### 1. Introduction

### 2. Materials and Methods

### 2.1. Transport Experiments

_{6}) previously presented in [4]. We used methane gas (99.99%) to mix with nitrogen gas (99.99%) to form a light gaseous mixture in the IC.

^{3}. Ten data sets were obtained for the vertically upward transport experiment and were grouped into two: the first to the fourth set with ${\rho}_{m}^{o}$ in a range of 0.0109 to 0.0118 kg/m

^{3}and the fifth to the tenth set with ${\rho}_{m}^{o}$ in a range of 0.0451 to 0.0523 kg/m

^{3}.

### 2.2. Mathematical Models

*i*(

*= 1,…, n-1) where n is the number of gas components and total molar concentration. These component and total molar balance equations for vertically one-dimensional system with a rigid porous matrix, single gas flow and no reactions and external sources/sinks are, respectively, written as [11]:*

_{i}*θ*

*is gas-filled porosity;*

_{a}*C*

*and $C(={\displaystyle \sum _{i}}{C}_{i})$ are the molar concentration of gas component*

_{i}*i*and total molar concentration, respectively (

*moleL*

*);*

^{−3}*t*is time coordinate (

*t*) ;

*z*is vertical coordinate which is directed downward (

*L*); ${N}_{i}^{T}$ and ${N}^{T}(={\displaystyle \sum _{i}}{N}_{i}^{T})$ represent the total molar flux of gas component

*i*and total molar flux, respectively (

*molL*

^{−2}*t*

*). ${N}_{i}^{T}$ is the sum of total diffusive, ${N}_{i}^{D}$, and viscous fluxes, ${N}_{i}^{\nu}$, of gas component*

^{−1}*i*.

*N*

*is the sum of total diffusive, ${N}^{\text{D}}(={\displaystyle \sum _{i}}{N}_{i}^{\text{D}})$ and viscous flux, ${N}^{\nu}(={\displaystyle \sum _{i}}{N}_{i}^{\nu})$. For a two-component DGM equations, ${N}_{i}^{D}$ can be expressed as*

^{T}##### (3)

$$\begin{array}{c}{N}_{i}^{D}=-\frac{{D}_{ij}^{e}{D}_{i}^{K}C}{{D}_{ij}^{e}+{D}_{i}^{K}(1-{a}_{i}{X}_{i})}\frac{\partial {X}_{i}}{\partial z}-\frac{{D}_{i}^{K}({D}_{ij}^{e}+{D}_{i}^{K}){X}_{i}}{{D}_{ij}^{e}+{D}_{i}^{K}(1-{a}_{i}{X}_{i})}\frac{\partial C}{\partial z}-\\ \frac{{D}_{i}^{K}({D}_{ij}^{e}{M}_{i}+{D}_{i}^{K}M){X}_{i}}{{D}_{ij}^{e}+{D}_{i}^{K}(1-{a}_{i}{X}_{i})}\frac{Cg}{RT}\end{array}$$*X*

*is the mole fraction of gas component*

_{i}*i*;

*R*is the universal gas constant (

*ML*

^{2}*t*

^{−2}*T*

^{−1}*mol*

*);*

^{−1}*T*is the absolute temperature (°

*K*);

*g*is the gravitational constant (

*Lt*

*); ${D}_{ij}^{e}$ and ${D}_{i}^{K}$ are the "effective" binary molecular diffusion coefficient and the Knudsen diffusion coefficient, respectively (*

^{−2}*L*

^{2}*t*

*). Note, ${D}_{ij}^{e}={Q}_{m}{D}_{ij}^{o}$, in which ${D}_{ij}^{o}$ is air filled binary molecular diffusion coefficient for gas pair*

^{−1}*i*and

*j*and

*Q*

*accounting for the influence of medium tortuosity is the obstruction factor defined in the DGM.*

_{m}*a*

_{1}= 1 − (

*M*

_{1}/

*M*

_{2})

^{1/2};

*M*=

*X*

_{1}

*M*

_{1}+

*X*

_{2}

*M*

_{2}, where

*M*

*is molar mass of gas component*

_{i}*i*. Note that, the third term in Eq. (2) is deleted for horizontal transport.

##### (4)

$${N}_{i}^{\nu}=-\frac{{k}_{e}}{{\mu}_{m}}{X}_{i}RTC\frac{\partial C}{\partial z}+\frac{{k}_{e}}{{\mu}_{m}}{X}_{i}{C}^{2}Mg$$*k*

*is the effective gas permeability of the medium*

_{e}*(L*

^{2}*)*;

*μ*

*is the dynamic viscosity of the gas mixture*

_{m}*(ML*

^{−1}*t*

*).*

^{−1}*ML*

*);*

^{−3}*q*is Darcy velocity and implicitly representative of the mass average velocity in the total mass balance equation. That is to say, the total mass flux in the second term on the left side of Eq. (5) includes viscous but excludes total diffusive part as compared to Eq. (2).

*i*and is usually related to ${D}_{ij}^{o}$ by a tortuosity factor (τ) with ${D}_{i}^{*}=\tau {\theta}_{a}{D}_{ij}^{o}$.

### 2.3. Model Parameters and Implementation

_{2}and ${\text{CH}}_{4}\hspace{0.17em}({D}_{{CH}_{4}-{N}_{2}}^{o})$ with the obstruction factor (

*Q*

*). The effective diffusion coefficient of ${\text{CH}}_{4}\hspace{0.17em}({D}_{{CH}_{4}}^{*})$ for the Fickian-type diffusion is approximated to be in magnitude the same as the ${D}_{{CH}_{4}-{N}_{2}}^{e}$ value, since ${D}_{{N}_{2}}^{K},\hspace{0.17em}{D}_{{CH}_{2}}^{K}$ are greater than ${D}_{{CH}_{4}-{N}_{2}}^{e}$ for more than two orders of magnitude (see Table 2 and [2]). It is not necessary to consider sorption of methane gas onto dry sea sand [3, 7]. However, the sorption can be arisen during the transport experiment due to the acrylic material that the soil column was made of [4]. Thus, a retardation factor was introduced into the models as a model calibration parameter for a better fit between the predicted methane gas density evolution profile and the measured data. As a result, only one parameter, the retardation factor, was calibrated in the modeling work.*

_{m}### 3. Results and Discussions

### 3.1. Variations of Pressure Differences

### 3.2. Methane Density Evolution

^{3}. This difference reflects variability of the source condition during the transport experiment and needs to be considered as a boundary condition in model simulation. At a higher methane source density (Fig. 4(b)), however, the methane density measured at the 0-cm point (and at the IC) during the experiment is not deviated from the averaged ${\rho}_{m}^{o}$ a lot for vertically upward transport. Besides, the methane densities measured at the OC were all very close to zero, except for the results obtained from the upward transport experiment at high methane source densities (see Fig. 4(b)).

### 4. Conclusions

^{3}, as the system responded with small positive pressure variations to methane transport and reached steady state in two hours. Besides, both models predicted similar methane density profiles for upward transport of methane from a pure methane source. It implies gravitation-induced viscous flux component dominates over the diffusive one under this scenario and a Fickian-based model is adequate for predicting upward transport of light gas in porous systems. For horizontal transport at a low source density DGPT and MISER over predicted the methane densities. We suspect it is due to incomplete mixing of gas mixture in the IC during the experiment since high pressure variations were observed in the horizontal transport experiments. This study further showed that the methane density profile predicted by DGPT is over the MISER result for at most 15% of difference in methane densities for horizontal transport of methane from a pure methane source. This difference is arisen from different diffusive components predicted by these models and implies that a Fickian-based model may under predict methane density for horizontal transport of a methane plume and the degree of under predictions increases with the increase of the methane source density.