Modelling facility for field-average 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)	$\begin{array}{l}\displaystyle \phi_n(p) = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} \cdot F_O +\frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot F_G +B_w \, F_W\end{array}$

(2)	$\begin{array}{l}\displaystyle \phi_n = \exp \left[ c_\phi \, (p-p_i) \right] \approx 1 + c_\phi \, (p-p_i) + 0.5 \, c^2_\phi \, (p-p_i)^2\end{array}$

(3)	$\begin{array}{l}\displaystyle F_O = V_e^{-1} \, \delta \, Q_O + \left[ \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}}\right]\end{array}$

(4)	$\begin{array}{l}\displaystyle \delta \, Q_O = - Q^{\uparrow}_O\end{array}$

(5)	$\begin{array}{l}\displaystyle F_G = V_e^{-1} \, \delta \, Q_G + \left[ \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}}\right]\end{array}$

(6)	$\begin{array}{l}\displaystyle \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP}\end{array}$

(7)	$\begin{array}{l}\displaystyle F_W = V_e^{-1} \, \delta \, Q_W + \frac{ s_{wi}}{B_{wi}}\end{array}$

(8)	$\begin{array}{l}\displaystyle \delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ}\end{array}$

where

$\begin{array}{l}p_i\end{array}$	initial formation pressure: $\begin{array}{l}p_i = p(0)\end{array}$	$\begin{array}{l}Q^{\uparrow}_O(t)\end{array}$	Cumulative oil production by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}V_e = V \cdot \phi_i\end{array}$	initial drainage volume of Main Pay (excluding the aquifer and gas cap)	$\begin{array}{l}Q^{\uparrow}_G(t)\end{array}$	Cumulative gas production by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}\phi_i = \phi(p_i)\end{array}$	initial porosity	$\begin{array}{l}Q^{\uparrow}_W(t)\end{array}$	Cumulative water production by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}c_\phi\end{array}$	pore compressibility	$\begin{array}{l}Q^{\downarrow}_W(t)\end{array}$	Cumulative water injection by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}s_{wi}\end{array}$	initial water saturation	$\begin{array}{l}Q^{\downarrow}_G(t)\end{array}$	Cumulative gas injection by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}s_{gi}\end{array}$	initial gas saturation	$\begin{array}{l}Q^{\downarrow}_{WAQ}(t)\end{array}$	Cumulative water influx from Aquifer Expansion by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}s_{oi}\end{array}$	initial oil saturation: $\begin{array}{l}s_{oi} = 1 - s_{wi} - s_{gi}\end{array}$	$\begin{array}{l}Q^{\downarrow}_{GCAP}t)\end{array}$	Cumulative gas influx from Gas Cap expansion by the time moment $\begin{array}{l}t\end{array}$

$\begin{array}{l}B_o(p)\end{array}$	Oil formation volume factor as functions of reservoir pressure $\begin{array}{l}p\end{array}$	$\begin{array}{l}R_s(p)\end{array}$	Solution GOR as functions of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}B_g(p)\end{array}$	Gas formation volume factor as functions of reservoir pressure $\begin{array}{l}p\end{array}$	$\begin{array}{l}R_v(p)\end{array}$	Vaporized Oil Ratio as functions of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}B_w(p)\end{array}$	Water formation volume factor as functions of reservoir pressure $\begin{array}{l}p\end{array}$

The MatBal equation (1) 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):

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

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

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

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

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

which closes equation (1) for the pressure $\begin{array}{l}p(t)\end{array}$ .

Approximations

In some specific cases equation (1) can be explicitly integrated:

Slightly compressibility flow

Low pressure dry gas

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

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

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

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

The above approximations sometime allow using simple graphical methods for rough estimation of drainage volume $\begin{array}{l}V_e\end{array}$ and associated Hydrocarbon Reserves.

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Approximations

See Also

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Material Balance Pressure @model

Approximations

See Also