A Numerical study on characteristics of fluid flow in a three-dimensional discrete fracture network with variation of length distributions of fracture elements

3차원 이산 균열망 흐름장에서 균열요소의 길이분포 변화에 따른 내 유체 흐름 특성에 관한 수치적 연구

Abstract

In this study, the effect of the fluid flow characteristics on the length distribution of the fracture elements composing the fracture network is analyzed numerically using the 3D fracture crack network model. The truncated power-law distribution is applied to generate the length distribution of the fracture elements and the simulations of fluid flow are carried out with the exponent ${\beta}_l$ from 1.0 to 6.0. As a result of simulations, when the exponent ${\beta}_l$ increases, the length distribution of the fracture elements gradually decreases, and the connectivity between the fracture elements affecting the permeability of the fracture network becomes weak. When we analyzed the distributions of flow rate calculated at each fracture element with the exponent ${\beta}_l$, the mean flow rate at ${\beta}_l=1.0$ was estimated to be about 447 times larger than that at ${\beta}_l=6.0$ and for the flow calculated at the outflow boundary of the fracture network, the case of ${\beta}_l=1.0$ was estimated to be 6,440 times larger than that of ${\beta}_l=6.0$.

본 연구에서는 3차원 이산 균열망 수치모형을 이용하여 균열망을 구성하는 균열요소의 길이분포가 유체 흐름 특성이 미치는 영향에 대해 수치적으로 분석하였다. 균열요소의 길이분포의 생성을 위해 절단멱분포법칙을 적용하였으며, 지수 ${\beta}_l$을 1.0에서부터 6.0까지 변화시키면서 유체 흐름 모의를 수행하였다. 모의결과 지수 ${\beta}_l$이 증가함에 따라 균열요소들의 길이분포는 점차적으로 작아지며, 이로 인해 균열망의 투수성에 영향을 미치는 균열요소들 간의 연결성은 취약해지는 것으로 나타났다. 각각의 지수 ${\beta}_l$에 대해 균열요소 각각에서 계산된 유량분포를 분석하였을 때 ${\beta}_l=1.0$에서의 평균유량이 ${\beta}_l=6.0$에 비해 약 447배 크게 산정되었으며, 균열망의 유출경계에서 계산된 유량의 경우 ${\beta}_l=1.0$일 때가 6.0에 비해 약 6,440배 크게 산정되었다.

Fig. 1. Orientation of a fracture element

Fig. 2. (a) Flow channeling on surfaces of natural fractures and (b) concept of equivalent flow channeling adapted in this study

Fig. 5. Comparison with numerical and analytical discharges at outlet (First case)

Fig. 6. (a) Second case with four single fracture elements and (b) calculated hydraulic head distributions for aperture of 2.0 mm

Fig. 9. Discrete fracture networks generated with power law exponent, β_l

Fig. 10. Variation of fracture length distribution with power law exponent, β_l

Fig. 11. Connectivity between fracture elements with power law exponent, β_l

Fig. 12. Number of fracture elements available to flow with power law exponent, β_l

Fig. 13. Head distributions calculated with power law exponent, β_l

Fig. 14. Flow rate distributions calculated at each fracture element with power law exponent, β_l(The flow rate is represented by the log exponent and values less than –10 are replaced by –10)

Fig. 15. Mean and maximal flow rates at each fracture element and number of –10 or less than -10 with power law exponent, β_l

Fig. 16. Flow rates at outlet boundary with power law exponent, β_l

Fig. 3. Rectangular cross-sectional area between two intersected fracture elements

Fig. 4. (a) First case of verification with one single fracture and simulation conditions and (b) calculated hydraulic head distributions for aperture of 2.0 mm

Fig. 7. Comparison with numerical and analytical discharges at outlet (Second case)

Fig. 8. (a) Two sets of fracture elements intersecting perpendicularly each other and (b) its parameters (N_tf : theoretical number of fracture elements)

Table 1. Basic statistical values of flow rates at each fracture element with β_l (log scale)