While the detailed leakage along a well is obviously very important information, it is unlikely that we will know the kinds of detailed information to predict leakage details when analyzing a field injection operation. As shown in Gasda et al. (2004), Celia et al. (2006), and Bachu and Celia (2008), among others, an injection operation in may produce a CO_{2} plume that intersects hundreds of existing wells, whose condition is very difficult to determine a *priori*. Therefore the tools needed for large-scale modeling must be able to describe three-dimensional multi-phase flow with leakage along many wells whose properties are highly uncertain.

## Semi-analytical models

To accommodate this situation, Michael Celia’s group has developed a set of semi-analytical models that are sufficiently fast to allow many realizations to be run, thereby providing Monte Carlo-type of stochastic analysis. The computational efficiency of the models allows thousands of simulations to be run in a probabilistic framework, so that we have a much richer output set for analysis and decision making. The result is a probability distribution for leakage amounts, given statistical parameters for well properties as inputs.

Current applications are focused on the trade-off between injection depth and risk reduction, as well as loss of injectivity as a function of depth of injection. For example, in the Wabamun Lake area of Alberta, where four large power plants produce about 35 Mt CO_{2}/yr, we have analyzed injectivity and leakage risk across a number of formations in the vertical sequence. The lowest permeable formation is the Keg River formation, which has only a handful of wells that penetrate its caprock. However, the formation is thin (thickness is about 15 meters) and not very permeable, so maximum injection rate, constrained by fracture pressure limits, is only about 0.25 Mt CO_{2}/yr. In this case, leakage risk is very low, but injectivity is also very low. The number of wells that penetrate caprock formations progresses from 5 (Keg River) to 40 (Nisku) to 58 (Wamamun) to 731 (Nordegg) to 883 (Lower Manville) to 895 (Viking) with distance toward the surface. Note that our earlier studies of well densities focused on the Viking formation (Gasda et al., 2004). Injectivities in all of these formations are determined largely by formation thicknesses, and in none of the formations do we find injectivities much above 1 Mt CO_{2}/yr. If the entire 35 Mt CO_{2}/yr is to be stored here, most or all of the formations would probably be used. As such, all wells in the area would likely be considered in the analysis. We also note more generally that while many formations have apparently large storage capacity, injectivity restrictions may limit their utility. The results from our semi-analytical model runs show that, as long as we assume an uncorrelated permeability structure along each potentially leaky well, we continue to see large reductions in the leakage fluxes as a function of distance toward the land surface. That is, the intervening permeable layers above the injection formation tend to trap most of the CO_{2} that leaks from the injection formation. While this is good news, it also highlights the need to generate field data on the effective permeability of well segments, and to analyze the statistical characteristics of the data sets.

We have also performed a series of analyses to identify the limits of applicability for these kinds of simplified models – see Gasda et al. (2008). One of the results of this analysis was to determine when the assumption that formation slope can be ignored during injection is valid. We find that for all of the cases using the moderate permeability value of 5 x 10^{-14} m^{2} (that is, 50 milli-Darcy, which is characteristic of, for example, the Alberta Basin), values of a dimensionless centroid measure, which provides a measure of the amount of asymmetry in the plume, are all less than 0.05. This means that the centroid has moved less than 5% of the overall distance that the plume has moved. We consider this to be a reasonable criterion for when the effect of slope can be neglected, and therefore when we can reasonably use the suite of semi-analytical models we have developed. For the high-permeability case, which we took to be a permeability of 3 Darcy, we observe centroid movements of between 5% and 30% of the plume extent. In those cases, slope can clearly be important, and should not be ignored in the analysis.

## Basin-scale simulations

In addition to these semi-analytical models, the Celia group is developing hybrid numericalanalytical approaches to allow for analysis of very large spatial domains. These techniques will allow simulation of an entire sedimentary basin, covering several hundred thousand square kilometers. This is motivated by the question of where displaced brine may move on a basinwide scale, especially under scenarios of full implementation of a CCS strategy.

Our basin-scale model is based on a multi-scale finite element method, tailored to the specific problem of CO_{2} injection and leakage estimation. In particular, we are developing algorithms based on a method referred to as the Variational MultiScale (VMS) method. The general idea is to use a numerical VMS-based approach to capture large-scale flows on regional scales of hundreds of kilometers, and to use versions of our semi-analytical approximations to form the sub-space Green’s functions which are needed to accurately capture the effect of injection and leakage on the local scale. This will allow us to combine the flexibility of considering complex heterogeneous large-scale flow fields with fast and accurate representations of local flow processes. In addition, the VMS methodology inherently leads to decoupled fine scale problems for the equations we consider, a significant advantage in the design and implementation of efficient parallel algorithms.

We continue to collaborate with Stefan Bachu at the Alberta Geological Survey and Alberta Energy and Utilities Board to use the Alberta Basin as a test case. We are interested in how the basin, at about 500,000 km^{2} , would behave under a scenario of full implementation of CCS across Alberta. If we can develop this model of the subsurface, then we could couple into it surface facilities, including pipelines and other infrastructure, to see what a fully developed CCS world might look like several decades into the future. Such a model allows for broad environmental as well as economic analysis, and could have a variety of optimization and general trade-off analyses included.

Finally, we note that to maximize the problems that can be analyzed, we are porting all of these codes to a parallel computing environment, where we expect to be able to run large basin-wide simulations. We have already accomplished this with most of our semi-analytical models, and we can now perform thousands of simulations over time periods of days. We know of no other computational tools that can achieve this kind of efficiency.

## Upscaling and other science issues

Celia’s group continues to work on basic issues associated with upscaling for both flow physics and geochemistry, and on other basic science issues associated with multi-phase flow in porous media. The recent work of Nordbotten et al. (2007; 2008a,b) focuses on how to define a macroscopic pressure in the presence of sub-scale variability. The definitions presented resolve some paradoxes with traditional averaging, including calculations of upscaled relative permeabilities that exceed unity. The work of Li et al. (2006; 2007a,b,c) addresses the question of upscaling geochemical reactions and the fundamental mathematical representation of reactions in the presence of sub-scale variability. This is being complemented by detailed studies of rock structure and mineralogy, led by our colleague Professor Peters and her students. Finally, we note the recent publication of a paper authored by Kyle Meng, Bob Williams, and Michael Celia, focused on early opportunities for CCS in China.