Thermal and Compositional Plumes in

Non-Reactive Porous Media

 

 

A numerical model was used to predict the movement of fluids of varying temperatures and concentrations, injected into a fluid saturated box of uniform and constant porosity and permeability.

 

 

 

Some important equations that are solved numerically by the model:

 

 

The stream function is obtained by solving this Poisson type equation:

 

                                                  , where  , ,      

                                                                                                 

 

 

 

 

Conservation of enthalpy equation is solved in order to update the temperature field with each time step:

 

                                             , where  ,  ,                                                                                                                                                                                                                                ,

                                                                                                 ,                                                                                                                                                                                                                                                             

 

 

 

Conservation of concentration equation is solved in order to update the concentration field with each time step:

 

                                 , where  ,                                                                                                                                                                                                                                                                      

 

            In this model,  was set at 0.60 and ε was set to 0.1.

 

 

 

The boundary conditions of the box do not allow for heat or mass flux through any of the sides.  The fluid initially saturating the medium had a dimensionless temperature of 0.0 and a dimensionless concentration of 0.0.  By varying the temperature and concentration of the injected fluid, the effects of density can be observed.  The injection of fluid is simulated by setting different temperature and concentration values on a small portion of the lower boundary. 

As the temperature is advected away from the source, both the matrix and the pore fluid are heated.  As the concentration is  advected away from the source, it moves through the pore spaces only and does not interact with the matrix.  Therefore, the concentration moves more quickly through the medium than does the temperature.

 

 

 

 

Case 1:  The concentration on the small section of the lower boundary was set to a negative value and the temperature was set so that it would not affect the buoyancy.  This simulates the injection of a fluid with a lighter concentration and the same temperature as the fluid already in the box.  The Rayleigh number was set to 10000.  The result is a plume of fluid that rises quickly to the top of the box with little lateral spreading.   

 

 A.  Concentration field                                                                                      B.  Temperature field 

           

 

Movies: Concentration, Temperature and Stream Function,   Stream Function

 

 

 

                                                                       

 

Case 2:  The temperature on the small section of the lower boundary was set to a positive value and the concentration was set so that it would not affect buoyancy (thus, the concentration acts like a passive tracer of the injected fluid).  This simulates the injection of a fluid with higher temperature and the same concentration as the original pore fluid.  The Rayleigh number was set to 10000.  This results in the temperature field advecting more slowly than the injected fluid, allowing the neutrally buoyant fluid to spread laterally as well as upwards.  The mushroom shape of the fluid plume is due to the heating of the original pore fluid near to the injection site, which then rises and becomes mixed in with the injected fluid.

 

A.  Concentration field                                                                                         B.  Temperature field

           

 

Movies: Concentration, Temperature and Stream Function,   Stream Function

 

                        

 

                    

 

Case 3:  The temperature and concentration on the small section of the lower boundary were both set to positive values.  The temperature is set to be three times more buoyant than the concentration.  This simulates the injection of a fluid with both greater temperature and concentration than the original pore fluid.  The Rayleigh number was set to 5000.  This results in the fluid initially rising upwards due to the high temperature.  As the injected fluid begins to cool further away from the source, the high concentration then creates a density contrast with the original pore fluid, and so the fluid begins to move downwards.  

 

A.  Concentration field                                                                                        B.  Temperature field

           

 

Movies:  Concentration, Temperature and Stream Function,   Stream Function

 

 

 

 

 

Some practical uses:

Knowledge of the movements of fluids of varying temperatures and concentrations through porous media can be used:

1)      To predict the movement of pollutants through aquifers.

2)      To predict the precipitation pattern of economical minerals.

3)      To determine whether the injection of fluid into a hydrocarbon reservoir will enhance recovery.