Porous Flow Tutorial Page 03. Adding heat advection and conduction
This Page adds heat conduction and advection with the fluid. The differential equation governing the temperature evolution is (1)
This equation is nonlinear since there are products of and , as well as the nonlinear function . Most of the nomenclature was used in Page 01, and the additional symbols introduced are:
is time (units s)
is the density of the rock grains (units kg.m)
is the specific heat capacity of the rock grains (units J.kg.K)
is the temperature (units K)
is the specific heat capacity of the fluid (units J.kg.K)
is the thermal conductivity of the rock-fluid system (units J.s.m.K). It is a tensor.
where the parameters , , , , and , have been used in the final expression. In Page 02 the choice Pa.m was made. Choosing K.m yields
Note that this is significantly greater than the for the fluid equation. In the main, MOOSE can handle these types of discrepancies, but it is good practise to scale the variables so that their residuals are of similar magnitude. Therefore, a scaling factor of is applied to the temperature variable.
To model this thermo-hydro system, the
PorousFlowBasicTHM action needs to be enhanced to read:
[Variables] [./porepressure] [../] [./temperature] initial_condition = 293 scaling = 1E-8 [../]  [PorousFlowBasicTHM] porepressure = porepressure temperature = temperature coupling_type = ThermoHydro gravity = '0 0 0' fp = the_simple_fluid 
and some extra properties need to be added to the
[Modules] [./FluidProperties] [./the_simple_fluid] type = SimpleFluidProperties bulk_modulus = 2E9 viscosity = 1.0E-3 density0 = 1000.0 thermal_expansion = 0.0002 cp = 4194 cv = 4186 porepressure_coefficient = 0 [../] [../] 
The boundary conditions used are the same as in Page 01 in addition to specifying a constant injection temperature of 313K:
[./constant_injection_temperature] type = PresetBC variable = temperature value = 313 boundary = injection_area [../]
Finally, some temperature-related
Materials need to be defined
[./thermal_expansion] type = PorousFlowConstantThermalExpansionCoefficient biot_coefficient = 0.8 drained_coefficient = 0.003 fluid_coefficient = 0.0002 [../] [./rock_internal_energy] type = PorousFlowMatrixInternalEnergy density = 2500.0 specific_heat_capacity = 1200.0 [../] [./thermal_conductivity] type = PorousFlowThermalConductivityIdeal dry_thermal_conductivity = '10 0 0 0 10 0 0 0 10' block = 'caps aquifer' [../]
An animation of the results is shown in Figure 1. Readers are encouraged pause and explore the effect of changing parameters such as the rock thermal conductivity.