Impact of the capillary fringe in vertically integrated models for CO2 storage

Authors


Abstract

[1] This paper investigates vertically integrated equilibrium models for CO2 storage. We pay particular attention to the importance of including the effect of fine-scale capillary forces in the integrated equations. This aspect has been neglected in previous work, where the fluids are segregated by a sharp interface. Our results show that the fine-scale capillary forces lead to qualitative and quantitative alterations of the integrated equations. Interestingly, while such forces are dispersive on the fine scale, they lead to self-sharpening of the solution on the integrated scale. We discuss these aspects for injection, leakage, and long-term migration through the application by comparison to common sharp interface models proposed in the literature.

1. Introduction

[2] Storage of carbon dioxide (CO2) in saline aquifers has emerged as a promising, and likely, option for offsetting a fraction of the global atmospheric emissions [Intergovernmental Panel on Climate Change, 2005; Pacala and Socolow, 2004]. Geological storage operations will typically handle the emissions from one to several industrial point sources such as power plants, indicating that injection rates of several million tons (Mt) per year localized in a small region will be of interest. For typical geological formations, where the vertical extent is measured in tens of meters, this will lead to plume extents on the order of kilometers during the injection phase and possibly much greater in a postinjection phase [Nordbotten and Celia, 2006c; Hesse et al., 2008; Celia et al., 2011; Juanes et al., 2010]. These extreme aspect ratios, together with a strong gravity override resulting from a significant density difference, argue for models in which flow in the vertical direction is neglected, such as vertically integrated models [Celia and Nordbotten, 2009]. While such models have historically received some consideration in petroleum extraction [Martin, 1968; Coats et al., 1971], they have received renewed attention in the context of geological storage.

[3] During the injection phase, when plume evolution can often be considered to have a radial character, vertically averaged, sharp interface models have been successfully applied to characterize plume spread in a confined formation [Nordbotten and Celia, 2006a, Nordbotten and Celia, 2006c]. If we further consider cases where there is limited dip in the geological formation, the radial symmetry may persist beyond the injection phase for some time. A particularly useful observation is that the governing equations linearize (in the sense of the fractional flow function) when the gravity tongue becomes thin, allowing for late time asymptotic analysis for sharp interface models [Barenblatt, 1996; Huppert and Woods, 1995]. These approaches are appealing in that they may also include additional physical effects [see, e.g., Gasda et al., 2009].

[4] In the postinjection phase, it is often necessary to consider the impact of either formation dip or a regional background flow. The emphasis on lateral movement causes line symmetry to be used as an alternative to radial symmetry for the purpose of obtaining analytically tractable equations. The study of plume evolution under these conditions often emphasizes the role of residual trapping [Hesse et al., 2008; Juanes et al., 2010; Farcas and Woods, 2009]. An important limit appears when the advective (e.g., the net lateral) movement dominates over the dispersive (redistributive) forces. In this case, the solution can be approximated from the hyperbolic part of the equation, and closed form expressions can be obtained both for the plume shape and its ultimate travel distance and time [see, e.g., Hesse et al., 2008; Juanes et al., 2010].

[5] It is natural to consider the accuracy of the approximations introduced in the analysis reviewed above. Indeed, the sharp interface formulations compare well to experiments, during both the injection phase and the redistribution phase [see, e.g., Huppert and Woods, 1995]. However, these experiments are primarily conducted with one phase dominating the mobility ratio (as is the case for groundwater mounds, where the viscosity of air is negligible). Furthermore, the materials are often chosen such that the capillary forces are weak, in accordance with the sharp interface assumption. In experiments with stronger capillary forces, the sharp interface approximation is less suited [see, e.g., Franz et al., 2011]. Similarly, numerical experiments, particularly when line symmetry is assumed, tend to show quantitative and qualitative differences between sharp interface [see, e.g., Hesse et al., 2008; Juanes et al., 2010] and resolved models (see, e.g., Riaz and Tchelepi [2006] and Pruess et al. [2009] for analysis of similar problems). A direct comparison indicates that for some processes, sharp interface models get the correct trend for the plume, although both the quantitative and qualitative properties of the tip are not captured [Gasda et al., 2008].

[6] Herein, we explore modeling frameworks that avoid the sharp interface assumption (see Coats et al. [1971], Martin [1968], Lake [1989], and Yortsos [1995] for earlier studies along these lines in the context of oil recovery). This is motivated by the observation that in real porous media, there will always be a transition zone, which for immiscible flow, will have a reduced effective permeability because of the nonlinear relative permeability functions. We formulate our model to account for hysteresis in both capillary and relative permeability curves. For this model we show explicitly the assumptions involved in a sharp interface approximation. The main result of the paper is to show that the capillary fringe has a potentially important impact on the quantitative and qualitative estimates of plume shape and spread, both during the injection and postinjection phases. Second, we emphasize the often-neglected point that the reduced mobility of brine in the region of residually trapped CO2 leads to a qualitative difference in upslope migration patterns during postinjection (this point is also discussed by Juanes and MacMinn [2008] and Pruess et al. [2009]). These two observations diminish the quantitative importance of analytic expressions for plume migration based on sharp interface assumptions derived by Hesse et al. [2008] and Juanes et al. [2010]. Similarly, the analytic expressions derived under sharp interface assumptions by Nordbotten and Celia [2006a] are shown to be inaccurate; however, for this problem we give new solutions that fully account for the effect of the capillary fringe.

2. Vertically Averaged Equations

[7] We give here the derivation of the vertically averaged equations, which account for the existence of a capillary fringe and allow for hysteretic relative permeability and capillary pressure functions. We also include some discussion, including both relevant dimensionless groups and the limits of relatively large and small capillary forces.

2.1. Derivation of a Two-Phase Pseudomodel

[8] 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 equation image with the horizontal plane, and the characteristic aerial extent L is much larger than the thickness, equation image, where equation image 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, equation image and equation image, where equation image, are taken to be constants. At typical reservoir conditions, equation image.

[9] Under the conditions given above we may state mass conservation equations in terms of fluid saturation equation image as

equation image

where volumetric flux equation image is determined by the extended Darcy's law

equation image

with equation image. In equations (1) and (2)equation image is the porosity, equation image denotes sink-source terms, k denotes the permeability, equation image is the phase pressure, g is the vector of gravitational acceleration, and equation image is the phase mobility, where

equation image

[10] Here the relative permeability equation image 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:

equation image

[11] 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:

equation image

[12] 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 equation image. 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.

[13] 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

equation image

Here g refers to the gravitational constant.

[14] We now decompose the velocity field into components equation image, which is parallel to the aquifer plane, and equation image, which is normal to the aquifer. Because of the low aspect ratio equation image, the normal velocity is negligible compared to the parallel velocity equation image. Inserting this assumption into Darcy's law, equation (2), gives an expression for hydrostatic equilibrium:

equation image

where equation image 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

equation image

where, from now on, equation image. We chose equation image to serve as the upscaled phase pressure. The upscaled capillary pressure then becomes

equation image

and relates to the fine-scale pc by

equation image

[15] Thus, if Pc = Pc(x, y) is known, the fine-scale saturation can be reconstructed using

equation image

[16] Recall that sc is the inverse capillary pressure function. Since equation image, sn will be a strictly increasing function of z when the fine-scale capillary pressure curve is independent of z.

[17] To proceed, introduce other upscaled variables as (weighted or simple) averages across the aquifer:

equation image
equation image
equation image
equation image
equation image

[18] 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 equation image and the capillary pressure function is homogeneous, we see that equation image. 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:

equation image

[19] 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:

equation image

[20] This allows us to integrate equations (1) and (2) using equations (11)(15) to obtain

equation image
equation image

and also Sn + Sw = 1.

[21] 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

equation image

and upscaled fractional flow functions as

equation image

[22] Here we have taken the mobilities as scalars to simplify the exposition.

[23] Following Chavent and Jaffre [1986], we define global pressure P as

equation image

and observe that

equation image
equation image

[24] 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:

equation image

[25] The saturation equation then becomes

equation image

2.2. Consideration of Dimensionless Groups

[26] 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 2.2.12.2.3.

2.2.1. Vertical Flow

[27] 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

equation image

[28] Recalling that the characteristic velocity is given by equation image, we can obtain a characteristic time by neglecting saturation variability and assuming a characteristic mobility equation image. This allows us to introduce a nondimensional time associated with vertical segregation:

equation image

[29] We use this dimensionless time as an indicator, and we do not expect vertically integrated models to be applicable for time scales equation image, in particular, if injection is preferentially in the lower part of the aquifer.

2.2.2. Capillary Fringe

[30] 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

equation image

[31] The equation governing the deviation equation image from the equilibrium capillary fringe distribution equation image is thus

equation image

[32] 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

equation image

[33] The vertical extent of the capillary fringe can be estimated as

equation image

from which it follows that the dimensionless time associated with the formation of a capillary fringe is defined as

equation image

[34] Again, we use this dimensionless time as an indicator for the applicability of vertically integrated models and, in particular, capillary fringe models. We consider equation image a condition for the applicability of capillary fringe models. When equation image, the vertical saturation profile is more strongly dependent on the vertical flow characteristics of the system.

[35] An alternative view is to consider the balance of the time scale equation image to the time scale of horizontal flow, given by

equation image

where equation image is the horizontal resolution of the model. Capillary fringe models can then be argued for if the dimensionless parameter group

equation image

2.2.3. Fringe Extent

[36] 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

equation image

[37] 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 equation image on the solution, it is sometimes convenient to consider the dimensionless parameter equation image.

2.3. Sharp Interface Approximation

[38] 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 equation image be a measure of the variation in the capillary pressure–saturation relationship: equation image. This number can be used to define an approximation to the characteristic derivative of the capillary pressure curve,

equation image

[39] Then we define the dimensionless number equation image associated with this particular choice of characteristic variation in capillary pressure as equation image. By the mean value theorem and the monotonicity of fine-scale saturation sn(z), there exists a unique equation image in [0, H] such that

equation image

with the trivial exception of sn being constant. Replacing z by equation image in equation (9) results in

equation image

[40] We now interpret the sharp interface as the limit equation image 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,

equation image

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

equation image
equation image
equation image
equation image
equation image

where we have defined equation image and equation image. Substituting equations (38)(41) into equations (23), (24), and (26) and omitting terms of equation image give a complete set of equations for p, u, and the interface location h = h(x, y).

2.4. Capillary Equilibrium Upscaling Limit

[41] A low density contrast, or, more precisely, a large capillary fringe, is defined as equation image. For this case, there is essentially no vertical variation in pressure:

equation image

[42] Therefore, saturation variations within the aquifer is due only to capillary forces. In particular, we deduce from equation (10) that

equation image

[43] 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, equation image. Furthermore, with homogeneous relative permeability functions, the coarse mobility functions also simplify significantly:

equation image

[44] 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)

equation image

[45] We obtain the following relationship:

equation image

and, in particular,

equation image

[46] These relationships will be useful when discussing the results presented in section 3.

2.5. Effect of Residual Trapping on Upscaled Equations

[47] 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.

[48] 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

equation image

[49] Here hmin satisfies the constraint hminh. Furthermore, we have the important observation that

equation image

[50] 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 [2008].

[51] For the current case, the upscaled saturation and mobility relationships become

equation image
equation image
equation image

where we define equation image and the mobility ratio equation image. Let the difference in brine mobility at pure brine and brine at residual CO2 saturation be denoted equation image, and introduce

equation image

[52] 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

equation image

[53] 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 equation image. 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. [1983] 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, equation image is correctly neglected.

3. Illustrative Results

[54] In this section, we have two objectives. First, we consider a specific choice of relative permeability and capillary pressure functions at the fine scale. For this choice of fine-scale constitutive relationships, we illustrate the associated pseudofunctions for the vertically integrated model.

[55] Our second objective in this section is to assess the importance of accounting for the capillary fringe in several applications that are typical for CO2 storage. In particular, we consider three cases: injection of CO2 from a vertical well, upconing near a leaky well, and long-term upslope migration of CO2.

3.1. Example Pseudofunctions

[56] We consider the CO2 phase to be the nonwetting phase and use the function [Dahle et al., 2009]

equation image

to model the capillary pressure as a function of the normalized brine saturation, defined as sw,N = (swswr)/(1 – swr). Associated with this capillary pressure function, the relative permeabilities are specified as

equation image
equation image

[57] Additionally, the aquifer is specified as 50 m thick and homogeneous, with a permeability of 10−13 m2, porosity of 15%, CO2 and brine densities of 733 and 1099 kg/m3, respectively, and correspondingly viscosities of 0.0611 and 0.511 mPa s.

[58] For this choice of fine-scale parameters, we can calculate the pseudofunctions for the vertically integrated model using equations (14) and (16); see Nordbotten and Dahle [2010] for the full calculation. The resulting curves are shown in Figures 1 and 2.

Figure 1.

Fine-scale capillary pressure curves together with coarse capillary pressure curves for the case of a capillary fringe and a sharp interface. The limiting case of negligible gravitational forces is identical (up to a constant) to the fine-scale capillary pressure curve.

Figure 2.

Fine-scale CO2 relative permeability curves together with coarse capillary pressure curves for the case of a capillary fringe and a sharp interface. The limiting case of negligible gravitational forces is identical (up to a constant) to the fine-scale capillary pressure curve.

[59] Note that the pseudofunctions are dependent on the aquifer thickness. We see that the pseudo relative permeabilities fall between the fine-scale and sharp interface relative permeability curves, as expected from equation (43). Thus, the effect of a capillary fringe is to make the pseudo relative permeabilities more nonlinear than for a sharp interface model and closer to their fine-scale counterparts. This will lead to self-sharpening in Buckley-Leverett–type displacements; thus, we have the interesting observation that although capillary forces disperse the solution on the fine scale, they lead to self-sharpening of the solution in the integrated model.

3.2. Injection Into a Confined Aquifer

[60] For horizontal and radial problems with constant injection rate into an initially brine-filled medium, it is well known that the 2-D vertically integrated equation (26) can be written in self-similar form as [Blunt and King, 1991; Nordbotten and Celia, 2006a]

equation image

[61] Here QT is the total injection rate, and the self-similar coordinate is given as

equation image

[62] This nonlinear boundary value problem is constrained by mass balance:

equation image

where Qc is the injection rate of CO2.

[63] We solve equation (52) by an iterative shooting method to obtain the curves given in Figure 3 The hyperbolic limit, corresponding to high injection rates, simplifies equation (52) to a first-order differential equation that can be integrated numerically.

Figure 3.

Modeled plume extent during injection from a fully penetrating well into a homogeneous formation. The x axis is normalized to equal total injected volume. The hyperbolic limit refers to the limiting case of high injection rates.

[64] We show four different cases in Figure 3 The base case is one with a moderate injection rate of 1000 cm3/s CO2, where we have used the fringe model to calculate plume extent. The corresponding reconstructed fine-scale saturation distribution is shown in Figure 4 The three other cases correspond to the hyperbolic limit (high injection rate and/or low permeability), the solution of the sharp interface model, and, finally, the CO2 plume that is predicted for mixed injection of 75% CO2 and 25% brine.

Figure 4.

Reconstruction of the saturation distribution according to the base case in Figure 3.

[65] From these results we see that including the fringe in the injection analysis has a large impact on the extent and shape of the plume tip. Thus, the predicted footprint of CO2 injection is strongly sensitive to the vertically integrated model. In contrast, we see that the bulk thickness of the plume is less dependent on the capillary fringe. This is consistent with the dependence of the dimensionless parameter equation image on the coarse saturation.

[66] We also make the observation that because of fluid segregation in the vertical direction, the integrated models predict that mixed injection of CO2 and brine will lead to lower sweep and thus lower storage in the form of residual trapping. This is in contrast to the findings obtained using only a 1-D representation of the system, where the opposite conclusion is drawn [Qi et al., 2009].

3.3. Upconing Near a Leaking Well

[67] Upconing is analyzed by considering the stationary solution to equation (26) in radial coordinates. We consider a case with a pumping rate of 1000 cm3/s, where we observe that 25% of the pumped fluid is brine. We assume that the fluid saturation is continuous into the well and that the fluids are hydrostatic within the aquifer portion of the well, so that the CO2-brine interface is basically at the top of the formation as we enter the well. Figure 5 shows the results using both a fringe model and an upconing model as we integrate equation (26) outward from the well. For comparison, we have also included a model that does not use the Dupuit approximation [Nordbotten and Celia, 2006b].

Figure 5.

Model of upconing near a leaking or pumping well. Note the small sensitivity to the presence of a capillary fringe, relative to the impact of adding a vertical flow component, illustrated by the vertically structured pressure model. The radial axis is in logarithmic coordinates to emphasize the near-well effects.

[68] We see that contrary to the injection problem, the inclusion of a capillary fringe makes little difference for the pumping problem. This is again related to the fact that equation image is inversely proportional to the coarse saturation Sn, and for the upconing problem, Sn does not approach zero in the same way as for the injection problem.

3.4. Migration Under a Sloping Caprock

[69] We consider a problem motivated by the Svalbard benchmark problem [Dahle et al., 2009]. Here we consider injection of 20 Mt CO2 into a homogeneous, tilting aquifer, with a dip angle of 1%. We create a lower-dimensional representation of the problem by assuming that the CO2 spreads independently of the aquifer slope during the injection phase. For simplicity and for conformity with previous studies such as the one by Hesse et al. [2008], we do not take into account the true shape of the injection plume but consider only a pulse initial condition. This implies that we are interested in a problem where the initial CO2 is located within a block with side lengths of 2.4 km, with a height filling the aquifer. Aquifer porosity is 15%, and the residual saturations of nonwetting and wetting phases are 20%. We model the problem assuming line symmetry, thus only solving for variation along the slope of the formation.

[70] To account for the residual saturations, we consider the fine-scale functions defined only for the normalized saturations between the residual saturations, thus ignoring the primary drainage curve. This is essentially equivalent to assuming an irreducible gas and brine saturation equal to the residual saturation for the whole injection period.

[71] We solve equation (50) for this problem using a standard first-order explicit finite volume discretization, with upwind treatment of the hyperbolic term. Figures 6 and 7 show results of our simulations, with and without a fringe, respectively. In Figure 6 we have shown the initial condition together with the solution after 5000 years, as well as the maximum extent of the CO2 plume over the whole simulation period. This corresponds to the region where CO2 is trapped because of hysteresis. There is a reduction of travel speed for the fringe solution compared to the sharp interface solution, as is expected because of the lower fractional flow functions, in particular, near the end points. However, the total time until trapping for the sharp interface and fringe models is comparable because of the shorter migration distance predicted by the fringe model. Interestingly, it is not only the quantitative estimates of migration speed and distance that differentiate the models; the qualitative shape of the plume is also different. While a sharp interface model predicts essentially a rarefaction followed by a shock, the opposite is predicted by the fringe model. This is a consequence of the nonlinear pseudo relative permeabilities, which lead to a self-sharpening at the plume tip.

Figure 6.

Simulation of migration under a sloping caprock. The simulation results including the effects of both a capillary fringe and trapping are shown.

Figure 7.

Simulation of migration under a sloping caprock. The simulation results including where the effects have been neglected, i.e., a sharp interface model, are shown.

[72] To understand the impact of the hysteresis models discussed in section 2.3, we also show the sharp interface results in Figure 7 with those obtained using the simplified hysteresis model that only accounts for the sign of the time rate of change of integrated saturation. For the relative permeability function given in equation (51) and the flow parameters of this example, we see essentially no difference between the two hysteresis models (not shown). We further investigated the sensitivity to the relative permeability formulation by applying the van Genuchten curve for liquid,

equation image

where the exponent was set to m = 0.457. This model was also used by Pruess et al. [2009] to model the Wilcox aquifer and gives a brine relative permeability of 0.075 in the residual region. Altering relative permeability models has no impact when considering the simplified hysteresis model. However, for the full model it may lead to a significant impact because of the dependence of equation image on the shape of the relative permeability curve. The resulting simulation is shown in Figure 8. We observe that the simulated plume migration now becomes strongly dependent on both the particular relative permeability model and the choice of coarse-scale hysteresis model. A more complete exploration of the parameter space for this problem is given by Nordbotten and Dahle [2010].

Figure 8.

Simulation of migration under a sloping caprock. The simulation results including the effects of both a capillary fringe and trapping using an alternative relative permeability model for the brine are shown.

[73] While we make no direct comparisons with full 2-D simulations herein, we note that fully resolved simulations display nonmonotonic behavior that cannot be captured by a simplified hysteresis model [Pruess et al., 2009].

[74] The implications of the capillary fringe in terms of storage security are noted here: The large difference in travel speed and distance imply a much greater aquifer sweep; thus, more CO2 can be stored in a given formation according to the model that includes a capillary fringe. Similarly, for monitoring, it is important to have a priori estimates of not only the plume location but also the plume shape. The qualitative difference in plume shapes may, in several cases, impact the optimal monitoring strategy.

4. Conclusions

[75] We show in this paper that even a modest capillary fringe has a first-order impact on the formation of plumes both near an injection point and on postinjection migration. We characterize the size of the capillary fringe with the dimensionless parameter equation image and show that the limiting values of this dimensionless parameter give sharp interface models and the fine-scale model. These calculations imply that we should be cautious about using previous analytical expressions, derived under an assumption of sharp interfaces, for quantitative assessment.

[76] In the case of injection into confined formations, the presence of a capillary fringe does not lead to added complexity, and we give the self-similar equation that characterizes the solution. For the case of long-term plume migration, the traditional hyperbolic analysis needs to be reconsidered. While it appears that this analysis can be extended to account for a capillary fringe, we also observe that in some cases the simplified hysteresis model may not be appropriate. Inclusion of both the capillary fringe and a full hysteresis model in the hyperbolic model represents a noteworthy increase in complexity.

Appendix A

[77] Our objective is to prove equation (42), which states that

equation image

under the stated conditions. In particular, there is no vertical variation in permeability or porosity, and we will therefore consistently remove these terms from the integrals.

[78] It is sufficient to prove the expression for equation image. We now proceed from equation (14) by changing the variables,

equation image

[79] Here we have used the shorthand notation equation image and equation image, and primes denote ordinary derivatives. A direct calculation now gives

equation image

[80] We now need two intermediate calculations. First, we note that

equation image
equation image

[81] Furthermore, we have from implicit function theory that

equation image

[82] Combining, we obtain the inequalities

equation image
equation image

[83] Also note that the preceding inequalities imply

equation image

[84] Returning to equation (A1), we are now equipped to prove the desired result,

equation image

[85] In this calculation, we have used the shorthand

equation image

which satisfies equation image. The second equality is the mean value theorem, which holds with equation image if equation image, and Jensen's inequality holds if

equation image

which was the stated requirement.

Acknowledgments

[86] The authors wish to thank Sarah Gasda and Michael Celia for many interesting discussions on this topic. This work was prepared in part in response to Norwegian Research Council, Statoil, and Norske Shell grant 178013/I30.

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