Inflow performance relationship of vertical wells in fractured vuggy media during semi-steady state flow regime

The inflow performance relationship (IPR) is one of the essential key parameters that control the fluids production/injection rate from wells. The presence of the vugs in addition to fractures in carbonate reservoirs introduce many difficulties related the estimation of IPR of future wells. In this study, a novel analytical IPR of dual-permeability, triple porosity model is derived for the first time. The introduced IPR formula is derived for the standard scenario model of a vertical and fully penetrating well in a circular reservoir under single-phase flow of slightly compressible fluid. The three governing flow equations (matrix flow equation, fracture flow equation and interaction between vugs and other two continuum equation) are solved simultaneously as a system of differential equations for semi-steady state (SSS) flow regime. The generated SSS IPR is the same as Darcy's straight line IPR multiplied by an error parameter (E₂₃) which is bounded between zero and one. A detailed investigation of the effect of the reservoir petro-physical parameters like storativities, inter-porosities and fracture to matrix permeability ratio are presented. The newly presented IPR of dual-permeability, triple porosity system can be reduced to sub-models IPRs by further modification of error parameter to fit sub-model as single permeability, triple porosity (E₁₃) or dual permeability, dual porosity (E₂₂). The presented IPR formula has the advantage of dividing the pressure drop of the fluid flow to two components, one for the internal resistance between different continuums (which is resented by E) and the other one is the radial flow in the same continuum toward/from well-bore. The generated IPR model is double verified using both cell pressure centred finite volume numerical simulation and conventional pressure transient analysis.