Joachim Moortgat

Critical Dynamics of Gravito-Convective Mixing in Geological Carbon Sequestration

When CO2 is injected in saline aquifers, dissolution causes a local increase in brine density that can cause Rayleigh-Taylor-type gravitational instabilities. Depending on the Rayleigh number, density-driven flow may mix dissolved CO2 throughout the aquifer at fast advective time-scales through convective mixing. Heterogeneity can impact density-driven flow to different degrees. Zones with low effective vertical permeability may suppress fingering and reduce vertical spreading, while potentially increasing transverse mixing. In more complex heterogeneity, arising from the spatial organization of sedimentary facies, finger propagation is reduced in low permeability facies, but may be enhanced through more permeable facies. The connectivity of facies is critical in determining the large-scale transport of CO2-rich brine. We perform high-resolution finite element simulations of advection-diffusion transport of CO2 with a focus on facies-based bimodal heterogeneity. Permeability fields are generated by a Markov Chain approach, which represent facies architecture by commonly observed characteristics such as volume fractions. CO2 dissolution and phase behavior are modeled with the cubic-plus-association equation-of-state. Our results show that the organization of high-permeability facies and their connectivity control the dynamics of gravitationally unstable flow. We discover new flow regimes in both homogeneous and heterogeneous media and present quantitative scaling relations for their temporal evolution.

Three-Phase Compositional Modeling With Capillarity in Heterogeneous and Fractured Media

We model for the first time capillarity in fully compositional three-phase flow, with higher-order finite-element (FE) methods. Capillary pressure gradients may be an important driving force, particularly in layered or fractured porous media, which exhibit sharp discontinuities in permeability. We introduce a simple local computation of the capillary pressure gradients, propose a fractional-flow formulation in terms of the total flux, and resolve complications arising from gravity and capillarity in the upwinding of phase fluxes. Fractures are modeled with the crossflow equilibrium concept, which allows large timesteps and includes all physical interactions between fractures and matrix blocks. The pressure and flux fields are discretized by the mixed hybrid finite-element method, and mass transport is approximated by a higher-order local discontinuous Galerkin (DG) method. Numerical-dispersion and grid-orientation effects are significantly reduced, which allows computations on coarser grids and with larger timesteps. The main advantages in the context of this work are the accurate pressure gradients and fluxes at the interface between regions of different permeabilities. The phase compositions are computed with state-of-the-art phase-splitting algorithms and stability analyses to guarantee the global minimum of Gibbs free energy. Accurate compositional simulation motivates the use of an implicitpressure/explicit-composition (IMPEC) scheme, and we discuss the associated Courant-Friedrichs-Lewy (CFL) condition on the timesteps. We present various numerical examples on both coreand large-scale, illustrating the capillary end effect, capillary-driven crossflow in layered media, and the importance of capillarity in fractured media for three-phase flow.

Viscous and gravitational fingering in multiphase compositional and compressible flow

Abstract Viscous and gravitational fingering refer to flow instabilities in porous media that are triggered by adverse mobility or density ratios, respectively. These instabilities have been studied extensively in the past for (1) single-phase flow (e.g., contaminant transport in groundwater, first-contact-miscible displacement of oil by gas in hydrocarbon production), and (2) multi-phase immiscible and incompressible flow (e.g., water-alternating-gas (WAG) injection in oil reservoirs). Fingering in multiphase compositional and compressible flow has received much less attention, perhaps due to its high computational complexity. However, many important subsurface processes involve multiple phases that exchange species. Examples are carbon sequestration in saline aquifers and enhanced oil recovery (EOR) by gas or WAG injection below the minimum miscibility pressure. In multiphase flow, relative permeabilities affect the mobility contrast for a given viscosity ratio. Phase behavior can also change local fluid properties, which can either enhance or mitigate viscous and gravitational instabilities. This work presents a detailed study of fingering behavior in compositional multiphase flow in two and three dimensions and considers the effects of (1) Fickian diffusion, (2) mechanical dispersion, (3) flow rates, (4) domain size and geometry, (5) formation heterogeneities, (6) gravity, and (7) relative permeabilities. Results show that fingering in compositional multiphase flow is profoundly different from miscible conditions and upscaling techniques used for the latter case are unlikely to be generalizable to the former.

Higher-order compositional modeling of three-phase flow in 3D fractured porous media based on cross-flow equilibrium

Numerical simulation of multiphase compositional flow in fractured porous media, when all the species can transfer between the phases, is a real challenge. Despite the broad applications in hydrocarbon reservoir engineering and hydrology, a compositional numerical simulator for three-phase flow in fractured media has not appeared in the literature, to the best of our knowledge. In this work, we present a three-phase fully compositional simulator for fractured media, based on higher-order finite element methods. To achieve computational efficiency, we invoke the cross-flow equilibrium (CFE) concept between discrete fractures and a small neighborhood in the matrix blocks. We adopt the mixed hybrid finite element (MHFE) method to approximate convective Darcy fluxes and the pressure equation. This approach is the most natural choice for flow in fractured media. The mass balance equations are discretized by the discontinuous Galerkin (DG) method, which is perhaps the most efficient approach to capture physical discontinuities in phase properties at the matrix-fracture interfaces and at phase boundaries. In this work, we account for gravity and Fickian diffusion. The modeling of capillary effects is discussed in a separate paper. We present the mathematical framework, using the implicit-pressure-explicit-composition (IMPEC) scheme, which facilitates rigorous thermodynamic stability analyses and the computation of phase behavior effects to account for transfer of species between the phases. A deceptively simple CFL condition is implemented to improve numerical stability and accuracy. We provide six numerical examples at both small and larger scales and in two and three dimensions, to demonstrate powerful features of the formulation.

Implicit finite volume and discontinuous Galerkin methods for multicomponent flow in unstructured 3D fractured porous media

Abstract We present a new implicit higher-order finite element (FE) approach to efficiently model compressible multicomponent fluid flow on unstructured grids and in fractured porous subsurface formations. The scheme is sequential implicit: pressures and fluxes are updated with an implicit Mixed Hybrid Finite Element (MHFE) method, and the transport of each species is approximated with an implicit second-order Discontinuous Galerkin (DG) FE method. Discrete fractures are incorporated with a cross-flow equilibrium approach. This is the first investigation of all-implicit higher-order MHFE-DG for unstructured triangular, quadrilateral (2D), and hexahedral (3D) grids and discrete fractures. A lowest-order implicit finite volume (FV) transport update is also developed for the same grid types. The implicit methods are compared to an Implicit-Pressure-Explicit-Composition (IMPEC) scheme. For fractured domains, the unconditionally stable implicit transport update is shown to increase computational efficiency by orders of magnitude as compared to IMPEC, which has a time-step constraint proportional to the pore volume of discrete fracture grid cells. However, when lowest-order Euler time-discretizations are used, numerical errors increase linearly with the larger implicit time-steps, resulting in high numerical dispersion. Second-order Crank–Nicolson implicit MHFE-DG and MHFE-FV are therefore presented as well. Convergence analyses show twice the convergence rate for the DG methods as compared to FV, resulting in two to three orders of magnitude higher computational efficiency. Numerical experiments demonstrate the efficiency and robustness in modeling compressible multicomponent flow on irregular and fractured 2D and 3D grids, even in the presence of fingering instabilities.

Geographic axes and the persistence of cultural diversity

Jared Diamond’s Guns, Germs, and Steel [Diamond J, (1997) Guns, Germs, and Steel (WW Norton, NY)] has provided a scientific foundation for answering basic questions, such as why Eurasians colonized the global South and not the other way around, and why there is so much variance in economic development across the globe. Diamond’s explanatory variables are: (i) the susceptibility of local wild plants to be developed for self-sufficient agriculture; (ii) the domesticability of large wild animals for food, transport, and agricultural production; and (iii) the relative lengths of the axes of continents with implications for the spread of human populations and technologies. This third “continental axis” thesis is the most difficult of Diamond’s several explanatory factors to test, given that the number of continents are too few for statistical analysis. This article provides a test of one observable implication of this thesis, namely that linguistic diversity should be more persistent to the degree that a geographic area is oriented more north-south than east-west. Using both modern states and artificial geographic entities as the units of analysis, the results provide significant confirmation of the relationship between geographic orientation and cultural homogenization. Beyond providing empirical support for one observable implication of the continental axis theory, these results have important implications for understanding the roots of cultural diversity, which is an important determinant of economic growth, public goods provision, local violence, and social trust.

Dissolution Trapping of Carbon Dioxide in Heterogeneous Aquifers

The geologic architecture in sedimentary reservoirs affects the behavior of density-driven flow and the dispersion of CO2-rich brine. The spatial organization and connectivity of facies types play an important role. Low-permeability facies may suppress fingering and reduce vertical spreading, but may also increase transverse mixing. This is more pronounced when geologic structures create preferential flow pathways through connected facies types. We perform high-resolution simulations of three-dimensional (3D) heterogeneous formations whose connectivity cannot be represented in two-dimensional models consistent with percolation theory. This work focuses on the importance of 3D facies-based heterogeneity and connectivity on advection-diffusion transport of dissolved CO2. Because the dissolution of CO2 and the subsequent density increase of brine are the driving force for gravitational instabilities, we model the phase behavior with the accurate cubic-plus-association equation-of-state, which accounts for the self-association of polar water molecules and the cross-association between CO2 and water. Our results elucidate how the spatial organization of facies affects the dynamics of CO2 convective mixing. Scaling relations for the evolution of a global dispersion-width provide insights that can be universally applied. The results suggest that the long-term evolution and scaling of dispersion are surprisingly similar for homogeneous and (binary and multiscale) heterogeneous porous media.

High-resolution finite element methods for 3D simulation of compositionally triggered instabilities in porous media

The formation and development of patterns in the unstable interface between an injected fluid and hydrocarbons or saline aqueous phase in a porous medium can be driven by viscous effects and gravity. Numerical simulation of the so-called fingering is a challenge, which requires rigorous representation of the fluid flow and thermodynamics as well as highresolution discretization in order to minimize numerical artifacts. To achieve such a high resolution, we present higherorder 3D finite element methods for the simulation of fully compositional, three-phase and multi-component flow. This is based on a combination of the mixed hybrid finite element (MHFE) method for total fluid velocity and discontinuous Galerkin (DG) method for the species transport. The phase behavior is described by cubic or cubic-plus-association (CPA) equations of state. We present challenging numerical examples of compositionally triggered fingering at both the core and the large scale. Four additional test cases illustrate the robustness and efficiency of the proposed methods, which demonstrate their power for problems of this complexity. Results reveal three orders of magnitude improvement in CPU time in our method compared with the lowest-order finite difference method for some of the examples. Comparison between 3D and 2D results highlights the significance of dimensionality in the flow simulation.

Hydrothermodynamic mixing of fluids across phases in porous media

We investigate the coupled dynamics of fluid mixing and viscously unstable flow under both miscible (single-phase) and partially miscible (two-phase) conditions, and in both homogeneous and heterogeneous porous media. Higher-order finite element methods and fine grids are used to resolve the small-scale onset of fingering and tip splitting. An equation of state determines the thermodynamic phase behavior and Fickian diffusion. We compute global quantitative measures of the spreading and mixing of a diluting slug to elucidate key differences between miscible and partially miscible systems. Hydrodynamic instabilities are the main driver for mixing in miscible flow. In partially miscible flow, however, we find that relative permeabilities spread the two-phase zone. Within this mixing zone dissolution and evaporation drive mixing thermodynamically while reducing mobility contrasts and thus fingering instabilities. The different mixing dynamics in systems involving multiple phases with mutual solubilities have important implications in hydrogeology and energy applications, such as geological carbon sequestration and gas transport in hydrocarbon reservoirs.

Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3D unstructured grids

Problems of interest in hydrogeology and hydrocarbon resources involve complex heterogeneous geological formations. Such domains are most accurately represented in reservoir simulations by unstructured computational grids. Finite element methods accurately describe flow on unstructured meshes with complex geometries, and their flexible formulation allows implementation on different grid types. In this work, we consider for the first time the challenging problem of fully compositional three-phase flow in 3D unstructured grids, discretized by any combination of tetrahedra, prisms, and hexahedra. We employ a mass conserving mixed hybrid finite element (MHFE) method to solve for the pressure and flux fields. The transport equations are approximated with a higher-order vertex-based discontinuous Galerkin (DG) discretization. We show that this approach outperforms a face-based implementation of the same polynomial order. These methods are well suited for heterogeneous and fractured reservoirs, because they provide globally continuous pressure and flux fields, while allowing for sharp discontinuities in compositions and saturations. The higher-order accuracy improves the modeling of strongly non-linear flow, such as gravitational and viscous fingering. We review the literature on unstructured reservoir simulation models, and present many examples that consider gravity depletion, water flooding, and gas injection in oil saturated reservoirs. We study convergence rates, mesh sensitivity, and demonstrate the wide applicability of our chosen finite element methods for challenging multiphase flow problems in geometrically complex subsurface media.