[material]

added a new thermal conductivity law (Johansen), with a better fit to
experimental data for lower water saturation

reviewed by martinb



git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@14548 2fb0f335-1f38-0410-981e-8018bf24f1b0

dumux/material/fluidmatrixinteractions/2p/thermalconductivityjohansen.hh

+ // -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+// vi: set et ts=4 sw=4 sts=4:
+/*****************************************************************************
+ *   See the file COPYING for full copying permissions.                      *
+ *                                                                           *
+ *   This program is free software: you can redistribute it and/or modify    *
+ *   it under the terms of the GNU General Public License as published by    *
+ *   the Free Software Foundation, either version 2 of the License, or       *
+ *   (at your option) any later version.                                     *
+ *                                                                           *
+ *   This program is distributed in the hope that it will be useful,         *
+ *   but WITHOUT ANY WARRANTY; without even the implied warranty of          *
+ *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the            *
+ *   GNU General Public License for more details.                            *
+ *                                                                           *
+ *   You should have received a copy of the GNU General Public License       *
+ *   along with this program.  If not, see <http://www.gnu.org/licenses/>.   *
+ *****************************************************************************/
+/*!
+ * \file
+ *
+ * \brief   Relation for the saturation-dependent effective thermal conductivity
+ */
+#ifndef THERMALCONDUCTIVITY_JOHANSEN_HH
+#define THERMALCONDUCTIVITY_JOHANSEN_HH
+
+#include <algorithm>
+#include <dumux/common/basicproperties.hh>
+#include <dumux/common/propertysystem.hh>
+
+namespace Dumux
+{
+
+struct JohansenIndices
+{
+    static const int wPhaseIdx = 0;
+    static const int nPhaseIdx = 1;
+};
+
+/*!
+ * \ingroup fluidmatrixinteractionslaws
+ *
+ * \brief Relation for the saturation-dependent effective thermal conductivity
+ *
+ * The Johansen method (Johansen 1975) computes the thermal conductivity of dry and the
+ * wet soil material and uses a root function of the wetting saturation to compute the
+ * effective thermal conductivity for a two-phase fluidsystem. The individual thermal
+ * conductivities are calculated as geometric mean of the thermal conductivity of the porous
+ * material and of the respective fluid phase.
+ * The material law is:
+ * \f[
+ \lambda_\text{eff} = \lambda_{\text{dry}} + \sqrt{(S_w)} \left(\lambda_\text{wet} - \lambda_\text{dry}\right)
+ \f]
+ *
+ * with
+ * \f[
+ \lambda_\text{wet} = \lambda_{solid}^{\left(1-\phi\right)}*\lambda_w^\phi
+ \f]
+ * and the semi-empirical relation
+ *
+ * \f[
+ \lambda_\text{dry} = \frac{0.135*\rho_s*\phi + 64.7}{\rho_s - 0.947 \rho_s*\phi}.
+ \f]
+ *
+ * Source: Phdthesis (Johansen1975) Johansen, O. Thermal conductivity of soils Norw. Univ. of Sci. Technol., Trondheim, Norway, 1975
+ */
+template<class Scalar, class Indices = JohansenIndices>
+class ThermalConductivityJohansen
+{
+public:
+    /*!
+     * \brief Returns the effective thermal conductivity \f$[W/(m K)]\f$ after Johansen (1975).
+     *
+     * \param volVars volume variables
+     * \param spatialParams spatial parameters
+     * \param element element (to be passed to spatialParams)
+     * \param fvGeometry fvGeometry (to be passed to spatialParams)
+     * \param scvIdx scvIdx (to be passed to spatialParams)
+     *
+     * \return Effective thermal conductivity \f$[W/(m K)]\f$ after Johansen (1975)
+     *
+     * This formulation is semi-empirical and fitted to quartz sand.
+     * This gives an interpolation of the effective thermal conductivities of a porous medium
+     * filled with the non-wetting phase and a porous medium filled with the wetting phase.
+     * These two effective conductivities are computed as geometric mean of the solid and the
+     * fluid conductivities and interpolated with the Kersten number.<br>
+     * Johansen, O. 1975. Thermal conductivity of soils. Ph.D. diss. Norwegian Univ.
+     *                    of Sci. and Technol., Trondheim. (Draft Transl. 637. 1977. U.S. Army
+     *                    Corps of Eng., Cold Regions Res. and Eng. Lab., Hanover, NH.)
+     */
+    template<class VolumeVariables, class SpatialParams, class Element, class FVGeometry>
+    static Scalar effectiveThermalConductivity(const VolumeVariables& volVars,
+                                               const SpatialParams& spatialParams,
+                                               const Element& element,
+                                               const FVGeometry& fvGeometry,
+                                               int scvIdx)
+    {
+        Scalar sw = volVars.saturation(Indices::wPhaseIdx);
+        Scalar lambdaW = volVars.fluidThermalConductivity(Indices::wPhaseIdx);
+        Scalar lambdaN = volVars.fluidThermalConductivity(Indices::nPhaseIdx);
+        Scalar lambdaSolid = volVars.solidThermalConductivity();
+        Scalar porosity = volVars.porosity();
+        Scalar rhoSolid = spatialParams.solidDensity(element, fvGeometry, scvIdx);
+
+        return effectiveThermalConductivity(sw, lambdaW, lambdaN, lambdaSolid, porosity, rhoSolid);
+    }
+
+    /*!
+     * \brief Returns the effective thermal conductivity \f$[W/(m K)]\f$ after Johansen (1975).
+     *
+     * \param sw The saturation of the wetting phase
+     * \param lambdaW the thermal conductivity of the wetting phase
+     * \param lambdaN the thermal conductivity of the non-wetting phase
+     * \param lambdaSolid the thermal conductivity of the solid phase
+     * \param porosity The porosity
+     *
+     * \return Effective thermal conductivity \f$[W/(m K)]\f$ after Johansen (1975)
+     */
+    static Scalar effectiveThermalConductivity(const Scalar Sw,
+                                               const Scalar lambdaW,
+                                               const Scalar lambdaN,
+                                               const Scalar lambdaSolid,
+                                               const Scalar porosity,
+                                               const Scalar rhoSolid)
+    {
+        const Scalar satW = std::max<Scalar>(0.0, Sw);
+
+        const Scalar kappa = 15.6; // fitted to medium quartz sand
+        const Scalar rhoBulk = rhoSolid*porosity;
+        const Scalar lSat = std::pow(lambdaSolid, (1.0 - porosity)) * std::pow(lambdaW, porosity);
+        const Scalar lDry = (0.135*rhoBulk + 64.7)/(rhoSolid - 0.947*rhoBulk);
+        const Scalar Ke = (kappa*satW)/(1+(kappa-1)*satW);// Kersten number, equation 13
+
+        return lDry + Ke * (lSat - lDry); // equation 14
+    }
+};
+}
+#endif