Practical Modeling Approaches for Geological Storage of Carbon Dioxide

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1 Issue Paper/ Practical Modeling Approaches for Geological Storage of Carbon Dioxide by Michael A. Celia 1,2 and Jan M. Nordbotten 2,3 Abstract The relentless increase of anthropogenic carbon dioxide emissions and the associated concerns about climate change have motivated new ideas about carbon-constrained energy production. One technological approach to control carbon dioxide emissions is carbon capture and storage, or CCS. The underlying idea of CCS is to capture the carbon before it emitted to the atmosphere and store it somewhere other than the atmosphere. Currently, the most attractive option for large-scale storage is in deep geological formations, including deep saline aquifers. Many physical and chemical processes can affect the fate of the injected CO 2, with the overall mathematical description of the complete system becoming very complex. Our approach to the problem has been to reduce complexity as much as possible, so that we can focus on the few truly important questions about the injected CO 2,mostof which involve leakage out of the injection formation. Toward this end, we have established a set of simplifying assumptions that allow us to derive simplified models, which can be solved numerically or, for the most simplified cases, analytically. These simplified models allow calculation of solutions to large-scale injection and leakage problems in ways that traditional multicomponent multiphase simulators cannot. Such simplified models provide important tools for system analysis, screening calculations, and overall risk-assessment calculations. We believe this is a practical and important approach to model geological storage of carbon dioxide. It also serves as an example of how complex systems can be simplified while retaining the essential physics of the problem. Introduction Anthropogenic emissions of CO 2 have increased atmospheric concentrations of CO 2 by about 35% over the past 200 years, from about 280 parts per million (ppm) to the current value of about 380 ppm (Intergovernmental Panel on Climate Change [IPCC] 2007). These emissions are largely associated with fossil fuel combustion, with the two largest contributors being electricity generation and 1 Corresponding author: Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544; ; celia@princeton.edu 2 Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ Department of Mathematics, University of Bergen, Johs Brunsgt. 12, N-5008 Bergen, Norway Received January 2009, accepted April Copyright 2009 The Author(s) Journal compilation 2009 National Ground Water Association. doi: /j x transportation (International Energy Agency [IEA] 2004). The anthropogenic emission rate continues to increase with time; in 2006, about 8.4 billion metric tonnes of carbon (1 billion metric tonnes = 1 gigatonne [Gt] = g) were emitted to the atmosphere (Canadell et al. 2007), and under many business as usual scenarios this rate is projected to double over the next 50 years (Pacala and Socolow 2004). There is now a broad consensus that these emissions are leading to significant climate change, especially in the high latitudes, and that these changes will become more significant with time (IPCC 2007). Because the underlying sources of CO 2 emissions are associated with technologies that provide modern services and conveniences, which humans are unlikely to give up, feasible solutions involve combinations of improved efficiencies of use coupled with development of technologies that produce energy with little or no atmospheric carbon emissions. As such, this becomes a problem of technology and implementation. In a seminal paper on available technological options, Pacala and NGWA.org Vol. 47, No. 5 GROUND WATER September-October 2009 (pages ) 627

2 Socolow (2004) presented a list of current technologies that could be used, in their current form, to solve the carbon problem. They also calculated the associated implementation effort needed to make a significant impact on the problem. One of these technologies is referred to as carbon capture and storage, or CCS. This technology involves use of fossil fuels for energy production, but instead of emitting the produced CO 2 to the atmosphere, it is instead captured and redirected to other storage compartments on the planet. Geological storage, which is the most likely option, involves injection of the captured CO 2 into deep geological formations. These can include depleted oil and gas reservoirs, deep unminable coal seams, and deep saline aquifers. Of these, deep saline aquifers are the most ubiquitous and offer the largest potential storage volume. Estimates of overall storage capacity indicate that geological storage can accommodate many decades of projected CO 2 emissions (IPCC 2005). Given that the current number of large coal-fired power plants worldwide is about 2100 (Kintisch 2007), and that each of these emits 4 to 6 million metric tonnes, or megatonnes (Mt) of CO 2 per year, implementation of CCS for the coal sector of electricity generation would account forupto12gtco 2, which is equivalent to about 3 Gt carbon, per year in emission avoidance. Therefore, CCS offers the possibility to have a major impact on the carbon problem. Leakage and Risk Assessment Injection would take place at sufficient depths for both temperature and pressure to exceed their critical points 31.1 C and Pascal (or 7.38 mega- Pascal [MPa]) so that the CO 2 would be in a supercritical state. Given typical geothermal gradients, these pressure and temperature values correspond to a depth of about 800 m. Across expected temperature and pressure ranges, supercritical CO 2 would have densities of about 250 to 750 kg/m 3, and viscosities between 2% and 20% of the viscosity of water (Nordbotten et al. 2005a). Furthermore, supercritical CO 2 is only slightly soluble with water, up to a few percent by volume (IPCC 2005). Therefore, most of the CO 2 will remain as a separate fluid phase for a significant period of time. For injection into deep saline aquifers, the separate-phase CO 2 will tend to rise toward the top of the formation in what is sometimes referred to as gravity override, and the outer tip of the injection plume will extend farther away from the injection well because of the unfavorable viscosity ratio. A schematic of a typical CO 2 plume associated with a vertical injection well is shown in Figure 1. An idealized sharp interface delineates the outer edge of the separatephase CO 2 plume. If the injection involves dry CO 2,then a second front, referred to as the drying front, can form, as indicated by the second front in the figure. This gives three regions of fluids: (1) a region of single-phase dry CO 2 near the well, (2) a two-phase region of CO 2 and brine, and (3) a region of single-phase brine. Given the mass exchange between phases, the CO 2 in region 2 will Figure 1. Schematic of CO 2 plume of thickness h(r, t). Also shown is the drying front, with dry CO 2 having thickness i(r, t). From Nordbotten and Celia (2006a). be wet with water evaporated into it at the level of a fraction of 1%, whereas the water in region 2 and some parts of region 3 will have CO 2 dissolved in it. In the system illustrated in Figure 1, the two most important basic features that are necessary for a good CO 2 injection operation are a host formation that has sufficient permeability and areal extent to allow for good injectivity, and a caprock overlying the injection formation that has sufficiently low permeability to keep the buoyant CO 2 in the injection formation. Assuming good injectivity allows for sufficient injection rates without exceeding fracture pressures, and an intact and areally extensive caprock, the storage operation should be a success. Under these circumstances, the injection will induce a pressure field that allows injected CO 2 to displace the resident brine. The buoyant CO 2 is kept at depth by the caprock, and the CO 2 plume spreads as injection proceeds. After the injection ceases, the CO 2 will slowly continue to dissolve into the brine, and some geochemical reactions may occur that might ultimately convert the injected carbon into solid carbonate precipitates. Dissolved CO 2 slightly increases the brine density, leading to gravitational stability with regard to upward leakage of CO 2.Oncedissolved,the CO 2 will move with regional flows in the system. Any solidified carbon would remain buried. With this overall picture of the system, two general concerns may be identified. The first is that when the CO 2 is injected, it will displace some amount of resident brine. Where that brine flows, and at what rate, must be estimated to be sure highly saline waters do not migrate into drinking water aquifers or other unwanted locations. The second is a more general concern that the caprock may not be perfect, allowing both CO 2 and brine to leak out of the injection formation. Three general kinds of leakage pathways may be identified. The first is diffusive leakage through the caprock. This can certainly occur for the brine the behavior being that of a standard leaky aquifer and could also occur for the CO 2 depending on the capillary entry pressure associated with the caprock. A second 628 M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5: NGWA.org

3 possible leakage pathway is associated with fault and fracture zones. These would lead to more concentrated leakage flow paths, with the possible interaction between fluid pressure and fault-zone permeability being an important factor. The third type of possible leakage pathway corresponds to human-made flow features, typically oil and gas wells. Figure 2, taken from the recent IPCC report, shows that this last pathway is of special concern in North America, where more than a century of oil and gas exploration and production has left many millions of wells that tend to perforate otherwise very good caprock. For cased holes, leakage pathways can be formed either inside the casing, for example, through improperly placed cement plugs, or along the outside of the casing through poorly or degraded cement zone or along vertical zones that lack cement. A subset of possible pathways is shown in Figure 3, where spatially connected small-scale flow paths may lead to large-scale leakage out of the injection formation. Note that the locations of many of the well fields represented in Figure 2 correlate strongly with the deep sedimentary basins that are most suitable for carbon injection (IPCC 2005). Therefore, the issue of leakage along old wells has become a topic of interest. In order to proceed with a large-scale carbon storage project, a quantitative risk assessment is likely to be required, with leakage estimation at its core. Given the uncertainties associated with large-scale injection and storage of CO 2, this can become an extremely challenging computational problem. The major features are an areally extensive CO 2 plume, with a much larger pressureperturbation footprint, both of which may intersect some number of existing wells. Those wells typically have materials such as well cements whose quality is largely unknown, and therefore quantitative parameters such as effective permeability will be highly uncertain. The injection is likely to take place in a typical vertical structure characterized by alternating layers of permeable (aquifers) and essentially impermeable (aquitard or caprock) formations. If leakage occurs along one or more of the wells contacted by the plumes, secondary plumes may form in permeable layers above the injection formation, with some of the leaking fluid possibly reaching potable water zones or the shallow soil and atmosphere. All of these need to be captured quantitatively in a risk-assessment framework over domains like the one shown schematically in Figure 4. Modeling Challenges The modeling challenges associated with a system like that illustrated in Figure 4 include the following. The horizontal domain will likely exceed several thousand square kilometers, and the vertical domain will extend from the land surface to the deepest injection layers, which are likely to be a few kilometers deep. Multiphase, multicomponent flow and transport will occur, with possibly important geochemical reactions, geomechanical effects, and nonisothermal effects with complex phase behavior. Numerical models will require fine discretization around all injection and potentially leaky wells as well as vertical resolution of the layer structure and the strongly buoyant (and unstable) flows within each layer. Given that a domain of several thousand kilometers can include more than a thousand old oil and gas wells (Gasda et al. 2004), the computational demands for this kind of simulation become overwhelming, with tens to hundreds of millions of grid cells involved. Given the high uncertainty associated with old wells, a risk assessment will also require a probabilistic analysis, leading to Monte Carlo approaches. With even a single simulation being computationally prohibitive, running many thousands of simulations becomes impossible, even on the largest supercomputers ,400 No wells / data ,400 61,000 Figure 2. Areal density of oil and gas wells. From IPCC (2005). NGWA.org M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5:

4 Figure 3. Schematic of abandoned well showing possible leakage pathways. From Gasda et al. (2004). These observations led us to consider a number of approaches to make the problem more tractable. For example, upscaling approaches (Gasda and Celia 2005) and domain decomposition (Gasda 2008), applied to the multiphase flow equations, proved interesting but ultimately ineffective because of limitations in the upscaling results and excessive computational demands for the domain decomposition algorithms. Therefore, we decided to take a different approach: identify the processes that are truly critical to a risk-assessment analysis and ignore all remaining processes. That is, we have attempted to develop models that are as simple as possible while still capturing what we believe to be the essential physics. We describe this approach in the remainder of the article. Simplifications Several dominant features of the system can be used to guide the development of a set of simplifying assumptions. These include the following: (1) the strong density difference between CO 2 and brine, (2) the hierarchy of distinct time scales over which important physical and chemical processes occur, (3) the relatively small changes in fluid properties for formations that are sufficiently deep, where both temperature and pressure are not close to the critical point, (4) the relatively constant temperature within a given formation, (5) the dominant large-scale effects of geological layering in the vertical direction, (6) the strong separation of spatial scales between the large-scale plumes and dominant leakage features, and (7) the dominant role that parameter uncertainty plays in these systems. We use each of these to develop a set of seven strong assumptions, which can be combined in a number of ways to greatly simplify the mathematical descriptions of the system. We do not address assumptions associated with numerical solutions. In particular, we accept without further comment the widely used assumption that numerical discretization errors in space and time can be controlled adequately in the cases where the governing partial differential equations are solved numerically. 630 M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5: NGWA.org

5 Figure 4. Schematic of CO 2 injection and possible leakage along existing wells. From Gasda et al. (2004). Simplification 1: Macroscopic Sharp Interface Given that the density difference between CO 2 and brine is about 500 kg/m 3, there is a very strong buoyancy drive in the system (for comparison, note that salt water and fresh water in coastal aquifers have density differences of about 25 kg/m 3 ). Therefore, CO 2 moves strongly toward the top of the formation, in what petroleum reservoir engineers refer to as gravity override. This has already been illustrated in Figure 1. Given this strong tendency for the phases to separate by gravity, we make the assumption that the system can be described well by a macroscopic sharp interface. Referring to Figure 1, this assumption means that all fluid beyond the interface h(x,y,t) is brine, all fluid between the injection well and the interface i(x,y,t) is CO 2, and between i(x,y,t) and h(x,y,t) both brine and CO 2 fill the pores. Note that this assumes that all the complex three-dimensional flows that occur initially around the injection well can be ignored when dealing with the larger-scale, developed CO 2 plume. This appears to be a reasonable assumption for practical CO 2 injections that will involve large amounts of CO 2 and commensurately large plume sizes. In addition, as a corollary to the sharp-interface assumption, we assume that in the two-phase region, brine exists at residual saturation. Therefore, its relative permeability is zero, and the only mobile fluid in the two-phase region is CO 2. Simplification 2: Vertical Equilibrium A simplification that complements a sharp interface is to assume that both fluid phases are in vertical equilibrium, such that the pressure in each phase changes in the vertical direction in proportion to phase density multiplied by the gravitational constant. A corollary assumption is that capillary pressure along the macroscopic interface can be neglected, meaning that the pressures in the two phases are equal along the macroscopic interface. Under this simplification, vertical flow components within the formation are negligible. Furthermore, knowledge of the pressure at one point in the vertical direction allows values of pressure to be calculated at all other vertical locations, assuming knowledge of the interface location h(x,y,t). Note that a nonzero capillary pressure could be assigned along the interface, corresponding to an average entry pressure. The vertical distribution can still be calculated, with a jump imposed at the interface. In addition, a less strict version of the vertical equilibrium assumption allows for vertical flows in the formation, but with the vertical functional form of the velocity specified. For example, vertical flows could be assumed to vary linearly (as a function of vertical distance away from the interface) in each phase, with the maximum vertical flow occurring at the interface. That is, in each phase, vertical flows change linearly from a maximum at the interface to zero at the top and bottom boundaries. In this case, both the interface location h(x,y,t) and of the maximum velocity must be known to construct the pressure distribution at all other vertical locations. The vertical permeability must also be specified for this calculation. We refer to this less strict assumption, which allows for some vertical flows, as Simplification 2a Structured Vertical Velocities, with this particular example referred to as a linear vertical structure. Note that this linear structure for velocity NGWA.org M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5:

6 implies a quadratic variation of pressure in the vertical direction. This linear vertical structure was used in Nordbotten and Celia (2006b) to derive solutions for interface upconing around wells. Simplification 3: Separation of Time Scales The general consensus (IPCC 2005) is that three important physical and chemical processes involved in the CO 2 injection problem occur on fairly distinct time scales. The three processes of interest are as follows: (1) twophase flow physics, (2) dissolution of CO 2 into brine, and (3) geochemical reactions leading to possible formation of solid carbonate minerals and associated changes in porosity. Although there are uncertainties in the reaction rate coefficients, there is general consensus that geochemical reactions will occur on a time scale of a thousand years or longer (Xu et al. 2006, 2007). Dissolution is limited by diffusion, and although enhancements may occur because of density instabilities associated with dissolved CO 2 (Riaz et al. 2006), the usual time scale observed from detailed simulations is on the order of hundreds of years to perhaps a thousand years. Both of these observations imply that the large majority of injected CO 2 will remain in a separate CO 2 -rich phase for hundreds of years. Its movement during this time will be driven by two-phase flow physics including buoyancy. A typical large-scale injection operation is likely to last for the lifetime of a power plant, which is on the order of 50 years. During that time, the CO 2 and brine movement will be dominated by the imposed pressure associated with the injection and the buoyant drive associated with the substantial density differences. In order to maximize injection rate, the injection pressure is expected to be at the maximum allowable, probably 90% of fracture pressure. Once injection ceases, the pressure perturbation will relax and the system will be dominated by buoyancy while dissolution slowly erodes the separatephase CO 2. Note that while the injection proceeds, the displacement process will be one of drainage, with the nonwetting fluid (CO 2 ) displacing the wetting fluid (brine). This displacement leads to residual saturations of brine in the invaded zone, corresponding to the region between i(x,y,t) and h(x,y,t) in Figure 1. Once the injection ceases, any net movement of the plume, for example, up-dip along the bottom of a sloping caprock, will involve movement of the trailing edge of the plume, which is an imbibition process. This leads to regions where the invading brine phase will trap some of the CO 2 by local capillary forces, forming a residual saturation of CO 2. Given that this saturation can be several tens of percent (Bennion and Bachu 2008), residual trapping of CO 2 can be an important mechanism to immobilize CO 2 over intermediate time scales. From this description of the system, we do sense that the highest risk for leakage is when the CO 2 remains in a separate, mobile phase and the imposed pressure remains high. This occurs during the injection process. Once injection ceases, the physical and chemical mechanisms of capillary trapping, dissolution, and geochemical precipitation Figure 5. Relative importance of different trapping mechanisms as a function of time. From IPCC (2005). all lead to reductions in risk of leakage. This is captured in Figure 5, taken from the 2005 IPCC report. With this viewpoint, the critical time for leakage risk is during and immediately after injection. For the purpose of risk assessment, this early time frame becomes the critical period of time. And during this time frame, two-phase flow physics is the dominant process. Therefore, we assume that both dissolution and geochemical reactions can be ignored for the purpose of risk modeling. This forms our second set of important assumptions: that the model should focus on two-phase flow and ignore geochemical reactions and dissolution. As corollary assumptions, we assume that by remaining below fracture pressure, geomechanical responses will be minor and can be ignored. Simplification 4: Locally Constant Fluid Properties and Isothermal Systems For deep sedimentary formations with both temperature and pressure well beyond the critical point, the changes in fluid properties within the formation during an injection operation do not appear to have a significant effect on the resulting flow field (see Nordbotten et al. 2005a; Nordbotten and Celia 2006a). This allows the density and viscosity for the brine and CO 2 to be estimated from the local pressure and temperature conditions (and salinity for the brine density). Therefore, within the mathematical description, we can assume that changes in density and viscosity as a function of changing pressure can be ignored. In addition, we assume that the dominant forcing in the system is the imposed pressure increase and the associated buoyant drive, and that any thermal effects are secondary and can be ignored. This means that no energy balance equation needs to be written. Simplification 5: Dominant Spatial Features Are Large-Scale Layering and Concentrated Leakage Pathways Although many locations for injection may be considered, most of them are expected to be in large sedimentary 632 M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5: NGWA.org

7 basins. These basins tend to have distinct layered structures, with permeable formations (reservoirs) alternating with essentially impermeable (caprock) formations. For large-scale injections, where CO 2 plumes will extend for many kilometers and pressure pulses potentially much farther, the dominant spatial features of the system are likely to be the large-scale layers. Smaller-scale heterogeneities within individual formations, which have been the focus of intensive research for much of the past three decades, are hypothesized to play a relatively minor role, with the most important role likely to be associated with definition of a set of large-scale residual saturations and the associated value of end point relative permeability. Therefore, we assume that the system is defined by alternating layers of reservoirs and caprocks, with the properties within each formation being homogenized. Leakage pathways of significance are taken to be diffuse brine leakage through the caprock formations, and concentrated leakage of both brine and CO 2 along fault zones as well as along old and abandoned wells. The diffuse leakage can affect the pressure pulse, but we assume it has relatively little practical importance in terms of mass transport. Therefore, it is accounted for only in the pressure solutions. The concentrated leakage pathways have characteristic length scales that are much smaller than the other important length scales in the problem, including plume size and formation thicknesses. This allows us to assume a strong scale separation among the dominant spatial features in the system. Simplification 6: Parameter Uncertainty Is a Dominant Characteristic of the System When considering possible leakage scenarios, parameters must be assigned to whatever equation set is used to model the system. The parameters must describe the formations as well as the leakage pathways. All these parameters are inherently uncertain. When pathways such as old and abandoned wells are considered, the lack of any reliable data on the current state of these potential leakage pathways makes their quantitative description highly uncertain. Uncertainty in characterization of old wells is often much larger than the uncertainty associated with more traditional parameters such as formation permeability and porosity. Therefore, we often assume that the level of uncertainty associated with leakage features, such as effective permeability of well cements, is much more significant than the uncertainties associated with formation parameters. As such, probabilistic analysis will often focus on uncertainty in leakage-pathway parameters, with concomitant assumptions that uncertainties in formation parameters are negligibly small by comparison. Overall, the existence of large uncertainties across the parameter space is a dominant feature and must be accommodated in simulation studies. Simplification 7: The Formations Are Horizontal and Homogeneous This strong assumption allows the set of equations to be simplified to the point that the resulting equations can be solved analytically. Although this is a very strong simplification, it may be reasonable for some number of practical calculations. For example, Gasda et al. (2008) analyzed the impact of formation slope on the resulting movement of a CO 2 plume during the injection phase of an operation and found that midcontinent sedimentary formations in North America may be reasonably approximated by ignoring the slope. Furthermore, as the size of the plume increases, its length scale may become large relative to the heterogeneities within the formation, and under these conditions an assumption of homogeneity can be reasonable, given that a few key parameters can be upscaled based on the subplume-scale heterogeneity. Those parameters that need to be upscaled include the residual saturation and the effective (or upscaled) relative permeability at residual saturation. Given the level of uncertainties expected in these systems, the lack of detailed information for many formations, and the need for practical estimation of system response, this may not be a bad approximation for many problems, and the resulting simplified solutions can provide important back of the envelope calculations that can be quite useful. Equations and Practical Applications By choosing some or all the simplifications listed, the set of governing equations can be simplified greatly, even to the point of allowing for analytical solutions. This gives a wide range of possibilities in terms of model complexities. And for some problems of practical importance, it provides feasible solutions when traditional numerical solutions are infeasible. In this section, we provide an overview of some of the possible equations, and associated solutions, that can be derived for the carbon injection problem. Numerical Models using the Sharp Interface Vertical Equilibrium Assumption In this class of simplified models, the general governing equations for two-phase flow in porous media are integrated vertically under the assumption of a sharp interface between the CO 2 and the more-dense brine. For the sake of exposition, we ignore the drying front as well as vertical variations of horizontal permeability, and impose simplifications 1, 2, 3, and 4. In this case, derivation of the governing equations follows directly from standard procedures for buoyant flows such as salt water intrusion into coastal aquifers (see, for example, Pinder and Celia 2006; Bear 1979), yielding the following set of equations: ϕ ( 1 SB res ) h t + [ ] q C [ ] x x + q C y y = q C leak (1) ϕ ( 1 SB res ) (H h) + [ ] q B x + y t x [ ] q B y = q B leak (2) NGWA.org M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5:

8 , q C = h kkc rel ( P bot ρ μ B g H C ) + ( ρ) g h + ρ C g z top (3) q B = (H h) k μ B ( P bot + ρ B g z bot ) (4) These equations can be seen as the vertically integrated forms of the mass balance equations and the Darcy equations for each phase. In these equations, the two primary unknowns are the CO 2 thickness, h(x,y,t),andthe pressure along the bottom of the formation, p bot (x,y,t). The overbar on quantities implies that the variable is vertically integrated, and all vector and tensor quantities are defined in the x and y directions. In the equations, ϕ denotes porosity, SB res denotes residual saturation of the brine, H(x,y) is the thickness of the formation, q denotes the vertically integrated Darcy flow vector for the appropriate phase, k is the intrinsic permeability, krel C is the relative permeability function evaluated at the CO 2 saturation equal to (1 SB res ), ρ denotes density of the appropriate phase, ρ = ρ B ρ C is the density difference, g is the gravitational constant, q leak denotes leakage of the appropriate phase out of the given formation, and z top (x, y) denotes the elevation of the top boundary of the formation. The equations must be supplemented with the usual initial and boundary conditions. These equations can be solved much more efficiently than the full three-dimensional equations because they have been reduced from three dimensions to two, with the associated reduction in the number of unknowns appearing in the numerical approximations. In addition, as shown in Gasda (2008) and Gasda et al. (2009), the nonlinearities in the system are also reduced significantly. For example, the macroscale effective (or pseudo ) relative permeability and capillary pressure are linear functions of the dependent variable h, which also serves as a measure of two-dimensional (effective) saturation. All these are significant computational advantages. An example application of this kind of numerical model has been presented recently by Gasda et al. (2009). Injection of CO 2 into a spatially heterogeneous formation deep under the North Sea was modeled as part of a code comparison exercise. Results from the model of Gasda et al. (2009) compared well to other models, which solved the full three-dimensional version of the governing equations (Class et al. 2009). Numerical results using the sharp-interface assumption are shown in Figure 6, where the problem domain is also shown. The domain involves a deep formation that is split by a large fault zone, with thickness of the sloping formation, permeability, and porosity all varying in space. This calculation shows that fairly complex geometries as well as spatial heterogeneities can be accommodated readily in this approach. Hybrid Numerical-Analytical Models When considering a problem like that illustrated in Figure 4, where important leakage pathways correspond to very small features such as abandoned wells, the vertical equilibrium numerical model requires substantial grid refinement in the vicinity of a leaky well to properly capture the flow dynamics. When many wells are present in the domain, this type of grid refinement becomes prohibitive in terms of computational effort, even with the reduction in dimensionality. Therefore, we are motivated to seek special kinds of multiscale models where the large-scale features and flows are captured with the numerical model on relatively coarse numerical discretizations, and the local effects associated with well leakage are treated with a different approach. One particularly attractive option is to use local analytical models in the grid blocks that contain leaky wells, where local corrections for both the pressure field and the location of the sharp interface are included in the overall largescale model. This constitutes one particular example of a hybrid numerical-analytical model where the two different kinds of models are applied over different length scales. The local analytical corrections can be formulated in several different ways. If the interface position is considered first, the problem involves determination of the interface location in the vicinity of a leaky well. This is a classic interface upconing problem, although it is a bit more challenging because the flow rate in the well is not known a priori. In Nordbotten and Celia (2006b), a new solution was derived that provides better representation of vertical flow components near the well. In that case, the linear version of Simplification 2a Structured Vertical Velocities, was used. Additional assumptions included Simplification 7 Homogeneous and Horizontal Formations, which is applied locally (within a grid cell) to allow for analytical solutions. The resulting expressions for interface location and flow rates in the leaky well were used in the model of Gasda et al. (2009) and applied to another test problem as described in Class et al. (2009). Gasda et al. also used a pressure correction within a grid cell containing a leaky well to relate the cell-average pressure to the local pressure at the leaky well. Results for the test problem, which involved one injection well and one leaky well, matched very well with other solutions that solved the full three-dimensional multiphase equations on refined grids. Those other solutions tended to take much more computational effort to compute. See Class et al. (2009) for details of the computational comparisons. Analytical Solutions In addition to local solutions to account for smallscale behavior around wells, analytical solutions can also be used to simulate the overall injection and transport processes. For example, to derive analytical solutions for the CO 2 injection problem, we invoke simplifications 1, 2, 3, 4, and 7. That is, we assume a sharp interface with vertical equilibrium, isothermal conditions with constant fluid properties, and a formation that is homogeneous and horizontal. In this case, the vertically averaged equations simplify because the top and bottom boundaries are 634 M.A. Celia, J.M. 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9 (a) (b) Figure 6. Formation showing geometry and porosity variation (a) and simulation results from sharp-interface numerical model (b). From Class et al. (2009) and Gasda et al. (2009). horizontal, and therefore their derivatives with respect to x and y are zero, and the total thickness H is constant so its gradient is also zero. As shown in detail in Nordbotten and Celia (2006a), the resulting simplified equations are of a self-similar form, and can be transformed into a set of ordinary differential equations where the single independent variable is the dimensionless similarity variable defined by χ 2πϕH ( 1 SB res ) r 2 Q well t (5) where r denotes the radial coordinate, radial symmetry is assumed, and the injection rate is assumed to be constant at the volumetric rate Q well.asdetailedinnordbotten and Celia (2006a), the set of ordinary differential equations can be solved numerically for h(r, t) and p bot (r, t) through solutions for h(χ) andp bot (χ). When a particular dimensionless Ɣ is sufficiently small, the solution simplifies to a closed-form analytical solution. The dimensionless group is defined as Ɣ 2π ( ) ρ b ρ c gkλb H 2 (6) Q well NGWA.org M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5:

10 where λ b is the brine mobility, defined in this case as the inverse of the brine viscosity. For values of Ɣ<0.5, the resulting solution for the shape of the CO 2 front is given by ( ) h H = 1 2λ (λ 1) χ 1 (7) where λ is the ratio of CO 2 mobility to brine mobility. Note that this last equation allows for very simple estimation of the maximum extent of the CO 2 plume: it is simply given by the point where h goes to zero, which is χ = 2λ. Nordbotten and Celia (2006a) present more general expressions involving mass transfer between fluid phases and calculation of the location of the drying front. An example of applications of these simplified equations to determine the extent of both the invading CO 2 front and the drying front, across a range of different formations, can be found in Nordbotten and Celia (2006c). Semianalytical Solutions If the general system of multiple layers and multiple potentially leaky wells is considered (Figure 4), then a more general solution framework needs to be developed. As an example of this type of system, consider the data illustrated in Figure 7. There a 50 km by 50 km region of Alberta is identified, within which four large power plants emit approximately 35 Mt CO 2 /year. Therefore, this is a reasonable location at which to consider large-scale injection of captured CO 2. Figure 7 shows that approximately 1250 oil and gas wells are located within this region a density of about 0.5 well/km 2. A layered sequence of reservoirs and caprocks also characterizes this region. The computational challenge is to simulate large-scale injection of CO 2 into one or more of the reservoirs layers, capturing the flow of both CO 2 and brine within all the formations and along all the wells. This computational challenge, whose data are detailed more fully at Alberta Geological Survey (2006), can only be solved with specialized formulations that take advantage of the simplifications outlined previously. Our current approach to this problem is to invoke all seven of the simplifications, so that all plumes in the system are described by solutions like those presented for the single plume. Flow across caprock formations is restricted to flow along wells, which is represented by the standard two-phase form of Darcy s equation. The solutions include the upconing solutions of Nordbotten and Celia (2006b) and a number of other specific solution approaches. The overall algorithm is presented in some detail in the recent paper by Nordbotten et al. (2009) as well as its predecessor, Nordbotten et al. (2005b). The overall algorithm is currently restricted to hydrostatic initial conditions for pressure with no initial saturation of CO 2 in the domain. Constant pressure conditions are applied along the outer boundary of the domain, which can be either finite or infinite in areal extent. The algorithms required to solve this kind of leakage problem use supplementary assumptions, including radial symmetry of all plumes, superposition of pressure fields, and linearization of all nonlinear parameters within a simple time-stepping algorithm. The time stepping is necessary because the nonlinearities in the problem are important and cannot be accommodated in any other way. As such, we refer to these solutions as semianalytical: analytical solutions are used in space, whereas finite time stepping is used to march through time. Our current implementation of these algorithms allow us to simulate 50 years of injection over the domain shown in Figure 7 is about 10 min of computational time on Figure 7. Wabamun Lake area showing location of coal-fired power plants and existing wells in the area. From Bachu and Celia (2009). 636 M.A. Celia, J.M. Nordbotten GROUND WATER 47, no. 5: NGWA.org

11 a laptop computer. This kind of simulation time shows the power of the simplifications listed herein. It also allows for the kinds of multiple-realization, probabilistic approaches required for this problem. For typical results, see Nordbotten et al. (2009), Celia et al. (2006, 2008), and Kavetski et al. (2006). Conclusions The complexity level of any modeling exercise is necessarily limited by both computational resources and data availability. What level of modeling is most appropriate for a given problem is often not obvious. The CO 2 injection problem is no exception. It is easy to argue that all the physical and chemical processes involved should be represented. Proceeding down this path, it is not difficult to convince yourself that a fully coupled simulator involving multicomponent, multiphase, nonisothermal flow and transport with coupled geochemistry and geomechanics is required to properly model the system. Heterogeneities at many different length scales could also be important, including concentrated leakage pathways such as wells and faults. The end result is a model that can handle almost any situation, but which may be practically useless because of not only the enormous computational cost associated with a converged discretizations in the numerical solution but also the lack of data at multiple length scales involving the many coupled processes. Therefore, in order to move forward with practical calculations, some set of simplifications must be made. Herein, we have proposed a set of seven simplifications, which taken in part or in whole allow for a range of reductions in complexity with concomitant simplifications in the resulting mathematical models. We have tried to craft these simplifications to be consistent with many of the major characteristics of the CO 2 injection system. This involves reliance on separations of scales in both space and time, and identification of the dominant transport mechanisms across different scales. Which, if any, of these assumptions will be appropriate for a given injection site requires the usual kinds of analysis with sound engineering and scientific judgments. For the CO 2 injection problem, we believe that simplified models can provide substantial insights into system behavior, can provide reliable back of the envelope estimates, and in some case can form the basis of an applicable analysis tool. Our current work involves development of methodologies that allow different parts of the numerical and analytical approaches to be put together within a seamless overall framework. We already use analytical solutions to represent subgrid-scale behavior associated with leaky wells (Gasda et al. 2009), and we are working on more general multiscale numerical/analytical hybrid methods to represent other kinds of local complexities (Nordbotten 2009). We are also working on allowing different layers in the system to be represented using different approximation types. Our approach continues to rely on macroscopic sharp-interface assumptions, because we believe this assumption is strongly justified. At this point, the only places where we would apply fully reactive and nonisothermal simulations would be along specific critical features such as leaking wells. There, we have flexibility in our general modeling framework to include more complex one-dimensional simulations, which may be justified owing to the critical part that leaky wells play in an overall risk assessment. We also run more complex general models, but those are usually motivated by the need to understand how we might simplify mathematical descriptions using upscaled variables. Overall, we have tried to follow the general philosophy that we want a model that is simple as possible while capturing the essential physics of the system. Although motivated by computational constraints, we also believe this is a very fruitful general philosophy that inevitably leads to enhanced understanding of the system. Acknowledgments The work described herein is part of a larger effort at Princeton University related to solutions of the carbon problem, which is called the Carbon Mitigation Initiative (CMI). CMI is supported by BP and Ford Motor Company. The opinions expressed herein are solely those of the authors. The authors wish to thank Mark Person and two additional anonymous reviewers for their helpful and constructive comments, which have improved the article. 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