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

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

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

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

where

$\begin{array}{l}\displaystyle s= E_D = \frac{s_w - s_{wi}}{1-s_{wi}-s_{or}}\end{array}$	water → oil 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
$\begin{array}{l}\Sigma(x) = h \, D\end{array}$	cross-section area available for flow
$\begin{array}{l}h(x)\end{array}$	reservoir thickness
$\begin{array}{l}D(x)\end{array}$	reservoir 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

Approximations

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

(4)	$\begin{array}{l}\displaystyle \frac{\partial s}{\partial t_D} +\frac{\partial f}{\partial x_D} = 0\end{array}$

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

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

where

$\begin{array}{l}\displaystyle t_D = \frac{V_\phi \, t}{q}\end{array}$	dimensionless time
$\begin{array}{l}\displaystyle x_D = \frac{x}{L}\end{array}$	dimensionless distance
$\begin{array}{l}L\end{array}$	reservoir length along $\begin{array}{l}x\end{array}$ -axis
$\begin{array}{l}V_\phi= \phi \cdot h \cdot D \cdot L\end{array}$	reservoir pore volume

The equation (4) can be explicitly integrated:

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

where

$\begin{array}{l}s^*\end{array}$	critical saturation where fractional flow function reaches inflection point: $\begin{array}{l}\ddot f(s^*) = 0\end{array}$
$\begin{array}{l}x^_D= f(s^) \cdot t_D^*\end{array}$	"inflection" distance
$\begin{array}{l}t_D^*\end{array}$	"inflection" time
$\begin{array}{l}\displaystyle \dot f(s) = \frac{d f}{ds}, \, \, \ddot f(s) = \frac{d^2 f}{ds^2}\end{array}$	first and second derivatives of the fractional flow function

Algebraic equation (7) can be used to find a solution of (4) in terms of saturation over time and distance: $\begin{array}{l}s(t,x)\end{array}$ (see Fig. 1).

Fig. 1 – Sample case of reservoir saturation profile $\begin{array}{l}s(t,x)\end{array}$ capturing themoment when water front is still in a mid-way t the producing well (sitting at $\begin{array}{l}x_D = 1 \Leftrightarrow x = L\end{array}$ )

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Approximations

See Also

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Buckley–Leverett @model

Approximations

See Also