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

(B_o - R_s \, B_g) \, G_O +(B_g - R_V \, B_o) \, G_G + (B_w  \, G_W - \phi_n )\, (1- R_s \, R_v)  = 0

G_O = V_e^{-1} \, \delta \, Q_O + \left[ \frac{s_{oi}}{B_{oi}}  + \frac{R_{vi}\, s_{gi}}{B_{gi}}\right]

\delta \, Q_O = - Q^{\uparrow}_O

G_G = V_e^{-1} \, \delta \, Q_G + \left[ \frac{R_{si}\, s_{oi}}{B_{oi}}  + \frac{ s_{gi}}{B_{gi}}\right]

\delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP}

G_W = V_e^{-1} \, \delta \, Q_W +  \frac{ s_{wi}}{B_{wi}}

\delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ}

\phi_n = 1 + c_\phi \, (p-p_i)  + 0.5 \, c^2_\phi \, (p-p_i)^2

where

	formation pressure at time moment		Cumulative oil production by the time moment
	initial formation pressure		Cumulative gas production by the time moment
	initial drainage volume		Cumulative water production by the time moment
	pore compressibility		Cumulative water injection by the time moment
	initial water saturation		Cumulative gas injection by the time moment
	initial gas saturation		Cumulative water influx from aquifer
	initial oil saturation		Cumulative gas influx from gas cap
	Oil formation volume factor as functions of reservoir pressure
	Gas formation volume factor as functions of reservoir pressure
	Water formation volume factor as functions of reservoir pressure
	Solution GOR and Vaporized Oil Ratio as functions of reservoir pressure

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 equation for the pressure .

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} \right]

where

drainage volume

This allows using simple graphical methods for estimating drainage volume .

Variations

See Also