Jan M. Nordbotten

Similarity solutions for fluid injection into confined aquifers

Fluid injection into the deep subsurface, such as injection of carbon dioxide (CO 2 ) into deep saline aquifers, often involves two-fluid flow in confined geological formations. Similarity solutions may be derived for these problems by assuming that a sharp interface separates the two fluids, by imposing a suitable no-flow condition along both the top and bottom boundaries, and by including an explicit solution for the pressure distribution in both fluids. When the injected fluid is less dense and less viscous than the resident fluid, as is the case for CO 2 injection into a resident brine, gravity override produces a fluid flow system that is captured well by the similarity solutions. The similarity solutions may be extended to include slight miscibility between the two fluids, as well as compressibility in both of the fluid phases. The solutions provide the location of the interface between the two fluids, as well as drying fronts that develop within the injected fluid. Applications to cases of supercritical CO 2 injection into deep saline aquifers demonstrate the utility of the solutions, and comparisons to solutions from full numerical simulations show the ability to predict the system behaviour.

Practical Modeling Approaches for Geological Storage of Carbon Dioxide

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 CO2, 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 CO2, most of 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.

Monotonicity of control volume methods

Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property. It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied. Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in terms of monotonicity.

Quantitative estimation of CO2 leakage from geological storage: Analytical models, numerical models, and data needs

Publisher Summary This chapter focuses on development of large-scale modeling tools to quantify potential CO2 leakage along existing wells. Geological storage of CO2 is emerging as one of the most promising options for carbon mitigation. While this approach appears to be technically feasible, a comprehensive risk assessment is required to determine the overall effectiveness and possible environmental consequences of this approach. One important part of such a risk assessment is an analysis of potential leakage of injected CO2 from the formation into which is injected, to other permeable formations or to the atmosphere. Such leakage is a concern because it may contaminate existing energy, mineral, and/or groundwater resources, it may pose a hazard at the ground surface, and it will contribute to increased concentrations of CO2 in the atmosphere.

An efficient multi-point flux approximation method for Discrete Fracture-Matrix simulations

We consider a control volume discretization with a multi-point flux approximation to model Discrete Fracture-Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions. Inspired by a recently introduced approach based on a two-point flux approximation, we explicitly account for the fractures by representing them as hybrid cells between the matrix cells. As well as simplifying the grid generation, our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time-step restrictions associated with small cells. Excluding the small cells also reduces the condition number of the discretization matrix. For examples involving realistic anisotropy ratios in the permeability, numerical results show significant improvement compared to existing methods based on two-point flux approximations. We also investigate the hybrid method by studying the convergence rates for different apertures and fracture/matrix permeability ratios. Finally, the effect of removing the cells in the intersections of the fractures are studied. Together, these examples demonstrate the efficiency, flexibility and robustness of our new approach.

Vertically averaged approaches for CO2 migration with solubility trapping

The long-term storage security of injected carbon dioxide (CO2) is an essential component of geological carbon sequestration operations. In the postinjection phase, the mobile CO2 plume migrates in large part because of buoyancy forces, following the natural topography of the geological formation. The primary trapping mechanisms are capillary and solubility trapping, which evolve over hundreds to thousands of years and can immobilize a significant portion of the mobile CO2 plume. However, both the migration and trapping processes are inherently complex, spanning multiple spatial and temporal scales. Using an appropriate model that can capture both large-and small-scale effects is essential for understanding the role of these processes on the long-term storage security of CO2 sequestration operations. Traditional numerical models quickly become prohibitively expensive for the type of large-scale, long-term modeling that is necessary for characterizing the migration and immobilization of CO2 during the postinjection period. We present an alternative modeling option that combines vertically integrated governing equations with an upscaled representation of the dissolution-convection process. With this approach, we demonstrate the effect of different modeling choices for typical large-scale geological systems and show that practical calculations can be performed at the temporal and spatial scales of interest. Abstract: Multiscale methods can in many cases be viewed as special types of domain decomposition preconditioners. The localisation approximations introduced within the multiscale framework are

Evaluation of the spread of acid-gas plumes injected in deep saline aquifers in western Canada as an analogue for CO2 injection into continental sedimentary basins

Publisher Summary Injection of CO 2 into deep saline aquifers in sedimentary basins appears to be an important means for reducing anthropogenic emissions of CO 2 into the atmosphere. In the design, approval and monitoring of such operations it is important to predict the evolution of the plume of injected CO 2 and identify potential leakage pathways. In mature sedimentary basins such as those in North America that underwent intense exploration for and production of hydrocarbons, the number and density of wells is extremely high, and a plume of injected CO 2 is likely to encounter many wells that have to be identified and monitored. Under these circumstances, running full-blown numerical models becomes impractical and resource intensive, and simpler and faster tools are needed. An analytical model has been developed that, under a set of simplifying assumptions, can provide a rapid estimate of the shape and extent of a plume of CO 2 injected into an aquifer. The method assumes constant gas properties, which is a valid assumption for a wide range of conditions found in sedimentary basins, and homogeneous and uniform aquifer properties, which, depending on scale, can be assumed for many aquifers at least in a statistical sense. The aquifer is assumed to be horizontal, and there is no mixing, diffusion or dissolution between the injected gas and formation water.

Status of CO2 storage in deep saline aquifers with emphasis on modeling approaches and practical simulations

Carbon capture and storage (CCS) is the only viable technology to mitigate carbon emissions while allowing continued large-scale use of fossil fuels. The storage part of CCS involves injection of carbon dioxide, captured from large stationary sources, into deep geological formations. Deep saline aquifers have the largest identified storage potential, with estimated storage capacity sufficient to store emissions from large stationary sources for at least a century. This makes CCS a potentially important bridging technology in the transition to carbon-free energy sources. Injection of CO2 into deep saline aquifers leads to a multicomponent, multiphase flow system, in which geomechanics, geochemistry, and nonisothermal effects may be important. While the general system can be highly complex and involve many coupled, nonlinear partial differential equations, the underlying physics can sometimes lead to important simplifications. For example, the large density difference between injected CO2 and brine may lead to relatively fast buoyant segregation, making an assumption of vertical equilibrium reasonable. Such simplifying assumptions lead to a range of simplified governing equations whose solutions have provided significant practical insights into system behavior, including improved estimates of storage capacity, easy-to-compute estimates of CO2 spatial migration and pressure response, and quantitative estimates of leakage risk. When these modeling studies are coupled with observations from well-characterized injection operations, understanding of the overall system behavior is enhanced significantly. This improved understanding shows that, while economic and policy challenges remain, CO2 storage in deep saline aquifers appears to be a viable technology and can contribute substantially to climate change solutions.

Adaptive Variational Multiscale Methods for Multiphase Flow in Porous Media

Porous media are characterized by strongly heterogeneous properties on all scales of observation. This has motivated a sustained research effort in upscaling methods and lately multiscale methods. However, due to the lack of scale separation for realistic problems, static upscaling and multiscale methods may not be suitable in general, and adaptive approaches are needed. In this paper, we outline the application of an adaptive variational multiscale method for two-phase flow in porous media. We give the symmetric and nonsymmetric forms of the mixed variational multiscale method and suggest a framework for adaptively selecting the support of the fine scale Greens operators in space and time. The applicability of our framework is validated through a set of numerical comparisons.

Sufficient criteria are necessary for monotone control volume methods

Control volume methods are prevailing for solving the potential equation arising in porous media flow. The continuous form of this equation is known to satisfy a maximum principle, and it is desirable that the numerical approximation shares this quality. Recently, sufficient criteria were derived guaranteeing a discrete maximum principle for a class of control volume methods. We show that most of these criteria are also necessary. An implication of our work is that no linear nine-point control volume method can be constructed for quadrilateral grids in 2D that is exact for linear solutions while remaining monotone for general problems.