Combination of Analytical, Numerical and Geostatistical Methods to Model Naturally Fractured Reservoirs
Authors: He Nanqun , Lee Seong H. , Jensen Clair L. 
Published: December 2007

Abstract
Fractures play a critical role in storing and distributing fluid in naturally fractured reservoirs. To fully quantify the effect of fractures on reservoir performance, flow simulation is required. To model these flow effects, the fracture system needs to be well characterized, and fracture properties such as length, orientation, aperture and intensity can then be used to calculate effective simulation gridblock properties. In this paper, we show additional techniques to further improve the computational efficiency of our previously published hierarchical fracture modeling process.
In the hierarchical modeling approach, we calculate an effective tensor permeability associated with the small length-scale fractures, which is then used as input to the calculation of the effective tensor permeability associated with the medium length-scale fractures. Herein, we show that an efficient analytical calculation can be correctly made for those small-scale fractures that have lengths less than 10% of the reservoir model cell-size.
We have shown that a Boundary Element Method (BEM) provides an accurate effective tensor permeability, which is then used in flow simulation. However due to high computational demands, it is not practical to use the BEM technique for models much larger than one hundred thousand cells, without using parallel processing methods. To overcome this high computational demand, we use geostatistical methods to estimate effective permeabilities between cells for which the more rigorous BEM has been applied. Both hard and soft data used in the geostatistical simulation are critical to obtain a meaningful result. The hard data include effective permeabilities calculated by the BEM for about 10% of the total cells in the simulation model. Soft data include a statistical analysis of the correlation between BEM calculated tensor permeability values and fracture intensity for each simulation model-cell. Sequential Gaussian Simulation with co-Kriging is then used to estimate the effective permeability for the entire model. A field case demonstrates that the proposed methodology is efficient and practical for a large fractured reservoir model.

Introduction
In a previous paper (Lee et al.1), we proposed a hierarchical approach for modeling fluid flow in a naturally fractured reservoir with multiple length-scale fractures. Based on fracture length (lf) relative to the finite-difference grid size (lg), fractures are classified as belonging to one of three groups:short disconnected fractures (lflg),medium-length fractures (lf~lg), andlong fractures (lf&gt;&gt;lg). It was assumed that short fractures are randomly distributed and contribute to increasing the effective matrix permeability.
An analytical solution representing the permeability contribution from short fractures was derived. With the short fracture contribution to permeability, the resulting effective matrix permeability could be expressed in a general tensor form. For medium-length fractures in a grid-block, a coupled system of Poisson equations with tensor permeability was solved numerically using a boundary element method. The grid-block effective permeabilities were then used with a finite difference simulator to compute flow through the coupled fracture and rock matrix system. The simulator was enhanced to use a control-volume finite difference formulation2–5 for general tensor permeability input (i.e., 9-pt stencil for 2-D and 27-pt stencil for 3-D). Long fractures are modeled explicitly in the reservoir simulator, using a transport equation that describes flow between long fractures and surrounding simulation grid-blocks.
The computational time for effective permeabilities using the BEM is intensive when the number of cells in the model is very large. Whereas, the analytical method used for short fractures is very efficient. However, it is asymptotically accurate only for short fractures that do not connect to or intersect with other fractures.