Geostatistical models of reservoir properties can be hundreds of millions of cells; it is impractical to use them directly in flow simulation due to computational cost. Upscaling techniques are applied to average fine scale permeability values onto coarser flow simulation blocks. In cases where unstructured grids are used or the geology inside the grid block is not aligned with the block geometry, full permeability tensors arise instead of a diagonal tensor. The focus of this work is on development of a method to characterize the full permeability tensor for an unstructured grid block using fine scale heterogeneity information. A single phase flow-based upscaling is performed and a prototype program called ptensor is developed based on the random boundary conditions and optimization technique. Full, symmetric and diagonal permeability tensors are calculated for 2-D and 3-D blocks and sensitivity analysis is performed.
Geostatistical modeling of petrophysical properties can generate fine scale models with hundreds of millions of cells. Using those fine scale models directly in flow simulation is computationally inefficient. Upscaling techniques scale the fine scale models to coarser scale models while preserving the fine scale heterogeneity. A simple averaging is sufficient and reasonable for variables that average linearly; however, in the case of permeability which does not average linearly, a simple arithmetic averaging is inadequate.
For complex cases with heterogeneity, flow-based upscaling techniques yield more accurate results (1). In this type of upscaling the flow equation is solved for pressure and the results are used to calculate the block permeability.
Commonly unstructured grids are used in order to better capture the flow response near complex reservoir features such as faults and wells. Usually cases that involve the use of irregular block or a heterogeneous permeability field at fine scale require calculation of the full permeability tensor. White and Horne(2) and Gomez-Hernandez(3) proposed different methods to calculate permeability tensor for regular coarse blocks. In recent years, some approaches are presented by Durlofsky(4), Prevost(5) and He(6) to calculate the full permeability tensor for irregular shape grid blocks.
This paper introduces a simple, fast and accurate method to calculate full, symmetric or diagonal permeability tensor for any corner point geometry grids. The unstructured grid is surrounded by a bounding box and the geometry is simplified with the fine resolution grid. The steady state flow equation is solved, via finite difference, for the input fine grid cells within a bounding box. The results are used to calculate the permeability tensor of corresponding coarse regular or irregular blocks. Randomly assigned boundary conditions are used and the results are optimized to get the desired full, symmetric or diagonal tensor.
Flow based upscaling is used to calculate effective permeability of coarse block. Consider a single rectangle (2-D) or a cube (3-D) imposed on a fine scale model. The idea here is to calculate the pressure at fine scale with specific boundary conditions applied at the boundary of the coarse block and then use the solution to calculate the full permeability tensor for that coarse block.
