Perturbative-Integro-Differential Solution for the Nonlinear Hydraulic Diffusivity Equation for Infinite-Acting-Oil Flow in a Permeability-Pressure-Sensitive Reservoir

Abstract

Summary The nonlinear hydraulic diffusivity equation (NHDE) models the isothermal single-phase Darcian flow through porous media considering the variation in the properties of the rock and the fluid present inside its pores. Typically, the dimensionless solution of the linear hydraulic diffusivity equation (LHDE) pD⁢(rD,tD) for constant permeability oil flow in porous media is computed through Laplace and Fourier transform or Boltzmann transformation. For these cases, the dimensionless general solution in cylindrical coordinates is expressed by the transcendental function exponential integral Ei(rD,tD). This work develops analytically a new coupled perturbative-integro-differential model to solve the NHDE for oil flow in a permeability-pressure-sensitive reservoir with source. The general solution is computed combining a first-order asymptotic series expansion, Green’s functions (GF), and a Volterra’s second kind integro-differential formulation. A set of pore pressure and permeability values for two sandstones samples in an offshore reservoir from Brazil is obtained experimentally using the geomechanical elastic parameters (e.g., the Young’s modulus and Poisson’s ratio in addition to a uniaxial cell). These data are used as input in the computational code to run the analytical model and evaluate the reservoir permeability change. After these data input, the model runs and it allows to compute the instantaneous reservoir permeability values over the well-reservoir life cycle. The model calibration is performed by comparing the developed solution with a numerical porous media oil flow simulator named IMEX®, widely used in reservoir engineering and well-testing field operations and scientific works. The general solution of the NHDE mD(rD,tD) is computed by the sum of the linear solution pD(rD,tD) (constant permeability) and the first-order term of the asymptotic series expansion mD(1)(rD,tD), composed of the nonlinearity present in solution caused by the reservoir permeability variation. The results have shown that the developed solution is accurate, when compared to the numerical simulator, providing to be an attractive mathematical tool to help the well-reservoir management due to its low computational costs, when compared to the numerical simulators acquisition costs.