@wikipedia


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

\frac{\partial s}{\partial t} + \frac{q}{\phi \, \Sigma} \cdot \frac{\partial f}{\partial x}  = 0
s(t=0,x) = 0
s(t,0) = 1

where

water → oil displacement efficiency

sandface injection rate, assumed equal to sandface liquid production rate

reservoir porosity

cross-section area available for flow

reservoir thickness

reservoir width = reservoir length transversal to flow

in-situ fractional flow function


relative oil mobility

relative water mobility


Approximations


In many practical applications (for example, laboratory SCAL tests and reservoir proxy-modeling) one can assume constant porosity and reservoir width: 

\frac{\partial s}{\partial t_D} +\frac{\partial f}{\partial x_D} = 0
s(t=0,x) = 0
s(t,0) = 1

where

dimensionless time

dimensionless distance between injector and producer

reservoir length along -axis

mobile reservoir pore volume


See Also

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Reservoir Flow Model (RFM)

[ Production / Subsurface Production / Reserves Depletion /  Recovery Methods ]


Reference

Buckley-Leverett.xlsx


The equation  can be explicitly integrated:

x_D(s) = \begin{cases}\dot f(s) \cdot t_D, & \mbox{if } s < s^*\\ 2 x^*_D- \dot f(s) \cdot t_D, & \mbox{if } s \geq s^*\end{cases}

where

critical saturation where fractional flow function reaches inflection point: 

 "inflection" distance   

"inflection" time

first and second derivatives of the fractional flow function


Algebraic equation  can be used to find a solution of  in terms of saturation over time and distance:  (see Fig. 1).


Fig. 1 – Sample case of  Buckley–Leverett reservoir saturation profile  capturing the moment,

when water front is still on mid-way towards the producing well, sitting at .