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

(2)	$\begin{array}{l}\displaystyle c_t(s,p) = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ]\end{array}$

(3)	$\begin{array}{l}\displaystyle s_w(t) = s_{wi} + (1-s_{wi}) \cdot \rm E_{Dow}(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFO(t)/E_S\end{array}$

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 gas influx 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 water influx from Aquifer Expansion
$\begin{array}{l}R_{sn}, \; R_{vn}\end{array}$	normalized cross-phase exchange ratios as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$	$\begin{array}{l}R_{sp}, \; R_{vp}\end{array}$	normalized cross-phase exchange derivatives as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

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):

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

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

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

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

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

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

Variations

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

Slightly compressibility flow

Low pressure dry gas

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

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

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

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

where

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

drainage volume

This allows using simple graphical methods for estimating drainage volume $\begin{array}{l}V_e\end{array}$ .

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Variations

See Also

Page tree

Material Balance Pressure @model

Variations

See Also