Membrane performance in gas separation is quantified by its selectivity, determined as a ratio of measured gas permeabilities of given gases at fixed pressure difference. In this manuscript a nonlinear dependence of gas permeability on pressure difference observed in the measurements of gas permeability of graphene oxide membrane on a manometric integral permeameter is reported. We show that after reasoned assumptions and simplifications in the mathematical description of the experiment, only static properties of any proposed governing equation can be studied, in order to analyze the permeation rate for different pressure differences. Porous Medium Equation is proposed as a suitable governing equation for the gas permeation, as it manages to predict a nonlinear behavior which is consistent with the measured data. A coefficient responsible for the nonlinearity, the polytropic exponent, is determined to be gas-specific—implications on selectivity are discussed, alongside possible hints to a deeper physical interpretation of its actual value.

The unit step response is a standard tool for experimental identification; its shape is equivalent to a solution of an appropriate governing equation. In this study, we investigate two variants of nonlinear first-order governing equation, $${y'=k(1-y^\beta )/\beta }$$y ′ = k ( 1 - y β ) / β , and mainly its rescaled version $${y'=k\left( 1-y^\beta \right) }$$y ′ = k 1 - y β . We explore a continuum of response shapes, ranging from constant zero to constant one, including standard shapes such as $$1-\exp (-t)$$1 - exp ( - t ) and $${\tanh (t)}$$tanh ( t ) , but also uncommon shapes such as a part of the logarithmic integral. Two methods for estimating the parameter $$\beta $$β are proposed: one based on analyzing the curvature of a single response and the other based on analyzing initial slopes for multiple responses, while varying the amplitude of the step. Our investigation is conducted in the context of gas permeation through a homogeneous porous medium, where nonlinear response shapes have been observed. This context allows for a meaningful interpretation of the results, consistent with expectations for specific thermodynamic processes. However, neither method is constrained by this context, and either might be considered for any first-order system that fails to have amplitude-scale invariance or for which the shapes seem to exhibit an effect of saturation.