Upscaling the Steam-Assisted-Gravity-Drainage Model for Heterogeneous Reservoirs

The effects of heterogeneities in the rock and fluid properties on the governing equations in modeling the steam-assisted-gravitydrainage (SAGD) process are investigated using a spectrum-based upscaling approach. In this approach, heterogeneity is included by assigning random perturbation fields to permeability and thermal diffusivity that hold the assumption of first and second orders of stationarity. Heat and mass-transport parameters and dependent variables (e.g., temperature and viscosity) that are affected by heterogeneity are represented by their mean and perturbations around the mean values. Substituting the perturbed variables and coefficients into the basic governing equations (heat-diffusion and Darcy equations) results in new sets of stochastic partial-differential equations that include mean and perturbation. Mean equations are essentially upscaled new governing equations that include the autocorrelations and cross correlations between different perturbed quantities. The cross correlations are derived by applying the Fourier-Stieltjes transform. Verification and validation of the developed results for the heat-diffusion equation are performed using numerical simulations with synthetic heterogeneities. The upscaled equations embrace the heterogeneity in permeability and thermal diffusivity and can predict the flow rate and shape of the steam chamber. The upscaled model is compared with the homogeneous model, developed by Butler (1991), to quantify the effects of heterogeneities in permeability and thermal diffusivity on the SAGD efficiency. A case is studied by assigning harmonic distribution to the perturbations of thermal diffusivity and permeability. Different cases with different relationships between permeability and thermal-diffusivity fields are considered to investigate their effects on the oil flow rate. The results show that local perturbations decrease the oil flow rate. Furthermore, steam-chamber growth is retarded in a reservoir with local perturbations compared with homogeneous reservoirs. The developed model can be used in the initial stages of SAGD development projects for long-range planning as a guide to find the upper and lower limits of production in reservoirs with heterogeneities.