Abstract
In this work, we study the behavior of saturation fronts for two-phase flow through a long homogeneous porous column. In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses traveling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include nonmonotone profiles, multiple shocks, and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we discuss the implication of our findings in the context of available experimental results.
| Original language | English |
|---|---|
| Pages (from-to) | 449-492 |
| Number of pages | 44 |
| Journal | Studies in Applied Mathematics |
| Volume | 144 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - May 2020 |
Funding
K. Mitra is supported by TU Dortmund University, and was partly supported by the Netherlands Organisation for Scientific Research (NWO) through the CSER programme (project 14CSER016) and by Hasselt University, Belgium through the project BOF17BL04. I.S. Pop is supported by the Research Foundation‐Flanders (FWO), Belgium through the Odysseus programme project G0G1316N and the project G051418N. The work of T. Köppl and R. Helmig is supported by the Cluster of Excellence in Simulation Technology (EXC 310/2) of Stuttgart University. Furthermore, C.J. van Duijn and R. Helmig acknowledge the support of the Darcy Center of Utrecht University and Eindhoven University of Technology and the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), SFB 1313, Project Number 327154368. K. Mitra is supported by TU Dortmund University, and was partly supported by the Netherlands Organisation for Scientific Research (NWO) through the CSER programme (project 14CSER016) and by Hasselt University, Belgium through the project BOF17BL04. I.S. Pop is supported by the Research Foundation-Flanders (FWO), Belgium through the Odysseus programme project G0G1316N and the project G051418N. The work of T. Köppl and R. Helmig is supported by the Cluster of Excellence in Simulation Technology (EXC 310/2) of Stuttgart University. Furthermore, C.J. van Duijn and R. Helmig acknowledge the support of the Darcy Center of Utrecht University and Eindhoven University of Technology and the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), SFB 1313, Project Number 327154368.
Keywords
- two-phase flow
- hysteresis and dynamic capillarity
- Riemann problem
- Travelling waves
- dynamic capillarity
- traveling waves
- hysteresis