Modelling facility for field-average formation pressure and Bottom-Hole Pressure ( for producers and for injectors) 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) + V^{\downarrow}_{GC}(t) + V^{\downarrow}_{AQ}(t)

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

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

where

	drainage area		effective formation thickness averaged over drainage area
	initial formation pressure		full-field cumulative reservoir fluid balance
	productivity index of -th producer		cumulative offtakes from -th producer by the time moment
	injectivity Index of -th injector		cumulative intakes to -th injector by the time moment
	field-average BHP in producers		full-field cumulative offtakes by the time moment
	field-average BHP in injectors		full-field cumulative intakes by the time moment
	effective porosity as function of formation pressure		full-field cumulative volumetric inflow from gas cap expansion
	total compressibility as function of formation pressure		full-field cumulative volumetric inflow from aquifer expansion

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 and Gas Cap drive models which are normally based on the relations:

Gas Cap Expansion @model

Aquifer Expansion @model

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

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

Variations

In some specific cases equation can be explicitly integrated:

Low compressibility rocks and fluids

Ideal 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