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

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

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

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

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
$\begin{array}{l}p^{\uparrow}_{wf}(t)\end{array}$	field-average BHP in producers	$\begin{array}{l}Q^{\uparrow}_t(t)\end{array}$	full-field cumulative offtakes by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}p^{\downarrow}_{wf}(t)\end{array}$	field-average BHP in injectors	$\begin{array}{l}Q^{\downarrow}_t(t)\end{array}$	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 $\begin{array}{l}p(t)\end{array}$	$\begin{array}{l}Q^{\downarrow}_{GC}(t)\end{array}$	full-field cumulative volumetric inflow from gas cap expansion
$\begin{array}{l}c_t(p)\end{array}$	total compressibility as function of formation pressure $\begin{array}{l}p(t)\end{array}$	$\begin{array}{l}Q^{\downarrow}_{AQ}(t)\end{array}$	full-field cumulative volumetric inflow from aquifer expansion
$\begin{array}{l}h_e\end{array}$	effective formation thickness averaged over drainage area	$\begin{array}{l}Q^{\uparrow}_t(t)\end{array}$	cumulative offtakes from $\begin{array}{l}k\end{array}$ -th producer by the time moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}A_e\end{array}$	drainage area	$\begin{array}{l}Q^{\downarrow}_t(t)\end{array}$	cumulative intakes to $\begin{array}{l}i\end{array}$ -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	$\begin{array}{l}J^{\downarrow}_i\end{array}$	injectivity Index of $\begin{array}{l}i\end{array}$ -th injector

For low compressibility rocks and fluids $\begin{array}{l}\{ \phi_e = {\rm const}, \ c_t = {\rm const} \}\end{array}$ the MatBal equation (1) can be explicitly integrated:

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

where

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

drainage volume

For ideal dry gas:

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

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MatBal Pressure @model

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