A spatial Markov model for upscaling transport of adsorbing-desorbing solutes

An alternative general model for the effective longitudinal diffusion in chromatographic beds filled with ordered porous particles

The two-zone moment-analysis method for the determination of the dispersion tensor in hierarchical retentive porous media has been adopted to compute and model the effective longitudinal diffusion Deff, or equivalently the B-term band broadening, in chromatographic beds filled with ordered porous particles. On the one hand, this approach offers accurate numerical results for Deff while keeping computational expenses low. On the other hand, it also gives direct insight for the analytical modelling, readily revealings the two main essential quantities (resp. referred to as the mobile-zone and stationary-zone effective diffusion factors γm and γs) that contribute to Deff. Modelling these two main parameters provided us with two new analytical models for Deff: a general one, valid for diluted and concentrated packings and accurate in the whole range of relevant intra-particle diffusion coefficient Dpz, and an approximate one, reliable for diluted packings and accurate also for concentrated packings with low to intermediate values of Dpz. The large advantage of both models is that they do not need any fitting parameter because all the required information is incorporated into the experimentally accessible geometric obstruction factor in the mobile phase originating from the tortuosity of the through-pore space (limiting case of fully solid particles without any retention). These models hence serve as an alternative to the Effective Medium Theory (EMT) models used so far in the literature. To validate the theory, five ordered geometries have been investigated. The accuracy of the general model proposed has been quantified and found to be comparable with that of the 3rd order approximate Torquato model for four geometries, even for macro-porosities close to the close-packing limit. The case of a 2-d triangular array of ellipsoidal particles with different elongations is also investigated to show the general validity and applicability of the models.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

Effects of large-scale heterogeneity and temporally varying hydrologic processes on estimating immobile pore space: A mesoscale-laboratory experimental and numerical modeling investigation

The advection-dispersion equation (ADE) often fails to predict solute transport, in part due to incomplete mixing in the subsurface, which the development of non-local models has attempted to deal with. One such model is dual-domain mass transfer (DDMT); one parameter that exists within this model type is called immobile porosity. Here, we explore the complexity of estimating immobile porosity under varying flow rates and density dependencies in a large-scale heterogeneous system. Immobile porosity is estimated experimentally and using numerical models in 3-D flow systems, and is defined by domains of comparatively low advective velocity instead of truly immobile regions at the pore scale. Tracer experiments were conducted in a mesoscale 3-D tank system with embedded large impermeable zones and the generated data were analyzed using a numerical model. The impermeable zones were used to explore how large-scale structure and heterogeneity affect parameter estimation of immobile porosity, assuming a dual-porosity model, and resultant characterization of the aquifer system. Spatially and temporally co-located fluid electrical conductivity (σ_f) and bulk apparent electrical conductivity (σ_b)-using geophysical methods-were measured to estimate immobile porosity, and numerical modeling (i.e., SEAWAT and R3t) was conducted to explore controls of the immobile zones on the experimentally observed flow and transport. Results showed that density-dependent flow increased the hysteresis between measured fluid and bulk electrical conductivity, resulting in larger interpreted immobile pore-space estimates. Increasing the dispersivity in the model simulations decreased the estimated immobile porosity; flow rate had no impact. Overall, the results of this study highlight the difficulty faced in determining immobile porosity values in field settings, where hydrogeologic processes may vary temporally. Our results also highlight that immobile porosity is an effective parameter in an upscaled model whose physical meaning is not necessarily clear and that may not align with intuitive interpretations of a porosity.

A review of spatial Markov models for predicting pre-asymptotic and anomalous transport in porous and fractured media

Heterogeneity across a broad range of scales in geologic porous media often manifests in observations of non-Fickian or anomalous transport. While traditional anomalous transport models can successfully make predictions in certain geological systems, increasing evidence suggests that assumptions relating to independent and identically distributed increments constrain where and when they can be reliably applied. A relatively novel model, the Spatial Markov model (SMM), relaxes the assumption of independence. The SMM belongs to the family of correlated continuous time random walks and has shown promise across a wide range of transport problems relevant to natural porous media. It has been successfully used to model conservative as well as more recently reactive transport in highly complex flows ranging from pore scales to much larger scales of interest in geology and subsurface hydrology. In this review paper we summarize its original development and provide a comprehensive review of its advances and applications as well as lay out a vision for its future development.