and environmental systems. In all cases, the interfacial tension acting between the ﬂuids gives rise to a two-phase region over which the saturation, or pore volume fraction, of each ﬂuid can vary. The model problem considered in § 4 is applicable to perched aquifers, which are formed when water in the subsurface accumulates on top of an impermeable barrier above the main water table in the partially saturated vadose zone (Fetter 2001). Another example is the migration, immobilization and reclamation of toxic substances such as chlorinated solvents and trichloroethylene, often referred to collectively as † Email address for correspondence: [email protected] Two-phase gravity currents in porous media 249 dense non-aqueous phase liquids (DNAPLs). A related problem is the migration of light non-aqueous phase liquids (LNAPLs) in the capillary zone above the water table. In this case, three phases (air, water and LNAPL) ﬂow in the pore space and similar concepts to those presented here apply (Parker & Lenhard 1989; Bear et al. 1996; Bear & Ryzhik 1998). Here we consider a two-phase gravity current resulting when a wetting phase, such as water, and a non-wetting phase, such as CO2 or oil, are conﬁned by an impermeable boundary in the porous medium and driven predominately horizontally by the density diﬀerence between the ﬂuids. One particularly timely and pressing example of two-phase ﬂows in porous media is the geological storage of CO2 . Carbon capture and storage (CCS) is currently the only proposed technological means to reduce CO2 emissions from fossil fuel power plants (Metz et al. 2005). The CO2 is captured at source, compressed and injected into porous geological formations at least 1 km underground. Potential storage sites include saline aquifers, depleted oil reservoirs and unproﬁtable coal seams. Here we focus on saline aquifers, which provide the largest potential storage volume. In siliciclastic aquifers, which are mostly composed of silicate minerals, the aqueous brine is typically the wetting phase and the injected supercritical CO2 is the non-wetting phase. The supercritical CO2 is less dense than the ambient brine, so the buoyant CO2 rises as a plume until it reaches a ﬂow barrier, at which point it spreads laterally as a gravity current. The CO2 continues to spread until either it is able to rise towards the surface, or it becomes permanently trapped by three main mechanisms listed here in order of increasing time scales (Metz et al. 2005): residual trapping where CO2 is immobilized in pore spaces by capillary forces once injection has ceased; solubility trapping where CO2 dissolves into brine to form a denser ﬂuid that sinks and mineral trapping where CO2 precipitates as minerals. The long-term security of geological CO2 storage is determined by the competition between leakage and trapping mechanisms (Nordbotten et al. 2005; Woods & Farcas 2009; Neufeld, Vella & Huppert 2009). In their study of the distribution of existing and abandoned wells in the oil-producing Alberta Basin, Gasda, Bachu & Celia (2004) highlight the large number of potential leakage pathways located in otherwise attractive CO2 storage sites. This emphasizes the importance of trapping mechanisms that reduce the amount of CO2 that is free to potentially escape, and the dependence of the security of long-term storage on trapping processes. Capillary forces can have signiﬁcant eﬀects on the migration and trapping of the CO2 injected into geological formations saturated with brine. Drainage occurs when the non-wetting ﬂuid displaces the wetting ﬂuid within the pore space and imbibition describes the opposite process. In both cases, a ﬁnite fraction of the displaced ﬂuid generally remains in the pores and may become immobile. During drainage, the wetting phase remains within the pores as continuous thin ﬁlms coating grain surfaces. During imbibition, the remaining fraction of the non-wetting phase is known as the residual saturation and is immobilized by capillary forces in the form of isolated ganglia. The residual trapping of the non-wetting ﬂuid can be a problem in situations such as oil extraction or chemical spills because it can be diﬃcult to remove much of the non-wetting phase eﬃciently. In contrast, capillary forces may have a positive inﬂuence on the long-term trapping of CO2 during geological storage. During injection of the non-wetting CO2 , capillary forces both reduce the CO2 saturation within the current and increase the vertical extent of the current. This increases the volume of reservoir and brine contacted by CO2 and is therefore beneﬁcial for dissolution and mineral trapping (Ennis-King & Paterson 2005; Riaz & Tchelepi 2006). When injection ceases, residual CO2 is trapped as the 250 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert current proceeds through the formation. The saturation of CO2 within the current during the injection stage greatly aﬀects the amount that is subsequently trapped, and experimental work quantiﬁes this dependence through initial–residual curves (Pentland et al. 2008). Previous gravity current models have assumed a constant, uniform CO2 saturation and hence a constant amount of residual trapping (Hesse, Orr & Tchelepi 2008; Farcas & Woods 2009; MacMinn & Juanes 2009). Whilst these models capture the dependence of residual trapping on the areal extent of the current, the observed relationship between initial and residual saturations is not captured in estimates obtained in this way. Many studies of gravity currents in porous media use a sharp-interface approximation, in which the regions within and without the current are fully saturated, and the two ﬂuids are separated by a macroscopically sharp interface (Huppert & Woods 1995; Hesse et al. 2007; Gasda, Nordbotten & Celia 2009; Juanes, MacMinn & Szulczewski 2010). Such models, which predict the evolution of the current height h, were originally developed and tested for miscible ﬂows in porous media and are applicable to two-phase gravity currents only when capillary forces are negligible compared to gravitational and viscous forces. Sharp-interface models have been developed for gravity currents in conﬁned geometries (h ∼ H , where H is the thickness of the reservoir), where the ﬂow of both ﬂuids is non-negligible, and hence the viscosity ratio plays a key role in their behaviour (Nordbotten & Celia 2006; Hesse et al. 2007; Neufeld & Huppert 2008; Juanes et al. 2010). For sharp-interface gravity currents in unconﬁned aquifers (h H ), there is negligible ﬂow of the ambient ﬂuid, and hence models of their propagation are independent of the ratio of viscosities between the ﬂuids (Huppert & Woods 1995; Vella & Huppert 2006; Golding & Huppert 2010). However, when the two ﬂuids are diﬀerent phases, capillary forces may lead to a saturation transition zone that is not negligible, thus rendering the sharp-interface assumption invalid. In the particular case of CO2 –brine systems, the potentially large unstable viscosity contrast could further increase the size of the transition zone, or capillary fringe. Furthermore, sharp-interface models do not capture the strong reduction in relative permeability experienced by the non-wetting phase current, which could greatly aﬀect predictions for its spreading velocity (Bickle et al. 2007). This provides an important motivation for the systematic development of a two-phase gravity current model and its analysis. In this paper, we consider the buoyancy-driven propagation of immiscible ﬂuids within a semi-inﬁnite porous medium, taking explicit account of the eﬀects of capillarity on the propagation and saturation within the current. In § 2 we introduce the standard formulation of two-phase ﬂow in porous media (Pinder & Gray 2008). We use the large aspect ratio of the current to show that vertical velocities are negligible and that the saturation distribution is determined by vertical gravity– capillary equilibrium (Lake 1996). Hence, in § 3 we derive a two-phase gravity current model from the full governing equations. We ﬁnd that the eﬀect of capillary forces on the behaviour of the current is controlled by two nonlinear functions, which describe the dependence of the vertically integrated phase volume and ﬂux on the thickness of the current. In § 4 we apply our model to steady-state gravity currents that form beneath discontinuous horizontal ﬂow barriers during the buoyant rise of CO2 through heterogeneous geological formations (Johnson, Nitao & Knauss 2004; Green & Ennis-King 2010; Hesse & Woods 2010) and estimate the magnitude of residual trapping. We conclude in § 5 with a discussion of the eﬀect of two-phase phenomena on the behaviour of buoyant currents within the subsurface and provide an outlook for future research. Two-phase gravity currents in porous media 251 2. The standard model of two-phase ﬂow in porous media We consider the invasion of a non-wetting ﬂuid into a porous medium that is initially saturated with a wetting ﬂuid in a drainage process. In the following section, we summarize the standard two-phase model, which is based on the concept of macroscopic capillarity (Leverett 1941). The model considers the eﬀects of interfacial tension and the reduction in ﬂuid ﬂow caused by the convoluted interface of the two ﬂuids within the porous matrix on a scale much larger than the typical pore scale. Both ﬂuid phases are considered to be immiscible and incompressible Newtonian liquids of constant, though unequal, density and viscosity. The action of capillary forces on the pore scale results in only a partial displacement of the wetting phase, and so the two phases co-exist within the same pore spaces. We denote quantities pertaining to the wetting and non-wetting phases by subscripts w and n respectively. The saturation, Si , is a macroscopic quantity that describes the average volume fraction, φi , of phase i in a representative elementary volume (REV), normalized by porosity φ, i.e. Si = φi /φ, i = w, n. (2.1) This deﬁnition implies the saturation constraint Sw + Sn = 1. (2.2) The irreducible wetting phase saturation, denoted by Swi , is the wetting phase saturation that always remains within the pores during drainage and is assumed to be constant. Two-phase ﬂow models are therefore generally written in terms of an eﬀective non-wetting phase saturation, s, and wetting phase saturation 1 − s, deﬁned by Sw − Swi Sn and 1 − s = . (2.3) s= 1 − Swi 1 − Swi The capillary pressure caused by the surface tension at the interface between two phases is expressed on the macroscopic scale by the diﬀerence in the pressure between phases, pn − pw = pc (s), (2.4) where the standard model assumes that the capillary pressure, pc , is a function of saturation only. In a drainage process, a minimum capillary pressure, called the capillary entry pressure, must be achieved for the non-wetting phase to enter a pore. The magnitude of the capillary entry pressure is inversely proportional to the size of the entrance to the pore. This entry pressure can create ﬂow barriers for the non-wetting phase in regions where the porosity is reduced. As the capillary pressure increases, successively smaller pores are invaded. Typical pc (s) curves for a primary drainage process, where the non-wetting phase is invading the porous medium for the ﬁrst time, are displayed in ﬁgure 1(a) for rock types with diﬀerent pore-size distributions. If the porous medium has a narrow pore-size distribution (solid curve), once the capillary pressure is above the entry threshold for the largest pore, a small increase will be suﬃcient for the non-wetting phase to enter the majority of the remaining pores, and so the wetting phase saturation decreases rapidly. A wider distribution of pore sizes will result in a more gradual transition, as demonstrated in ﬁgure 1(a) (dash-dotted curve). Two commonly used empirical models for capillary pressure curves are the Brooks– Corey and van Genuchten models (Brooks & Corey 1964; van Genuchten 1980). The derivation of the two-phase gravity current model presented here is independent of the speciﬁc parameterization of the capillary pressure curve. However, we choose the 252 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert Sw 1.0 1.0 Λ =1 Λ = 10 0 (b) Relative permeability Capillary pressure (a) Swi Swi Sw 1.0 0.5 1.0 0.8 0.6 0.4 0.2 pe 0.5 1.0 0 1− s 1− s Figure 1. (a) Typical capillary pressure curves for drainage in porous media with narrow (solid curve) and wide (dash-dotted curve) pore-size distributions. (b) Non-wetting (+) and wetting (×) phase relative permeability data for Ellerslie Sandstone in the Alberta Basin, Canada, compared to typical model values with α = 2 and krn0 = 0.116 for the non-wetting (dash-dotted curve) and β = 2 for the wetting (dashed curve) phases. Data source: Bennion & Bachu (2005). simpler Brooks–Corey curve to illustrate the model in order to minimize the number of parameters and because it is the more appropriate curve for consolidated rocks with a well-deﬁned entry pressure. The Brooks–Corey capillary pressure function is given by (2.5) pc = pe (1 − s)−1/Λ , where Λ is a ﬁtting parameter related to the pore-size distribution. A small value of Λ corresponds to a wide distribution of pore sizes, whereas in the limit Λ → ∞, all pores are of the same size. Examples of this curve are plotted in ﬁgure 1(a) for Λ = 1 and 10. The parameter pe can be understood as the capillary pressure required for the non-wetting phase to enter the largest pores. Local mass conservation in each phase requires ∂ (2.6) (Si ) + ∇ · ui = 0, i = w, n, ∂t where ui is the volumetric ﬂux of phase i. The extension of Darcy’s law to two-phase ﬂow (Leverett 1941) assumes that φ ui = −kλi [∇pi − ρi g], i = w, n, (2.7) where k is the intrinsic permeability of the porous medium, g is the gravitational acceleration, ρi is the density of phase i and pi is the pressure averaged over a REV of phase i. The mobility of each phase, λi (s) = kri (s)/µi , (2.8) is inversely proportional to the dynamic viscosity, µi , and proportional to the relative permeability, kri , of each phase, which is assumed to be a unique nonlinear function of s only, during primary drainage. The presence of two phases alters both the pore space geometry and microscopic boundary conditions for each ﬂuid, thereby changing the ﬂuid ﬂux in each phase which results from the application of a pressure gradient. Two-phase gravity currents in porous media 253 To account for this on the macro-scale, the relative permeability, kri , of each phase is a dimensionless scaling factor that multiplies the intrinsic permeability of the medium, k (Leverett 1939). An approximate relationship for kri (s) is commonly based on a power law (Corey 1954) of the eﬀective saturation (2.3). Hence, we write the relative permeability of the non-wetting phase as krn = krn0 s α , (2.9) and of the wetting phase as krw = (1 − s)β , (2.10) for some powers α and β, where krn0 < 1 is the end point relative permeability of the non-wetting phase at its maximum saturation s = 1. When s = 1, the wetting phase is at the irreducible saturation, and is still hydraulically connected in the sense that Darcy ﬂow could still occur via thin ﬁlms. However, the eﬀective permeability of the wetting phase is approximately zero since the ﬂuid ﬁlms are too thin for ﬂow to be induced by the application of a typical pressure gradient. A review of various models used for the constitutive relations for consolidated rocks is given by Li & Horne (2006). Laboratory data for CO2 in Ellerslie standstone, obtained by Bennion & Bachu (2005), are plotted in ﬁgure 1(b) along with the model values with α = β = 2 and krn0 = 0.116, which show excellent agreement. In this paper we consider ﬂow in unconﬁned reservoirs for which ﬂuid motion in the wetting phase can be neglected. Therefore, we require an expression only for the non-wetting phase relative permeability, and for this we set α = 2 where necessary to enable analytical and numerical calculations. However, we emphasize that the model requires only krn = krn (s) and is by no means restricted to this value of α. Equations (2.2), (2.4) and (2.6)–(2.8), along with the constitutive relations (2.5), (2.9) and (2.10), form a complete description of two-phase ﬂow. The capillary entry pressure, pe , and pore-size distribution parameter, Λ, in the Brooks–Corey capillary pressure model, along with the relative permeability parameters α, β and krn0 , are determined by ﬁtting (2.5), (2.9) and (2.10) to experimental data. The irreducible wetting phase saturation, Swi , embedded in the deﬁnition of eﬀective saturation (2.3), must also be determined experimentally. 3. Development of a two-phase gravity current model We now consider the buoyancy-driven ﬂow of non-wetting ﬂuid along a horizontal barrier and develop a vertically integrated model for a two-phase gravity current. As the current propagates, the non-wetting phase displaces the wetting phase which initially ﬁlls the pores in a primary drainage process. Fed by a constant ﬂux per unit width q, the gravity current propagates beneath an impermeable upper boundary in an inﬁnite reservoir initially fully saturated with a denser, wetting, ﬂuid. This scenario is applicable to CO2 spreading beneath an upper boundary of impermeable rock in a CO2 storage reservoir as well as the migration of non-aqueous phase contaminants. The current height, h(x, t), denotes the depth below which no non-wetting phase is present and where the non-wetting phase saturation drops to zero, s = 0 (see ﬁgure 2). The depth of the reservoir is much greater than the height of the current, and so the horizontal pressure gradient and the velocity in the wetting phase are negligible below the current. We additionally assume that the ﬂuids are simply connected which implies that, to leading order, there are no horizontal pressure gradients driving ﬂow of the wetting phase within the current. 254 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert s x h (x, t) q ρn, µn z φ, k s=0 h ρw, µw z Figure 2. Sketch showing a gravity current with a non-uniform saturation proﬁle propagating beneath an upper impermeable boundary into a porous medium saturated by ﬂuid of a diﬀerent phase, density and viscosity. The plot on the left shows a representative proﬁle of saturation with depth. 3.1. Vertical gravity–capillary equilibrium and saturation proﬁle The dominant balance governing two-phase gravity currents can be found by applying scaling arguments in a manner analogous to previous studies of single-phase gravity currents (Huppert & Woods 1995, for example). For a current whose length L is much greater than its height H , i.e. = H /L 1, local mass conservation, as expressed by (2.6), implies that the vertical velocities are much less than the horizontal non-wetting phase velocity. Thus, the pressure within each phase is nearly hydrostatic and given by ∂pw ∂pn (3.1) = ρn g and = ρw g. ∂z ∂z Pressures pn and pw are plotted in ﬁgure 3(a), where the diﬀerence at the current boundary z = h is equal to the capillary entry pressure pe . Consequently, (2.4) implies that ∂pc = −ρg, (3.2) ∂z where ρ = ρw −ρn , which mathematically captures the balance between gravitational and capillary forces within the current, known as vertical gravity–capillary equilibrium. We integrate (3.2) between z and h to obtain the expression for capillary pressure as a function of height pc [h(x, t), z] = pe − ρg(z − h), (3.3) where we note that the capillary pressure at the edge of the current pc (z = h) = pe . The assumption of gravity–capillary equilibrium is commonly invoked in studies of multiphase ﬂow (Yortsos 1995; Lake 1996, for example). In this study, it follows from the large aspect ratio of the gravity current and the subsequent assumption that pressure within each phase is hydrostatic. We expand on this assumption in the Appendix and show that the time scale over which gravity–capillary equilibrium is recovered during the motion of the current is much smaller than the time scale for the motion itself. In this way, we show that the assumption remains valid as the current propagates. We note that the assumption breaks down in areas where saturation is very low (Lake 1996). In this region, which is located along the current boundary, the relative permeability of the non-wetting phase is very small, leading to potentially long time scales for recovering gravity–capillary equilibrium as the current propagates. However, this region is thin, and we anticipate that the behaviour within it has little inﬂuence on the overall motion of the current predicted by our vertically integrated model. 255 Two-phase gravity currents in porous media (a) 0 (b) 0 pw pn = pc + pw Capillary fringe z Λ=1 z 2 10 h h 0 Pressure 1 Saturation Figure 3. (a) Graph illustrating the hydrostatic pressure within each ﬂuid. The diﬀerence between the non-wetting and wetting phase pressures at z = h is equal to the capillary entry pressure pe . (b) Saturation proﬁles within the current are plotted for Λ = 1, 2 and 10. The size of the capillary fringe or transition zone, where the non-wetting phase saturation varies considerably, is indicated by arrows for each case. Parker & Lenhard (1989) combine an expression for capillary pressure resulting from the gravity–capillary equilibrium assumption in three-phase ﬂow, analogous to (3.3), with the Brooks–Corey model, to ﬁnd the implied wetting phase saturation as a function of vertical position. Here we treat the simpler two-phase case similarly by equating (2.5) and (3.3), to obtain an expression for the non-wetting phase saturation, s, within the gravity current as a function of h and z given by −Λ h−z s[h(x, t), z] = 1 − 1 + , (3.4) he where the length scale he is deﬁned by he ≡ pe /ρg. (3.5) The saturation proﬁle given by (3.4), which is a function of h and z, is plotted in ﬁgure 3(b). Physically, the capillary entry height, he , represents the height of a column of non-wetting ﬂuid such that the hydrostatic pressure equals the entry pressure of the largest pore throat found in the porous medium. It is therefore a measure of the relative strength of capillary forces to gravitational forces. The region in which the non-wetting phase saturation varies considerably is referred to as the transition zone, or capillary fringe. The width of the capillary fringe increases with the magnitude of capillary forces, i.e. with the inverse of the pore-size distribution parameter 1/Λ (as shown in ﬁgure 3b) and with the capillary entry height he . Bear et al. (1996) deﬁne a similar height to he , which is often used as a characteristic length scale for the capillary fringe in a three-phase system. 256 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert 3.2. Vertically integrated equation for the current height proﬁle Vertical integration of (2.6) for the non-wetting phase between z = 0 and z = h yields ϕ ∂ ∂t 0 h ∂h s dz − s(h) ∂t ∂ + ∂x h un dz − [un ]z=h 0 ∂h + [wn ]h0 = 0, ∂x (3.6) where ϕ = φ(1 − Swi ). This reduces to ∂ ϕ ∂t 0 h ∂ s dz + ∂x h un dz = 0, (3.7) 0 since s(h) = 0 and λn [s(h)] = 0 at the edge of the current, and therefore, by (2.7), the non-wetting phase velocities un = wn = 0 at z = h. We impose the condition wn = 0 on the upper impermeable boundary at z = 0. This equation was derived by Bear & Ryzhik (1998) to describe a DNAPL lens moving over an impermeable boundary. The ﬁrst integral in (3.7) is calculated directly using (3.4). We substitute (2.7) into the second integral and therefore ﬁnd that (3.7) becomes −Λ ∂h ∂ ∂h h h λn dz = 0. − kρg ϕ 1− 1+ he ∂t ∂x ∂x 0 (3.8) This can be further simpliﬁed by noting that the non-wetting phase saturation at the upper boundary z = 0 can be expressed by the saturation function −Λ h . s0 (h/ he ) ≡ s(h/ he , z = 0) = 1 − 1 + he (3.9) Hence, we obtain the principal new equation that governs the ﬂow of a two-phase buoyancy-driven current, ϕs0 (h/ he ) ∂h ρg k̃ ∂ ∂h − h F(h/ he ) = 0, ∂t µn ∂x ∂x (3.10) where k̃ = kkrn0 . Here, the eﬀect of the vertical saturation distribution on ﬂow due to variation in relative permeability is encapsulated in the ﬂux function h µn λn [s(z)] dz krn0 h 0 s0 (h/ he ) he S α (1 − S)−(Λ+1)/Λ dS. = Λh 0 F(h/ he ) ≡ (3.11) (3.12) To obtain (3.12), we change variables from z to s (deﬁned by (3.4)) by noting that ∂s/∂z = − (Λ/ he )(1 − s)(Λ+1)/Λ . It may be interesting to note that the integral in (3.12) x is, in fact, the incomplete beta function, deﬁned as B(x : a, b) ≡ 0 ua−1 (1 − u)b−1 du, with x = s0 (h/ he ), a = 1 + α and b = − 1/Λ. We are able to integrate (3.12) directly Two-phase gravity currents in porous media 257 when α = 2 to ﬁnd an analytical expression for F given by ⎧ 2 3s0 − 2s0 ⎪ ⎪ ⎪ (Λ = 0.5) ⎪ (1 − s )2 − 2 log(1 − s0 ) ⎪ 0 ⎪ ⎪ he ⎨ s0 (2 − s0 ) + 2 log(1 − s0 ) (Λ = 1) F(h/ he ) = h ⎪ 1 − s0 ⎪ ⎪ ⎪ ⎪ (1 − s0 )−1/Λ 2 2Λ(s0 − Λ) 2Λ2 ⎪ ⎪ s0 + + (Λ = 0.5, 1), ⎩ 1 − 2Λ Λ−1 (Λ − 1)(1 − 2Λ) (3.13) where s0 = s0 (h/ he ), and we will use this in § 4.1. The explicit framework encapsulated by (3.9), (3.10) and (3.12) represents a signiﬁcant advance in our understanding of the eﬀects that two-phase phenomena have on the propagation of gravity currents in porous media, accounting for the non-uniform saturation distribution of the ﬂuid in the current. Equation (3.10) is the two-phase analogue of the sharp-interface models considered by a number of authors (Huppert & Woods 1995). The two-phase behaviour is captured by two functions of h/ he , deﬁned in (3.9) and (3.12). The ﬂux function, F, and the saturation function, s0 , are investigated further in § 3.3. We show that in the appropriate limits, h/ he → ∞ and Λ → ∞, the two-phase functions s0 (h/ he ) → 1 and F(h/ he ) → 1, and thus the single-phase governing equation is recovered. It is worth emphasizing the generality of (3.11), for which any function of saturation can be supplied for the non-wetting phase mobility, λn . We use a power relation for the nonwetting phase relative permeability in this study to illustrate the two-phase behaviour (Li & Horne 2006). To complete the mathematical description of the problem, we specify two boundary conditions. The ﬂux at the origin can be written as ρg k̃ ∂h h F(h/ he ) = −q. µn ∂x (3.14) The second boundary condition recognizes that the height of the current at the nose, located at x = xN (t), is h(xN ) = 0. (3.15) 3.3. The ﬂux function, F(h/ he ), and saturation function, s0 (h/ he ) The ﬂux function F(h/ he ), deﬁned by (3.12), accounts for the depth-integrated relative permeability of the non-wetting phase and depends on the pore-size distribution parameter Λ, the capillary entry height he and the power α in the constitutive relationship (2.9). We focus on the eﬀects of varying Λ and he in this paper in order to constrain the parameter space, and because the value α = 2 is reasonable for many CO2 /brine systems, as demonstrated in ﬁgure 1(b) (Bennion & Bachu 2005). Figure 4(a) shows a graph of F(h/ he ) for Λ = 0.5, 1, 2 and 10. For ﬁnite values of Λ, F is a monotonically increasing function of h/ he , which tends to 1 as h/ he → ∞. This is due to the fact that when the current is much thicker than the capillary fringe, the non-wetting ﬂuid nearly saturates the pores in most of the current, and so capillary eﬀects become less important to its overall motion. The limit also corresponds to the case of negligible capillary entry pressure, i.e. pe 1. When h is comparable to or smaller than the capillary fringe, non-wetting phase saturations are lower, which results in reduced local relative permeabilities and hence lower values of F. 258 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert (a) 1.0 0.6 0.8 2 0.4 1 0.2 0 s0 (h/he) F(h/he) 0.8 (b) 1.0 Λ = 10 4 6 h/he 1 0.6 0.5 0.4 0.2 0.5 2 Λ = 10 2 8 10 0 2 4 6 8 10 h/he Figure 4. The ﬂux function F(h/ he ), plotted in (a), and the saturation function s0 (h/ he ), plotted in (b), are shown for Λ = 0.5, 1, 2 and 10. The saturation function, s0 (h/ he ), plotted in ﬁgure 4(b) for Λ = 0.5, 1, 2 and 10, indicates the thickness of the current with respect to the capillary fringe, which is deﬁned as the depth over which the non-wetting ﬂuid saturation in the current increases from s = 0 to some cut-oﬀ value s = sc ∼ 1. The width of the capillary fringe is the value of h for which s0 (h/ he ) = sc . We can see clearly from ﬁgure 4(b) that the capillary fringe width increases with 1/Λ and he . Thus, for all ﬁnite values of Λ, the current approaches the single-phase result, where F ≡ 1 and s0 ≡ 1, when h/ he 1. The rate of convergence to the single-phase limit increases with Λ because the capillary fringe is thinner. In the limit Λ → ∞, which describes a medium of uniform pore size with a single capillary entry pressure, there are no capillary eﬀects on the gravity current and s0 ≡ 1 and F ≡ 1 for all values of h/ he . Conversely, in the limit Λh/ he 1, the saturation function is approximately linear in h, given by (3.16) s0 (h/ he ) ≈ Λh/ he , and F can be approximated as α 1 Λh . (3.17) F(h/ he ) ≈ α + 1 he 4. An application to geological CO2 storage Many geological systems identiﬁed as potential sites for CO2 storage are heterogeneous, containing nearly horizontal, discontinuous barriers to ﬂow. Such barriers may include shale layers, which have a greatly reduced permeability, are regions of high capillary entry pressure, and are therefore eﬀectively impermeable to the non-wetting phase. Johnson et al. (2004) identiﬁed these ubiquitous shale layers as having a crucial eﬀect on migration and trapping of CO2 in storage reservoirs such as Sleipner. More recently, Saadatpoor, Bryant & Sepehrnoori (2010) showed that capillary barriers can have a dominant eﬀect on CO2 migration and trapping. The enhanced lateral dispersion due to discontinuous shales is important for residual trapping because it causes the CO2 to contact a larger volume of the reservoir (Johnson et al. 2004; Hesse & Woods 2010). The ‘inject low and let rise’ strategy for geological CO2 storage (Kumar et al. 2005; Bryant, Lakshminarasimhan & Pope 2008) tries to maximize the dispersion and trapping of CO2 due to ﬂow barriers, by injecting it through a horizontal well at the base of the storage formation (ﬁgure 5). 259 Two-phase gravity currents in porous media Injection (a) Post-injection (c) (b) ‘Inject low and let rise’ Steady state Imbibition and residual trapping Figure 5. Sketch showing the three stages of injecting CO2 into a storage reservoir. Figure (a) depicts the early stages of injection where CO2 starts to rise through a geological formation containing shale layers. In ﬁgure (b) the gravity currents that form beneath each shale layer have reached a steady state. Figure (c) sketches the post-injection stage where CO2 is residually trapped in regions previously occupied by the non-wetting phase gravity currents. We envisage the following idealized scenario, shown schematically in ﬁgure 5. Injection at the base of a heterogeneous formation begins with the rise of buoyant CO2 through the permeable pore space (ﬁgure 5a). As the CO2 rises, it encounters horizontal barriers to ﬂow and spreads laterally (ﬁgure 5b), which has important implications for the lateral dispersal and numerical upscaling of the reservoir, as discussed in Hesse & Woods (2010) and Green & Ennis-King (2010). The CO2 perched beneath each barrier forms a steady-state gravity current. At the end of the injection period (ﬁgure 5c), ﬂuid continues to rise throughout the system, leaving residual CO2 trapped in its wake. In this scenario, residual trapping occurs most notably in the regions beneath ﬂow barriers, previously occupied by steady-state CO2 plumes that can be modelled as gravity currents. The volume of CO2 that is residually trapped during this ﬁnal stage is of primary importance for the long-term safe storage of CO2 within the subsurface because it leads to rapid immobilization of CO2 after the end of injection. The quantity trapped depends on two key factors, both of which are inﬂuenced by two-phase phenomena: the volume of rock invaded by CO2 in the reservoir and the saturation within the contacted region. Experimental data indicate that the residual saturation, Sn,res , depends on the initial saturation of the non-wetting phase, Sn,init , at the time of ﬂow reversal (Pentland et al. 2008). Therefore, the amount of CO2 immobilized by capillary forces and the rate of trapping depends on the saturation distribution of the non-wetting CO2 at the end of the injection phase. At leading order, the buoyant rise of CO2 through a series of ﬂow barriers can be understood by studying the steady current beneath a single layer (Hesse & Woods 2010). Therefore, we focus on the buoyancy-driven ﬂow of non-wetting ﬂuid beneath a single barrier (ﬁgure 6). For continuous injection, we anticipate that the gravity currents will reach a steady state after suﬃcient time, in which the ﬂux into the current equals the ﬂux rising at the ends. In § 4.1 we investigate how capillary forces aﬀect the vertical extent and saturation distribution within a gravity current beneath a single impermeable barrier. The height and saturation proﬁles obtained in § 4.1 are then used as inputs to a trapping model, which captures the relationship between initial and residual saturations. In this way, we are able to gain valuable insight into how capillary forces aﬀect the magnitude of residual trapping in the ‘inject low and let rise’ strategy for CO2 storage. 260 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert q q −L L 0 x h(x) z 2q Figure 6. Sketch of a steady-state two-phase gravity current beneath a barrier of ﬁnite extent. Injection of a ﬂux 2q at x = 0 is balanced exactly by vertical ﬂow at x = ± L, resulting in a steady-state current whose saturation we model with depth. 4.1. Steady-state two-phase current beneath a ﬁnite boundary We consider a steady-state gravity current, under a barrier of ﬁnite length, 2L, where L is much greater than the current height. Non-wetting ﬂuid is injected with constant ﬂux 2q at the centre of the barrier and spreads to the edges at x = ± L, where it rises vertically from the system and so the edges eﬀectively act as sinks. Due to the symmetry of the problem, we consider only half of the current sketched in ﬁgure 6 and solve for a current fed by a ﬂux of magnitude q at x = 0. We see from (3.10) that the proﬁle of the steady state is given by d dh (4.1) h F (h/ he ) = 0, dx dx with boundary conditions ρg k̃ dh h F(h/ he ) = −q (x = 0), µn dx h = 0 (x = L). (4.2a) (4.2b) Unlike the case for unbounded spreading, there exists a natural horizontal length scale to our steady-state problem, L, and we deﬁne the corresponding vertical height scale 1/2 qµn L H = (4.3) ρg k̃ introduced by Huppert & Woods (1995). We deﬁne dimensionless variables x = x/L and h = h/H, (4.4) but drop the primes for ease of notation in the subsequent analysis. The existence of a third length scale, the capillary rise height he , is expressed in terms of a Bond number ρgH B≡ = H / he , (4.5) pe which is the ratio of the hydrostatic pressure for a scale height H and the capillary entry pressure. It therefore indicates the relative importance of capillary forces compared to gravitational and viscous forces. The single-phase case is recovered in the limit B → ∞ in which either the capillary entry pressure pe → 0 or the height of the current is suﬃciently large that the capillary fringe is small in comparison. Two-phase gravity currents in porous media 261 Hence, after integration and application of the boundary condition at x = 0, the dimensionless governing equation is dh F (hB) = −1. dx The remaining boundary condition is h h(x = 1) = 0, and the ﬂux function and saturation function are given by s0 (hB) 1 F(hB) = S α (1 − S)−(Λ+1)/Λ dS BΛh 0 (4.6) (4.7) (4.8) and s0 (hB) = 1 − (1 + hB)−Λ (4.9) respectively. When α = 2, we can use the analytical expressions for the function F(h), (3.13), to greatly simplify the governing equation for the current height. We make the expressions dimensionless using (4.4), substitute into the steady-state governing equation (4.6), integrate and rearrange, and hence obtain implicit algebraic expressions for h, displayed here for the values Λ = 0.5 and 1 used in ﬁgures 7(c–f ), ⎧ ⎪ ⎪h2 + 6 h + 16 [1 − (1 + hB)3/2 ] + 2 (1 + hB) log(1 + hB) (Λ = 0.5), ⎨ B 3B 2 B2 2(1 − x) = ⎪ h 2 ⎪ ⎩h2 + 6 − (3 + 2hB) log(1 + hB) (Λ = 1). B B2 (4.10) In a similar manner, equations for other values of Λ can be derived, but due to their intricate nature, they are not displayed here. Hence, the steady-state height proﬁle of a two-phase gravity current is the solution to a nonlinear algebraic equation. We note that the solution for the height proﬁle in this steady-state problem in the sharp-interface limit is (4.11) h = 2(1 − x), which has been shown to ﬁt experimental data in a Hele-Shaw cell very well (Hesse & Woods 2010). It is worth noting that when B → ∞, the steady-state single-phase solution is recovered. 4.1.1. The eﬀect of pore geometry and capillary forces By studying this steady-state problem, much insight can be gained into the general inﬂuence of capillarity and pore-size distribution on two-phase gravity currents, as captured in our model by parameters B and Λ respectively. The pore-size distribution parameter Λ determines how quickly the saturation increases with distance from the current boundary. A series of plots in ﬁgure 7 shows non-wetting phase saturation proﬁles and distributions for Λ = 0.5, 1 and 10, where the Bond number B = 1 is held constant. When Λ = 10, the current approaches the sharp interface limit. In this case, ﬁgures 7(a) and 7(b) show that there is a small transition zone at the edge of the current. As Λ decreases and capillary eﬀects become more important (ﬁgures 7c–f ), the transition zone becomes larger. As this happens, the average saturation decreases, and consequently the current thickens in order to maintain the constant ﬂux of injected ﬂuid. 262 (a) M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert s (x = 0, z) 0 1.0 0.5 x (b) 0 z 1 0.2 0.4 0.6 0.8 1.0 1 2 Λ = 10 (c) Λ = 10 2 (d) 0 0.5 1.0 0 1 0.2 0.4 0.6 0.8 1.0 1 z 2 2 3 (e) 0 Λ=1 0.5 1.0 (f) 0 1 1 2 2 z 3 0.2 0.4 0.6 0.8 1.0 3 4 5 Λ=1 3 4 Λ = 0.5 Λ = 0.5 5 Non-wetting phase saturation scale 0 0.2 0.4 0.6 0.8 1.0 Figure 7. Saturation proﬁles and distributions for B = 1, with Λ = 10 (a, b), Λ = 1 (c, d ) and Λ = 0.5 (e, f ). Figures (a), (c) and (e) show the saturation at x = 0, and ﬁgures (b), (d ) and (f ) illustrate the shape and saturation within each current. In each case the solid line indicates h(x), where s(h) = 0, and the dashed line indicates the sharp-interface limit. The height of the current at the origin, h0 ≡ h(x = 0), as a function of Λ and B, is plotted in ﬁgures 8(a) and 8(b) respectively. We use the current height at the origin to give an indication of the areal extent of the current, which strongly inﬂuences the extent of residual trapping that can occur, as discussed in § 4.2. Both graphs demonstrate how increasing capillary eﬀects, either by decreasing Λ or B, thicken the current. The eﬀect of varying B on the average saturation within a current where Λ = 1 is summarized in ﬁgure 8(c). It is not clear, a priori, how the opposing eﬀects of a lower average saturation and a larger areal extent will aﬀect the total volume of non-wetting ﬂuid contained within the current. The graph in ﬁgure 8(d ) shows the total volume of the non-wetting phase 263 Two-phase gravity currents in porous media (a) 5 (b) 7 B=1 4 Λ=1 6 5 3 h0 h0 2 4 3 2 1 0 1 2 4 6 8 0 10 20 40 Λ 0.8 0.6 0.4 0.2 0 80 100 (d) 1.15 1.0 Λ=1 20 40 60 B 80 100 Dimensionless volume of non-wetting fluid Average saturation (c) 60 B Λ=1 1.10 1.05 1.00 0.95 0.90 0 20 40 60 80 100 B Figure 8. Figures (a) and (b) plot the current height at the origin, h0 , against Λ, with B = 1 and against B, with Λ = 1, respectively. Figures (c) and (d ) plot respectively the average saturation and the total volume of non-wetting ﬂuid contained within the steady-state current as functions of B, with Λ = 1. The dashed lines indicate the values in the sharp-interface limit. as a function of B, for Λ = 1. We see that the volume always increases with increasing capillary forces. We investigate in § 4.2 how both the volume of ﬂuid contained in the currents and the saturation distribution aﬀect estimates for residual trapping within the system. 4.2. Residual trapping Residual trapping occurs when ganglia of the non-wetting phase become disconnected from the main bulk of the ﬂuid and therefore immobilized during imbibition. It is caused by surface tension acting within the pore geometry, making it impossible for the wetting ﬂuid to displace all of the non-wetting ﬂuid. For detailed explanations of the mechanisms on the pore scale which cause residual trapping, the reader is directed towards Lenormand, Zarcone & Sarr (1983) and Pinder & Gray (2008). At the macro-scale, the key experimental observation is the dependence of the non-wetting phase residual saturation on the initial saturation at the time of ﬂow reversal (Land 1968). An initial–residual graph summarizing these data for a number of two-phase-rock systems studied in the literature can be found in ﬁgure 2 in Pentland et al. (2008). All the data sets have a common feature that a higher fraction of ﬂuid is trapped in regions where the initial saturation is lower. Several models have been developed to represent these empirical measurements and a useful summary of these results is contained in ﬁgure 6 of Pentland et al. (2008). The magnitude of residual trapping in the gravity currents considered in § 4.1 is primarily dependent on the areal extent of the current and the maximum saturation reached during the drainage stage, i.e. the steady-state height and saturation proﬁle. This allows us to avoid modelling the details of the imbibition process when inferring 264 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert 0.8 Trapped CO2/initial CO2 A =1 0.7 0.6 0.5 0 20 40 60 80 100 B Figure 9. Fraction of ﬂuid trapped residually, for Λ = 1, as a function of B estimated using Land’s model with C = 1. The dashed line represents the estimate obtained by applying Land’s model to a sharp-interface gravity current. Note that this limit is approached more quickly with respect to increasing B for higher values of Λ. the volume of the residually trapped non-wetting phase and instead apply a trapping model directly to the steady-state saturations. As an example, we use Land’s model (Land 1968) to estimate the fraction of ﬂuid that is trapped residually in the steady-state currents considered in § 4.1. Land’s model relates initial and residual saturations by Sn,res = Sn,init , 1 + CSn,init (4.12) where C is known as Land’s coeﬃcient, and for application to a CO2 –brine–rock system, has a value of order 1 (Juanes et al. 2006). We apply this relation across the initial saturation proﬁle calculated from the steady-state problem, integrate over the height and length of the current and thus calculate how much ﬂuid would be trapped if the constant ﬂux were turned oﬀ. Figure 9 presents estimates for the fraction of initial ﬂuid in the steady-state current that would be residually trapped as a function of B, with Λ = 1. We compare this to estimates obtained using a sharp interface model, with Sn,init = 1 everywhere and ﬁnd that including capillary eﬀects can increase estimates for the fraction of CO2 residually trapped by more than 50 %. We see this increase in the proportion of trapped ﬂuid because the average saturation in the currents is lower. Furthermore, the actual volume of ﬂuid contained in a current is larger with increased capillary forces, due to its greater areal extent (ﬁgure 8d ), which means that a greater absolute volume of ﬂuid is trapped. 5. Conclusions and discussion One of the major diﬃculties of modelling the ﬂow of CO2 during geological storage is the range of scales over which processes occur. Full numerical simulations of ﬂow (Kumar et al. 2005; Mo et al. 2005; Ide, Jessen & Orr 2007) are able to incorporate two-phase phenomena, but the computational intensity requires a compromise on spatial resolution. On the other hand, sharp-interface models lose small-scale details such as the saturation distribution but are able to model ﬂow over much larger distances (Nordbotten & Celia 2006; Hesse, Orr Jr & Tchelepi 2008; Juanes et al. Two-phase gravity currents in porous media 265 2010). Many theoretical models of gravity currents in porous media to date have assumed a sharp interface separating the ﬂuid in the current from the ambient (Huppert & Woods 1995; Hesse et al. 2007). This is appropriate in situations where capillary forces are negligible compared to viscous and gravitational forces. However, when the intruding and ambient ﬂuids are diﬀerent phases, capillary forces acting between them can have a signiﬁcant inﬂuence on their motion. In this study, we considered the propagation of a gravity current of a non-wetting ﬂuid, along an impermeable horizontal boundary, into a porous medium initially saturated by a wetting ﬂuid. We assume that suﬃcient time has elapsed for the current to become long and thin. We model the resulting non-uniform saturation distribution and derived a vertically integrated, time-dependent equation that governs the height of the gravity current. The two-phase governing equation is analogous to its single-phase counterpart (Huppert & Woods 1995), with the two-phase eﬀects incorporated by two new functions describing the saturation at the impermeable boundary and the vertically integrated ﬂux through the current as a function of its height. We show that our two-phase model recovers the sharp-interface case in appropriate parameter limits. The two-phase framework is attractive because it captures the eﬀects introduced by surface tension between the two ﬂuids, whilst maintaining the form of the sharpinterface model. The two key features to consider when modelling two-phase ﬂow on the macroscale are capillary pressure and relative permeability. The model developed in this paper captures the general functional dependence of both features on saturation. We illustrate this dependence by using the Brooks–Corey model for capillary pressure and power law relations for relative permeability. The central assumption is that of gravity–capillary equilibrium, where gravitational and capillary forces balance in the vertical direction, which we show to be valid for long, thin gravity currents. Using this assumption of gravity–capillary equilibrium, along with the capillary pressure as a function of saturation, we obtained an explicit expression for the saturation proﬁle within the current. This proﬁle depends primarily on the pore-size distribution, characterized by Λ, and the ratio of capillary to gravitational forces, characterized by he . We found that as both 1/Λ → 0 and he → 0, capillary forces become negligible and the capillary fringe reduces to the sharp-interface limit. The eﬀect of the saturation proﬁle on the ﬂow of a two-phase gravity current was encapsulated in the ﬂux function F(h/ he ), which itself is a function of Λ and he . The value of the height scale he in geological settings varies widely depending on the rock and ﬂuid types. For example, Bennion (2006) measured capillary entry pressures pe = 3.5 kPa in a sandstone formation and pe = 493.6 kPa in a carbonate formation in the Alberta Basin, Canada, which could potentially be suitable as sequestration sites. Using these values, along with g = 9.8 m s−2 and representative densities ρw = 1020 kg m−3 and ρn = 550 k gm−3 (Bickle et al. 2007), we calculate representative capillary entry heights of he = 0.76 m and 107 m respectively. We used our model of a two-phase gravity current to assess the magnitude of residual trapping as a buoyant plume of CO2 rises through a heterogeneous formation. By considering CO2 ponded beneath a single, ﬁnite barrier, we were able to investigate how the main parameters inﬂuence the areal extent and non-uniform saturation proﬁle of a two-phase gravity current. We demonstrated how capillary forces simultaneously thicken the current, i.e. increase its vertical extent, and reduce the average saturation within it. These two eﬀects have opposing inﬂuences on the volume of ﬂuid that accumulates in steady-state currents. We have shown that the net result in all cases studied is an increase with capillary forces in the volume of 266 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert non-wetting ﬂuid contained within a current, indicating that the increased vertical extent dominates. Residual trapping happens once injection of CO2 ceases and is one of the key mechanisms enabling the success of long-term storage of CO2 as part of the immense challenge of climate change mitigation. Some steps have been made towards incorporating the eﬀects of residual trapping in sharp interface models, by assuming that the volume of ﬂuid in the current decreases at a constant rate, controlled by a trapping parameter (Kochina, Mikhailov & Filinov 1983; Hesse et al. 2008; Juanes et al. 2010). With the two-phase model developed in this paper, we are able to investigate the extent of residual trapping more accurately because we know the height and saturation proﬁles at the end of the injection period. Whilst we use Land’s trapping model to obtain estimates, it is important to note that any model relating initial–residual saturations could be used once the saturation proﬁle is known. The essential feature that at lower saturations, the fraction of ﬂuid trapped is higher is the reason why, as capillary eﬀects are increased and the two-phase currents have lower average saturations, the fraction of ﬂuid residually trapped is greater. Moreover, the amount of ﬂuid contained in the steady-state current at the start of imbibition is larger when capillary forces are higher, so the overall volume of ﬂuid trapped is also higher, not just the fraction. This result is important given that residual trapping post-injection is proportional to the volume of ﬂuid per unit length in the steady current (Hesse et al. 2008). Furthermore, the increased height means a greater volume of rock is invaded by CO2 as capillary eﬀects are increased, and it is therefore able to contact a larger volume of brine, which could increase further trapping by dissolution (Johnson et al. 2004). Whilst we have considered steady-state gravity currents in this paper, the model developed is fully time-dependent and therefore provides a theoretical framework in which to investigate how capillary forces aﬀect the motion of transient two-phase gravity currents. For example, it is not clear how the reduced relative permeability at the nose of the current, caused by the lower saturation when the height decreases to zero, aﬀects the steepness of the nose and the thickness and velocity of the entire current. Other future work will extend the methods outlined here to two-phase gravity currents propagating in a conﬁned aquifer, where the viscosity ratio of the ﬂuids plays an important role. Our framework also holds the promise of being able to model drainage and imbibition in a self-consistent manner, using diﬀering capillary pressure and relative permeability relationships, thus improving the assessment of residual trapping. Furthermore, the simpliﬁcations introduced by the vertical gravity– capillary equilibrium assumption allow eﬃcient calculations of two-phase currents over distances required at the basin scale during geological sequestration of CO2 . In all these systems, and in a number of related areas, the incorporation of capillary eﬀects provides new insight into the dynamics of two-phase gravity currents. The authors would like to gratefully acknowledge the artistic expertise of Mark Hallworth. The research of M.J.G. is funded by the EPSRC, J.A.N. by the Leverhulme Trust and Lloyds Tercentenary Foundation, M.A.H. partly by a David Crighton Fellowship and the work of H.E.H. is partially supported by a Royal Society Wolfson Research Merit Award. Appendix. Justiﬁcation for vertical gravity–capillary equilibrium Vertical gravity–capillary equilibrium is the key assumption of the two-phase gravity current model presented in this paper because it enables us to resolve the saturation Two-phase gravity currents in porous media 267 distribution within the current. In a long, thin gravity current, where ﬂow is assumed to be predominately horizontal, we showed in § 3.1 that gravity–capillary equilibrium is implied by assuming that each phase is in hydrostatic equilibrium. However, the height of the current rises during propagation, so the saturation distribution, which depends on current height, must keep re-equilibrating in order for the assumption to remain valid. We ﬁnd expressions for the time scale, tm , over which the current height changes due to gravity-driven horizontal motion and the time scale, tc , required for the saturation distribution to return to gravity–capillary equilibrium after a small change in current height. By showing that the ratio of time scales rt = tc /tm is small, we demonstrate that to a good approximation, gravity–capillary equilibrium is maintained as the current propagates. It should be noted that the assumption breaks down at the front of the current, where ﬂow in the vertical direction is not negligible compared to the horizontal motion. However, this is the case in all gravity current models which consider long, thin currents and rely on the assumption of hydrostatic pressure. For example, Huppert (1982) showed experimentally for viscous gravity currents that it makes no diﬀerence to the overall motion of the current. In order to ﬁnd expressions for the time scales tm and tc , we vertically integrate the non-wetting phase local mass conservation equation (2.6) between z and h. This yields h ∂ ∂V (h, z) un dz − [wn ]z = 0, (A 1) + ϕ ∂t ∂x z h where V (h, z) = z s(h, z ) dz is the volume of ﬂuid per unit length contained in a column of current that has height h, and we have used s(z = h) = un (z = h) = wn (z = h) = 0. The time scale, tm , over which the current height changes due to the horizontal velocity of the current ﬂuid during propagation is obtained from the balance between the ﬁrst and second terms in (A 1). We make the assumption, as explained in § 3.1, that the current is in gravity–capillary equilibrium, and therefore we ﬁnd that ∂V /∂t = s(h, z)∂h/∂t by using (3.4) and integrating directly. Darcy’s law (2.7) provides an expression for horizontal velocity un = −kλn (s) ∂pn ∂h = −ρgkλn (s) , ∂x ∂x (A 2) where the second equality comes from the expression for the hydrostatic non-wetting phase pressure, given by pn (h, z) = pe + ρw gh − ρn g(h − z). (A 3) Note that this uses the condition pn = pe + ρw gh at the current boundary z = h. We scale t ∼ tm , z, h ∼ H and x ∼ L, where H and L are the characteristic length scales for the current height and horizontal extent, respectively, and write = H /L. Hence, the ﬁrst two terms of (A 1) yield the scaling ϕs H ∼ ρgkλn (s) 2 , tm (A 4) which we can rearrange to ﬁnd an expression for the time scale, tm , given by tm ∼ ϕH s −2 . ρgk λn (s) (A 5) 268 M. J. Golding, J. A. Neufeld, M. A. Hesse and H. E. Huppert Now we consider a current at height h(x, t), which is in gravity–capillary equilibrium and therefore has saturation distribution s(h, z), given by (3.4), and suppose that its height increases by δh. The time scale tc indicates how quickly the recovery of vertical gravity–capillary equilibrium is achieved due to the vertical velocity of the nonwetting phase and is therefore given by the balance between the ﬁrst and third terms in (A 1). Once gravity–capillary equilibrium is recovered, the saturation distribution is s(h + δh, z), and therefore we ﬁnd that δV = V (h + δh, z) − V (h, z) = s(h, z)δh to ﬁrst order in δh. From (2.7) the vertical velocity, wn , is given by ∂pn − ρn g , wn = −kλn (s) ∂z (A 6) (A 7) where the vertical gradient of the hydrostatic pressure pn (h + δh, z) after a change in height of δh can be written as ∂pn pn (h + δh, h + δh) − pn (h + δh, 0) (h + δh) = = ρn g . ∂z h h (A 8) Hence, (A 7) becomes δh . (A 9) h We use (A 6) and (A 9) to scale ∂V /∂t ∼ s(h, z)δh/tc and wn ∼ ρn gkλn (s)δh/ h in (A 1) to ﬁnd δh δh ∼ ρn gkλn (s) , (A 10) ϕs tc H which can be rearranged to give wn = −ρn gkλn (s) tc ∼ ϕH s . ρn gk λn (s) (A 11) Thus, we ﬁnd, using (A 5) and (A 11), that the ratio of time scales rT ∼ ρ 2 . ρn (A 12) For a long, thin gravity current, 1, and so we have shown that vertical gravity– capillary equilibrium is recovered for a given change in current height on a time scale much smaller than the time scale of the overall motion of the current. 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