Pietrzyk Research Group

Despite the ubiquity of heterogeneous porous media in Earth and energy engineering systems, the effects of heterogeneity on macroscopic flow and transport phenomena continue to elude our fundamental understandings. Traditionally, rigorous upscaling theories (e.g., Homogenization Theory and the Method of Volume Averaging) have provided robust models for quantifying, characterizing, and predicting coarse-scale behaviors when fine-scale and coarse-scale phenomena (i.e., structural features and physical dynamics) exist on different spatiotemporal scales (i.e., when there is a clear separation of scales). As heterogeneities intertwine fine- and coarse-scale phenomena, traditional upscaling theories tend to lose validity, and new methodologies for studying flow and transport in heterogeneous porous media are needed. To better understand and quantify the effects of heterogeneity and model macroscopic physics in porous media, we have developed the Method of Finite Averages (MoFA). This rigorous upscaling methodology can accommodate general heterogeneous porous media, a wide range of physical regimes, and nonlocal effects (i.e., system-scale boundary effects). So far, we have demonstrated the advantages of MoFA in modeling flow and advective-diffusive transport in heterogeneous porous media. In our future work, we plan to continue advancing, applying, and studying MoFA to better understand how the effects of heterogeneous porous media can be quantified, characterized, and predicted.

In traditional upscaling techniques, differential equations describing a system's fine-scale dynamics (e.g., equations describing pore-scale physics) are used to systematically derive upscaled models, which predict the system's coarse-scale dynamics (e.g., equations for the macroscopic physics in a porous medium) with known modeling error thresholds. The derivations associated with these techniques are often time-consuming and become quickly intractable for practical systems (e.g., multi-physical systems with many equations). To make upscaling feasible for practical utilization, we developed a symbolic-computational framework called Symbolica for automating the implementation of upscaling techniques. In other words, Symbolica accepts differential equations describing the pore-scale physics in a porous medium and automatically derives the corresponding upscaled model in seconds. By automating and accelerating the upscaling procedure, Symbolica democratizes rigorous multiscale modeling techniques for broad utilization and real-world applications.

Traditional upscaling techniques, such as Homogenization Theory and the Method of Volume Averaging, largely provide the foundation for our work in multiscale modeling and porous media theory. In addition to deriving new methodologies from such techniques (e.g., the Method of Finite Averages) and developing automated upscaling frameworks (e.g., Symbolica), we are broadly interested in advancing upscaling theory for multiscale model discovery, and applying our advancements to Earth and energy systems for engineering analysis and design. For example, our previous work extended the applicability of Homogenization Theory to more reactive physical regimes in advection-diffusion-reaction systems. Under this extension, we discovered novel terms in our upscaled models describing emergent phenomena. We then applied the mathematical strategy behind our extension to model heat transfer in electric vehicle battery packs under practical thermal runaway conditions that challenged previous models. Oftentimes, our work in advancing upscaling theory is integrated into Symbolica, our automated upscaling framework, so that it can be applied to diverse problems in an expedited and democratized fashion.

A variety of mesoscale phenomena in Earth and energy systems exhibit dynamic interfaces and evolving structures (e.g., multi-phase flows, precipitation and dissolution, crack formation and propagation, phase transitions, and grain growth). Such complexities can be modeled using a combination of phase-field models (PFMs) and spectral numerical methods (SNMs), as the computational expense of solving PFMs is offset by the computational efficiency of SNMs. However, applying PFMs and SNMs simultaneously can be challenging: while PFMs are often highly nonlinear, SNMs require additional analytical preparations and numerical-solving algorithms, particularly when handling nonlinearities. To facilitate combined implementations of PFMs and SNMs, we have developed Fouriera, a symbolic-numerical computing framework that fully automates the implementation of PFMs with SNMs. In other words, given a phase-field model in symbolic form, Fouriera automatically completes the analytical preparations required to solve the model via the Fourier spectral method, and solves the model. In our future work, we hope to use Fouriera to study how the aforementioned mesoscale phenomena influence macroscale behaviors and effective material properties. This work has been performed in collaboration with Bo Wang, who is currently at The Pennsylvania State University.

We are currently setting up a microfluidics laboratory to study the multiscale behaviors of flow, reactive transport, and heat transfer through porous media. Our capabilities will include PDMS microfluidic chip fabrication, 3D-printing model porous media, performing tracer-tracking and micro-PIV experiments, and performing thermal imaging of model porous media. We anticipate focusing our experimental efforts on 1.) discovering novel flow, reactive transport, and heat transfer phenomena that inspire multiscale model and methodology development, and 2.) calibrating and validating the multiscale models and methodologies developed in our group. If you are interested in combining theory with experiments, and have experience in microfluidics or the aforementioned capabilities, we encourage you to get in touch with Prof. Pietrzyk via email (see Contact tab).