Subdiffusive flow in a composite medium with a communicating (absorbing) interface

Abstract

Two-dimensional subdiffusion in media separated by a partially communicating interface is considered. Starting with the appropriate Green’s functions, solutions are developed in terms of the Laplace transformation reflecting two circumstances at the interface: situations where there is perfect contact and situations where the interface offers a resistance. Asymptotic solutions are derived; limiting forms of the expressions reduce to known solutions for both classical diffusion and subdiffusion. Specifics are analyzed in depth with reference to flow in porous media with potential applications to the evaluation of the role of subsurface faults and flow in fractured rocks. Characteristics of the derivative responses are documented extensively as they are the linchpin for evaluation of pressure tests. Results given here may be used for evaluation at the Theis (1935; Eos Trans. AGU 2, 519–524) scale along with geological and geophysical properties, and production statistics. Yet a subdiffusive model does not imply a single value for properties. The method presented here may be extended to multiple contiguous media and to subdiffusive transport in many contexts (complex wellbores such as inclined, fractured and horizontal wells, situations such as sequestration, production of geothermal systems, etc.).

Time‐Fractional Flow Equations (t‐FFEs) to Upscale Transient Groundwater Flow Characterized by Temporally Non‐Darcian Flow Due to Medium Heterogeneity

“…Water flux with the delayed transfer of water pressure due to low‐ K islands violates the standard Darcy's law‐based flow equation (with constant parameters) which is a classical (Fickian) diffusion equation. This mechanism is more like a sub‐diffusive transfer of water pressure, similar to the concept of “sub‐diffusive flow in a composite medium” proposed first by Raghavan and Chen (2020), which is opposite to the super‐diffusive flow defined by the s‐FFE ().…”

An application of a multiindex, time-fractional differential equation to evaluate heterogeneous, fractured rocks

“…Subdiffusive flow is represented through a fractional derivative defined by Caputo (1967) that may be deduced through the fractional engine discussed in Metzler and Klafter (2000). Detailed and elaborate citations that include theoretical considerations and experimental observations of the fractional flux law may be found in Raghavan and Chen (2020). This flux law is also particularly suitable for considering problems in 2D such as those considered in Beier (1994).…”