Modelling facility for field-average saturation $\begin{array}{l}s(t) = \{ s_w(t), \, s_o(t), \, s_g(t) \}\end{array}$ and formation pressure $\begin{array}{l}p(t)\end{array}$ at any time moment $\begin{array}{l}t\end{array}$ as response to production flowrates history:

(1)	$\begin{array}{l}\displaystyle \frac{ds_w}{dt} =\frac{1}{A_e \, h_e \, \phi_e(p)} \left[ q^{\downarrow}_w(t) - q^{\uparrow}_w(t) + q^{\downarrow}_{WAQ}(t) \right] - \left[ c_r(p) s_w +c_w(p) s_w \right] \frac{dp}{dt}\end{array}$

(2)	$\begin{array}{l}\displaystyle \frac{ds_o}{dt} =\frac{1}{A_e \, h_e \, \phi_e(p)} \left[ q^{\downarrow}_o(t) - q^{\uparrow}_o(t) \right] - \left[ c_r(p) s_o +c_o(p) s_o \right] \frac{dp}{dt}\end{array}$

(3)	$\begin{array}{l}\displaystyle \frac{ds_g}{dt} =\frac{1}{A_e \, h_e \, \phi_e(p)} \left[ q^{\downarrow}_g(t) - q^{\uparrow}_g(t) + q^{\downarrow}_{GC}(t) \right] - \left[ c_r(p) s_w +c_g(p) s_g \right] \frac{dp}{dt}\end{array}$

(4)	$\begin{array}{l}\displaystyle s_w + s_o + s_g = 1\end{array}$

where

$\begin{array}{l}p_i = p(0)\end{array}$	initial formation pressure	$\begin{array}{l}\Delta Q (t)\end{array}$	full-field cumulative reservoir fluid balance
$\begin{array}{l}A_e\end{array}$	drainage area	$\begin{array}{l}Q^{\uparrow}_t(t)\end{array}$	full-field cumulative offtakes by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}h_e\end{array}$	effective formation thickness averaged over drainage area	$\begin{array}{l}Q^{\downarrow}_t(t)\end{array}$	full-field cumulative intakes by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}\phi_e(p)\end{array}$	effective porosity as function of formation pressure $\begin{array}{l}p(t)\end{array}$	$\begin{array}{l}Q^{\downarrow}_{GC}(t)\end{array}$	cumulative volumetric inflow from Gas Cap Expansion
$\begin{array}{l}c_t(p)\end{array}$	total compressibility as function of formation pressure $\begin{array}{l}p(t)\end{array}$	$\begin{array}{l}Q^{\downarrow}_{AQ}(t)\end{array}$	cumulative volumetric inflow from Aquifer Expansion

The direct consequence of the above equations:

(5)	$\begin{array}{l}\displaystyle A_e \, h_e \int_{p_i}^p \phi_e(p) \, c_t(p) \, dp = \Delta Q (t) = Q^{\downarrow}_t(t) - Q^{\uparrow}_t(t) + Q^{\downarrow}_{GC}(t) + Q^{\downarrow}_{AQ}(t)\end{array}$

The MatBal equation (5) is often complemented by constant PI model of Bottom-Hole Pressure ( $\begin{array}{l}p^{\uparrow}_{wf}(t)\end{array}$ for producers and $\begin{array}{l}p^{\downarrow}_{wf}(t)\end{array}$ for injectors):

(6)	$\begin{array}{l}\displaystyle p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt}\end{array}$

(7)	$\begin{array}{l}\displaystyle p^{\downarrow}_{wf, \, j}(t) = p(t) - {J^{\downarrow}_j}^{-1} \cdot \frac{dQ^{\downarrow}_j}{dt}\end{array}$

where

$\begin{array}{l}p^{\uparrow}_{wf, \, k}(t)\end{array}$

BHP in $\begin{array}{l}k\end{array}$ -th producer

$\begin{array}{l}p^{\downarrow}_{wf, \, j}(t)\end{array}$

BHP in $\begin{array}{l}j\end{array}$ -th injector

$\begin{array}{l}Q^{\uparrow}_k(t)\end{array}$

cumulative offtakes from $\begin{array}{l}k\end{array}$ -th producer by the time moment $\begin{array}{l}t\end{array}$

$\begin{array}{l}Q^{\downarrow}_j(t)\end{array}$

cumulative intakes to $\begin{array}{l}j\end{array}$ -th injector by the time moment $\begin{array}{l}t\end{array}$

$\begin{array}{l}J^{\uparrow}_k\end{array}$

productivity index of $\begin{array}{l}k\end{array}$ -th producer

$\begin{array}{l}J^{\downarrow}_j\end{array}$

injectivity Index of $\begin{array}{l}j\end{array}$ -th injector

In practice there is no way to measure the external influx $\begin{array}{l}q^{\downarrow}_{GC}(t)\end{array}$ and $\begin{array}{l}q^{\downarrow}_{AQ}(t)\end{array}$ so that one need to model them and calibrate model parameters to fit available data on production flowrates history and formation pressure data records.

There is a list of various analytical Aquifer Drive and Gas Cap Drive models which are normally related to pressure dynamics $\begin{array}{l}p(t)\end{array}$ :

Gas Cap Drive @model

Aquifer Drive @model

(8)	$\begin{array}{l}\displaystyle q^{\downarrow}_{GC}(t) = q^{\downarrow}_{GC}(p(t))\end{array}$

(9)	$\begin{array}{l}\displaystyle q^{\downarrow}_{AQ}(t) = q^{\downarrow}_{AQ}(p(t))\end{array}$

which closes a set of equations (1)– (4) for the pressure $\begin{array}{l}p(t)\end{array}$ and saturations $\begin{array}{l}\{ s_w, \, s_o, \, s_g \}\end{array}$ .

Variations

In some specific cases equation (5) can be explicitly integrated:

Slightly compressibility flow

Low pressure dry gas

$\begin{array}{l}\{ \phi_e = {\rm const}, \ c_t = {\rm const} \}\end{array}$

$\begin{array}{l}c_t = c_r + \frac{1}{p} \sim \frac{1}{p}\end{array}$

(10)	$\begin{array}{l}\displaystyle p(t) = p_i + \frac{\Delta Q(t)}{V_e \cdot c_t}\end{array}$

(11)	$\begin{array}{l}\displaystyle p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_e \cdot c_t} \right]\end{array}$

where

$\begin{array}{l}V_e = A_e \, h_e \, \phi_e\end{array}$

drainage volume

Page tree

Variations

See Also

Page tree

MatBal Flow @model

Variations

See Also