Advection of Methane in the Hydrate Zone: Model, Analysis and Examples

TL;DR

This paper develops a mathematical model for methane transport in sediments, incorporating hydrate formation and dissolution, and provides analytical solutions and estimates for the complex free-boundary problem involved.

Contribution

It introduces a novel two-phase, two-component model with a rigorous mathematical analysis, including existence proofs and maximum principle estimates, for methane advection in hydrate zones.

Findings

Unique generalized solution in $L^1$ for the model

Maximum principle estimates for phase-change relations

Limitations identified through pure advection example

Abstract

A two-phase two-component model is formulated for the advective-diffusive transport of methane in liquid phase through sediment with the accompanying formation and dissolution of methane hydrate. This free-boundary problem has a unique generalized solution in $L^1$; the proof combines analysis of the stationary semilinear elliptic Dirichlet problem with the nonlinear semigroup theory in Banach space for an m-accretive multi-valued operator. Additional estimates of maximum principle type are obtained, and these permit appropriate maximal extensions of the phase-change relations. An example with pure advection indicates the limitations of these estimates and of the model developed here. We also consider and analyze the coupled pressure equation that determines the advective flux in the transport model.