Modeling flow and transport in fracture networks using graphs

Fractures form the main pathways for flow in the subsurface within low-permeability rock. For this reason, accurately predicting flow and transport in fractured systems is vital for improving the performance of subsurface applications. Fracture sizes in these systems can range from millimeters to kilometers. Although modeling flow and transport using the discrete fracture network (DFN) approach is known to be more accurate due to incorporation of the detailed fracture network structure over continuum-based methods, capturing the flow and transport in such a wide range of scales is still computationally intractable. Furthermore, if one has to quantify uncertainty, hundreds of realizations of these DFN models have to be run. To reduce the computational burden, we solve flow and transport on a graph representation of a DFN. We study the accuracy of the graph approach by comparing breakthrough times and tracer particle statistical data between the graph-based and the high-fidelity DFN approaches, for fracture networks with varying number of fractures and degree of heterogeneity. Due to our recent developments in capabilities to perform DFN high-fidelity simulations on fracture networks with large number of fractures, we are in a unique position to perform such a comparison. We show that the graph approach shows a consistent bias with up to an order of magnitude slower breakthrough when compared to the DFN approach. We show that this is due to graph algorithm's underprediction of the pressure gradients across intersections on a given fracture, leading to slower tracer particle speeds between intersections and longer travel times. We present a bias correction methodology to the graph algorithm that reduces the discrepancy between the DFN and graph predictions. We show that with this bias correction, the graph algorithm predictions significantly improve and the results are very accurate. The good accuracy and the low computational cost, with $O\left({10}^4\right)$ times lower times than the DFN, makes the graph algorithm an ideal technique to incorporate in uncertainty quantification methods.

Figure 1

Discrete fracture network made up of 6330 circular fracture whose radii are sampled from three independent truncated power-law distributions. Fractures are colored by family. There are about $13\times {10}^6$ grid cells in this model.

Figure 2

The general work flow in our proposed method involves building an equivalent graph for a given DFN. The connectivity of DFN is transformed into the graph connectivity. (Left) Eight fracture DFN with a mesh that is used for performing the high-fidelity flow and transport calculations. (Right) Equivalent graph with nodes (red spheres) representing fracture intersections. The geometric information of the fractures such as distance between intersections, apertures, etc., are stored in weights of the edges between the graph nodes. Properties such as permeability, porosity, and viscosity are also stored in these weights. The mesh to resolve the full network has 179792 triangular elements with 88200 vertices, while the graph representation has 15 nodes.

Figure 3

Illustration of a single fracture plane showing how the geometrical information of fractures is used to map the intersections $i,j$ to nodes of an equivalent graph.

Figure 4

Comparison between DFN and graph approaches for eight fractures with homogeneous permeability (Case 1). (Top) Shows the breakthrough curve comparison. Time is in seconds. (Bottom) Shows the particle statistics between fracture intersections. The four subplots on the bottom left side are individual particle statistics with all the particles traveling through the same connection shown with the same color. The four subplots on the bottom right side are the average statistics of all the particles traveling through the same connection.

Figure 5

Comparison between DFN and graph approaches for eight fractures with heterogeneous permeability (Case 2). (Top) Shows the breakthrough curve comparison. Time is in seconds. (Bottom) Shows the particle statistics between fracture intersections. The four subplots on the bottom left side are individual particle statistics with all the particles traveling through the same connection shown with the same color. The four subplots on the bottom right side are the average statistics of all the particles traveling through the same connection.

Figure 6

Comparison between DFN and graph approaches for 150 fractures with homogeneous permeability (Case 3). (Top) Shows the breakthrough curve comparison. Time is in seconds. (Bottom) Shows the particle statistics between fracture intersections. The four subplots on the bottom left side are individual particle statistics with all the particles traveling through the same connection shown with the same color. The four subplots on the bottom right side are the average statistics of all the particles traveling through the same connection.

Figure 7

Comparison between DFN and graph approaches for 500 fractures with heterogeneous permeability (Case 4). (Top) Shows the breakthrough curve comparison. Time is in seconds. (Bottom) Shows the particle statistics between fracture intersections. The four subplots on the bottom left side are individual particle statistics with all the particles traveling through the same connection shown with the same color. The four subplots on the bottom right side are the average statistics of all the particles traveling through the same connection.

Figure 8

Breakthrough curves for ten realizations of 500 fracture networks with heterogeneous permeability. Blue curves are for graph and red is for DFN. The graph breakthrough is consistently slower than DFN.

Figure 9

Breakthrough curves for four realizations of 500 fracture networks with heterogeneous permeability. Blue curves are for the DFN, orange is for the graph, and green is the graph utilizing the bias correction procedure (called “Graph++” in the legend).

Figure 10

Plot comparing the CPU times for various steps in the graph and DFN methods. Note that the $y$ axis is in logarithmic scale. The star marker shows the ratio of graph method to DFN times.