Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

We study the combined effect of spatial heterogeneity and temporal pressure fluctuations on the dispersion of the saturation of a displacing fluid during injection into an immiscible one. In a stochastic modeling framework, we define two different dispersion quantities, one which measures the front uncertainty due to temporal pressure fluctuations and another one which quantifies the actual spreading and dispersion of the saturation front in a typical medium realization. We derive an effective large‐scale flow equation for the saturation of the displacing fluid that is characterized by a nonlocal dispersive flux term. From the latter we derive measures for the spread of the saturation front due to spatiotemporal fluctuations. Our analysis demonstrates that temporal fluctuations enhance the front uncertainty but not the average spreading of the saturation front. The analytical developments are complemented by numerical simulations of the full two‐phase flow problem. Correct assessment of the spread of the saturation front is of importance to several applications, including the assessment of the sequestration potential of a carbon dioxide storage site, for example. Spreading enhances the surface area between the fluids, which in turn enhances the dissolution and entrapment efficiency. The latter facilitates reactions and thus the sequestration of CO 2 in stable forms. Our study provides a theoretical basis for the design of injection strategies to optimize dispersion and to minimize uncertainty of the saturation front.