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

 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)

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) ]

s_w(t) = s_{wi} + (1-s_{wi}) \cdot \rm E_{Dow}(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFO(t)/E_S

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 gas influx from Gas Cap Expansion
	total compressibility as function of formation pressure		cumulative water influx from Aquifer Expansion
	normalized cross-phase exchange ratios as functions of reservoir pressure and temperature		normalized cross-phase exchange derivatives as functions of reservoir pressure and temperature

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 \cdot c_t} \right]

where

drainage volume

This allows using simple graphical methods for estimating drainage volume .

Variations

See Also