Ideally balanced water + dead oil 1D waterflood model without gravity and capillary effects.

(1)	$\begin{array}{l}\displaystyle \frac{\partial s}{\partial t} + q \, \frac{\partial }{\partial x} \left( \frac{f}{\phi \, A } \right) = 0\end{array}$

(2)	$\begin{array}{l}\displaystyle s(t=0,x) = 0\end{array}$

(3)	$\begin{array}{l}\displaystyle s(t,0) = 1\end{array}$

where

$\begin{array}{l}\displaystyle s= E_D = \frac{s_w - s_{wi}}{1-s_{wi}-s_{or}}\end{array}$	displacement efficiency
$\begin{array}{l}q\end{array}$	sandface injection rate, assumed equal to sandface liquid production rate
$\begin{array}{l}\phi(x)\end{array}$	reservoir porosity
$\begin{array}{l}A(x) = h \, L_D\end{array}$	cross-section area available for flow
$\begin{array}{l}h(x)\end{array}$	reservoir thickness
$\begin{array}{l}L_D(x)\end{array}$	rerservoir width = reservoir length transversal to flow
$\begin{array}{l}\displaystyle f = \frac{1}{1+M_{ro}/M_{rw}}\end{array}$	in-situ fractional flow function
$\begin{array}{l}M_{ro}= k_{ro}(s_o)/\mu_o\end{array}$	relative oil mobility
$\begin{array}{l}M_{wo} = k_{rw}(s_w)/\mu_w\end{array}$	relative water mobility

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