We present a network flow model to compute transport, through a pore network, of a compositional fluid consisting of water with a dissolved hydrocarbon gas. The model captures single-phase flow (below local bubble point conditions) as well as the genesis and migration of the gas phase when bubble point conditions are achieved locally. Constant temperature computational tests were run on simulated 2D and 3D micro-networks near bubble point pressure conditions. In the 2D simulations which employed a homogeneous network, negligible capillary pressure, and linear relative permeability relations, the observed concentration of CO2 dissolved in the liquid phase throughout the medium was linearly related to the liquid pressure. In the case of no gravity, the saturation of the gas phase throughout the medium was also linearly related to the liquid pressure; under gravity, the relationship became nonlinear in regions where buoyancy forces were significant. The 3Dheterogeneous network model had non-negligible capillary pressure and nonlinear relative permeability functions. While 100 % of the CO2 entered the 3D network dissolved in the liquid phase, 25 % of the void space was occupied by gas phase and 47 % of the CO2 exiting the outlet face did so via the gaseous phase after 500 s of simulation time.

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The study of the present paper is inspired by Gangl, Kaneko and Zagier's result of the connection with double zeta values and modular forms. We introduce double Eisenstein series $E_{r,s}$ in positive characteristic with double zeta values $\zeta_A(r,s)$ as their constant term and compute the t-expansions of the double Eisenstein series. Moreover, we derive the shuffle relations of double Eisenstein series which match the shuffle relations of double zeta values in \cite{Huei2015}.

We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the Möbius band, and the products of$R$with the total spaces of flat vector bundles over closed infranilmanifolds.

For each closed, positive (1,1)-current$ω$on a complex manifold$X$and each$ω$-upper semicontinuous function$φ$on$X$we associate a disc functional and prove that its envelope is equal to the supremum of all$ω$-plurisubharmonic functions dominated by$φ$. This is done by reducing to the case where$ω$has a global potential. Then the result follows from Poletsky’s theorem, which is the special case$ω$=0. Applications of this result include a formula for the relative extremal function of an open set in$X$and, in some cases, a description of the$ω$-polynomial hull of a set.