From 3fc738c0fa629e1cdc3183c3ddccfc59440b788c Mon Sep 17 00:00:00 2001
From: Leon Keim <leon.keim@googlemail.com>
Date: Tue, 27 Sep 2022 16:25:48 +0200
Subject: [PATCH] Adding description of Forchheimerslaw
---
doc/handbook/dumux-handbook.bib | 12 ++++++
dumux/flux/darcyslaw.hh | 2 +-
dumux/flux/forchheimerslaw.hh | 63 +++++++++++++++++++++++++++++--
dumux/flux/forchheimervelocity.hh | 62 ++----------------------------
4 files changed, 75 insertions(+), 64 deletions(-)
diff --git a/doc/handbook/dumux-handbook.bib b/doc/handbook/dumux-handbook.bib
index 8213ef14e3..e2f7c103f8 100644
--- a/doc/handbook/dumux-handbook.bib
+++ b/doc/handbook/dumux-handbook.bib
@@ -1996,3 +1996,15 @@ url={https://doi.org/10.1007/s10765-012-1254-5}
year = {1975},
url = {https://link.springer.com/content/pdf/10.1007/BF00867119.pdf}
}
+
+@article{Fichot2006,
+title = {The impact of thermal non-equilibrium and large-scale 2D/3D effects on debris bed reflooding and coolability},
+journal = {Nuclear Engineering and Design},
+volume = {236},
+number = {19},
+pages = {2144-2163},
+year = {2006},
+issn = {0029-5493},
+doi = {10.1016/j.nucengdes.2006.03.059},
+author = {F. Fichot and F. Duval and N. Trégourès and C. Béchaud and M. Quintard},
+}
diff --git a/dumux/flux/darcyslaw.hh b/dumux/flux/darcyslaw.hh
index d9fe5e76da..f07e3a7648 100644
--- a/dumux/flux/darcyslaw.hh
+++ b/dumux/flux/darcyslaw.hh
@@ -21,7 +21,7 @@
* \ingroup Flux
* \brief Advective fluxes according to Darcy's law
*
- * Darcy's law describes the advective flux in porous media on the macro-scale and is valid in the creeping flow regime (Reynolds number << 1).
+ * Darcy's law describes the advective flux in porous media on the macro-scale and is valid in the creeping flow regime (Reynolds number << 1, Forchheimer extensions is also implemented->see forcheimerslaw.hh).
* The advective flux characterizes the bulk flow for each fluid phase including all components in case of compositional flow.
* It is driven by the potential gradient \f$\textbf{grad}\, p - \varrho {\textbf g}\f$,
* accounting for both pressure-driven and gravitationally-driven flow.
diff --git a/dumux/flux/forchheimerslaw.hh b/dumux/flux/forchheimerslaw.hh
index 021cdb5601..14d68b9d96 100644
--- a/dumux/flux/forchheimerslaw.hh
+++ b/dumux/flux/forchheimerslaw.hh
@@ -19,10 +19,65 @@
/*!
* \file
* \ingroup Flux
- * \brief Forchheimer's law specialized for different discretization schemes
- * This file contains the data which is required to calculate
- * volume and mass fluxes of fluid phases over a face of a finite volume by means
- * of the Forchheimer approximation. Specializations are provided for the different discretization methods.
+ * \brief Advective fluxes according to Forchheims's law (extends Darcy's law by an nonlinear drag)
+ *
+ * Darcy’s law is linear in the seepage velocity v. As described in "darcyslaw.hh", this is valid for creeping flow (Re<<1).
+ * As v increases further, a nonlinear drag arises. The nonlinear drag arises, because the friction becomes comparable to surface drag due to friction.
+ * This additional friction is added as the forchheimer term to the Darcy's law. Resulting in the Forchheimer's law.
+ * see e.g. Nield & Bejan: Convection in Porous Media \cite nield2006
+ *
+ * For multiphase flow, the relative passability \f$ \eta_r\f$ is the "Forchheimer-equivalent" to the relative
+ * permeability \f$ k_r\f$.
+ * We use the same function for \f$ \eta_r\f$ as for \f$ k_r \f$ (Van-Genuchten, Brooks-Corey, linear), other authors use a simple
+ * power law e.g.: \f$\eta_{rw} = S_w^3\f$
+ *
+ * This leads to the equation in the form:
+ * \f[ \mathbf{v_\alpha} + c_F \sqrt{\mathbf{K}} \frac{\rho_\alpha}{\mu_\alpha }
+ * |\mathbf{v_\alpha}| \mathbf{v_\alpha}
+ * + \frac{k_{r \alpha}}{\mu_\alpha} \mathbf{K} \nabla \left(p_\alpha
+ * + \rho_\alpha g z \right)= 0
+ * \f]
+ * This already includes the assumption \f$ k_r(S_w) = \eta_r(S_w)\f$:
+ * - \f$\eta_{rw} = S_w^x\f$ looks very similar to e.g. Van Genuchten relative permeabilities
+ * - Fichot et al. (2006) \cite Fichot2006 state that several authors claim
+ * that \f$ k_r(S_w), \eta_r(S_w)\f$ can be chosen equal
+ * - It leads to the equation not degenerating for the case of \f$S_w=1\f$, because I do not
+ * need to multiply with two different functions, and therefore there are terms not being
+ * zero.
+ * - If this assumption is not to be made: Some regularization needs to be introduced ensuring
+ * that not all terms become zero for \f$S_w=1\f$.
+ *
+ * This non-linear equations is solved for \f$\mathbf{v_\alpha}\f$ using Newton's method
+ * and an analytical derivative w.r.t. \f$\mathbf{v_\alpha}\f$.
+ *
+ * The gradient of the Forchheimer relations looks as follows (mind that \f$\sqrt{\mathbf{K}}\f$
+ * is a tensor):
+ *
+ * \f[ f\left(\mathbf{v_\alpha}\right) =
+ * \left(
+ * \begin{array}{ccc}
+ * 1 & 0 &0 \\
+ * 0 & 1 &0 \\
+ * 0 & 0 &1 \\
+ * \end{array}
+ * \right)
+ * +
+ * c_F \frac{\rho_\alpha}{\mu_\alpha} |\mathbf{v}_\alpha| \sqrt{\mathbf{K}}
+ * +
+ * c_F \frac{\rho_\alpha}{\mu_\alpha}\frac{1}{|\mathbf{v}_\alpha|} \sqrt{\mathbf{K}}
+ * \left(
+ * \begin{array}{ccc}
+ * v_x^2 & v_xv_y & v_xv_z \\
+ * v_yv_x & v_{y}^2 & v_yv_z \\
+ * v_zv_x & v_zv_y &v_{z}^2 \\
+ * \end{array}
+ * \right)
+ * \f]
+ *
+ * \note We restrict the use of Forchheimer's law to diagonal permeability tensors so far. This might be changed to
+ * general tensors using eigenvalue decomposition to get \f$\sqrt{\mathbf{K}}\f$
+ * \note Forchheimer's law specialized for different discretization schemes (e.g. Box, CCTpfa).
+ * \note Forchheimer's law works with every model that contains Darcy's law.
*/
#ifndef DUMUX_FLUX_FORCHHEIMERS_LAW_HH
#define DUMUX_FLUX_FORCHHEIMERS_LAW_HH
diff --git a/dumux/flux/forchheimervelocity.hh b/dumux/flux/forchheimervelocity.hh
index 7ea9ae2974..188696fe1f 100644
--- a/dumux/flux/forchheimervelocity.hh
+++ b/dumux/flux/forchheimervelocity.hh
@@ -41,6 +41,8 @@ namespace Dumux {
/*!
* \ingroup Flux
* \brief Forchheimer's law
+ * For a detailed description see dumux/flow/forchheimerslaw.hh
+ *
* This file contains the calculation of the Forchheimer velocity
* for a given Darcy velocity
*/
@@ -60,65 +62,7 @@ class ForchheimerVelocity
using DimWorldVector = Dune::FieldVector<Scalar, dimWorld>;
using DimWorldMatrix = Dune::FieldMatrix<Scalar, dimWorld, dimWorld>;
- /*! \brief Compute the Forchheimer velocity of a phase at
- * the given sub-control volume face using the Forchheimer equation.
- *
- *
- * see e.g. Nield & Bejan: Convection in Porous Media \cite nield2006
- *
- * The relative passability \f$ \eta_r\f$ is the "Forchheimer-equivalent" to the relative
- * permeability \f$ k_r\f$.
- * We use the same function as for \f$ k_r \f$ (VG, BC, linear) other authors use a simple
- * power law e.g.: \f$\eta_{rw} = S_w^3\f$
- *
- * Some rearrangements have been made to the original Forchheimer relation:
- *
- * \f[ \mathbf{v_\alpha} + c_F \sqrt{\mathbf{K}} \frac{\rho_\alpha}{\mu_\alpha }
- * |\mathbf{v_\alpha}| \mathbf{v_\alpha}
- * + \frac{k_{r \alpha}}{\mu_\alpha} \mathbf{K} \nabla \left(p_\alpha
- * + \rho_\alpha g z \right)= 0
- * \f]
- *
- * This already includes the assumption \f$ k_r(S_w) = \eta_r(S_w)\f$:
- * - \f$\eta_{rw} = S_w^x\f$ looks very similar to e.g. Van Genuchten relative permeabilities
- * - Fichot, et al. (2006), Nuclear Engineering and Design, state that several authors claim
- * that \f$ k_r(S_w), \eta_r(S_w)\f$ can be chosen equal
- * - It leads to the equation not degenerating for the case of \f$S_w=1\f$, because I do not
- * need to multiply with two different functions, and therefore there are terms not being
- * zero.
- * - If this assumption is not to be made: Some regularization needs to be introduced ensuring
- * that not all terms become zero for \f$S_w=1\f$.
- *
- * This non-linear equations is solved for \f$\mathbf{v_\alpha}\f$ using Newton's method
- * and an analytical derivative w.r.t. \f$\mathbf{v_\alpha}\f$.
- *
- * The gradient of the Forchheimer relations looks as follows (mind that \f$\sqrt{\mathbf{K}}\f$
- * is a tensor):
- *
- * \f[ f\left(\mathbf{v_\alpha}\right) =
- * \left(
- * \begin{array}{ccc}
- * 1 & 0 &0 \\
- * 0 & 1 &0 \\
- * 0 & 0 &1 \\
- * \end{array}
- * \right)
- * +
- * c_F \frac{\rho_\alpha}{\mu_\alpha} |\mathbf{v}_\alpha| \sqrt{\mathbf{K}}
- * +
- * c_F \frac{\rho_\alpha}{\mu_\alpha}\frac{1}{|\mathbf{v}_\alpha|} \sqrt{\mathbf{K}}
- * \left(
- * \begin{array}{ccc}
- * v_x^2 & v_xv_y & v_xv_z \\
- * v_yv_x & v_{y}^2 & v_yv_z \\
- * v_zv_x & v_zv_y &v_{z}^2 \\
- * \end{array}
- * \right)
- * \f]
- *
- * \note We restrict the use of Forchheimer's law to diagonal permeability tensors so far. This might be changed to
- * general tensors using eigenvalue decomposition to get \f$\sqrt{\mathbf{K}}\f$
- */
+
template<class ElementVolumeVariables>
static DimWorldVector velocity(const FVElementGeometry& fvGeometry,
const ElementVolumeVariables& elemVolVars,
--
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