Vertically Averaged Equations with Variable Density for $$\hbox {CO}_2$$ CO 2 Flow in Porous Media

Abstract

Carbon capture and storage has been proposed as a viable option to reduce CO 2 emissions. Geological storage of CO 2 where the gas is injected into geological formations for practically indefinite storage, is an integral part of this strategy. Mathematical models and numerical simulations are important tools to better understand the processes taking place underground during and after injection. Due to the very large spatial and temporal scales involved, commercial 3D-based simulators for the petroleum industry quickly become impractical for answering questions related to the long-term fate of injected CO 2 . There is an interest in developing simplified modeling tools that are effective for this type of problem. One approach investigated in recent years is the use of upscaled models based on the assumption of vertical equilibrium (VE). Under this assumption, the simulation problem is essentially reduced from 3D to 2D, allowing much larger models to be considered at the same computational cost. So far, most work on VE models for CO 2 storage has either assumed incompressible CO 2 or only permitted lateral variations in CO 2 density (semi-compressible). In the present work, we propose a way to fully include variable CO 2 density within the VE framework, making it possible to also model vertical density changes. We derive the fine-scale and upscaled equations involved and investigate the resulting effects. In addition, we compare incompressible, semi-compressible, and fully compressible CO 2 flow for some model scenarios, using an in-house, fully-implicit numerical code based on automatic differentiation, implemented using the MATLAB reservoir simulation toolkit.

Fully-implicit simulation of vertical-equilibrium models with hysteresis and capillary fringe

“…Early methods assumed a sharp interface between CO 2 and the resident brine and were successfully applied to study long-term CO 2 migration in the large-scale Utsira [23] and Johansen [24] aquifers. Since then, the VE formulation has been extended to include many relevant physical effects, such as compressibility [25], convective dissolution [26], capillary fringe [27], smallscale caprock topography variations [28][29][30], as well as various hysteretic effects [31,32]. Presently, VE formulations can be used to efficiently simulate CO 2 migration and resolve the delicate balances governing structural, residual, and solubility trapping over thousand of years.…”

“…Likewise, the model equations include rock compressibility but disregard fluid compressibility. The effect of variable density in vertical-equilibrium models was investigated by Andersen et al [25], who concluded that for the vast majority of long-term, large-scale storage systems it is sufficient to account for CO 2 density variations only in the lateral direction (i.e., a semi-compressible model). Both fully compressible and semi-compressible models have been implemented in MRST, but for brevity we herein disregard fluid compressibility when discussing how to derive expressions for effective properties by vertical integration.…”

“…Herein, we will make the simplifying assumption that the density is constant within each vertical column, which is reasonable in most cases in which VE-models are suited for modeling CO 2 storage [25]. Moreover, to simplify the notation, we will integrate from 0 to H instead of from Z to Z + H when considering a single column.…”

“…The aquifer is described by the top surface Z(ξ, η) and its thickness H (ξ, η) in the ζ -direction without hysteresis, h is equivalent to the maximum depth with nonzero CO 2 saturation s n . The formula (4) assumes constant vertical density, but an extension of the VE formulation without this assumption is studied in [25], in which case the vertical pressure profile is reconstructed by solving an ordinary differential equation. The left diagram in Fig.…”

Robust simulation of sharp-interface models for fast estimation of CO2 trapping capacity in large-scale aquifer systems

“…Early studies focused on VE models with a sharpinterface assumption [24][25][26], and models that only account for the basic effects of buoyant migration were successfully used to simulate long-term migration in the Utsira [27] and Johansen [28] aquifers. Later, the class of VE models has been extended to incorporate most of the flow effects that are pertinent to large-scale migration, including compressibility [29], convective dissolution [30,31], capillary fringe [32], small-scale caprock topography variations [33][34][35], various hysteretic effects [36][37][38], multiple geological layers [39,40], and heat transfer [41]. In particular, several studies show that vertical equilibrium simulations compare well with 3D simulators on case studies of the Johansen aquifer [42] and the 9 th layer of the Sleipner injection [27,43].…”