Abstract
Abstract A stochastic Galerkin formulation for the transport of $$\hbox {CO}_2$$ CO 2 in a tilted aquifer with uncertain heterogeneous properties is presented. We consider a simplified physics model assuming capillary pressure to be negligible compared to hydrostatic and viscous pressure. The flow is dominated by buoyancy and capillary trapping. We assume a stochastic permeability field and a stochastic model for the uncertain relative permeabilities. We prove that the proposed stochastic Galerkin formulation results in a hyperbolic system of equations, and we devise a numerical method that captures the expected solution discontinuities. The shock-capturing solver for the flux function is combined with an adaptive quadrature method for discontinuous isosurfaces that is used to compute the discontinuous stochastic accumulation coefficient. The stochastic solver is validated against Monte Carlo sampling of an analytical solution for the deterministic problem. The sharp features of the statistics of the solution are accurately captured by the numerical solver. The polynomial chaos framework admits low-cost post-processing of the output to obtain statistics of interest. By construction of an accurate polynomial chaos surrogate model of the output, fast sampling admits calculation of risk for leakage and failure probabilities.
Highlights
Permanent storage of CO2 in subsurface saline aquifers is a potentially effective means to reduce anthropogenic CO2 emission to the atmosphere (Benson and Cook 2005)
The polynomial chaos framework (Ghanem and Spanos 1991; Xiu and Karniadakis 2002) for solution of partial differential equations (PDEs) subject to uncertainty has been applied in the context of subsurface flow and CO2 storage, and may be an efficient alternative to Monte Carlo methods
We present a stochastic Galerkin formulation and a numerical solver for a simplified physics model of subsurface CO2 storage
Summary
Permanent storage of CO2 in subsurface saline aquifers is a potentially effective means to reduce anthropogenic CO2 emission to the atmosphere (Benson and Cook 2005). MacMinn et al (2010) presented an analytical solution for a sloping aquifer with groundwater flow and capillary trapping in one spatial dimension and evaluated the model for storage efficiency at the basin scale (Juanes et al 2010). The polynomial chaos framework (Ghanem and Spanos 1991; Xiu and Karniadakis 2002) for solution of PDEs subject to uncertainty has been applied in the context of subsurface flow and CO2 storage, and may be an efficient alternative to Monte Carlo methods. We present a stochastic Galerkin formulation and a numerical solver for a simplified physics model of subsurface CO2 storage. The stochastic Galerkin projection is based on the polynomial chaos framework but results in a single extended coupled system of equations that are solved only once to obtain all statistical information. To determine the accumulation coefficient efficiently everywhere in discrete space and time, we adjust the adaptive quadrature method for discontinuous interfaces presented in Müller et al (2012)
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