2.1. Derivation of a Two-Phase Pseudomodel
 For completeness we will derive vertically averaged equations for a two-phase CO2-brine system in an aquifer and then state a so-called fractional flow formulation. Similar derivations are known in the literature [e.g., Lake, 1989; Yortsos, 1995]. For clarity of exposition, lowercase symbols are used for three-dimensional variables, referred to as fine scale, while uppercase symbols are reserved for the associated upscaled variables. We will also assume, for simplicity of exposition, an aquifer with uniform cross-sectional thickness H; the extension to more general cases is straightforward [Gasda et al., 2009]. The aquifer is enclosed above and below by impermeable formations and makes a constant dip angle with the horizontal plane, and the characteristic aerial extent L is much larger than the thickness, , where is the horizontal to vertical anisotropy ratio in the permeability [Lake, 1989; Yortsos, 1995]. This scale constraint implies that we will resolve the impact of capillary forces on the vertical variation of saturation but that the corresponding horizontal variation will be below the resolution of our model. We assume that CO2 and resident brine appear as two immiscible fluids such that CO2 is the nonwetting phase and brine is the wetting phase, denoted by subscripts n andw, respectively. Furthermore, the viscosity and density of each fluid, and , where , are taken to be constants. At typical reservoir conditions, .
 Under the conditions given above we may state mass conservation equations in terms of fluid saturation as
where volumetric flux is determined by the extended Darcy's law
with . In equations (1) and (2) is the porosity, denotes sink-source terms, k denotes the permeability, is the phase pressure, g is the vector of gravitational acceleration, and is the phase mobility, where
 Here the relative permeability is a known, possibly hysteretic, function of saturation. We assume that one of the main directions of anisotropy of permeability is aligned with the third coordinate, so that we can decompose the permeability into a horizontal permeability tensor and a scalar vertical permeability, formally written as a block matrix:
 The difference between the nonwetting and wetting phase pressures may be expressed as an algebraic and possibly hysteretic function of saturation. Since the pore space is completely saturated by the two fluids, the following constraint on the saturations always applies: sw + sn = 1. Hence, we may express the relative permeability and capillary pressure–saturation relationships in terms of the CO2 (nonwetting) saturation:
 The parameterization of hysteresis is not given explicitly in these expressions. Without being physically restrictive, assume that these algebraic relationships are monotone and differentiable. In particular, it follows that the capillary pressure–saturation relationship has a hysteretic inverse function . This inverse function is dependent on the hysteresis parameterization, just like the capillary function itself. Equations (1)–(4) form a complete model for two-phase, immiscible, and incompressible flow if appropriate initial and boundary conditions are provided.
 We introduce a Cartesian coordinate system such that (x, y) are coordinates in the aquifer plane and 0 < z < H is the coordinate across the aquifer. The corresponding basis vectors are ei, i = 1, 2, 3, directed such that
Here g refers to the gravitational constant.
 We now decompose the velocity field into components , which is parallel to the aquifer plane, and , which is normal to the aquifer. Because of the low aspect ratio , the normal velocity is negligible compared to the parallel velocity . Inserting this assumption into Darcy's law, equation (2), gives an expression for hydrostatic equilibrium:
where is the pressure at the lower aquifer plane. Defining the coarse pressure at the bottom of the formation is an arbitrary choice and may seem unphysical when sn (z = 0) = 0. For this case, the pressure at some known elevation must be extrapolated. However, this is of no consequence for the model and, indeed, an identical governing equation would be obtained if any other constant elevation was chosen as the datum for the coarse pressure. The pressure gradients are constant across the aquifer such that
where, from now on, . We chose to serve as the upscaled phase pressure. The upscaled capillary pressure then becomes
and relates to the fine-scale pc by
 Thus, if Pc = Pc(x, y) is known, the fine-scale saturation can be reconstructed using
 Recall that sc is the inverse capillary pressure function. Since , sn will be a strictly increasing function of z when the fine-scale capillary pressure curve is independent of z.
 To proceed, introduce other upscaled variables as (weighted or simple) averages across the aquifer:
 Observe from equation (10) that dynamic fine-scale information can be eliminated from the definition of the upscaled variables. In general, however, upscaled capillary pressure and relative permeability are functions of the spatial locations x and y even though their fine-scale analogs are functions of fine-scale saturation only. However, if porosity can be written as and the capillary pressure function is homogeneous, we see that . Because of the properties of the fine-scale capillary pressure this expression is invertible, so that the upscaled capillary pressure can be given as a function of the upscaled nonwetting saturation without dependence on x or y:
 If, in addition, we have the decomposition k = k⊥(z)k‖(x, y), then the upscaled relative permeability and mobility also become functions only of upscaled saturation:
 This allows us to integrate equations (1) and (2) using equations (11)–(15) to obtain
and also Sn + Sw = 1.
 Beyond differences in interpretation of variables, equations (18) and (19) take the exact same form as equations (1) and (2). To rewrite these equations in so-called global pressure–fractional flow form, define total flow and mobility-weighted average density as
and upscaled fractional flow functions as
 Here we have taken the mobilities as scalars to simplify the exposition.
 Following Chavent and Jaffre , we define global pressure P as
and observe that
 Together equations (23) and (24) form the pressure-velocity equations for global pressure and total velocity. To get the saturation equation, rewrite the phase velocity in terms of the total velocity:
 The saturation equation then becomes
2.2. Consideration of Dimensionless Groups
 Three dimensionless groups are of particular significance for the vertically integrated models. These represent the time scale of vertical fluid migration, the time scale for establishing a capillary fringe, and the dimensionless extent of the capillary fringe. We will state these dimensionless groups in sections 2.2.1–2.2.3.
2.2.1. Vertical Flow
 While vertical flow of CO2 in a 1-D column forces countercurrent flow of brine, vertical flow in a 3-D domain will likely lead to reduced perturbations of the pressure in the brine phase. The flux of CO2 in a porous medium away from the top boundary can consequently be estimated assuming a hydrostatic pressure distribution by Darcy's law as
 Recalling that the characteristic velocity is given by , we can obtain a characteristic time by neglecting saturation variability and assuming a characteristic mobility . This allows us to introduce a nondimensional time associated with vertical segregation:
 We use this dimensionless time as an indicator, and we do not expect vertically integrated models to be applicable for time scales , in particular, if injection is preferentially in the lower part of the aquifer.
2.2.2. Capillary Fringe
 From a sharp (segregated) fluid distribution, the equation governing the evolution of the capillary fringe in a 1-D system can be derived from Darcy's law and mass conservation, analogously to equation (26), as
 The equation governing the deviation from the equilibrium capillary fringe distribution is thus
 As in section 2.2.1, we denote characteristic values with a dagger, and we see that the characteristic diffusion parameter for the parabolic equation (30) is given as
 The vertical extent of the capillary fringe can be estimated as
from which it follows that the dimensionless time associated with the formation of a capillary fringe is defined as
 Again, we use this dimensionless time as an indicator for the applicability of vertically integrated models and, in particular, capillary fringe models. We consider a condition for the applicability of capillary fringe models. When , the vertical saturation profile is more strongly dependent on the vertical flow characteristics of the system.
 An alternative view is to consider the balance of the time scale to the time scale of horizontal flow, given by
where is the horizontal resolution of the model. Capillary fringe models can then be argued for if the dimensionless parameter group
2.2.3. Fringe Extent
 The extent of the capillary fringe has already been estimated. The natural normalization, when flow of the CO2 plume is of interest, is, with respect to the “thickness” of the CO2 plume, SnH; thus, we obtain
 We see that the relative extent of the capillary fringe is therefore dependent on the coarse solution Sn itself. This indicates that capillary fringe models are increasingly important for thin CO2 plumes, which are expected to be prevailing during long-time migration. To avoid the dependence of on the solution, it is sometimes convenient to consider the dimensionless parameter .
2.3. Sharp Interface Approximation
 The extent of the capillary fringe is the vertical length l by which saturation goes from residual gas saturation snr to residual brine saturation sn = 1 − swr. Let be a measure of the variation in the capillary pressure–saturation relationship: . This number can be used to define an approximation to the characteristic derivative of the capillary pressure curve,
 Then we define the dimensionless number associated with this particular choice of characteristic variation in capillary pressure as . By the mean value theorem and the monotonicity of fine-scale saturation sn(z), there exists a unique in [0, H] such that
with the trivial exception of sn being constant. Replacing z by in equation (9) results in
 We now interpret the sharp interface as the limit of a smooth saturation distribution with sn(0) = snr and sn(H) = 1 − swr. Physically, this is equivalent to the case of negligible capillary transition, which will be the case for high density contrasts and small interfacial tension. Then,
where z = h(x, y) is the unknown location of the interface. Note that when the system is initially filled with the wetting fluid, as is the case for CO2 injection, then snr = 0 below the interface h. If porosity and absolute permeability are independent of z and are isotropic, the sharp interface limit gives
where we have defined and . Substituting equations (38)–(41) into equations (23), (24), and (26) and omitting terms of give a complete set of equations for p, u, and the interface location h = h(x, y).
2.4. Capillary Equilibrium Upscaling Limit
 A low density contrast, or, more precisely, a large capillary fringe, is defined as . For this case, there is essentially no vertical variation in pressure:
 Therefore, saturation variations within the aquifer is due only to capillary forces. In particular, we deduce from equation (10) that
 We see that the vertical variation in saturation comes from vertical heterogeneity in the capillary pressure function. In the case of homogeneous capillary pressure function, the vertical fluid distribution is constant, . Furthermore, with homogeneous relative permeability functions, the coarse mobility functions also simplify significantly:
 Together with the sharp interface limit, the low density contrast limit forms the end-members of possible vertical equilibrium models. Indeed, for porous media with no vertical variation in parameters and where the relative permeabilities satisfy (see Appendix A for proof)
 We obtain the following relationship:
and, in particular,
 These relationships will be useful when discussing the results presented in section 3.
2.5. Effect of Residual Trapping on Upscaled Equations
 An important trapping mechanism for CO2 is the immobilization of fluid due to interfacial forces, leading to residual saturation. This is modeled by vanishing relative permeability for saturations below the corresponding residual saturations at the fine scale. Consider the following situation: Initially, the aquifer is completely saturated by resident brine and sn = 0 everywhere. When supercritical CO2 is injected into the aquifer, the resident brine is displaced, corresponding to primary drainage. As the plume starts to migrate, brine starts to imbibe into parts of the CO2 region. This leaves immobile CO2 at residual saturation sn = snr 0, in a region bounded by the lowest extent of the CO2 fringe, denoted hmin, and the interface h separating mobile CO2 from the immobile phase. If the fine-scale relative permeability and capillary pressure are given by a hysteresis model, we may generate corresponding upscaled quantities on the basis of equations (11)–(15). This follows because equation (7) guarantees that a change in flow direction at the fine scale has to occur simultaneously everywhere in the vertical z direction. Thus, depending on the particular hysteresis model given, we may have to keep track of the fine-scale saturation distribution at the times of flow reversals in addition to hmin and h to obtain appropriate coarse-scale models.
 In general, we cannot generate closed form expressions for upscaled relative permeability relationships and capillary pressure, and we need to rely on numerics. However, for the sharp interface approximation and the situation where there is residual trapping but no fine-scale hysteresis in relative permeabilities as described above, we have
 Here hmin satisfies the constraint hmin ≤ h. Furthermore, we have the important observation that
 We point out these implications since it has been common to use this time rate of change as a simplified hysteresis model in the literature [Farcas and Woods, 2009; Hesse et al., 2008; Juanes et al., 2010; Kochina et al., 1983]. One study that accounts for the full hysteresis model in a sharp interface setting is reported by Juanes and MacMinn .
 For the current case, the upscaled saturation and mobility relationships become
where we define and the mobility ratio . Let the difference in brine mobility at pure brine and brine at residual CO2 saturation be denoted , and introduce
 An important special case arises when flow is considered only in the x direction. Then, in the absence of source and sink terms, equation (26) reduces to
 This model is similar to the model used by various authors [Hesse et al., 2008; Juanes et al., 2010] to investigate gravity currents with residual trapping; however these authors have used the simplified hysteresis model obtained by considering only the sign of time rate of change of h, as pointed out in equation (45). For simple flow patterns, this is equivalent to neglecting . As shown in section 3, these terms may have an impact on the solution because of the reduced relative permeability of the brine in the region defined by trapped CO2. We note in conclusion that the simplified hysteresis model was originally proposed by Kochina et al.  for an air-water system, where the system is initially filled with air. There it is a reasonable approximation to let the air be inviscid; thus, is correctly neglected.