From b7bf093868f59a84714facbd695e877db7f332c3 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 16:32:49 +0100 Subject: [PATCH 01/21] Readme.md for thermal element is added, update 01 --- .../custom_elements/README.md | 170 ++++++++++++++++++ 1 file changed, 170 insertions(+) create mode 100644 applications/GeoMechanicsApplication/custom_elements/README.md diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md new file mode 100644 index 000000000000..af1e0070976c --- /dev/null +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -0,0 +1,170 @@ +# Transient Thermal Element + +## Introduction +The extraction of geothermal energy relies on a profound understanding of heat transport mechanisms within the Earth's subsurface. As heat traverses geological formations, its efficient transfer is governed by mathematical equations encapsulating principles of thermal dynamics. Central to this understanding is the heat transport equation, a fundamental tool that elucidates how thermal energy moves through the Earth's crust, influencing the feasibility and optimization of geothermal applications. In this context, the heat transport equation becomes a linchpin for designing sustainable and efficient geothermal energy systems. + +## 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. + +$$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = -\rho^w c^w q_i \frac{\partial T}{\partial x_i} + \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega \label{eq:t1} $$ + +where, + +\begin{table}[H] + \begin{tabular}{ll} + $c^w$ &specific heat capacity liquid phase $\mathrm{[J/kgC]}$\\ + $c^s$ &specific heat capacity solid phase $\mathrm{[J/kgC]}$\\ + $D_{ij}$ &hydrodynamic thermal dispersion $\mathrm{[W/mC]}$\\ + $T$ &temperature $\mathrm{[C]}$\\ + $\rho^s$ &density solid phase $\mathrm{[kg/m^3]}$ + \end{tabular} +\end{table} + +The hydrodynamic thermal dispersion is defined as: + +$$ + D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho^w \left( (\alpha_l - \alpha_t) \frac{q_i q_j}{q} + \delta_{ij} \alpha_t q \right) + \label{eq:t2} +$$ + +where + +\begin{table}[H] + \begin{tabular}{ll} + $\alpha_l$ &longitudinal dispersivity $\mathrm{[m]}$\\ + $\alpha_t$ &transverse dispersivity $\mathrm{[m]}$\\ + $\delta_{ij}$ &Kronecker delta $\mathrm{[-]}$\\ + $\lambda^w$ &thermal conductivity water $\mathrm{[W/mC]}$\\ + $\lambda^s$ &thermal conductivity solid matrix $\mathrm{[W/mC]}$ + \end{tabular} +\end{table} + +In the absence of ground water flow, equations \ref{eq:t1} is simplified to, + +$$ + \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) + \qquad \text{on} \quad \Omega + \label{eq:t3} +$$ + +$$ + D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + \label{eq:t4} +$$ + +## Boundary Conditions + +### Dirichlet boundary condition +$$ + T = \overline T \qquad \text{on} \quad \Gamma_{1}^T + \label{eq:t5} +$$ + +where $\overline T \left[ C \right]$ is a prescribed temperature + +### Neumann Boundary Condition +$$ + D_{ij} \frac{\partial T}{\partial x_j} n_i = + %\tilde{c}^w \tilde{\rho}^w \overline q_n \tilde{T} \qquad \text{on} \quad \Gamma_{2}^T + \overline{f} \qquad \text{on} \quad \Gamma_{2}^T + \label{eq:t6} +$$ + +where $\overline f \left[ C \right]$ is a prescribed conductive heat flux. + +### Robin bounday condition +$$ + D_{ij} \frac{\partial T}{\partial x_j} n_i = + %c \tilde \rho \overline q_n \left(\tilde T-T\right) \qquad \text{on} \quad \Gamma_{3}^T + \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T + \label{eq:t6} +$$ + +where $\overline g \left[ C \right]$ is a prescribed convective-conductive heat flux. + + +## Derived Properties +% +The density of the bulk material $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calculated as, +% +\begin{equation} + \rho = n S \rho^w + \left( 1 - n \right) \rho^s +\end{equation} +% +And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^3 C \right]}$ is: +% +\begin{equation} + C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s +\end{equation} +% +The thermal conductivity of the bulk material $\lambda$ $\mathrm{\left[ W/mC \right]}$ +% +\begin{equation} + \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s +\end{equation} +% +dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temperature (\cite{Diersch1}) is given by: +\begin{equation} + \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} + \label{eq:t10} +\end{equation} + +density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and \cite{Diersch1} proposes a six order Taylor expansion which is approximated here by: +\begin{multline} + \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \; T - 8.993699 \cdot 10^{-3} \; T^2 + \\ 9.143518 \cdot 10^{-5} \; T^3 - 8.907391 \cdot 10^{-7} \; T^4 + 5.291959 \cdot 10^{-9} \; T^5 - \\ 1.359813 \cdot 10^{-11} \; T^6 + \label{eq:t11} +\end{multline} + + + +## Finite Element Formulation + +Kratos solves the equations based on incremental method. In The fram of Generelized Newmark method (GN11), the fully implicit incremental temperature formulation read as, +% +\begin{multline} + \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) + \label{eq:t15b} +\end{multline} + +Compressibility matrix +% +$$ + \boldsymbol{S} = \int_{\Omega^e} \left( n S \rho^w c^w + \left(1-n\right) \rho^s c^s \right)^{n+1} \boldsymbol{N}^T \boldsymbol{N} d \Omega + \label{eq:t16} +$$ + +Convectivity matrix +% +$$ + \boldsymbol{A} = \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega + \label{eq:t18} +$$ + +Conductivity matrix +% +$$ + \boldsymbol{H} = \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega + \label{eq:t17} +$$ + +Neumann condition (dispersive boundary) +% +$$ + \boldsymbol{V} = \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma + \label{eq:t19} +$$ + +Robin condition (convective boundary) +% +$$ + \boldsymbol{W^r} = \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma + \label{eq:t19a} +$$ +% +$$ + \boldsymbol{W^l} = + \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma + \label{eq:t19b} +$$ \ No newline at end of file From fb5a07c3601d3039cb90390e21ed78779c7ba13e Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 16:38:59 +0100 Subject: [PATCH 02/21] Readme.md for thermal element is added, update 02 --- .../GeoMechanicsApplication/custom_elements/README.md | 9 ++------- 1 file changed, 2 insertions(+), 7 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index af1e0070976c..e46dc2299dea 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -6,7 +6,7 @@ The extraction of geothermal energy relies on a profound understanding of heat t ## 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. -$$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = -\rho^w c^w q_i \frac{\partial T}{\partial x_i} + \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega \label{eq:t1} $$ +$$\left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = -\rho^w c^w q_i \frac{\partial T}{\partial x_i} + \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega$$ where, @@ -22,10 +22,7 @@ where, The hydrodynamic thermal dispersion is defined as: -$$ - D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho^w \left( (\alpha_l - \alpha_t) \frac{q_i q_j}{q} + \delta_{ij} \alpha_t q \right) - \label{eq:t2} -$$ +$$D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho^w \left( (\alpha_l - \alpha_t) \frac{q_i q_j}{q} + \delta_{ij} \alpha_t q \right)$$ where @@ -44,12 +41,10 @@ In the absence of ground water flow, equations \ref{eq:t1} is simplified to, $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega - \label{eq:t3} $$ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s - \label{eq:t4} $$ ## Boundary Conditions From c0cd2938c51c735eb373a47ce9a9fa655749f072 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 16:48:27 +0100 Subject: [PATCH 03/21] Readme.md for thermal element is added, update 03 --- .../custom_elements/README.md | 93 ++++++------------- 1 file changed, 28 insertions(+), 65 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index e46dc2299dea..24acaa76e37d 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -6,7 +6,7 @@ The extraction of geothermal energy relies on a profound understanding of heat t ## 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. -$$\left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = -\rho^w c^w q_i \frac{\partial T}{\partial x_i} + \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega$$ +$$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = -\rho^w c^w q_i \frac{\partial T}{\partial x_i} + \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega $$ where, @@ -22,7 +22,7 @@ where, The hydrodynamic thermal dispersion is defined as: -$$D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho^w \left( (\alpha_l - \alpha_t) \frac{q_i q_j}{q} + \delta_{ij} \alpha_t q \right)$$ +$$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho^w \left( (\alpha_l - \alpha_t) \frac{q_i q_j}{q} + \delta_{ij} \alpha_t q \right) $$ where @@ -38,84 +38,54 @@ where In the absence of ground water flow, equations \ref{eq:t1} is simplified to, -$$ - \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) - \qquad \text{on} \quad \Omega -$$ +$$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega $$ -$$ - D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s -$$ +$$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s $$ ## Boundary Conditions ### Dirichlet boundary condition -$$ - T = \overline T \qquad \text{on} \quad \Gamma_{1}^T - \label{eq:t5} -$$ +$$ T = \overline T \qquad \text{on} \quad \Gamma_{1}^T $$ where $\overline T \left[ C \right]$ is a prescribed temperature ### Neumann Boundary Condition -$$ - D_{ij} \frac{\partial T}{\partial x_j} n_i = - %\tilde{c}^w \tilde{\rho}^w \overline q_n \tilde{T} \qquad \text{on} \quad \Gamma_{2}^T - \overline{f} \qquad \text{on} \quad \Gamma_{2}^T - \label{eq:t6} -$$ +$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \tilde{c}^w \tilde{\rho}^w \overline q_n \tilde{T} \qquad \text{on} \quad \Gamma_{2}^T \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ where $\overline f \left[ C \right]$ is a prescribed conductive heat flux. ### Robin bounday condition -$$ - D_{ij} \frac{\partial T}{\partial x_j} n_i = - %c \tilde \rho \overline q_n \left(\tilde T-T\right) \qquad \text{on} \quad \Gamma_{3}^T - \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T - \label{eq:t6} -$$ +$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \tilde \rho \overline q_n \left(\tilde T-T\right) \qquad \text{on} \quad \Gamma_{3}^T \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T \label{eq:t6} $$ where $\overline g \left[ C \right]$ is a prescribed convective-conductive heat flux. ## Derived Properties -% + The density of the bulk material $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calculated as, -% -\begin{equation} - \rho = n S \rho^w + \left( 1 - n \right) \rho^s -\end{equation} -% + +$$ \rho = n S \rho^w + \left( 1 - n \right) \rho^s $$ + And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^3 C \right]}$ is: -% -\begin{equation} - C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s -\end{equation} -% + +$$ C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s $$ + The thermal conductivity of the bulk material $\lambda$ $\mathrm{\left[ W/mC \right]}$ -% -\begin{equation} - \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s -\end{equation} -% + +$$ \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s $$ + dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temperature (\cite{Diersch1}) is given by: -\begin{equation} - \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} - \label{eq:t10} -\end{equation} +$$ \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} $$ density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and \cite{Diersch1} proposes a six order Taylor expansion which is approximated here by: -\begin{multline} - \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \; T - 8.993699 \cdot 10^{-3} \; T^2 + \\ 9.143518 \cdot 10^{-5} \; T^3 - 8.907391 \cdot 10^{-7} \; T^4 + 5.291959 \cdot 10^{-9} \; T^5 - \\ 1.359813 \cdot 10^{-11} \; T^6 - \label{eq:t11} -\end{multline} +$$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \; T - 8.993699 \cdot 10^{-3} \; T^2 + 9.143518 \cdot 10^{-5} \; T^3 - 8.907391 \cdot 10^{-7} \; T^4 + 5.291959 \cdot 10^{-9} \; T^5 - 1.359813 \cdot 10^{-11} \; T^6 $$ ## Finite Element Formulation Kratos solves the equations based on incremental method. In The fram of Generelized Newmark method (GN11), the fully implicit incremental temperature formulation read as, -% + \begin{multline} \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = \\ @@ -124,42 +94,35 @@ Kratos solves the equations based on incremental method. In The fram of Genereli \end{multline} Compressibility matrix -% + $$ \boldsymbol{S} = \int_{\Omega^e} \left( n S \rho^w c^w + \left(1-n\right) \rho^s c^s \right)^{n+1} \boldsymbol{N}^T \boldsymbol{N} d \Omega \label{eq:t16} $$ Convectivity matrix -% + $$ \boldsymbol{A} = \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega \label{eq:t18} $$ Conductivity matrix -% + $$ \boldsymbol{H} = \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega \label{eq:t17} $$ Neumann condition (dispersive boundary) -% + $$ \boldsymbol{V} = \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma \label{eq:t19} $$ Robin condition (convective boundary) -% -$$ - \boldsymbol{W^r} = \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma - \label{eq:t19a} -$$ -% -$$ - \boldsymbol{W^l} = - \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma - \label{eq:t19b} -$$ \ No newline at end of file + +$$ \boldsymbol{W^r} = \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ + +$$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ \ No newline at end of file From a6b76563b594b0e0cb5902a7fdb2b52eeb9dacb9 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 17:07:10 +0100 Subject: [PATCH 04/21] Readme.md for thermal element is added, update 04 --- .../custom_elements/README.md | 29 +++++++------------ 1 file changed, 11 insertions(+), 18 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 24acaa76e37d..b49e54ec6b95 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -26,15 +26,11 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho where -\begin{table}[H] - \begin{tabular}{ll} - $\alpha_l$ &longitudinal dispersivity $\mathrm{[m]}$\\ - $\alpha_t$ &transverse dispersivity $\mathrm{[m]}$\\ - $\delta_{ij}$ &Kronecker delta $\mathrm{[-]}$\\ - $\lambda^w$ &thermal conductivity water $\mathrm{[W/mC]}$\\ - $\lambda^s$ &thermal conductivity solid matrix $\mathrm{[W/mC]}$ - \end{tabular} -\end{table} +| $\alpha_l$ | longitudinal dispersivity | $\mathrm{[m]}$ | +| $\alpha_t$ | transverse dispersivity | $\mathrm{[m]}$ | +| $\delta_{ij}$ | Kronecker delta | $\mathrm{[-]}$ | +| $\lambda^w$ | thermal conductivity water | $\mathrm{[W/mC]}$ | +| $\lambda^s$ | thermal conductivity solid matrix | $\mathrm{[W/mC]}$ | In the absence of ground water flow, equations \ref{eq:t1} is simplified to, @@ -50,12 +46,12 @@ $$ T = \overline T \qquad \text{on} \quad \Gamma_{1}^T $$ where $\overline T \left[ C \right]$ is a prescribed temperature ### Neumann Boundary Condition -$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \tilde{c}^w \tilde{\rho}^w \overline q_n \tilde{T} \qquad \text{on} \quad \Gamma_{2}^T \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ +$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ where $\overline f \left[ C \right]$ is a prescribed conductive heat flux. ### Robin bounday condition -$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \tilde \rho \overline q_n \left(\tilde T-T\right) \qquad \text{on} \quad \Gamma_{3}^T \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T \label{eq:t6} $$ +$$ 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 \label{eq:t6} $$ where $\overline g \left[ C \right]$ is a prescribed convective-conductive heat flux. @@ -74,10 +70,12 @@ The thermal conductivity of the bulk material $\lambda$ $\mathrm{\left[ W/mC \ri $$ \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s $$ -dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temperature (\cite{Diersch1}) is given by: +dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temperature is given by: + $$ \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} $$ density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and \cite{Diersch1} proposes a six order Taylor expansion which is approximated here by: + $$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \; T - 8.993699 \cdot 10^{-3} \; T^2 + 9.143518 \cdot 10^{-5} \; T^3 - 8.907391 \cdot 10^{-7} \; T^4 + 5.291959 \cdot 10^{-9} \; T^5 - 1.359813 \cdot 10^{-11} \; T^6 $$ @@ -86,12 +84,7 @@ $$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \; T - 8.993699 \cdot Kratos solves the equations based on incremental method. In The fram of Generelized Newmark method (GN11), the fully implicit incremental temperature formulation read as, -\begin{multline} - \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) - \label{eq:t15b} -\end{multline} +$$ \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = \left( \frac{1}{\theta} - 1 \right) \boldsymbol{S} \frac{dT^n}{dt} - \left(\boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{T}^{n} + \left( \boldsymbol{V} + \boldsymbol{W}^r \right) $$ Compressibility matrix From 616eb848ca3563b282c2f6c6a95399d0e0f67482 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 17:14:02 +0100 Subject: [PATCH 05/21] Readme.md for thermal element is added, update 05 --- applications/GeoMechanicsApplication/custom_elements/README.md | 1 + 1 file changed, 1 insertion(+) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index b49e54ec6b95..4b2afab928de 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -26,6 +26,7 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho where +|:--- |:--- |:--- | | $\alpha_l$ | longitudinal dispersivity | $\mathrm{[m]}$ | | $\alpha_t$ | transverse dispersivity | $\mathrm{[m]}$ | | $\delta_{ij}$ | Kronecker delta | $\mathrm{[-]}$ | From 98af7b80ee9f2f8a3a2e0dfc99bd276694c38a21 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 17:18:51 +0100 Subject: [PATCH 06/21] Readme.md for thermal element is added, update 06 --- applications/GeoMechanicsApplication/custom_elements/README.md | 1 + 1 file changed, 1 insertion(+) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 4b2afab928de..645c6ff1f2f8 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -26,6 +26,7 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho where +|||| |:--- |:--- |:--- | | $\alpha_l$ | longitudinal dispersivity | $\mathrm{[m]}$ | | $\alpha_t$ | transverse dispersivity | $\mathrm{[m]}$ | From fd177df6d329de7c777abb2fd8d83c4d1d9f02d1 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 17:25:34 +0100 Subject: [PATCH 07/21] Readme.md for thermal element is added, update 07 --- .../custom_elements/README.md | 23 +++++++++++-------- 1 file changed, 14 insertions(+), 9 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 645c6ff1f2f8..547759118b3e 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -10,15 +10,20 @@ $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t where, -\begin{table}[H] - \begin{tabular}{ll} - $c^w$ &specific heat capacity liquid phase $\mathrm{[J/kgC]}$\\ - $c^s$ &specific heat capacity solid phase $\mathrm{[J/kgC]}$\\ - $D_{ij}$ &hydrodynamic thermal dispersion $\mathrm{[W/mC]}$\\ - $T$ &temperature $\mathrm{[C]}$\\ - $\rho^s$ &density solid phase $\mathrm{[kg/m^3]}$ - \end{tabular} -\end{table} + + +|||| +|----------|-------------------------------------|--------------------| +| $c^w$ | specific heat capacity liquid phase | \mathrm{[J/kgC]}$ | +| $c^s$ | specific heat capacity solid phase | \mathrm{[J/kgC]}$ | +| $D_{ij}$ | hydrodynamic thermal dispersion | \mathrm{[W/mC]}$ | +| $T$ | temperature | \mathrm{[C]}$ | +| $\rho^s$ | density solid phase | \mathrm{[kg/m^3]}$ | + The hydrodynamic thermal dispersion is defined as: From 84c08a1aeb9ddefccca668ca3a17cbdf7403a456 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Thu, 25 Jan 2024 17:27:48 +0100 Subject: [PATCH 08/21] Readme.md for thermal element is added, update 08 --- .../custom_elements/README.md | 23 ++++++++++++------- 1 file changed, 15 insertions(+), 8 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 547759118b3e..ac0e2414c0d8 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -11,18 +11,25 @@ $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t where, |||| -|----------|-------------------------------------|--------------------| -| $c^w$ | specific heat capacity liquid phase | \mathrm{[J/kgC]}$ | -| $c^s$ | specific heat capacity solid phase | \mathrm{[J/kgC]}$ | -| $D_{ij}$ | hydrodynamic thermal dispersion | \mathrm{[W/mC]}$ | -| $T$ | temperature | \mathrm{[C]}$ | -| $\rho^s$ | density solid phase | \mathrm{[kg/m^3]}$ | +|----------|--------------------------------------------------|--------------------| +| $c^w$ | specific heat capacity liquid phase | $\mathrm{[J/kgC]}$ | +| $c^s$ | specific heat capacity solid phase | $\mathrm{[J/kgC]}$ | +| $D_{ij}$ | hydrodynamic thermal dispersion | $\mathrm{[W/mC]}$ | +| $T$ | temperature | $\mathrm{[C]}$ | +| $\rho^s$ | density solid phase | $\mathrm{[kg/m^3]}$ | The hydrodynamic thermal dispersion is defined as: From 45fd2ee7e259845c94077acf3fe36028988eea07 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Fri, 26 Jan 2024 11:11:36 +0100 Subject: [PATCH 09/21] Readme.md for thermal element is added, update 09 --- .../custom_elements/README.md | 73 ++++++------------- 1 file changed, 22 insertions(+), 51 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index ac0e2414c0d8..530f597bd155 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -10,27 +10,11 @@ $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t where, - - -|||| -|----------|--------------------------------------------------|--------------------| -| $c^w$ | specific heat capacity liquid phase | $\mathrm{[J/kgC]}$ | -| $c^s$ | specific heat capacity solid phase | $\mathrm{[J/kgC]}$ | -| $D_{ij}$ | hydrodynamic thermal dispersion | $\mathrm{[W/mC]}$ | -| $T$ | temperature | $\mathrm{[C]}$ | -| $\rho^s$ | density solid phase | $\mathrm{[kg/m^3]}$ | - +- $c^w$ = specific heat capacity liquid phase $\mathrm{[J/kg ^{\circ}C]}$ +- $c^s$ = specific heat capacity solid phase $\mathrm{[J/kg ^{\circ}C]}$ +- $D_{ij}$ = hydrodynamic thermal dispersion $\mathrm{[W/m ^{\circ}C]}$ +- $T$ = temperature $\mathrm{[ ^{\circ}C]}$ +- $\rho^s$ = density solid phase $\mathrm{[kg/m^3]}$ The hydrodynamic thermal dispersion is defined as: @@ -38,15 +22,13 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho where -|||| -|:--- |:--- |:--- | -| $\alpha_l$ | longitudinal dispersivity | $\mathrm{[m]}$ | -| $\alpha_t$ | transverse dispersivity | $\mathrm{[m]}$ | -| $\delta_{ij}$ | Kronecker delta | $\mathrm{[-]}$ | -| $\lambda^w$ | thermal conductivity water | $\mathrm{[W/mC]}$ | -| $\lambda^s$ | thermal conductivity solid matrix | $\mathrm{[W/mC]}$ | +- $\alpha_l$ = 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 $\mathrm{[W/m ^{\circ}C]}$ -In the absence of ground water flow, equations \ref{eq:t1} is simplified to, +In the absence of ground water flow, these equations are simplified to, $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x_i} \left( D_{ij} \frac{\partial T}{\partial x_j} \right) \qquad \text{on} \quad \Omega $$ @@ -57,17 +39,17 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s $$ ### Dirichlet boundary condition $$ T = \overline T \qquad \text{on} \quad \Gamma_{1}^T $$ -where $\overline T \left[ C \right]$ is a prescribed temperature +where $\overline T \left[ ^{\circ}C \right]$ is a prescribed temperature ### Neumann Boundary Condition $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ -where $\overline f \left[ C \right]$ is a prescribed conductive heat flux. +where $\overline f \left[ ^{\circ}C \right]$ is a prescribed conductive heat flux. ### Robin bounday condition $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T \label{eq:t6} $$ -where $\overline g \left[ C \right]$ is a prescribed convective-conductive heat flux. +where $\overline g \left[ ^{\circ}C \right]$ is a prescribed convective-conductive heat flux. ## Derived Properties @@ -90,7 +72,8 @@ $$ \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} $$ density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and \cite{Diersch1} proposes a six order Taylor expansion which is approximated here by: -$$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \; T - 8.993699 \cdot 10^{-3} \; T^2 + 9.143518 \cdot 10^{-5} \; T^3 - 8.907391 \cdot 10^{-7} \; T^4 + 5.291959 \cdot 10^{-9} \; T^5 - 1.359813 \cdot 10^{-11} \; T^6 $$ +$$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \cdot T - 8.993699 \cdot 10^{-3} \cdot T^2 + 9.143518 \cdot 10^{-5} \cdot T^3$$ +$$ - 8.907391 \cdot 10^{-7} \cdot T^4 + 5.291959 \cdot 10^{-9} \cdot T^5 - 1.359813 \cdot 10^{-11} \cdot T^6 $$ @@ -102,34 +85,22 @@ $$ \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol Compressibility matrix -$$ - \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 - \label{eq:t16} -$$ +$$ \boldsymbol{S} = \sum_e \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 $$ Convectivity matrix -$$ - \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 - \label{eq:t18} -$$ +$$ \boldsymbol{A} = \sum_e \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega $$ Conductivity matrix -$$ - \boldsymbol{H} = \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega - \label{eq:t17} -$$ +$$ \boldsymbol{H} = \sum_e \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ Neumann condition (dispersive boundary) -$$ - \boldsymbol{V} = \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma - \label{eq:t19} -$$ +$$ \boldsymbol{V} = \sum_e \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ Robin condition (convective boundary) -$$ \boldsymbol{W^r} = \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ +$$ \boldsymbol{W^r} = \sum_e \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ -$$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ \ No newline at end of file +$$ \boldsymbol{W^l} = \sum_e \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ \ No newline at end of file From 46d4a0832a1c04f4352df2d010b44bcdf6c6a4e8 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Fri, 26 Jan 2024 11:20:46 +0100 Subject: [PATCH 10/21] Readme.md for thermal element is added, update 10 --- .../custom_elements/README.md | 17 ++++++++--------- 1 file changed, 8 insertions(+), 9 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 530f597bd155..a6a8ed7fc4d9 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -47,7 +47,7 @@ $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \q where $\overline f \left[ ^{\circ}C \right]$ is a prescribed conductive heat flux. ### Robin bounday condition -$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T \label{eq:t6} $$ +$$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T $$ where $\overline g \left[ ^{\circ}C \right]$ is a prescribed convective-conductive heat flux. @@ -70,10 +70,9 @@ dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temper $$ \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} $$ -density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and \cite{Diersch1} proposes a six order Taylor expansion which is approximated here by: +density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and Diersch proposes a six order Taylor expansion which is approximated here by: -$$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \cdot T - 8.993699 \cdot 10^{-3} \cdot T^2 + 9.143518 \cdot 10^{-5} \cdot T^3$$ -$$ - 8.907391 \cdot 10^{-7} \cdot T^4 + 5.291959 \cdot 10^{-9} \cdot T^5 - 1.359813 \cdot 10^{-11} \cdot T^6 $$ +$$ \rho^w = 9.998396 \cdot 10^2 + 6.764771 \cdot 10^{-2} \cdot T - 8.993699 \cdot 10^{-3} \cdot T^2 + 9.143518 \cdot 10^{-5} \cdot T^3 - 8.907391 \cdot 10^{-7} \cdot T^4 + 5.291959 \cdot 10^{-9} \cdot T^5 - 1.359813 \cdot 10^{-11} \cdot T^6 $$ @@ -83,23 +82,23 @@ Kratos solves the equations based on incremental method. In The fram of Genereli $$ \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = \left( \frac{1}{\theta} - 1 \right) \boldsymbol{S} \frac{dT^n}{dt} - \left(\boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{T}^{n} + \left( \boldsymbol{V} + \boldsymbol{W}^r \right) $$ -Compressibility matrix +###Compressibility matrix $$ \boldsymbol{S} = \sum_e \int_{\Omega^e} \left( n S \rho^w c^w + \left(1-n\right) \rho^s c^s \right)^{n+1} \boldsymbol{N}^T \boldsymbol{N} d \Omega $$ -Convectivity matrix +###Convectivity matrix $$ \boldsymbol{A} = \sum_e \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega $$ -Conductivity matrix +###Conductivity matrix $$ \boldsymbol{H} = \sum_e \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ -Neumann condition (dispersive boundary) +###Neumann condition (dispersive boundary) $$ \boldsymbol{V} = \sum_e \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ -Robin condition (convective boundary) +###Robin condition (convective boundary) $$ \boldsymbol{W^r} = \sum_e \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ From 3f91c8e78941c34cc241fb66f92596b40752de6f Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Fri, 26 Jan 2024 11:32:45 +0100 Subject: [PATCH 11/21] Readme.md for thermal element is added, update 11 --- .../custom_elements/README.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index a6a8ed7fc4d9..7fb4322285b1 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -44,12 +44,12 @@ where $\overline T \left[ ^{\circ}C \right]$ is a prescribed temperature ### Neumann Boundary Condition $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ -where $\overline f \left[ ^{\circ}C \right]$ is a prescribed conductive heat flux. +where $\overline f \left[ W/m^2 \right]$ is a prescribed conductive heat flux. ### Robin bounday condition $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T $$ -where $\overline g \left[ ^{\circ}C \right]$ is a prescribed convective-conductive heat flux. +where $\overline g \left[ W/m^2 \right]$ is a prescribed convective-conductive heat flux. ## Derived Properties @@ -58,11 +58,11 @@ The density of the bulk material $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calc $$ \rho = n S \rho^w + \left( 1 - n \right) \rho^s $$ -And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^3 C \right]}$ is: +And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^3 ^{\circ}C \right]}$ is: $$ C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s $$ -The thermal conductivity of the bulk material $\lambda$ $\mathrm{\left[ W/mC \right]}$ +The thermal conductivity of the bulk material $\lambda$ $\mathrm{\left[ W/m ^{\circ}C \right]}$ $$ \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s $$ @@ -82,23 +82,23 @@ Kratos solves the equations based on incremental method. In The fram of Genereli $$ \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = \left( \frac{1}{\theta} - 1 \right) \boldsymbol{S} \frac{dT^n}{dt} - \left(\boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{T}^{n} + \left( \boldsymbol{V} + \boldsymbol{W}^r \right) $$ -###Compressibility matrix +### Compressibility matrix $$ \boldsymbol{S} = \sum_e \int_{\Omega^e} \left( n S \rho^w c^w + \left(1-n\right) \rho^s c^s \right)^{n+1} \boldsymbol{N}^T \boldsymbol{N} d \Omega $$ -###Convectivity matrix +### Convectivity matrix $$ \boldsymbol{A} = \sum_e \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega $$ -###Conductivity matrix +### Conductivity matrix $$ \boldsymbol{H} = \sum_e \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ -###Neumann condition (dispersive boundary) +###8 Neumann condition (dispersive boundary) $$ \boldsymbol{V} = \sum_e \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ -###Robin condition (convective boundary) +### Robin condition (convective boundary) $$ \boldsymbol{W^r} = \sum_e \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ From 93cd1795348d84e95e5ff579aaf8e0c80a476b2e Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Fri, 26 Jan 2024 11:39:37 +0100 Subject: [PATCH 12/21] Readme.md for thermal element is added, update 12 --- .../GeoMechanicsApplication/custom_elements/README.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 7fb4322285b1..e70e2b210fd1 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -39,17 +39,17 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s $$ ### Dirichlet boundary condition $$ T = \overline T \qquad \text{on} \quad \Gamma_{1}^T $$ -where $\overline T \left[ ^{\circ}C \right]$ is a prescribed temperature +where $\overline T$ $\mathrm{\left[ ^{\circ}C \right]}$ is a prescribed temperature ### Neumann Boundary Condition $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \quad \Gamma_{2}^T $$ -where $\overline f \left[ W/m^2 \right]$ is a prescribed conductive heat flux. +where $\overline f$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed conductive heat flux. ### Robin bounday condition $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{g} - \rho^w c^w q_n T \qquad \text{on} \quad \Gamma_{3}^T $$ -where $\overline g \left[ W/m^2 \right]$ is a prescribed convective-conductive heat flux. +where $\overline g$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed convective-conductive heat flux. ## Derived Properties @@ -58,7 +58,7 @@ The density of the bulk material $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calc $$ \rho = n S \rho^w + \left( 1 - n \right) \rho^s $$ -And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^3 ^{\circ}C \right]}$ is: +And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m{^3 \circ}C \right]}$ is: $$ C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s $$ @@ -94,7 +94,7 @@ $$ \boldsymbol{A} = \sum_e \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \bolds $$ \boldsymbol{H} = \sum_e \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ -###8 Neumann condition (dispersive boundary) +### Neumann condition (dispersive boundary) $$ \boldsymbol{V} = \sum_e \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ From 794bd07c67ff5f9898cb182c536aac69324cec6d Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Fri, 26 Jan 2024 11:44:11 +0100 Subject: [PATCH 13/21] Readme.md for thermal element is added, update 13 --- .../GeoMechanicsApplication/custom_elements/README.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index e70e2b210fd1..0afd34a09d91 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -82,23 +82,23 @@ Kratos solves the equations based on incremental method. In The fram of Genereli $$ \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) $$ -### Compressibility matrix +***Compressibility matrix*** $$ \boldsymbol{S} = \sum_e \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 $$ -### Convectivity matrix +***Convectivity matrix*** $$ \boldsymbol{A} = \sum_e \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega $$ -### Conductivity matrix +***Conductivity matrix*** $$ \boldsymbol{H} = \sum_e \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ -### Neumann condition (dispersive boundary) +***Neumann condition (dispersive boundary)*** $$ \boldsymbol{V} = \sum_e \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ -### Robin condition (convective boundary) +***Robin condition (convective boundary)*** $$ \boldsymbol{W^r} = \sum_e \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ From c40a2ce98d1958d9063748637bae4cdb3661666a Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Tue, 30 Jan 2024 13:15:40 +0100 Subject: [PATCH 14/21] Readme.md for thermal element is added, update 14 Modified based of reviews --- .../custom_elements/README.md | 61 +++++++++++++------ 1 file changed, 41 insertions(+), 20 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 0afd34a09d91..3ff8362de631 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -1,7 +1,7 @@ # Transient Thermal Element ## Introduction -The extraction of geothermal energy relies on a profound understanding of heat transport mechanisms within the Earth's subsurface. As heat traverses geological formations, its efficient transfer is governed by mathematical equations encapsulating principles of thermal dynamics. Central to this understanding is the heat transport equation, a fundamental tool that elucidates how thermal energy moves through the Earth's crust, influencing the feasibility and optimization of geothermal applications. In this context, the heat transport equation becomes a linchpin for designing sustainable and efficient geothermal energy systems. +The calculation of geothermal processes relies on a profound understanding of heat transport mechanisms within the Earth's subsurface. Central to this understanding is the heat transport equation. In this context, the heat transport equation becomes a linchpin for designing sustainable and efficient geothermal systems. ## Governing Equations The primary governing equation for heat transport in geothermal applications is the heat convection-conduction equation, which describes how heat flows through a porous medium. @@ -10,11 +10,13 @@ $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t where, -- $c^w$ = 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 $\mathrm{[ ^{\circ}C]}$ +- $c^w$ = 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 $\mathrm{[kg/m^3]}$ +- $q$ = specific discharge $\mathrm{[m/s]}$ +- $S$ = degree of saturation $\mathrm{[-]}$ The hydrodynamic thermal dispersion is defined as: @@ -36,7 +38,7 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s $$ ## Boundary Conditions -### Dirichlet boundary condition +### Dirichlet Boundary Condition $$ T = \overline T \qquad \text{on} \quad \Gamma_{1}^T $$ where $\overline T$ $\mathrm{\left[ ^{\circ}C \right]}$ is a prescribed temperature @@ -46,19 +48,19 @@ $$ D_{ij} \frac{\partial T}{\partial x_j} n_i = \overline{f} \qquad \text{on} \q where $\overline f$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed conductive heat flux. -### Robin bounday condition +### Robin Boundary Condition $$ 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 $$ -where $\overline g$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed convective-conductive heat flux. +where $\overline g$ $\mathrm{\left[ W/m^2 \right]}$ is a prescribed convective-conductive heat flux and $q_n$ is the specific discharge $\mathrm{\left[ m/s \right]}$ at the boundary. ## Derived Properties -The density of the bulk material $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calculated as, +The density of the porous media $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calculated as, $$ \rho = n S \rho^w + \left( 1 - n \right) \rho^s $$ -And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m{^3 \circ}C \right]}$ is: +And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^{3 \circ}C \right]}$ is: $$ C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s $$ @@ -70,36 +72,55 @@ dynamic viscosity $\mu$ $[\mathrm {Pas}$] of pure water as a function of temper $$ \mu = 2.4318 \cdot 10^{-5} \cdot 10^{{247.8} / {\left(T+133.0\right]}} $$ -density of water $\rho^w$ $[\mathrm {kg/m^3}]$ relates to temperature and Diersch proposes a six order Taylor expansion which is approximated here by: +density of water $\rho^w$ $[\mathrm {kg/m^3}]$ is a function of temperature and Diersch (2014) proposes a sixth order Taylor expansion which is approximated here by: $$ \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 $$ - +Note: the dependency of water density and viscosity to temperature is user defined and the user can turn this feature on or off in the JSON file. ## Finite Element Formulation -Kratos solves the equations based on incremental method. In The fram of Generelized Newmark method (GN11), the fully implicit incremental temperature formulation read as, +Kratos solves the equations based on an incremental method. In the frame of Generelized Newmark method (GN11), the fully implicit incremental temperature formulation read as, $$ \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) $$ +with $\theta = 1$ this formulation leads to backward Euler formulation. It reads, + +$$ \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) $$ + +Note: the user can choose either Newmark or backward Euler from the JSON file. + ***Compressibility matrix*** -$$ \boldsymbol{S} = \sum_e \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 $$ +$$ \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 $$ ***Convectivity matrix*** -$$ \boldsymbol{A} = \sum_e \int_{\Omega^e} \left(\rho^w c^w\right)^{n+1} \boldsymbol{N}^T \boldsymbol{q}^{T,n+1} \boldsymbol{\nabla N} d \Omega $$ +$$ \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 $$ ***Conductivity matrix*** -$$ \boldsymbol{H} = \sum_e \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ +$$ \boldsymbol{H} = \int_{\Omega^e} \boldsymbol{\nabla N}^T \boldsymbol{D}^{n+1} \boldsymbol{\nabla N} d \Omega $$ -***Neumann condition (dispersive boundary)*** +***Neumann condition (conductive boundary)*** -$$ \boldsymbol{V} = \sum_e \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ +$$ \boldsymbol{V} = \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ ***Robin condition (convective boundary)*** -$$ \boldsymbol{W^r} = \sum_e \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ +$$ \boldsymbol{W^r} = \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ + +$$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ + +where + +- $\Delta t$ = time step $\mathrm{\left[s]}$ +- $\Delta T$ = temperature increment $T^{n+1} - T^n$ $\mathrm{\left[^\circ C \right]}$ +- $\boldsymbol N$ = shape function array $\mathrm{\left[ - \right]}$ +- $\theta$ = a coefficient in Newmark time integration $\mathrm{\left[ - \right]}$ +- $\Omega$ = domain region +- $\Gamma$ = boundary region + -$$ \boldsymbol{W^l} = \sum_e \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ \ No newline at end of file +## Bibliography +Diersch, H.-J. G., 2014. FEFLOW; Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer. From e363f24ac8a8e8127b8be5452a44e6e52b68fe08 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Tue, 30 Jan 2024 13:17:46 +0100 Subject: [PATCH 15/21] Readme.md for thermal element is added, update 15 Modified based of reviews --- .../GeoMechanicsApplication/custom_elements/README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 3ff8362de631..72dd3b49048b 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -60,11 +60,11 @@ The density of the porous media $\rho$ $\mathrm{\left[ kg/m^3 \right]}$ is calcu $$ \rho = n S \rho^w + \left( 1 - n \right) \rho^s $$ -And the heat capacity of the bulk material $C$ $\mathrm{\left[ J/m^{3 \circ}C \right]}$ is: +And the heat capacity of the porous media $C$ $\mathrm{\left[ J/m^{3 \circ}C \right]}$ is: $$ C = n S \rho^w c^w + \left( 1 - n \right) \rho^s c^s $$ -The thermal conductivity of the bulk material $\lambda$ $\mathrm{\left[ W/m ^{\circ}C \right]}$ +The thermal conductivity of the porous media $\lambda$ $\mathrm{\left[ W/m ^{\circ}C \right]}$ $$ \lambda = n S \lambda^w + \left( 1 - n \right) \lambda^s $$ @@ -114,7 +114,7 @@ $$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} where -- $\Delta t$ = time step $\mathrm{\left[s]}$ +- $\Delta t$ = 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 $\mathrm{\left[ - \right]}$ From b6007c6f19eb4d6a958f048af5c72756af44e6fb Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Tue, 30 Jan 2024 14:29:19 +0100 Subject: [PATCH 16/21] Readme.md for thermal element is added, update 16 Modified based of reviews --- .../GeoMechanicsApplication/custom_elements/README.md | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 72dd3b49048b..58cb0c0b6997 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -15,8 +15,10 @@ where, - $D_{ij}$ = 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 $\mathrm{[-]}$ +- $S$ = degree of saturation $\mathrm{[-]}$ +- $n$ = porosity $\mathrm{[-]}$ The hydrodynamic thermal dispersion is defined as: @@ -108,7 +110,7 @@ $$ \boldsymbol{V} = \int_{\Gamma_2^{ep}} f^{n+1} \boldsymbol{N}^T d \Gamma $$ ***Robin condition (convective boundary)*** -$$ \boldsymbol{W^r} = \int_{\Gamma_2^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ +$$ \boldsymbol{W^r} = \int_{\Gamma_3^{ep}} g^{n+1} \boldsymbol{N}^T d \Gamma $$ $$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} \boldsymbol{N}^T \boldsymbol{I} d \Gamma $$ @@ -121,6 +123,7 @@ where - $\Omega$ = domain region - $\Gamma$ = boundary region +The supercripts "$^l$" and "$^r$" for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts "$^e$" and "$^{ep}$" for $\Omega$ and $\Gamma$ indicat valunes at element and perpendicular on element interfaces, respectively. ## Bibliography Diersch, H.-J. G., 2014. FEFLOW; Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer. From 6c87e76f1aa1d175fbfee7f4ced751176fce680d Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Tue, 30 Jan 2024 14:34:37 +0100 Subject: [PATCH 17/21] Readme.md for thermal element is added, update 17 Modified based of reviews --- applications/GeoMechanicsApplication/custom_elements/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 58cb0c0b6997..c1f165efd9ad 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -123,7 +123,7 @@ where - $\Omega$ = domain region - $\Gamma$ = boundary region -The supercripts "$^l$" and "$^r$" for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts "$^e$" and "$^{ep}$" for $\Omega$ and $\Gamma$ indicat valunes at element and perpendicular on element interfaces, respectively. +The supercripts "$l$" and "$r$" for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts "$e$" and "${ep}$" for $\Omega$ and $\Gamma$ indicat valunes at element and perpendicular to element interfaces, respectively. ## Bibliography Diersch, H.-J. G., 2014. FEFLOW; Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer. From 24551a796bfa0a8ee30e9bf21e123b5d59215c74 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Tue, 30 Jan 2024 14:36:06 +0100 Subject: [PATCH 18/21] Readme.md for thermal element is added, update 18 Modified based of reviews --- applications/GeoMechanicsApplication/custom_elements/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index c1f165efd9ad..0c3cdc101ad8 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -123,7 +123,7 @@ where - $\Omega$ = domain region - $\Gamma$ = boundary region -The supercripts "$l$" and "$r$" for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts "$e$" and "${ep}$" for $\Omega$ and $\Gamma$ indicat valunes at element and perpendicular to element interfaces, respectively. +The supercripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicat valunes at element and perpendicular to element interfaces, respectively. ## Bibliography Diersch, H.-J. G., 2014. FEFLOW; Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer. From ba601ac451a030d0711d2c484a579be545c0c946 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Tue, 30 Jan 2024 14:39:30 +0100 Subject: [PATCH 19/21] Readme.md for thermal element is added, update 19 Modified based of reviews --- applications/GeoMechanicsApplication/custom_elements/README.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 0c3cdc101ad8..21f605c1f269 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -123,7 +123,8 @@ where - $\Omega$ = domain region - $\Gamma$ = boundary region -The supercripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicat valunes at element and perpendicular to element interfaces, respectively. +The supercripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicate values in the element volume and perpendicular to element boundaries, respectively. + ## Bibliography Diersch, H.-J. G., 2014. FEFLOW; Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer. From 59d892c1823ab3bb0e097c923d54a38a115b90fa Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Mon, 5 Feb 2024 15:25:33 +0100 Subject: [PATCH 20/21] Readme.md for thermal element is added, update 20 Modified based of reviews --- .../GeoMechanicsApplication/custom_elements/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index 21f605c1f269..a08633da9b69 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -82,7 +82,7 @@ Note: the dependency of water density and viscosity to temperature is user defin ## Finite Element Formulation -Kratos solves the equations based on an incremental method. In the frame of Generelized Newmark method (GN11), the fully implicit incremental temperature formulation read as, +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, $$ \left(\frac{1}{\theta \Delta t} \boldsymbol{S} + \boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{\Delta T} = \left( \frac{1}{\theta} - 1 \right) \boldsymbol{S} \frac{dT^n}{dt} - \left(\boldsymbol{A} + \boldsymbol{H} + \boldsymbol{W}^l \right) \boldsymbol{T}^{n} + \left( \boldsymbol{V} + \boldsymbol{W}^r \right) $$ @@ -123,7 +123,7 @@ where - $\Omega$ = domain region - $\Gamma$ = boundary region -The supercripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and righ hand side (vector), respectively. The supercripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicate values in the element volume and perpendicular to element boundaries, respectively. +The superscripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and right hand side (vector), respectively. The supercripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicate values in the element volume and perpendicular to element boundaries, respectively. ## Bibliography From c48c0bf0a0673159ab31a029cf8f4fbbbd113cb9 Mon Sep 17 00:00:00 2001 From: mnabideltares Date: Wed, 7 Feb 2024 11:48:58 +0100 Subject: [PATCH 21/21] Readme.md for thermal element is added, update 21 Modified based on Codacy --- .../custom_elements/README.md | 42 +++++++++---------- 1 file changed, 21 insertions(+), 21 deletions(-) diff --git a/applications/GeoMechanicsApplication/custom_elements/README.md b/applications/GeoMechanicsApplication/custom_elements/README.md index a08633da9b69..2f566903d3b6 100644 --- a/applications/GeoMechanicsApplication/custom_elements/README.md +++ b/applications/GeoMechanicsApplication/custom_elements/README.md @@ -10,15 +10,15 @@ $$ \left(n S \rho^w c^w + (1- n) \rho^s c^s \right) \frac{\partial T}{\partial t where, -- $c^w$ = specific heat capacity liquid phase $\mathrm{[J/kg ^{\circ}C]}$ -- $c^s$ = specific heat capacity solid phase $\mathrm{[J/kg ^{\circ}C]}$ -- $D_{ij}$ = hydrodynamic thermal dispersion $\mathrm{[W/m ^{\circ}C]}$ -- $T$ = temperature $\mathrm{[ ^{\circ}C]}$ -- $\rho^s$ = density solid phase $\mathrm{[kg/m^3]}$ -- $\rho^w$ = water density $\mathrm{[kg/m^3]}$ -- $q$ = specific discharge $\mathrm{[m/s]}$ -- $S$ = degree of saturation $\mathrm{[-]}$ -- $n$ = porosity $\mathrm{[-]}$ +- $c^w$ = specific heat capacity liquid phase $\mathrm{[J/kg ^{\circ}C]}$ +- $c^s$ = specific heat capacity solid phase $\mathrm{[J/kg ^{\circ}C]}$ +- $D_{ij}$ = hydrodynamic thermal dispersion $\mathrm{[W/m ^{\circ}C]}$ +- $T$ = temperature $\mathrm{[ ^{\circ}C]}$ +- $\rho^s$ = density solid phase $\mathrm{[kg/m^3]}$ +- $\rho^w$ = water density $\mathrm{[kg/m^3]}$ +- $q$ = specific discharge $\mathrm{[m/s]}$ +- $S$ = degree of saturation $\mathrm{[-]}$ +- $n$ = porosity $\mathrm{[-]}$ The hydrodynamic thermal dispersion is defined as: @@ -26,11 +26,11 @@ $$ D_{ij}= nS \lambda^w \delta_{ij} + \left(1-n\right) \lambda_{ij}^s + c^w \rho where -- $\alpha_l$ = 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 $\mathrm{[W/m ^{\circ}C]}$ +- $\alpha_l$ = 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 $\mathrm{[W/m ^{\circ}C]}$ In the absence of ground water flow, these equations are simplified to, @@ -116,14 +116,14 @@ $$ \boldsymbol{W^l} = \int_{\Gamma_3^{ep}} \left( \rho^w c^w q_n \right)^{n+1} where -- $\Delta t$ = 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 +- $\Delta t$ = 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 -The superscripts $^l$ and $^r$ for Robin boundary condition indicate the left hands side (matrix) and right hand side (vector), respectively. The supercripts $^e$ and $^{ep}$ for $\Omega$ and $\Gamma$ indicate values in the element volume and perpendicular to element boundaries, respectively. +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. ## Bibliography