# Upscaling Theories and Predictive Multiscale Formulations

Developing predictive models for multiscale multiphysics systems remains an open challenge in science and engineering. One of the core difficulties to effectively and predictively model complex multiscale multiphysics systems is the derivation of the mathematical models necessary to track the spatiotemporal dynamics of systems exhibiting a large range of scales, spanning many orders of magnitude both in space and time.

Upscaling (or coarse-graining) theories encompass a broad set of mathematical techniques ubiquitously used across many fields of computational physics. Such tools are employed to rigorously build model representations (in form of PDEs or ODEs) of the average response of multiscale physical systems from known dynamics at a fine-scale. One typical example is the use of such theories to derive effective (macroscopic, continuum) scale models of flow and transport in porous media from pore-scale equations. Typical applications may include reactive transport in geologic media, electrochemical transport in battery electrodes, heat transfer in battery packs, flow through membranes, etc. The use of effective (coarse-grained) models to represent the average behavior of a physical system over relevant spatial and temporal scales is critical in both engineering and environmental systems, where predictions, optimization and design are generally needed at scales much larger than the scales at which processes are well understood. Although coarse-graining methods provide a rigorous framework to mathematically traverse multiple scales (by iterative deployment of the technique between adjacent scales), their use is still generally limited to academic settings.

Our group is interested in (i) democratizing the use of such techniques to systems with realistic complexities and (ii) in advancing the theoretical development of fully predictive multiscale models (with minimum to no fitting parameters) within self-consistent coarse-graining frameworks, where upscaling errors can be a priori estimated and model performance (i.e. predictivity) can be consistently defined within the framework. We aim at developing a multiscale framework where model *develop*ment (bottom-up) and model *deploy*ment (top-down) are fully integrated, and model selection is guided by the tradeoff between predictivity and computational cost (Figure 1).

**Figure 1**: A sketch of the workflow that connects (1) rigorous model development based on upscaling methods (e.g., volume averaging, thermodynamically constrained averaging, homogenization, etc.) to (3) optimal model deployment at different scales through (2) diagnosis criteria (e.g., applicability conditions), which guide algorithmic refinement strategies within the validity of the different upscaling approximations being used. [Reproduced from Pietrzyk et al., TIPM (2021)].

We work primarily with homogenization theory, although similar approaches can be used with other upscaling methods. Our work in these areas include, but is not limited to:

- Elucidating the impact of fine-scale dynamic regimes on the validity of coarse-grained models;
- Performing closed-cycle validation of upscaled models through a priori error estimates;
- Developing numerical schemes that honor the mathematical requirements of upscaling theories;
- Advancing/generalizing the analytical treatment of closure formulations;
- Developing computer algebra algorithms which automate upscaling theories, speed up model generation and allow one to handle problems with realistic complexities.

Some relevant references are listed below.

## References

**Pietrzyk, K., Battiato, I.**, “Automated Symbolic Upscaling: Model Generation for Extended Applicability Regimes, Part 1”, Submission code: 2022WR033600, Water Resour. Res. (2022)

**Pietrzyk, K., Battiato, I.**, “Automated Symbolic Upscaling: Model Generation for Extended Applicability Regimes, Part 2”, Submission code: 2022WR033602, Water Resour. Res. (2022).

**K. Pietrzyk, S. Korneev, M. Behandish, I. Battiato**, Upscaling and Automation: Pushing the Boundaries of Multiscale Modeling through Symbolic Computing, *Transp Porous Med., 10.1007/s11242-021-01628-9 (2021).*

**D. Picchi, Battiato, I.**, ‘Scaling of water-steam relative permeability and thermal fluxes in geothermal reservoirs’, *Volume 129, 103257, Int. J. Multiphase Flow (2020).*

**B. Ling, Battiato, I.**, ‘*τ *-SIMPLE algorithm for the closure problem in homogenization of Stokes flows’ *144, 103712, Adv. Water Resour. (2020)*

**D. Picchi, Battiato, I.**, ‘Relative permeability scaling from pore-scale flow regimes’, *Water Resour. Res., 55 (4), pp. 3215-3233 (2019).*

**Battiato, I., D. O’ Malley, C. T. Miller, P. S. Takhar, F. Valdes-Parada, B. D. Wood**, ‘Theory and applications of macroscopic models in porous media’, *Transp. Porous Med., 13 (1) pp. 1-72, (2019).*

**D. Picchi, Battiato, I.** ‘The impact of pore-scale flow regimes on upscaling of immiscible two-phase flow in porous media’, *Water Resour. Res. 54 (9), pp. 6683-6707, (2018).*

**S. Korneev, Battiato, I.**, ‘Sequential homogenization of reactive transport in poly- disperse porous media’, *SIAM Multiscale Model. Sim., 14, 4, pp. 1301-1318 (2016).*

**Boso, F., Battiato, I.**, ‘Homogenizability conditions of multicomponent reactive transport processes’, *Adv. Water Resour., 62, pp. 254-265, (2013).*

**Battiato, I., Tartakovsky, D. M.**, ‘Applicability regimes for macroscopic models of reactive transport in porous media’. *J. Contam. Hydrol., 120-121, pp.18–26, (2011).*

**Battiato, I., Tartakovsky, D. M., Tartakovsky, A. M., Scheibe T. D.**, ‘On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media’. *Adv. Water Resour., 32, 11, pp.1664–1673, (2009).*