Juan J. Hidalgo

Dissolution patterns and mixing dynamics in unstable reactive flow

We study the fundamental problem of mixing and chemical reactions under a Rayleigh-Taylor-type hydrodynamic instability in a miscible two-fluid system. The dense fluid mixture, which is generated at the fluid-fluid interface, leads to the onset of a convective fingering instability and triggers a fast chemical dissolution reaction. Contrary to intuition, the dissolution pattern does not map out the finger geometry. Instead, it displays a dome-like, hierarchical structure that follows the path of the ascending fluid interface and the regions of maximum mixing. These mixing and reaction hot spots coincide with the flow stagnation points, at which the interfacial mixing layer is compressed and deformed. We show that the deformation of the boundary layer around the stagnation points controls the evolution of the global scalar dissipation and reaction rates and shapes the structure of the reacted zones. The persistent compression of the mixing layer explains the independence of the mixing rate from the Rayleigh number when convection dominates.

Non-Fickian Transport Under Heterogeneous Advection and Mobile-Immobile Mass Transfer

We study the combined impact of heterogeneous advection and mobile–immobile mass transfer on non-Fickian transport using the continuous-time random walk (CTRW) approach. The CTRW models solute transport in heterogeneous media as a random walk in space and time. Our study is based on a d-dimensional CTRW model that accounts for both heterogeneous advection and mass transfer between mobile and immobile zones, to which we also refer as solute trapping. The flow heterogeneity is mapped into the distribution of advective transition times over a characteristic heterogeneity scale. Mass transfer into immobile zones is quantified by a trapping rate and the distribution of particle return times. The total particle transition time over a characteristic heterogeneity scale then is given by the advective time and the sum of trapping times over the number of trapping events. We establish explicit integro-partial differential equations for the evolution of the concentration and discuss the relation to the multirate mass transfer approach, specifically the relation between the trapping time distribution and the memory function. We then analyze the signatures of anomalous transport due to advective heterogeneity and trapping in terms of spatial moments and first passage times or breakthrough curves. The behaviors for different disorder scenarios are analyzed analytically and through random walk particle tracking simulations. Assuming that advective mass transfer is faster than diffusive, we identify three regimes of distinct transport behaviors, which are separated by the characteristic trapping rate and trapping times. (1) At early times, we identify a pre-asymptotic time regime that is fully determined by advective heterogeneity and which is characterized by superlinear growth of longitudinal dispersion. (2) For longitudinal dispersion, we identify an intermediate regime of strong superlinear diffusion. This regime is determined by the combined effect of advective heterogeneity and trapping. (3) At larger time, the asymptotic trapping-driven regime shows the signatures of diffusion in immobile zones, which leads to both sub- and superlinear dispersion. These results shed some new light on the mechanism of non-Fickian transport and their manifestation in spatial and temporal solute distributions.

Upscaling and Scale Effects

Chapters 3 and 4 address the mathematical and numerical modeling of CO2 geological storage. This chapter, in turn, focuses on a specific important aspect of modeling, namely that of scale effects and upscaling. The geological systems are heterogeneous, with heterogeneity occurring at various scales. This gives rise to what is commonly named the “scale effect”. Certain process are critical at the scale of pores, while some of the effects of CO2 injection may have an effect and need to be modeled at the scale of tens and even hundreds of kilometers. Furthermore, various processes may be important at different scales. This requires understanding and methods of linking processes over a span of the scales. This is the topic of the current chapter.

Mixing across fluid interfaces compressed by convective flow in porous media

We study the mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyze the behavior of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyze a double-gyre synthetic velocity field. Second, a Rayleigh-Benard instability (the Horton-Rogers-Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh-Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated, and convection shutdown. The interface compression model correctly predicts the behavior of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behavior of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behavior between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns.

Controlling Factors of Wormhole Growth in Karst Aquifers

Flow and water discharge in karst aquifers are controlled by the conduit network. Therefore, understanding karst conduit formation is important to conjecture the aquifer topology, i.e., conduit density and size, and to predict the aquifer dynamics. Conduits are generated by preferential pathways to flow known as wormholes that grow competing with each other. The success of a wormhole is determined by its ability to drive water away from its neighbors. Once a wormhole forms, water tends to flow along this preferential path thus reducing the availability of water for the enlargement of less developed wormholes. Wormhole growth is then controlled by the flow rate, the dissolution mechanisms and the heterogeneity of the hydraulic conductivity field. In this work, we propose two conceptual models to describe the geometry of the wormhole capture zone and its effect on the surrounding wormholes. First, we consider a cross-section intersecting the wormhole longitudinally. Second, we consider a radial model centered in the wormhole. These models are representative of field (fracture) and laboratory (tube), respectively. We perform a series of steady state simulations to obtain the dependence of the capture zone on the wormhole’s geometry. This naturally leads to a relation between the wormhole’s geometry and the density of wormholes because only one wormhole grows within a capture zone.