This paper describes a fully coupled geochemical compositional Equation-of-State (EOS) compositional simulator for the simulation of CO2 storage in saline aquifers. The simulator (GEM-GHG) models the following phenomena:convective and dispersive flow in porous media;phase equilibrium between the oil, gas and aqueous phase;chemical equilibrium for reactions between the aqueous components andmineral dissolution and precipitation kinetics. For numerical robustness and stability, all equations are solved simultaneously. The simulator is applied to the simulation of typical field-scale CO2 sequestration processes, showing the migration of CO2(g) and CO2(aq), the dissociation of CO2(aq) into HCO3 and its subsequent conversion into carbonate minerals. Convection of high-density plumes of CO2-rich brine in conjunction with CO2 mineralization around the plumes is illustrated. Introduction Because of the climatic warming effect of CO2, CO2 storage is essential for reducing greenhouse effects. Gunter et al.1 provided a critical look at capacities, retention times, rates of uptake and costs for CO2 disposal in different classes of CO2 sinks in Canada. Sedimentary basins such as depleted oil and gas reservoirs and aquifers are potential sites for storage. Deep aquifers seem to be the most promising sites for CO2 storage2,3 as they are widely distributed, underlie most point sources of CO2 emission and are not limited by the reservoir size as in the case of depleted oil and gas reservoirs. Tanaka et al.4 discussed several structures for CO2 storage in Japan. These consist of (1) oil and gas reservoirs with neighboring aquifers, (2) aquifers in anticlinal structures, (3) aquifers in monoclinal structures on land and (4) aquifers in monoclinal structures offshore. Oil and gas reservoirs with neighboring aquifers in category 1 are still active and will be producing for some time in the future. When depleted, these reservoirs can be used for underground natural gas storage, instead of CO2 storage. Consequently, aquifers in categories (3) and (4) are the most attractive candidates for CO2 sequestration. Koide et al.5–6 provided additional discussions of the merit of storing CO2 in deep saline aquifers around the world in general and in Japan in particular. Baklid et al.7, Kongsjorden et al.8, and Chatwick et al.9 described the Sleipner Vest CO2 storage project in the North Sea. The rich gas of the Sleipner Vest Field contains sizable amounts of CO2 (9%). CO2 is removed using an activated amine and reinjected into an aquifer in the Utsira formation. Emberley et al.10 discussed the CO2 storage process in the CO2-EOR injection project in Weyburn, Saskatchewan, Canada. van der Meer11 reviewed significant milestones and successes achieved in underground CO2-storage technology over the past few years. All underground options including aquifer storage, EOR processes, CO2 storage in depleted gas and oil fields, and Enhanced Coalbed Methane are reviewed. He noted that Sleipner project has proven to be a successful storage project. CO2 has high density and high solubility in the aqueous phase at the high pressures that exist in deep aquifers. There are two ways in which CO2 can be trapped in aquifers:structural (or hydrodynamic) trapping andmineral trapping. The first process consists of trapping CO2 into a flow system with low flow velocity over geological periods of time. The second process converts CO2 to carbonate minerals and renders it immobile. The latter is very desirable as CO2 is sequestered in a form that is harmless to the environment. Wawersik et al.12 provide a comprehensive review of the physics and research needs related to the terrestrial sequestration of CO2 that highlight the importance of structural and mineral trapping. Geomechanics also plays an important role as the pressure increase due to the injection of CO2 may exceed the yield point of the cap rock or sealing faults. This will result in the undesirable leakage of CO2 into the environment.

San e e f f i c i e n t , high order methods are discussel for approximating the solution of an initial boundary value problem for a homogeneous parabolic

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Summary. A complete convergence analysis is carried out for upstream differencing as applied to one-dimensional (1D) tilted reservoirs experiencing gravity-segregation effects and possible countercurrent flows. It is shown that the standard upstream method converges to the correct physical solution of this problem, provided that an appropriate stability physical solution of this problem, provided that an appropriate stability condition holds. Results from the theory of monotone-difference schemes are used. it is concluded that in addition to the practical advantages of the upstream formulation, there is also a strong theoretical justification for use of upstream techniques in multidimensional reservoir simulation. Introduction Interest has been shown recently in the application of computational methods drawn from the study of conservation laws for the simulation of multiphase reservoir flow. Such methods are often based on schemes for advancing saturation profiles in a 1D reservoir. When coupled with auxiliary techniques, such as operator splitting, the ability to advance multidimensional saturation profiles is recovered. Then, when used with implicit-pressure/ profiles is recovered. Then, when used with implicit-pressure/ explicit-saturation (IMPES) techniques and an associated method for calculating pressure and total fluid velocities, a complete reservoir simulation model results. These techniques are somewhat complicated, especially when such effects as gravity or modeling of sources and sinks must be included. Difficulties often arise even in the solution of the appropriate 1D problems, as evidenced by the complications encountered by Gudunov's method or the random-choice methods as applied to the general nonconvex, nonmonotonic flux functions that describe simple two-phase problems with gravity effects. Complicated schemes for 1D problems are used in an attempt to control front smearing introduced by numerical diffusion. It is hoped that reduction of numerical diffusion will produce multidimensional fronts with little smearing when such schemes are incorporated into multidimensional IMPES simulators. Of course, because the fluid velocities in a 1D problem are easily determined, calculation of diffusion-free solutions for 1D flow is a realistic goal. However, because the velocities and saturation distributions can interact in multidimensional adverse-mobility-ratio flow problems to give physical instabilities (fingering), which must be damped by diffusive effects, complete removal of diffusion effects (numerical or otherwise) is not realistic in multidimensional flow. For instance, grid-orientation difficulties, which result from the uncontrolled growth of small numerical errors, are typically controlled by introducing rotationally invariant diffusion. Thus, the case for diffusion-free 1D schemes as components of multidimensional simulaters can be overstated. Standard upstream techniques, although somewhat prone to front smearing, are almost universally applied in multiphase, mul-tidimensional simulators. Formulations involving gravity osource/sink modeling offer no difficulties, and the schemes are mass conservative and physically reasonable. They can be easily applied physically reasonable. They can be easily applied to both implicit and IMPES simulators. This paper shows that when applied in an IMPES fashion as a method for simulating 1D reservoirs undergoing gravity segregation and possible countercurrent flows, standard upstream techniques are simple to compute, stable with no over/undershoot, and converge to the correct physical solution under a reasonable IMPES throughput condition. The analysis ignores compressibility effects because an oil/water situation is envisioned and relies on the theory of monotone-difference schemes. Thus, it follows that there is as much mathematical justification for using standard upstream techniques as for using more complicated schemes. It should be noted that Crandall and Majda's analysis applies not only to 1D schemes, but also to multidimensional schemes and to operator-split schemes in particular. Unfortunately, their analysis is based on conservation-law techniques and requires knowledge of the underlying velocity field (more or less constant). Under these conditions, diffusion-free solutions can be discussed. In multidimensional reservoir situations, however, a possibility exists fo real velocity-field-saturation distribution interactions that could result in unstable phenomena, such as fingering. Thus, such conservation-law analysis attacks only some of the problems inherent in multi-dimensional reservoir simulation. The next section describes the standard upstream-difference scheme and some of its properties. The description of the throughput condition is given in the Appendix. Convergence Properties of Upstreamed Schemes As noted before, a 1D reservoir situation will be envisaged involving two incompressible, immiscible fluids: oil and water. No capillary effects will be considered. For convenience, constant porosity, phi, and permeability, k, wig be used throughout the reservoir, porosity, phi, and permeability, k, wig be used throughout the reservoir, where × = 0 marks the inlet end and the reservoir extends in the positive × direction. The reservoir is assumed to have a constant crosssectional area, A, and to be undergoing a flood from the inlet (x = 0) end at a (positive) volumetric rate, q. Because the pressure level for such a problem is indeterminate, a single pressure value can be specified. This situation can be modeled with a uniform computational grid with cells of volume V=alpha A, where the ith cell is centered at × = i alpha × and i is any integer. Fluid injection is specified by attaching an infinitely long source of invading fluid associated with Cells i where i less than 0. Thus, the saturations S, in all Cells i at time level n=0 forj=o, w are assumed known. Two IMPES mass-conservation equations for S, j=o, w, in the ith cell can now be written (1) where S + S =1 must hold. In Eq. 1, n and n + 1 denote the time level, t =m alpha t; p denotes the (oil) pressure in Cell i; and p and mu i denote the (constant) density and viscosity of Fluidj, p and mu i denote the (constant) density and viscosity of Fluidj, re-spectivlely. The parameter g represents the component of the ac celeration caused by gravity in the positive × direction: g = 0 if the reservoir is flat; g less than 0 if the × = 0 injection end is the low end; and g is greater than 0 otherwise. It will be assumed, for convenience, that the fluids have been labeled so that g(p -p ) less than 0. Thus, if g less than 0, the heavier fluid should be labeled "water" and if g >0, the lighter fluid should be similarly labeled. This naming convention covers the case of greatest interest: injection of (heavier) water from an aquifer (or well) downdip in the reservoir. The functions 0 less than k (Sj) less than 1 are the given (increasing) relative permeability functions. The superscripts L and R indicate that the correct upstream direction must be used for the relative permeability evaluation. (This L/R-superscript convention will also apply later to the total mobility, fractional flow, and flux functions built from the relative permeability functions.) permeability functions.) SPERE P. 1053