Exercise 3

Warning

This section is under construction!

We will now try to reproduce the results presented by [Driesner, 2010b], the case files Driesner2010.

The magma heat source is simulated by a heat flux boundary condition, which can be set by customized boundary condition type of hydrothermalHeatFlux (see doc ). The model in [Driesner, 2010b] is 1 m thick, 3 km wide, 1 km height, the heat source is simulated by a Gaussian-shaped heat flux profile with total heat input 86 km, half-width 500 m and center 1500 m.

Let’s see how to set this boundary condition in HydrothermalFoam. hydrothermalHeatFlux supports Gaussian-shape (shape gaussian2d;) distribution and the shape is defined by four parameters x0, qmax, qmin, c, the equation of the shape is

(26)\[q_h(x) = q_{min} + (q_{max}-q_{min})e^{-\frac{(x-x_0)^2}{2c^2}}\]

for a normal Gaussian-shape profile in this model, qmin is set to zero and x0 is set to 1500. From equation equation (26), we can get the half-width and total heat flux (approximately) are \(c\sqrt{2ln2}\) and \((q_{max}-q_{min})\sqrt{2\pi}c\), respectively. According to the model setup, it’s easy to get \(c = \frac{500}{\sqrt{2ln2}} = 424.661\) and \(q_{max} = \frac{86}{c\sqrt{2\pi}} = 0.0808\ kw/m^2\). There the temperature boundary condition at the bottom patch can be set as,

Code snippet

Listing 19 Temperature boundary condition of bottom patch

bottom
{
    type        hydrothermalHeatFlux;
    q           uniform 0.05; //placeholder
    value       uniform 0; //placeholder
    shape       gaussian2d;
    x0          1500;
    qmax        80.8;
    qmin        0;
    c           424.661;
}

Heat flux profile

(Source code, svg, pdf)

../../_images/ventT_time_perm.jpg — Fig. 25 Maximum vent temperature changes with time.

Fig. 26 Vent temperature as a function of permeability.

Jupyter Notebook of case results