Numerical Model of Pressure and Velocity Fields in Axisymmetric Filtration in an Imperfectly Penetrated Stratum

A mathematical model of pressure and velocity fields resulting from the inflow of hydrocarbons to a well of finite radius from partially penetrated reservoir beds has been presented. It is assumed that two-dimensional flow initiated by an assigned depression in an imperfectly penetrated stratum is axisymmetric. A mathematical formulation of the problem in cylindrical geometry on the pressure field in an isolated isotropic homogeneous stratum whose boundaries do not coincide with the boundary points of a perforation interval has been given. A finite-diff erence model has been described on whose basis a program was made up for calculation of pressure and velocity fields in the stratum and computational experiments were consducted which allowed identifying new regularities of flow of practical importance that arise in actual oil and gas reservoirs. Space–time dependences of the pressure field in an oil and gas stratum and of the radial components and modulus of filtration velocity of hydrocarbons have been shown, which illustrate the distinctive features of two-dimensional flow with imperfect penetration of the stratum and basic differences from the well-studied case of plane radial onedimensional flow to the well of finite radius. Based on an analysis of the contour lines, new physical regularities of the flow have been identified that are associated with incomplete perforation of the thickness of the reservoir bed. By the numerical experiments, it has been established that in axisymmetric flow in the stratum, vertical flows inevitably arise which vanish at exit from the stratum into the well. It has been confirmed that in homogeneous incompletely penetrated reservoirs, intralayer crossflows arise in the near wellbore zone, and the radial component of filtration velocity in two-dimensional axisymmetric flow in the imperfectly homogeneous isotropic stratum depends on the vertical coordinate. This means that the inflow to the well is not uniformly distributed over the stratum thickness, and the modulus of the horizontal velocity component in all the curves peaks on the boundaries of the perforation interval. In the case of two-dimensional flow at the center of the perforation interval equally distant from the upper and lower boundaries of the oil and gas stratum, a minimum specific inflow is observed.