Modelling facility for field-average saturation and formation pressure at any time moment as response to production flowrates history:

\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}

\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}

\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}

s_w + s_o + s_g = 1

where

	initial formation pressure		full-field cumulative reservoir fluid balance
	drainage area		full-field cumulative offtakes by the time moment
	effective formation thickness averaged over drainage area		full-field cumulative intakes by the time moment
	effective porosity as function of formation pressure		cumulative volumetric inflow from Gas Cap Expansion
	total compressibility as function of formation pressure		cumulative volumetric inflow from Aquifer Expansion

The direct consequence of the above equations:

 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)

The MatBal equation is often complemented by constant PI model of Bottom-Hole Pressure ( for producers and for injectors):

p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt}

p^{\downarrow}_{wf, \, j}(t) = p(t) -  {J^{\downarrow}_j}^{-1} \cdot \frac{dQ^{\downarrow}_j}{dt}

where

BHP in -th producer

BHP in -th injector

cumulative offtakes from -th producer by the time moment

cumulative intakes to -th injector by the time moment

productivity index of -th producer

injectivity Index of -th injector

In practice there is no way to measure the external influx and 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 :

Gas Cap Drive @model

Aquifer Drive @model

q^{\downarrow}_{GC}(t) = q^{\downarrow}_{GC}(p(t))

q^{\downarrow}_{AQ}(t) = q^{\downarrow}_{AQ}(p(t))

which closes a set of equations – for the pressure and saturations .

Variations

In some specific cases equation can be explicitly integrated:

Slightly compressibility flow

Low pressure dry gas

p(t)  = p_i + \frac{\Delta Q(t)}{V_e \cdot c_t}

p(t)  = p_i \exp \left[ \frac{\Delta Q(t)}{V_e \cdot c_t} \right]

where

drainage volume

Variations

See Also