1. (3 points) If the aperture of a fracture is 0.001 m, what are the permeabilit
ID: 283883 • Letter: 1
Question
1. (3 points) If the aperture of a fracture is 0.001 m, what are the permeability and hydraulic
conductivity in the fracture? For a network of equally spaced fractures with fracture porosity of 1.0×10^-4, calculate the permeability and permeability and hydraulic conductivity of the fractured media.
2. (3 points) If the permeability of fracture and matrix blocks is 10^-4 and 10^-8 m^2, respectively, what is the effective permeability if an effective continuum model is used to describe the fracture medium? Assume that the ratio of cross-sectional contact areas between the fracture and the matrix blocks is 10^-3.
3. (4 points) Briefly describe the discrete fracture, the double-porosity, and the effective continuum approaches for dealing with water flow in fractured geological media.
Explanation / Answer
C DOUBLE POROSITY . A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter epsilon representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as epsilon tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as epsilon tends to zero.
The geological complexity of fractured reservoirs requires the use of simplified models for flow simulation. This is often addressed in practice by using flow modeling procedures based on the dual-porosity, dual-permeability concept. However, in most existing approaches, there is not a systematic and quantitative link between the underlying geological model [in this case, a discrete fracture model (DFM)] and the parameters appearing in the flow model. the effective continuum approaches for dealing with water flow in fractured geological media.
In this work, a systematic upscaling procedure is presented to construct a dual-porosity, dual-permeability model from detailed discrete fracture characterizations. The technique, referred to as a multiple subregion (MSR) model, represents an extension of an earlier method that did not account for gravitational effects. The subregions (or subgrid) are constructed for each coarse block using the iso-pressure curves obtained from local pressure solutions of a discrete fracture model over the block. The subregions thus account for the fracture distribution and can represent accurately the matrix-matrix and matrix-fracture transfer. The matrix subregions are connected to matrices in vertically adjacent blocks (as in a dual-permeability model) to capture phase segregation caused by gravity. Two-block problems are solved to provide fracture-fracture flow effects. All connections in the coarse-scale model are characterized in terms of upscaled transmissibilities, and the resulting coarse model can be used with any connectivity-based reservoir simulator.
The method is applied to simulate 2D and 3D fracture models, with viscous, gravitational, and capillary pressure effects, and is shown to provide results in close agreement with the underlying DFM.
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