Modelling facility for field-average formation pressure at any time moment as response to production flowrates history, which in case of MBO fluid takes form:

\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

\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

F_O = V_\phi^{-1} \, \delta \, Q_O + F_{Oi}

F_{Oi} = \frac{s_{oi}}{B_{oi}}  + \frac{R_{vi}\, s_{gi}}{B_{gi}}

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

F_G = V_\phi^{-1} \, \delta \, Q_G + F_{Gi}

F_{Gi} = \frac{R_{si}\, s_{oi}}{B_{oi}}  + \frac{ s_{gi}}{B_{gi}}

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

F_W = V_\phi^{-1} \, \delta \, Q_W + F_{Wi}

F_{Wi} = \frac{ s_{wi}}{B_{wi}}

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

where

	initial formation pressure:		Cumulative oil production by the time moment
	initial open pore volume of the main pay (excluding the aquifer and gas cap)		Cumulative gas production by the time moment
	initial porosity		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 Expansion by the time moment
	initial oil saturation:		Cumulative gas influx from Gas Cap expansion by the time moment

	Oil formation volume factor as functions of reservoir pressure		Solution GOR as functions of reservoir pressure
	Gas formation volume factor as functions of reservoir pressure		Vaporized Oil Ratio as functions of reservoir pressure
	Water formation volume factor as functions of reservoir pressure

The MatBal equation can be 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 .

Approximations

In some specific cases equation can be explicitly integrated with the accuracy sufficient for practical applications:

Slightly compressibility flow

Low pressure dry gas

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

p(t)  = p_i \exp \left[ \frac{\Delta Q(t)}{V_\phi} \right]

where is Cumulative Voidage Replacement Balance (CVRB):

\Delta Q = -  \frac{B_o - R_s \, B_g}{1- R_s \, R_v} \cdot  \, Q^{\uparrow}_O + \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot \, \left( Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} \right) + B_w \, \left( Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} \right)

The above approximations sometime allow using simple graphical methods for rough estimation of drainage volume and associated Hydrocarbon Reserves.

Approximations

See Also