# Math Modeling Seminar: Effective Models of Flow in Highly Heterogeneous Multiscale Naturally Fractured Porous Media

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Effective Models of Flow in Highly Heterogeneous Multiscale Naturally Fractured Porous Media

Assistant Professor of Applied Mathematics
Department of Mathematics
The University of Alabama

Abstract:
Given the contrast in properties of the multiscale network of fractures and the matrix in naturally fractured reservoirs, several flow regimes may form at microscale. For low contrasts, the conductivities of the matrix and fractures are close enough that the medium behaves as a homogenous medium. For higher contrasts, the matrix will not be conductive enough to be a part of the overall flow process in the reservoir, so it behaves as a source of fluid to fractures. For very high contrasts, the matrix is almost not conductive, consequently, the overall flow will be governed by the flow in the fracture network. Other intermediate cases may also occur.
The macroscopic pressure behavior of multi-scale fractured media is investigated using homogenization technique. This approach captures the details of the flow both within and between the porous matrix and fracture. A general equivalent macroscopic model is proposed. The equivalent porosity and the equivalent permeability of the averaged medium are derived. The memory effect in the medium is included in the averaged model through an integro-differential term. The memory effect represents the difference between the response time of matrix and fracture to the same pressure drop. Depending on the ratio of the fracture to matrix diffusivity, the fracture to matrix volume ratio, and the number of scales of nonhomogeneity in the medium, several memory terms are obtained. The impact of the presence of additional scale of the fracture network, such as secondary and tertiary fractures on the equivalent model, and pressure transient behavior is highlighted. The macroscale model for an arbitrary number of scales and its limit for the infinite hierarchy is obtained. The kernel of the memory operator is the solution of a nonlinear integro-differential equation. Pressure transient behavior of the multi-scale model is compared to classic double-porosity model.

Speaker Bio:
Dr. Mojdeh Rasoulzadeh is an assistant professor in the Department of Mathematics, adjunct faculty in the Department of Mechanical Engineering, and affiliated to the Center for Complex Hydro Systems at The University of Alabama, Tuscaloosa. She received her Ph.D. in Mechanics and Energetics from Lorraine University in France. Before joining the University of Alabama, she was a research engineer and postdoc in the oil and gas industry, Schlumberger and Total France, working on flow and geomechanical investigation of highly heterogeneous formations. She has about ten years of experience in the research focused on obtaining effective models of flow in highly heterogeneous reservoirs such as multiscale fractured and vuggy carbonates. She obtained the closed-form of transient flow behavior and transfer functions for multiscale fractured porous media as an integro-differential equation averaged over all the scales of heterogeneities. Dr. Rasoulzadeh has obtained novel models to investigate the role of meso-scale inhomogeneities such as vugs and cavities on the overall flow and geochemical behavior of the subsurface reservoirs. She is currently conducting research on coupled Hydro-Mechanical-Chemical processes and particulate flow in carbonate reservoirs and karst formations.

Intended Audience:

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The Math Modeling Seminar will recur each week throughout the semester on the same day and time. Find out more about upcoming speakers on the Mathematical Modeling Seminar Series webpage.

Contact
Nathan Cahill
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###### When and Where
April 27, 2021
2:00 pm - 2:50 pm