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applications/GeoMechanicsApplication/custom_elements/README.md
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| # Transient Thermal Element | ||
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| ## Introduction | ||
| The calculation of geothermal processes relies on a profound understanding of heat transport mechanisms within the Earth's subsurface. Central to this understanding is the heat transport equation. In this context, the heat transport equation becomes a linchpin for designing sustainable and efficient geothermal systems. | ||
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| ## Governing Equations | ||
| The primary governing equation for heat transport in geothermal applications is the heat convection-conduction equation, which describes how heat flows through a porous medium. | ||
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| $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = -\rho^w c^w q_i \frac{\partial T}{\partial x_i} + \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega $$ | ||
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| where, | ||
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| - $c^w$ = specific heat capacity liquid phase $\mathrm{[J/kg ^{\circ}C]}$ | ||
| - $c^s$ = specific heat capacity solid phase $\mathrm{[J/kg ^{\circ}C]}$ | ||
| - $D_{ij}$ = hydrodynamic thermal dispersion $\mathrm{[W/m ^{\circ}C]}$ | ||
| - $T$ = temperature $\mathrm{[ ^{\circ}C]}$ | ||
| - $\rho^s$ = density solid phase $\mathrm{[kg/m^3]}$ | ||
| - $\rho^w$ = water density $\mathrm{[kg/m^3]}$ | ||
| - $q$ = specific discharge $\mathrm{[m/s]}$ | ||
| - $S$ = degree of saturation $\mathrm{[-]}$ | ||
| - $n$ = porosity $\mathrm{[-]}$ | ||
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| The hydrodynamic thermal dispersion is defined as: | ||
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| $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho^w \left( (\alpha_l - \alpha_t) \frac{q_i q_j}{q} + \delta_{ij} \alpha_t q \right) $$ | ||
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| where | ||
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| - $\alpha_l$ = longitudinal dispersivity $\mathrm{[m]}$ | ||
| - $\alpha_t$ = transverse dispersivity $\mathrm{[m]}$ | ||
| - $\delta_{ij}$ = Kronecker delta $\mathrm{[-]}$ | ||
| - $\lambda^w$ = thermal conductivity water $\mathrm{[W/m ^{\circ}C]}$ | ||
| - $\lambda^s$ = thermal conductivity solid matrix $\mathrm{[W/m ^{\circ}C]}$ | ||
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| In the absence of ground water flow, these equations are simplified to, | ||
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| $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega $$ | ||
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| $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s $$ | ||
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| ## Boundary Conditions | ||
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| ### Dirichlet Boundary Condition | ||
| $$ T = \overline T \qquad \text{on} \quad \Gamma_{1}^T $$ | ||
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| where $\overline T$ $\mathrm{\left[ ^{\circ}C \right]}$ is a prescribed temperature | ||
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| ### Neumann Boundary Condition | ||
| $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ | ||
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| where $\overline f$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed conductive heat flux. | ||
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| ### Robin Boundary Condition | ||
| $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T $$ | ||
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| where $\overline g$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed convective-conductive heat flux and $q_n$ is the specific discharge $\mathrm{\left[ m/s \right]}$ at the boundary. | ||
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| ## Derived Properties | ||
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| The density of the porous media $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calculated as, | ||
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| $$ \rho = n S \rho^w + \left( 1 - n \right) \rho^s $$ | ||
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| And the heat capacity of the porous media $C$ $\mathrm{\left[ J/m^{3 \circ}C \right]}$ is: | ||
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| $$ C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s $$ | ||
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| The thermal conductivity of the porous media $\lambda$ $\mathrm{\left[ W/m ^{\circ}C \right]}$ | ||
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| $$ \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s $$ | ||
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| dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temperature is given by: | ||
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| $$ \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} $$ | ||
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| density of water $\rho^w$ $[\mathrm {kg/m^3}]$ is a function of temperature and Diersch (2014) proposes a sixth order Taylor expansion which is approximated here by: | ||
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| $$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \cdot T - 8.993699 \cdot 10^{-3} \cdot T^2 + 9.143518 \cdot 10^{-5} \cdot T^3 - 8.907391 \cdot 10^{-7} \cdot T^4 + 5.291959 \cdot 10^{-9} \cdot T^5 - 1.359813 \cdot 10^{-11} \cdot T^6 $$ | ||
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| Note: the dependency of water density and viscosity to temperature is user defined and the user can turn this feature on or off in the JSON file. | ||
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| ## Finite Element Formulation | ||
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| Kratos solves the equations based on an incremental method. In the frame of Generalized Newmark method (GN11), the fully implicit incremental temperature formulation read as, | ||
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| $$ \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = \left( \frac{1}{\theta} - 1 \right) \boldsymbol{S} \frac{dT^n}{dt} - \left(\boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{T}^{n} + \left( \boldsymbol{V} + \boldsymbol{W}^r \right) $$ | ||
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| with $\theta = 1$ this formulation leads to backward Euler formulation. It reads, | ||
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| $$ \left(\frac{1}{\Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = - \left(\boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{T}^{n} + \left( \boldsymbol{V} + \boldsymbol{W}^r \right) $$ | ||
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| Note: the user can choose either Newmark or backward Euler from the JSON file. | ||
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| ***Compressibility matrix*** | ||
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| $$ \boldsymbol{S} = \int_{\Omega^e} \left( n S \rho^w c^w + \left(1-n\right) \rho^s c^s \right)^{n+1} \boldsymbol{N}^T \boldsymbol{N} d \Omega $$ | ||
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| ***Convectivity matrix*** | ||
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| $$ \boldsymbol{A} = \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega $$ | ||
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| ***Conductivity matrix*** | ||
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| $$ \boldsymbol{H} = \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ | ||
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| ***Neumann condition (conductive boundary)*** | ||
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| $$ \boldsymbol{V} = \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ | ||
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| ***Robin condition (convective boundary)*** | ||
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| $$ \boldsymbol{W^r} = \int_{\Gamma_3^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ | ||
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| $$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ | ||
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| where | ||
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| - $\Delta t$ = time step $\mathrm{\left[s \right]}$ | ||
| - $\Delta T$ = temperature increment $T^{n+1} - T^n$ $\mathrm{\left[^\circ C \right]}$ | ||
| - $\boldsymbol N$ = shape function array $\mathrm{\left[ - \right]}$ | ||
| - $\theta$ = a coefficient in Newmark time integration $\mathrm{\left[ - \right]}$ | ||
| - $\Omega$ = domain region | ||
| - $\Gamma$ = boundary region | ||
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| The superscripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and right hand side (vector), respectively. The superscripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicate values in the element volume and perpendicular to element boundaries, respectively. | ||
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| ## Bibliography | ||
| Diersch, H.-J. G., 2014. FEFLOW; Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer. |
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