Applicability of vertical-equilibrium and sharp-interface assumptions in CO2 sequestration modeling
Highlights
► Vertical-equilibrium assumption is valid when fluids segregation has occurred. ► Sharp-interface assumption is valid when capillary transition zone is negligible. ► Vertically integrated model closely matches 3D model results including pcap. ► Wide variety of pcap-s and kr-s relationships used in CCS lit. with large impacts. ► Model complexity can be reduced for many CO2 injection problem.
Introduction
Geologic sequestration of carbon dioxide (CO2) is a promising option to reduce anthropogenic carbon emissions (Metz et al., 2005). When analyzing the viability of a proposed sequestration site, mathematical models are typically used to determine available sequestration volumes, sequestration safety (i.e., leakage risk), and migration of the injected CO2 and displaced resident brine. These models range in complexity from simple pore volume calculations (Bachu et al., 2007, Kopp et al., 2009, Szulczewski and Juanes, 2009) to three-dimensional multi-component, multi-phase reservoir simulators such as ECLIPSE (Schlumberger, 2010), NUFT (Nitao, 1998), TOUGH2 (Pruess, 2004), and STOMP (White and Oostrom, 1997). Several useful review papers and benchmark code comparison have been completed by (Michael et al., 2009, Pruess et al., 2009, Schnaar and Digiulio, 2009), and by (Class et al., 2009, Ebigbo et al., 2007, Nilsen et al., 2011, Pruess et al., 2002, Pruess et al., 2004, Pruess and Nordbotten, 2011) respectively. Fig. 1 presents a conceptual model of the considered system showing a section of a single supercritical CO2 plume in a deep saline injection aquifer bounded by impermeable aquitards above and below.
A family of models of intermediate complexity can be derived by assuming that the strong buoyant drive in the system leads to vertical segregation of the injected CO2 and resident brine on a time scale that is fast compared to the time scale of the simulation (Celia and Nordbotten, 2011, Nordbotten and Celia, 2012). This leads to a system where each fluid has a pressure distribution that is essentially hydrostatic, which is often referred to as vertical equilibrium. While the applicability of the vertical-equilibrium assumption has been analyzed based on spatial scales (see, for example, Lake, 1989 or Yortos, 1995), herein we consider the problem in terms of temporal scales. When the local-scale capillary pressure forces are negligible, the stratified fluids may be assumed to be separated by a macroscopic sharp-interface with constant fluid saturations on either side. These vertical-equilibrium sharp-interface models are sometimes used in the petroleum industry and, more recently, have been used to analyze CO2 injection and migration problems (Celia and Nordbotten, 2009, Celia and Nordbotten, 2011, Coats et al., 1971, Hesse et al., 2007, Hesse et al., 2008, Huppert and Woods, 1995, Juanes et al., 2010, Nordbotten and Celia, 2006, Nordbotten and Celia, 2012, Woods and Mason, 2000).
In a recent publication, Lu et al. (2009) compared fully resolved numerical simulations to simple analytical solutions based on the vertical-equilibrium and sharp-interface assumptions. The examples were based on injection of supercritical CO2 from a single vertical well into a hypothetical formation that was considered to be homogeneous, isotropic, and horizontal with no leakage along the top or bottom boundaries, which leads to a radially symmetric solution. Lu et al. concluded that the vertical-equilibrium sharp-interface analytical approaches of this simple radial solution presented by (Dentz and Tartakovsky, 2009a, Dentz and Tartakovsky, 2009b, Nordbotten et al., 2005) were unable to accurately predict the CO2 front position. As such, they concluded that these simplified models were not appropriate to model CO2 injection. In this paper, we look more closely at the conditions under which vertical-equilibrium models are appropriate, and the conditions under which sharp-interface models are appropriate. We show that for many cases vertical-equilibrium models are reasonable, whether or not the sharp-interface assumption applies. We also explain the results from the work of Lu et al. in light of this broader analysis.
The objective of this paper is to study the conditions under which both vertical-equilibrium and sharp-interface models are appropriate to model CO2 injection and migration. We first examine different timescales to understand when vertical-equilibrium of phase pressures can reasonably be assumed. We focus our calculations on the injection time period, as this is the most stringent test of the vertical equilibrium assumption – the assumption becomes increasingly appropriate for longer time scales. We then investigate the spatial scale associated with the capillary pressure – saturation function to understand when sharp-interface models are appropriate and when models including capillary forces (with no sharp-interface) are necessary. To put this into context, we consider a number of publications from the CO2 sequestration literature that have reported both capillary pressure-saturation (pcap-s) and relative permeability-saturation (kr-s) relationships, whether using them in simulations or measuring them experimentally. Several pcap-s functions representative of the range found in the literature are selected, and used to study a generic injection scenario of an industrial-scale CO2 capture and sequestration (CCS) operation. We also use the parameters and associated kr-s and pcap-s functions applied by Lu et al. (2009). Direct comparisons are made between results from the commercial simulator ECLIPSE and results from a relatively simple vertical-equilibrium, sharp-interface model named ELSA (Celia and Nordbotten, 2011, Dobossy et al., 2011, Nordbotten et al., 2009) and the more general vertical-equilibrium model called VESA (Gasda, 2008, Gasda et al., 2009, Gasda et al., 2010, Gasda et al., 2011, Janzen, 2010). Our overall purpose is to identify the limits of vertically-integrated numerical models and of the sharp-interface assumption, thereby providing some practical guidance for the level of mathematical model complexity required to model CO2 injection and migration.
The paper proceeds as follows. In the next section, we briefly describe the two vertical-equilibrium models used in this study: a numerical model without the sharp-interface assumption, and an analytical model with the sharp-interface assumption. Both are based on an assumption of vertical equilibrium. In Section 3, we present a review of the different kr-s and pcap-s functions that have been reported in the CCS literature. We include an analysis of the capillary pressure functions that provides a length-scale estimate of the “capillary transition zone”, which guides the analysis of the sharp-interface assumption. In Section 4 we use selected functions from this range of injection formation characteristics (pcap and kr) to explore how the system behaves. We begin by presenting a time evolution of the plume in a generic injection scenario modeled with ECLIPSE. This provides an illustration of time-scaling arguments associated with the establishment of vertical equilibrium. We then combine this with the spatial-scale analysis associated with capillary pressure, and consider other kinds of problems including those from Lu et al. (2009). We end the paper with a set of conclusions.
Section snippets
Model description
Injection of supercritical CO2 into the subsurface involves the flow of both the injected fluid and the resident brine. The movement of CO2 and brine, assumed to be immiscible, may be modeled using two-phase porous media flow equations, assuming CO2 (c) is the non-wetting phase and brine (b) is the wetting phase. The governing equations are based on the multi-phase extension of Darcy's Law and the mass balance equation for each phase (see Bear, 1972, Lake, 1989; or other classic references).
Literature review
Capillary pressure and relative permeability functions can have a significant impact on the CO2 plume shape and outer extent. Coupled with intrinsic (absolute) permeability, they also have important impacts on pressure build-up during injection. To investigate the impacts of parameter choices, specifically relative permeability and capillary pressure, we reviewed the range of pcap-s and kr-s relationships reported in the CCS literature. We found that pcap-s and kr-s relationships reported are
Results
In this section we use numerical experiments to explore conditions under which vertically-integrated models with and without the sharp-interface assumption are applicable. First, three-dimensional numerical simulations are used to investigate the time scales for vertical equilibrium to develop. This is followed by a comparison of three-dimensional model results to results using the vertically-integrated sharp-interface model for several different kr-s relationships. Finally, we explore cases
Summary and conclusions
CO2 sequestration safety evaluation will involve mathematical models of various complexity levels to answer different practical questions. While full three-dimensional numerical models are useful to answer many of these questions, simpler models can often provide similar information at a fraction of the computational cost, and with data requirements that are more comparable with the data sparsity often seen for injection operations into deep saline aquifers. One family of simpler models is
Acknowledgements
This work was supported in part by the Environmental Protection Agency under Cooperative Agreement RD-83438501 as well as the National Science Foundation under Grant EAR-0934722; the Department of Energy under Award No. DE-FE0001161, CFDA No. 81,089; and the Carbon Mitigation Initiative at Princeton University.
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