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 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

See Also