Tiddens

$-\Pi _{yx}=\eta ({\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}})$.

Gravitationskraft $F_{g}=\int _{V}{\vec {g}}\rho dV$.

Hydrostatische Druckkraft $F_{p}=-\int _{\partial V}pdA=-\int _{V}\Delta pdV$.

Reibungskraft $F_{v}=-\int _{\partial V}\Pi _{ij}dA=-\int _{V}\nabla \Pi _{ij}dV=\eta \int _{V}\nabla ^{2}{\vec {v}}dV$.

Massenerhaltung ${\frac {\partial }{\partial t}}m={\frac {\partial }{\partial t}}\int _{V}\rho dV=-\int _{\partial V}\rho {\vec {v}}d{\vec {A}}=-\int _{V}\Delta (\rho {\vec {v}})dV\Rightarrow {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {v}})=0$ mit rho const$\nabla \cdot {\vec {v}}=0$

Impulserhaltung ${\dot {p}}={\dot {m{\vec {v}}}}=m{\dot {\vec {v}}}+{\vec {v}}{\dot {m}}=m{\dot {\vec {v}}}=\int _{V}\rho dV{\frac {d{\vec {v}}}{dt}}=F={\vec {F}}_{g}+{\vec {F}}_{p}+{\vec {F}}_{v}=\int _{V}(\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}})dV\Rightarrow$

Navier-Stokes-Gleichung $\rho {\frac {d{\vec {v}}}{dt}}=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

Aus $df={\frac {\partial f}{\partial t}}dt+{\frac {\partial f}{\partial r}}dr\Rightarrow {\frac {df}{dt}}={\frac {\partial f}{\partial t}}+v{\frac {\partial f}{\partial r}}=\rho {\frac {d\theta }{dp}}{\frac {\partial p}{\partial t}}+\theta ({\frac {\partial \rho }{\partial p}}{\frac {\partial p}{\partial t}}+{\frac {\partial \rho }{\partial C}}{\frac {\partial C}{\partial t}})$ folgt:

Navier-Stokes-Gleichung $\rho {\frac {\partial {\vec {v}}}{\partial t}}+\rho ({\vec {v}}\cdot \nabla ){\vec {v}}=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

Falls (kompressibles Fluid) $\nabla \cdot {\vec {v}}\neq 0\Rightarrow \eta \nabla ^{2}{\vec {v}}\rightarrow \eta \nabla ^{2}{\vec {v}}+{\frac {1}{3}}\eta \nabla (\nabla \cdot {\vec {v}})$

laminarer Fluss $\rho {\frac {\partial {\vec {v}}}{\partial t}}=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

stationärer Fluss $0=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

Porosität $\Phi ={\frac {V_{p}}{V}}$

Trockendichte $\rho _{b}=\rho _{s}(1-{\frac {V_{V}}{V}})=\rho _{s}(1-\Phi )$

Volumetrischer Fluidgehalt $\theta ={\frac {V_{Fluid}}{V}}$

gravimetrischer Fluidgehalt $\theta ={\frac {M_{Fluid}}{M_{Matrix}}}$

Sättigung: $S={\frac {V_{w}}{V_{p}}}={\frac {\theta }{\Phi }}$

Potentielle Energie eines Fluidelements: $\Psi =p-p_{0}+\rho g(z-z_{0})=\Psi _{p}+\Psi _{g}$

Potentielle Energie pro Einheitsgewicht $h:=\Psi {\frac {V}{gm}}={\frac {\Psi }{\rho g}}={\frac {p-p_{0}}{\rho g}}+(z-z_{0})$

Darcy Gesetz: ${\frac {Q}{A}}=K{\frac {h_{1}-h_{2}}{L}}=q=-K\nabla h$

laminarer, langsamer, stationärer Fluss mit g=0 $\nabla p=\eta \nabla ^{2}{\vec {v}}\Rightarrow \nabla ^{2}{\vec {v}}^{\mu }={\frac {1}{\eta }}\nabla p^{\mu }$

$\rightarrow$(Haagen-Poisuille)$\Rightarrow {\vec {v}}^{\mu }=-{\frac {k^{\mu }}{\eta }}\nabla p^{\nu }$

makroskopische Betrachtung (Darcy-Gesetz für Newtonsche Fluide) ${\vec {q}}A=<{\vec {v}}^{\mu }>A_{W}$ mit ${\frac {A_{W}}{A}}={\frac {V_{W}}{V}}=\theta$$\Rightarrow {\vec {q}}=\theta <{\vec {v}}^{\mu }>=-{\frac {\theta }{\eta }}<k^{\mu }\nabla p^{\mu }>$and$\theta <k^{\mu }\nabla p^{\mu }>\rightarrow k\nabla p$ and $<{\vec {v}}^{\mu }>\rightarrow {\vec {v}}$$\Rightarrow q=-{\frac {k}{\eta }}\nabla p$

${\vec {q}}=\theta {\vec {v}}\rightarrow _{(gesaettigt)}{\vec {q}}=\Phi {\vec {v}}$

allgemeine Gleichung für g≠0${\vec {q}}=-{\frac {k}{\eta }}(\nabla p-\rho {\vec {g}})=_{g=const}-{\frac {k}{\eta }}\nabla (p+\rho gz)=-{\frac {k}{\eta }}\nabla \Psi =-{\frac {k}{\eta }}\nabla (h\rho g)=-{\frac {k\rho g}{\eta }}\nabla (h)$

Reynoldszahl: ${\mathit {Re}}={\frac {\varrho \cdot v\cdot d}{\eta }}$

Massenkontinuitätsgleichung ${\frac {\partial }{\partial t}}(V_{W}\rho )={\frac {\partial }{\partial t}}\int _{V}\rho dV=-\int \nabla (\rho {\vec {v}}^{\mu })dV=-\nabla \cdot \int \rho {\vec {v}}^{\mu }dV=-\nabla V_{W}\rho <{\vec {v}}^{\mu }>\Rightarrow {\frac {\partial }{\partial t}}(\theta \rho )+\nabla \cdot (\rho q)=0$

für gesättigte Verhältnisse ${\frac {\partial }{\partial t}}(\theta \rho )=_{\rho =const}=\rho {\frac {\partial \theta (p)}{\partial t}}=\rho {\frac {d\theta }{dp}}{\frac {\partial p}{\partial t}}$ wobei ${\frac {d\theta }{dp}}=c_{W}(p)$: Wasserkapazitätsfunktion; integriert man diese erhält man die Wasserretentionskurve $\sigma (p)$

für ungesättigte Verhältnisse ${\frac {\partial }{\partial t}}(\theta \rho )=\rho (\theta \kappa _{p,W}+(1-\theta )\kappa _{p,s}){\frac {\partial p}{\partial t}}$ wobei $\theta \kappa _{p,W}+(1-\theta )\kappa _{p,s}=s$; $\kappa _{p,W}$:Wasser-Kompressibilitätskoeffizient; $\kappa _{p,s}$:Matrix-Kompressibilitätskoeffizient; $\Rightarrow {\frac {\partial }{\partial t}}(\theta \rho )=\rho s{\frac {\partial p}{\partial t}}$

Richardsgleichung ${\frac {\partial }{\partial t}}(\theta \rho )+\nabla \cdot (\rho q)=0$ und ${\vec {q}}=-K\nabla h$ und ${\vec {q}}=-{\frac {k}{\eta }}(\nabla p-\rho q)$ $\Rightarrow \nabla \cdot (\rho {\frac {k}{\eta }}(\nabla p-\rho {\vec {g}}))={\frac {\partial }{\partial t}}(\theta \rho )$ und $\nabla \cdot (K\nabla h)={\frac {\partial \theta }{\partial t}}$

Spezialfälle der Richardsgleichung

für gespannten Aquifer: Kontinuitätsgleichgung mit s $\nabla \cdot (K\nabla h)=s_{s}{\frac {\partial h}{\partial t}}$ mit $s_{s}:=s\rho g$

mit einem zusätzlichen Quell- bzw. Senkenterm $\nabla \cdot (K\nabla h)={\frac {\partial \theta }{\partial t}}+\gamma$

für Anisotropie $K\rightarrow {\vec {\vec {K}}}$ d.h. $\nabla \cdot ({\vec {\vec {K}}}\nabla h)=s_{s}{\frac {\partial h}{\partial t}}$

stationäre Bedingungen, isotroper aquifer $\nabla \cdot (K\nabla )=0$

homogener, aber anisotroper Aquifer ${\vec {\vec {K}}}\rightarrow K_{x},K_{y},K_{z}$ d.h. $K_{x}{\frac {\partial ^{2}h}{\partial x^{2}}}+K_{y}{\frac {\partial ^{2}h}{\partial y^{2}}}+K_{z}{\frac {\partial ^{2}h}{\partial z^{2}}}=s_{s}{\frac {\partial h}{\partial t}}$

homogener, isotroper Aquifer ${\vec {\vec {K}}}\rightarrow K$ d.h. ${\frac {K}{s_{s}}}({\frac {\partial ^{2}h}{\partial x^{2}}}+{\frac {\partial ^{2}h}{\partial y^{2}}}+{\frac {\partial ^{2}h}{\partial z^{2}}})={\frac {K}{s_{s}}}\nabla ^{2}h={\frac {\partial h}{\partial t}}$

homogener, isotroper Aquifer mit stationären Bedingungen $\nabla ^{2}h=0$

homogener, isotroper Aquifer mit stationären Bedingungen mit Quell-Term (Laplace Gleichung) $\nabla ^{2}h=\gamma$

Laplace Gleichung (Oberflächenspannung): $p-p_{Luft}=\Delta p={\frac {2\sigma }{r}}$

Buckingham-Darcy $q=-K(\theta )(\nabla h_{p}+{\hat {\vec {z}}})$

Kappilares Steigen $H={\frac {2\sigma \cos \gamma }{\rho gR}}$

Matrixpotential $p-p_{0}=\Psi _{p}=\Psi _{m}={\frac {2\theta }{r}}$

Wassergehalt ist abhängig vom Matrix Potential $\theta =\theta (\Psi _{m})$

Richards-Gleichung mit Wasserkapazitätfunktion

1-${\frac {\partial }{\partial t}}(\theta \rho )=_{\rho =const}=\rho {\frac {d\theta }{dp}}{\frac {\partial p}{\partial t}}\Rightarrow {\frac {\partial \theta }{\partial t}}={\frac {d\theta }{dp}}{\frac {\partial p}{\partial t}}$

2- ${\frac {\partial }{\partial t}}(\theta \rho )+\nabla (\rho q)=0\Rightarrow {\frac {\partial \theta }{\partial t}}=-\nabla q$

$-\nabla \cdot q={\frac {d\theta }{dp}}{\frac {\partial p}{\partial t}}=C_{W}(p){\frac {\partial p}{\partial t}}=C_{W}(\Psi _{m}){\frac {\Psi _{w}}{t}}$

1. Fick'sches Gesetz im freien Fluid ${\vec {J}}_{dif}=-D_{W}\nabla C$

1. Fick'sches Gesetz im porösen Medium ${\vec {J}}_{dif}=-\theta D'_{W}\nabla C$ wobei $D'_{w}={\frac {D_{w}}{\tau }}<D_{w}$ wobei $\tau$: die Toruosität ist

Herleitung 2.Fick'sches Gesetz (Massenerhaltung) ${\frac {\partial }{\partial t}}V_{w}C={\frac {\partial }{\partial t}}\int _{V}CdV=-\int _{\partial V}{\vec {J}}'_{dif}d{\vec {A}}=-\int _{V}\nabla \cdot {\vec {J}}'_{dif}dV=-v_{w}D'_{w}\nabla C\Rightarrow$ mit ${\frac {v_{W}}{V}}=\theta \Rightarrow$${\frac {\partial }{\partial t}}(\theta C)+\nabla \cdot {\vec {J}}'_{dif}=0$$\Rightarrow _{mitD'_{w}und\theta =\psi =const}$${\frac {\partial C}{\partial t}}=-D'_{w}\nabla ^{2}C$

Advektion ${\vec {J}}_{adv}={\vec {v}}C=<{\vec {v}}^{\mu }>C$

hydrondynamischer Dispersionstensor (2. Stufe) $D_{ij}=\alpha _{ijkl}{\frac {v_{k}v_{l}}{|{\vec {v}}|}}+D'_{w}$ $\alpha _{ijkl}$ $\alpha _{ijkl}\rightarrow \alpha _{L};\alpha _{T}$$D_{ij}$

Stofffluss aufgrund von hydrodynamischer Dispersion ${\vec {J}}_{dis}=-{\vec {\vec {D}}}\nabla C$

Advektions-Dispersions-Gleichung ${\frac {\partial }{\partial t}}\int _{V}CdV=-\int _{V}\nabla \cdot {\vec {J}}_{dis}dV-\int _{V}\nabla \cdot {\vec {J}}_{adv}dV\Rightarrow {\frac {\partial }{\partial t}}V_{w}C=-\nabla \cdot \int _{V}{\vec {J}}_{dis}dV-\nabla \cdot \int _{V}{\vec {J}}_{adv}dV=-\nabla \cdot (-V_{w}{\vec {\vec {D}}}\nabla C)-\nabla \cdot (V_{w}{\vec {v}}C)$

${\frac {\partial }{\partial t}}(\theta C)=\nabla \cdot (\theta {\vec {\vec {D}}}\nabla C)-\nabla \cdot (qC)$

Beispiel gesättigt/konstante Porosität/inkompresibles Medium ${\frac {\partial C}{\partial t}}=\nabla \cdot ({\vec {\vec {D}}}\nabla C)-\nabla \cdot ({\vec {v}}C)=\nabla \cdot ({\vec {\vec {D}}}\nabla C)-({\vec {v}}\nabla C+C\nabla \cdot {\vec {v}})=\nabla \cdot ({\vec {\vec {D}}}\nabla C)-{\vec {v}}\nabla C$

1D Stofftransport in 1D Fließfeld bei konst. longitudinaler Dispersivität ${\frac {\partial C}{\partial t}}=D_{L}{\frac {\partial ^{2}C}{\partial x^{2}}}-v_{x}{\frac {\partial C}{\partial x}}$

Stofftransport mit Dichte Abhängigem Fluss $\rho =\rho _{0}+\beta (C-C_{0})$

aus $\nabla \cdot (\rho {\frac {k}{\eta }}(\nabla p-\rho {\vec {g}}))={\frac {\partial }{\partial t}}(\theta \rho )=\rho {\frac {\partial }{\partial t}}(\theta )+\theta {\frac {\partial }{\partial t}}(\rho )=\rho {\frac {\partial \theta }{\partial t}}+\theta {\frac {\partial \rho }{\partial t}}=\rho ({\frac {\partial \theta }{\partial p}}{\frac {\partial p}{\partial t}})+\theta ({\frac {\partial \rho }{\partial p}}{\frac {\partial p}{\partial t}}+{\frac {\partial \rho }{\partial C}}{\frac {\partial C}{\partial t}})$

Wärmefluss Fourier Gesetz: $q_{T}=-\lambda \nabla T$

Energieerhaltung $\rho c_{p}{\frac {\partial T}{\partial t}}+\nabla \cdot q_{T}=0$

Stromfluss Ohm'sches Gestetz ${\vec {j}}=-\sigma \nabla \theta$

Ladungserhaltung ${\frac {\partial \rho _{c}}{\partial }}+\nabla \cdot {\vec {j}}=0$

Innere Oberfläche Volumenbezogene spezifische Oberfläche

Korn(Kugel):$S_{tot}={\frac {S}{V}}={\frac {4\pi R^{2}}{(2R)^{3}}}={\frac {\pi }{2R}};R\rightarrow \mu m$ Tonplättchen:$S_{tot}={\frac {S}{V}}={\frac {2\pi R^{2}+2\pi RD}{(2R)^{2}D}}\approx _{(mitD<<R)}{\frac {\pi }{2D}};D\rightarrow nm$ Massenbezogene spezifische Oberfläche $S_{M}={\frac {S}{M}}={\frac {S}{\rho V}}\approx {\frac {\pi }{2}}{\frac {1}{\rho D}}\approx _{beiTon}{\frac {cm^{3}}{gnm}}={\frac {10^{3}m^{2}}{g}}\approx _{beiSand}0.1{\frac {m^{2}}{g}}$

Effektive Porosität $\Phi _{effektiv}=\Phi _{total}-\Phi _{Haftwasser}$

Porosität im Porenkanalmodell $\Phi ={\frac {\pi r^{2}l}{L^{3}}}=\pi \tau ({\frac {r}{L}})^{2}$ wobei $\tau ={\frac {l}{L}}$: Tortuisität

Für kreisförmige Poren:$k={\frac {\Phi r^{2}}{8\tau ^{2}}}={\frac {\Phi }{2S_{por}^{2}\tau ^{2}}};S_{por}={\frac {2}{r}}$ Für allgmeine Porenform:$k={\frac {\Phi }{\chi S_{por}^{2}\tau ^{2}}}$

Elektrische Leitfähigkeit/spezifischer elektrischer Wiederstand $\sigma ={\frac {1}{\rho }}$

$\sigma$$\rho$

$\epsilon =\epsilon _{r}\epsilon _{0}$;

$\epsilon$$\epsilon _{r}$$\epsilon _{0}$

Ohmsche Gesetz: ${\vec {j}}=\sigma {\vec {E}}$

Maxwell ${\vec {D}}=\epsilon {\vec {E}}$

Archie's Law $F={\frac {\rho _{b,s}}{\rho _{w}}}={\frac {a}{\phi ^{m}}}$

Leitfähigkeit des Wassers $\sigma _{w}=nq\mu$; $\sigma _{w}$

Leitfähigkeit aufgrund von Grenzflächenleitfähigkeit $\sigma _{b}=\sigma _{electrolyt}+\sigma _{grenzleitf}=\sigma _{b,s}+\sigma _{s}={\frac {\sigma _{w}}{F}}+\sigma _{s}$

4. Maxwellgleichung $\nabla \times {\vec {H}}={\vec {j}}+{\frac {\partial {\vec {D}}}{\partial t}}{\stackrel {\mathrm {({\vec {j}}=\sigma {\vec {E}})} }{=}}\sigma {\vec {E}}+{\frac {\partial }{\partial t}}(\epsilon {\vec {E}}){\stackrel {\mathrm {(mit{\vec {E}}\sim e^{i\omega t})} }{=}}\sigma {\vec {E}}+i\omega \epsilon {\vec {E}}=(\sigma +i\omega \epsilon ){\vec {E}}=\sigma ^{*}{\vec {E}}(=i\omega \epsilon ^{*}{\vec {E}})$

CRIM - (complex refractive index method) Modell ${\frac {1}{V_{ef}}}={\frac {1-\phi }{V_{m}}}+{\frac {\phi }{V_{f}}}$ $V={\frac {c}{\sqrt {\epsilon _{r}}}}$ $\Rightarrow {\sqrt {\epsilon _{ef}}}=(1-\phi ){\sqrt {\epsilon _{m}}}+\phi {\sqrt {\epsilon _{f}}}$

Grenzflächenleitfähigkeit $\sigma _{s}={\frac {2\mu _{s}Q_{s}}{F\Lambda }}$

Membranpolarisation $\tau \sim {\frac {r^{2}}{D}}$

komplexe (frequenzabhängige) Leitfähigkeit $\sigma ^{*}=|\sigma |e^{i\phi }=[\sigma _{el}+\sigma '_{surf}(\omega )]+i\sigma _{surf}''(\omega )$