Effects of Fracture Tensor Component and First Invariant on Block Hydraulic Characteristics of the 2-D Discrete Fracture Network Systems

절리텐서의 성분 및 일차불변량이 2-D DFN 시스템의 블록수리전도 특성에 미치는 영향

Abstract

In this study, the effects of fracture tensor component and first invariant on block hydraulic behaviors are evaluated in the 2-D DFN(discrete fracture network) systems. A series of regression analysis is performed between connected fracture tensor components and block hydraulic conductivities estimated at every $30^{\circ}$ hydraulic gradient directions for a total of 36 DFN systems having various joint density and size distribution. The directional block hydraulic conductivity seems to have strong relation with the fracture tensor component estimated in direction perpendicular to it. It is found that an equivalent continuum approach could be acceptable for the 2-D DFN systems under condition that the first invariant of fracture tensor is more than 2.0~2.5. The first invariant of fracture tensor seems highly correlated with average block hydraulic conductivity and can be used to evaluate hydraulic characteristics of the 2-D DFN systems. Also, a possibility of upscaling using the first invariant of fracture tensor for the DFN system is addressed through this study.

본 연구는 이차원 DFN(discrete fracture network) 시스템에서 절리의 빈도 및 길이분포에 따른 절리텐서의 성분 및 일차불변량이 DFN의 블록수리전도 특성에 미치는 영향을 평가하였다. 확정적인 두 방향의 절리군을 사용하여 절리군의 빈도와 길이분포에 따라 생성된 총 36개의 DFN 시스템에서 각각 매 $30^{\circ}$ 간격으로 설정된 수두경사에 따른 블록수리전도도와 절리텐서의 성분 간의 상관성 분석이 수행되었다. DFN 블록의 블록수리전도도는 이에 직교하는 방향의 절리텐서 성분과 강한 상관관계를 갖는다. 본 연구의 연결성을 유지한 DFN 시스템은 절리텐서의 일차불변량이 2.0~2.5 이상일 때 등가의 연속체 해석이 가능한 것으로 평가되었다. 절리텐서의 일차불변량은 평균 블록수리전도도와 매우 강한 함수관계를 갖으며 DFN 시스템의 블록수리전도 특성을 평가하는 데에 사용될 수 있다. 또한, 본 연구를 통하여 절리텐서의 일차불변량을 이용한 DFN 시스템의 업스케일링 가능성이 논의되었다.

Fig. 1. Solid angle, dΩ, in an upper hemisphere to estimate fracture tensor.

Fig. 2. Comparison between directional block conductivity(k^(p)) and connected fracture tensor component(CFTC) for different joint configurations of selected DFN systems; (a) & (b) D-1-1-*, (c) & (d) D2-2-*, (e) & (f) L1-3-*.

Fig. 3. Relation between CFTC and directional conductivity factor(K_f) for DFN systems.

Fig. 4. Effect of the first invariant(F₀) of connectedfracture tensor on ER.

Fig. 5. Relation between block conductivity factor and the first invariant of connected fracture tensor for DFN systems.

Fig. 6. The connected flow network on a square window of size 20 m showing (a) before and (b) after correction using the first invariant of 2-D fracture tensor.

Table 1. Summary of input parameters for the generated DFN systems having different joint orientation, density and size.(after Han et al., 2017)

Table 2. An example of fracture geometry before and after correction using the first invariant of 2-D fracture tensor

Table 3. Estimated directional block conductivity in different direction and average block conductivity for the DFN systems having fracture geometry in Table 2