The law of the conservation of energy can be summarised using the two variables:

 

c_w = specific heat capacity of the fluid [J/(kg K)]

c_s = specific heat capacity of the matrix [J/(kg K)]

 

can be mathematically formulated as follows, taking into account the pore space n:

 

The right-hand side of the equation represents the sum of all energy sources and sinks and T* the temperature of inflowing and outflowing rates.

The heat flows for convection, dispersion and heat conduction are used in this equation. Using the product rule for the convection term and the continuity condition (see mass transport):

 

results in the transient energy transport equation:

 

The heat diffusion parameters [m²/s] and [m²/s] are determined by:

 

and

 

With:

ρ_w,s = density of the fluid (w) or matrix (s) [kg/m³]

c_w,s = specific heat capacity [Ws /(kg K)]

λ_w,s = thermal conductivity [W / (m K)]

σ_w,s = Diffusion parameter = λ_w,s / (ρ_w,s c_w,s) = [m²/s]

I = Unit matrix [-]

D = Dispersion tensor [m²/s]

Sr = Degree of saturation [-]

n = flow effective pore space [-]

v = distance velocity [m/s]

q = div v (from continuity condition) [1/s]

T = temperature [K]

= temperature gradient [K/m]

T* = variable temperature inflow/outflow [K]

 

Like the heat production rate (QKON), the energy transport equation has the unit [W/m³] = [J/(m³s)].

 

Boundary and initial conditions