Study On Well Test Model Of Hydraulic Fractured Well Considering Complex Fractures

Fractured vertical wells(FVWs)and multi-stage fractured horizontal wells(MFHWs)are an important measure to improve production of oil and gas wells.Transient pressure analysis is a vital method of reservoir evaluation and formation parameter determination.Point source function is employed and a large number of domestic and international literature about well test model fractured vertical wells and multi-stage fractured horizontal wells are consulted in this paper.The 3-D diffusivity equation of homogeneous and dual porosity reservoirs is established by combing with motion equation,state equation and continuity equation.Green function is obtained by employing Laplace transform.The continuous point source solution of the top-bottom closed columnar reservoir and the box closed reservoir in Laplace domain is obtained by using Fourier transform,pressure drop superposition and mirror reflection principle.The uniform rate surface source solution is obtained by integrating towards the point source solution,and the wellbore pressure of FVWs and MFHWs with symmetric fractures and complex fractures are obtained by coupling the hydraulic fracture solution with the reservoir solution.Pressure response curves are drawn in log-log plot and the flow regimes are divided.The influence factor on typical curves is analyized.The main conclusion in this paper are listed as follows:(1).Combing with motion equation,state equation and continuity equation,the continuous point source solution of isotropic homogeneous and dual-medium reservoir is given.Based on equivalent permeability substitution,the continuous point source solution of anisotropic reservoirs is also obtained.According to the mirror reflection,Laplace transform and Fourier transform,the point source solutions of the top-bottom closed boundary are obtained in the column reservoir and the box closed reservoir.(2).Reservoir model and hydraulic fracture model are established and solved to obtain the conductivity function.Based on the model solution of infinite-conductivity fractured well and combing with conductivity function,the wellbore pressure solution of finite-conductivity FVWs and MFHWs with symmetric fractures is obtained.The solution of this paper is validated by comparing with the that of commercial software Saphir.The results reach a great agreement both cylindrical and box-closed reservoirs.The computing speed has been greatly improved.(3).Diffusivity equations of asymmetric and multi-wing hydraulic fractures are established and solved.Linear equations are formed by coupling hydraulic fracture solutions and uniform rate surface source solutions,and discretizing hydraulic fractures and pressure drop.The wellbore pressure solutions of asymmetric hydraulic fracture and multi-wing hydraulic fracture are obtained by solving linear equations.The solutions of asymmetric hydraulic fracture,which calculated by two different methods in this paper,are validated by Bermuen’s result that is calculated by numerical solution.The results are in good agreement with each other.(4).Based on the solution of single-wing fracture,MFHWs with complex hydraulic fractures are established and solved in both cylindrical and box-closed reservoirs.Linear equations are formed by Coupling hydraulic fracture solutions and uniform rate surface source solutions,and discretizing hydraulic fractures and pressure drop.The wellbore pressure solutions of MFHWs with complex hydraulic fractures are obtained by solving linear equations.(5).Based on Duhamel’s principle,the wellbore pressure solutions considering well storage and skin in Laplace domain are obtained.Wellbore pressure of real time is calculated by Stehfest numerical inversion.Pressure reponse characteristic curves are drawn in log-log plot and the characteristics of each flow regime are analyzed.Influence of some paramaters(such as conductivity,fracture angle,fracture distribution,and fracture number etc.)on pressure and pressure derivative curves is analyzed in detail.Based on study on well test model of FVWs and MFHWs with complex hydraulic fractures.Characteristic curves of FVWs and MFHWs with complex hydraulic fractures are understanded better in homogeneous and dual porosity reservoirs.Establishment of model and solution provide theoretical guidance for the actual well test data interpretation.