From 00aa1e9cd44ee331bb1d1ab665d30f08576f9384 Mon Sep 17 00:00:00 2001 From: Omotayo Omosebi Date: Wed, 7 Aug 2019 18:04:14 -0400 Subject: [PATCH] Explains formulation of the saturation equation --- examples/step-21/doc/intro.dox | 26 +++++++++++++++++--------- 1 file changed, 17 insertions(+), 9 deletions(-) diff --git a/examples/step-21/doc/intro.dox b/examples/step-21/doc/intro.dox index d1e082b700..543c69f6a7 100644 --- a/examples/step-21/doc/intro.dox +++ b/examples/step-21/doc/intro.dox @@ -76,8 +76,22 @@ can solve right away with the techniques of the first few tutorial programs similar). However, we have not said anything yet about the saturation, which of course is going to change as the fluids move around. -The second part of the equations is a therefore description of the -dynamics of the saturation. We model this as an advected quantity: +The second part of the equations is the description of the +dynamics of the saturation. The saturation equation for the displacing fluid (water) is: +@f{eqnarray*} + S_{t} + \nabla \cdot (F(S) \mathbf{u}) = \{q}_{w}, + \\ + S_{t} + F(S) \nabla \mathbf{u} + \mathbf{u} \cdot \nabla F(S) = S_{t} + F(S) * \q + \mathbf{u} \cdot \nabla F(S) = \{q}_{w}. +@f} +where $\{q}_{w}$ is the flow rate of the displacing fluid (water) and is related to the fractional flow F(S) through: +@f[ + \{q}_{w} = F(S) * \q, + \\ + F(S) + = + \frac{k_{rw}(S)/\mu_{w}}{k_{rw}(S)/\mu_{w} + k_{ro}(S)/\mu_{o}}. +@f] +Thus, we obtain the saturation equation in the following advected form: @f{eqnarray*} S_{t} + \mathbf{u} \cdot \nabla F(S) = 0, @f} @@ -86,12 +100,6 @@ where $\mathbf u$ is the total velocity \mathbf{u} = \mathbf{u}_{o} + \mathbf{u}_{w} = -\lambda(S) \mathbf{K}\cdot\nabla p. @f] -In addition, -@f[ - F(S) - = - \frac{k_{rw}(S)/\mu_{w}}{k_{rw}(S)/\mu_{w} + k_{ro}(S)/\mu_{o}} -@f] Note that the advection equation contains the term $\mathbf{u} \cdot \nabla F(S)$ rather than $\mathbf{u} \cdot \nabla S$ to indicate that the saturation is not simply transported along; rather, since the two phases move with @@ -108,7 +116,7 @@ In summary, what we get are the following two equations: - \nabla \cdot (\mathbf{K}\lambda(S) \nabla p) &=& q \qquad \textrm{in}\ \Omega\times[0,T], \\ - S_{t} + \nabla (F(S) \cdot \mathbf{u}) &=& 0 + S_{t} + \mathbf{u} \cdot \nabla F(S) &=& 0 \qquad \textrm{in}\ \Omega\times[0,T]. @f} Here, $p=p(\mathbf x, t), S=S(\mathbf x, t)$ are now time dependent -- 2.39.5