MODFLOW-based analysis on seepage in discrete fissure networks

Studies on karst bedrock fissures face challenge in modeling due to the heterogeneity and anisotropy of fissures. Groundwater primarily flows through fissure and pipe networks within karst aquifer systems, while the bedrock pores and micro-fissures mainly serve as water storage. Therefore, the ideal hydrological model for karst areas is the continuum–fissure–pipe model. Fissures, as the main pathway for groundwater flow, significantly influence seepage in fissure media due to their connectivity. The fissure–continuum model (FC), which integrates the advantages of continuum models and discrete fissure network (DFN) models, can realistically characterize the dual texture of karst aquifers. The FC model can also account for matrix diffusion and water exchange between the matrix and fissures. However, this model remains challenging to accurately determine water exchange between rock matrix and fissure media.

This study was based on the statistical distribution of fissure geometry and hydraulic parameters. It utilized the Monte Carlo stochastic simulation technique and MATLAB program to generate a two-dimensional fissure network diagram that mirrored the distribution of actual rock fissures. The adjacency matrix of an undirected graph of graph theory was employed to represent the intersection relationships between fissures in the fissure network. In addition, this study utilized percolation theory to eliminate isolated fissures and clusters of isolated fissures that cannot conduct fluid. This approach facilitated the generation of a first-level connectivity diagram. Fissures with single nodes were then eliminated to create a second-level connectivity diagram, and dead-ends in fissures were eliminated to generate a third-level connectivity diagram. Based on the connectivity diagrams of fissure networks, the cubic law and the continuity equation for seepage were applied to establish a single-phase, saturated, and stable DFN model. Each fissure in the connectivity diagram was mapped onto a finite difference grid to develop an FC model that combines DFN with MODFLOW. When eliminating the fissure dead-ends, computational rounding may result in the loss of some nodes. Therefore, the second-level and third-level connectivity diagrams were separately used for DFN seepage simulation and FC seepage simulation, respectively. Simulation programs were developed to reproduce cross-fissure test scenarios, and two multi-fissure cases were analyzed to investigate the effects of finite difference grid resolution and fissure dead-ends on the accuracy of stable seepage simulations. The DFN was used as a metric to evaluate the correctness and validity of FC model code writing. This study was based on the statistical distribution of fissure geometry and hydraulic parameters. It utilized the Monte Carlo stochastic simulation technique and MATLAB program to generate a two-dimensional fissure network diagram that mirrored the distribution of actual rock fissures. The adjacency matrix of an undirected graph of graph theory was employed to represent the intersection relationships between fissures in the fissure network. In addition, this study utilized percolation theory to eliminate isolated fissures and clusters of isolated fissures that cannot conduct fluid. This approach facilitated the generation of a first-level connectivity diagram. Fissures with single nodes were then eliminated to create a second-level connectivity diagram, and dead-ends in fissures were eliminated to generate a third-level connectivity diagram. Based on the connectivity diagrams of fissure networks, the cubic law and the continuity equation for seepage were applied to establish a single-phase, saturated, and stable DFN model. Each fissure in the connectivity diagram was mapped onto a finite difference grid to develop an FC model that combines DFN with MODFLOW. When eliminating the fissure dead-ends, computational rounding may result in the loss of some nodes. Therefore, the second-level and third-level connectivity diagrams were separately used for DFN seepage simulation and FC seepage simulation, respectively. Simulation programs were developed to reproduce cross-fissure test scenarios, and two multi-fissure cases were analyzed to investigate the effects of finite difference grid resolution and fissure dead-ends on the accuracy of stable seepage simulations. The DFN was used as a metric to evaluate the correctness and validity of FC model code writing. Results show that the total flow simulation values and head simulation of the fine grid under the two test conditions are in good agreement with the theoretical values of DFN. However, the node heads and total grid flow simulated by the FC model in two cases are either overestimated or underestimated in the coarse grid. Although elongated paths of grid flow Lg between cells are corrected by increasing the permeability coefficients of the cells, the coarse grid based on MODFLOW cannot accurately analyze the length of inclined fissures Lf on the grid. If the size of selected coarse grid is larger than the distance between two adjacent fissures, part of the two adjacent fissures will be merged in the grid, thereby shortening the actual flow path. Secondly, the length of inclined fissure Lf. is either overestimated or underestimated. The inaccurate analysis of flow paths will lead to an increase in the estimation error of flow rates and heads, especially in dense fissure zones near the center of the model domain. This effect can be greatly alleviated by refining the grid. The fissure dead-ends have non-negligible effects on MODFLOW grid flow. In this study, the MODFLOW-based FC model can be used to solve the pressure distributions between interconnected network of fissures and the rock matrix, overcoming the difficulty of determining water exchange between rock matrix and fissure media. This study can realize the expression of heterogeneity and anisotropy of fissure permeability in numerical computational units, which will promote the understanding of MODFLOW simulation of fissure flow based on connectivity.

Furthermore, The size of the MODFLOW grid should be determined in combination with the size of the study area, which should be small enough to capture the detailed features in the flow, and large enough to reduce the computational time and cost. Generally, the grid size is determined by a trial algorithm. When FC approach is applied to the actual watershed, the spatial variability of topography and landform factors should also be considered, such as coupling surface elevation DEM data.