@@ -90,7 +90,7 @@ The governing equations are derived in exercise two.
...
@@ -90,7 +90,7 @@ The governing equations are derived in exercise two.
\item When does the gas phase reach the lower boundary? \\
\item When does the gas phase reach the lower boundary? \\
{\em You can see how the gas quickly expands due to capillary diffusion and then stays immobile afterwards. The initial capillary pressure at the interface has to be larger than the entry pressure of the lower permeable layer. The capillary pressure depends on the saturation, i.e., only above a specific non-wetting phase saturation can the gas enter the layer. In this layer the capillary forces lead to a further radial distribution. If there exists a residual saturation the gas will probably not reach the lower boundary.}
{\em You can see how the gas quickly expands due to capillary diffusion and then stays immobile afterwards. The initial capillary pressure at the interface has to be larger than the entry pressure of the lower permeable layer. The capillary pressure depends on the saturation, i.e., only above a specific nonwetting phase saturation can the gas enter the layer. In this layer the capillary forces lead to a further radial distribution. If there exists a residual saturation the gas will probably not reach the lower boundary.}
\item How does a change of permeability, porosity, residual saturation or the Brooks-Corey parameters affect the breakthrough time? \\
\item How does a change of permeability, porosity, residual saturation or the Brooks-Corey parameters affect the breakthrough time? \\
{\em see above}
{\em see above}
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@@ -169,7 +169,7 @@ Alltogether this leads to 15 unknowns and 13 equations plus the two transport eq
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@@ -169,7 +169,7 @@ Alltogether this leads to 15 unknowns and 13 equations plus the two transport eq
as long as $0 < S_\alpha < 1$.
as long as $0 < S_\alpha < 1$.
\item Only wetting phase is present: The mass fraction of, e.g., N$_2$ in the wetting phase $X^n_l$ is used,
\item Only wetting phase is present: The mass fraction of, e.g., N$_2$ in the wetting phase $X^n_l$ is used,
as long as the maximum mass fraction is not exceeded ($X^n_l<X^n_{l,max}$)
as long as the maximum mass fraction is not exceeded ($X^n_l<X^n_{l,max}$)
\item Only non-wetting phase is present: The mass fraction of, e.g., water in the non-wetting phase, $X^w_g$, is used,
\item Only nonwetting phase is present: The mass fraction of, e.g., water in the nonwetting phase, $X^w_g$, is used,
as long as the maximum mass fraction is not exceeded ($X^w_g<X^w_{g,max}$)
as long as the maximum mass fraction is not exceeded ($X^w_g<X^w_{g,max}$)
@@ -18,9 +18,9 @@ FineBrooksCoreyEntryPressure = 400 # entry pressure for the Brooks-Core
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@@ -18,9 +18,9 @@ FineBrooksCoreyEntryPressure = 400 # entry pressure for the Brooks-Core
CoarseBrooksCoreyLambda = 2.0 # pore size distribution parameter for the Brooks-Corey capillary pressure-saturation relationship in the coarse soil [-]
CoarseBrooksCoreyLambda = 2.0 # pore size distribution parameter for the Brooks-Corey capillary pressure-saturation relationship in the coarse soil [-]
CoarseBrooksCoreyEntryPressure = 200 # entry pressure for the Brooks-Corey capillary pressure-saturation relationship in the coarse soil [Pa]
CoarseBrooksCoreyEntryPressure = 200 # entry pressure for the Brooks-Corey capillary pressure-saturation relationship in the coarse soil [Pa]
FineResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the fine soil [-]
FineResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the fine soil [-]
FineResidualSaturationNonWetting = 0.3 # residual saturation of the non-wetting phase in the fine soil [-]
FineResidualSaturationNonwetting = 0.3 # residual saturation of the nonwetting phase in the fine soil [-]
CoarseResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the coarse soil [-]
CoarseResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the coarse soil [-]
CoarseResidualSaturationNonWetting = 0.1 # residual saturation of the non-wetting phase in the coarse soil [-]
CoarseResidualSaturationNonwetting = 0.1 # residual saturation of the nonwetting phase in the coarse soil [-]
[Boundary]
[Boundary]
LowerPressure = 2.0e5 # Dirichlet pressure value for the boundary condition at the lower boundary [Pa]
LowerPressure = 2.0e5 # Dirichlet pressure value for the boundary condition at the lower boundary [Pa]
@@ -24,9 +24,9 @@ FineBrooksCoreyEntryPressure = 400 # entry pressure for the Brooks-Core
...
@@ -24,9 +24,9 @@ FineBrooksCoreyEntryPressure = 400 # entry pressure for the Brooks-Core
CoarseBrooksCoreyLambda = 2.0 # pore size distribution parameter for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [-]
CoarseBrooksCoreyLambda = 2.0 # pore size distribution parameter for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [-]
CoarseBrooksCoreyEntryPressure = 200 # entry pressure for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [Pa]
CoarseBrooksCoreyEntryPressure = 200 # entry pressure for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [Pa]
FineResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the fine soil [-]
FineResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the fine soil [-]
FineResidualSaturationNonWetting = 0.3 # residual saturation of the non-wetting phase in the fine soil [-]
FineResidualSaturationNonwetting = 0.3 # residual saturation of the nonwetting phase in the fine soil [-]
CoarseResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the coarse soil [-]
CoarseResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the coarse soil [-]
CoarseResidualSaturationNonWetting = 0.1 # residual saturation of the non-wetting phase in the coarse soil [-]
CoarseResidualSaturationNonwetting = 0.1 # residual saturation of the nonwetting phase in the coarse soil [-]
@@ -19,9 +19,9 @@ FineBrooksCoreyEntryPressure = 500 # entry pressure for the Brooks-Core
...
@@ -19,9 +19,9 @@ FineBrooksCoreyEntryPressure = 500 # entry pressure for the Brooks-Core
CoarseBrooksCoreyLambda = 2.0 # pore size distribution parameter for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [-]
CoarseBrooksCoreyLambda = 2.0 # pore size distribution parameter for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [-]
CoarseBrooksCoreyEntryPressure = 200 # entry pressure for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [Pa]
CoarseBrooksCoreyEntryPressure = 200 # entry pressure for the Brooks-Corey capillary pressure - saturation relationship in the coarse soil [Pa]
FineResidualSaturationWetting = 0.18 # residual saturation of the wetting phase in the fine soil [-]
FineResidualSaturationWetting = 0.18 # residual saturation of the wetting phase in the fine soil [-]
FineResidualSaturationNonWetting = 0.0 # residual saturation of the non-wetting phase in the fine soil [-]
FineResidualSaturationNonwetting = 0.0 # residual saturation of the nonwetting phase in the fine soil [-]
CoarseResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the coarse soil [-]
CoarseResidualSaturationWetting = 0.05 # residual saturation of the wetting phase in the coarse soil [-]
CoarseResidualSaturationNonWetting = 0.0 # residual saturation of the non-wetting phase in the coarse soil [-]
CoarseResidualSaturationNonwetting = 0.0 # residual saturation of the nonwetting phase in the coarse soil [-]
[Boundary]
[Boundary]
LowerPressure = 1.19612e5 # Dirichlet pressure value for the boundary condition at the lower boundary [Pa]
LowerPressure = 1.19612e5 # Dirichlet pressure value for the boundary condition at the lower boundary [Pa]
@@ -20,7 +20,7 @@ strongly dependent on the linearity.}
...
@@ -20,7 +20,7 @@ strongly dependent on the linearity.}
\item What is the influence of viscosity on the front? Explain!
\item What is the influence of viscosity on the front? Explain!
{\em The larger the ratio of the water viscosity to the NAPL viscosity
{\em The larger the ratio of the water viscosity to the NAPL viscosity
($\frac{\mu_w}{\mu_n}$), the slower travels the front and the higher are the
($\frac{\mu_w}{\mu_n}$), the slower travels the front and the higher are the
water saturations. See Figure \ref{Visc}. The ``middle'' curve shows always the case with equal viscosities. The more bulky, slower front belongs in both figures to the case where the wetting-phase (injected) has a higher viscosity than the non-wetting-phase.}
water saturations. See Figure \ref{Visc}. The ``middle'' curve shows always the case with equal viscosities. The more bulky, slower front belongs in both figures to the case where the wetting-phase (injected) has a higher viscosity than the nonwetting-phase.}
\begin{figure}[h]
\begin{figure}[h]
\begin{minipage}[b]{0.45\linewidth}
\begin{minipage}[b]{0.45\linewidth}
\centering
\centering
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@@ -32,7 +32,7 @@ water saturations. See Figure \ref{Visc}. The ``middle'' curve shows always the
...
@@ -32,7 +32,7 @@ water saturations. See Figure \ref{Visc}. The ``middle'' curve shows always the
\caption{{\it Influence of viscosity on the problem. Wetting phase is injected. Left: $\mu_w=const=10^{-3}$ Pa s, non-wetting phase viscosity is varied. Right: $\mu_n=const=10^{-3}$ Pa s, wetting phase viscosity is varied. }}
\caption{{\it Influence of viscosity on the problem. Wetting phase is injected. Left: $\mu_w=const=10^{-3}$ Pa s, nonwetting phase viscosity is varied. Right: $\mu_n=const=10^{-3}$ Pa s, wetting phase viscosity is varied. }}