1D heat and mass transport

Description

An analytical solution to the problem of 1D coupled heat and mass transport was initially developed by !citet, and later by !citet.

The problem consists of a 1D model where cold water is injected into a warm semi-infinite reservoir at a constant rate. The top and bottom surfaces of the reservoir are bounded by caprock which is neglected in the modelling to simplify the problem. Instead, these boundaries are treated as no-flow and adiabatic boundary conditions.

For the simple case of a 1D Cartesian model bounded on the upper and lower surfaces by no-flow and adiabatic boundaries, a simplified solution for the temperature profile $T(x, t)$ can be obtained !citep

\begin{equation} \frac{T(x, t) - T(x, 0)}{T(0, t) - T(x, 0)} = \frac{1}{2} \left[\mathrm{erfc}\left(\frac{x - vt} {\sqrt{4 D t}}\right) + \exp\left(\frac{vx}{D}\right) \mathrm{erfc}\left(\frac{x+vt} {\sqrt{4 D t}}\right)\right], \label{eq:avdonin} \end{equation}

where $T(x, 0)$ is the initial temperature in the reservoir, $T(0, t)$ is the temperature of the injected water,

\begin{equation} v = \frac{u_w \rho_w cp_w}{\rho_m cp_m}, \end{equation}

and

\begin{equation} D = \frac{\lambda_m}{\rho_m cp_m}, \end{equation}

where $u_w$ is the Darcy velocity of the water, $\rho_w$ is the density of water, $cp_w$ is the specific heat capacity of water, $\rho_m$ is the density of the fully saturated medium ($\rho_m = \phi \rho_w + (1 - \phi \rho_r)$ where $\phi$ is porosity and $\rho_r$ is the density of the dry rock), $cp_m$ is the specific heat capacity of the fully saturated porous medium, and $\lambda_m$ is the thermal conductivity of the fully saturated reservoir.

Model

This problem was considered in a code comparison by !citet, so we use identical parameters in this verification problem, see [tab:res].

!table id=tab:res caption=Model properties

Property	Value
Length	50 m
Pressure	5 MPa
Temperature	170 $^{\circ}$C
Permeability	$1.8 \times 10^{-11}$ m$^2$
Porosity	0.2
Saturate density	2,500 kg m$^{-3}$
Saturated thermal conductivity	25 W m$^{-1}$ K
Saturated specific heat capacity	1,000 J kg$^{-1}$ K

Following !citet, a constant fluid flow through the left boundary is obtained by applying a constant pressure gradient over the model by fixing porepressure at the boundaries. The temperature of the water entering the model is fixed at 160 $^{\circ}$C by fixing enthalpy at the left boundary.

Input file

The input file used to run this problem is

!listing modules/porous_flow/test/tests/fluidstate/coldwater_injection.i

Note that the test file is a reduced version of this problem. To recreate these results, follow the instructions at the top of the input file.

Results

The results for the temperature profile after 13,000 seconds are shown in [fig:avdonin]. Good agreement is shown, however some numerical diffusion is obvserved. Similar results are obtained using TOUGH2 for upstream weighting, see !citet.

!media media/porous_flow/1d_avdonin.png id=fig:avdonin style=width:60%;margin-left:10px; caption=Comparison between !citet result and MOOSE at t = 13,000 s.

!bibtex bibliography