Simulations of dissolution on fracture enlargement by dissolution were done to explore how conduit cross section size is distributed on a network. The FEFLOW simulations were done using the module for fracture evolution described on Chapter 4. Mineral dissolution is simulated using the model from Dreybrodt et al. (2005b) for laminar flow presented in
Shuanghe
radius(m)
Proportion
Sieben Hengste
radius(m)
Proportion
Wakulla
radius(m)
Proportion
Figure 6.1: Geometries and distribution of conduit radius in three mapped karst systems.
Chapter 4 and that accounts both for 1st and 4th order dissolution kinetics. Equilibrium concentration is assumed as 2 mmol L−1 represented as 2 mg L−1 in FEFLOW.
Figure 6.2 presents the first simulation. The model is a fracture network of rectangular cross section with homogeneous distribution of initial aperture equal to 4× 10−4 m and 1 m width. The fractures are created on the element boundaries of a FEM mesh generated by Delaunay triangulation. The hydraulic boundary condition is constant recharge of 50
m3d−1 on the top boundary and constant head on the bottom of the simulation domain.
Mesh elements have a high transmissivity on the top of the model to distribute flow evenly in fracture and a transmissivity several orders of magnitude less than fractures in the rest of the model. The mass boundary condition is fully unsaturated water at the injection nodes. The dissolution kinetics rate is assumed to be the same in all fractures representing homogeneous mineral composition. The simulation domain extent is 500∗ 500 m. Fractures were not created in the model boundaries because boundaries are the path with less resistance to flow between the injection points and the outlets. If fractures were set in the boundaries, these fractures would be the only ones that develop after the simulation starts. The number of simulated fractures is 2000.
Figure 6.2: Geometry of simulated fracture network evolution and initial fracture aperture.
The edges of triangles in 2D mesh represent fractures.
Figure 6.3 presents the results of the first model after a simulation time of 17, 000 yr.
Since fracture distribution is homogeneous, dissolved fractures do not resemble a cave. The development of fracture aperture after dissolution resembles more the geometry of wormholes developed by dissolution porous media (Petrus and Szymczak, 2016). The histogram shows the accumulated length of fractures of a certain aperture. The homogeneous distribution of apertures evolved into an exponential distribution.
A second simulation was done after setting the initial fracture apertures to an exponential distribution with mean µ = 4× 10−4 m. The exponential distribution is defined as y = f (x|µ) = 1/µ exp(−x/µ) . Figure 6.4 shows that the exponential distribution of aperture is randomly located in space.
Figure 6.3: Fracture network geometry after dissolution at simulation time t = 17, 000 yr.
Figure 6.4: Initial fracture aperture with exponential distribution at simulation time t = 0.
Figure 6.5 shows the geometry of the network after 19, 600 yr of simulated dissolution.
The initial distribution of aperture apparently had no effect on the geometry of the resulting network. The developed fractures also resemble a wormhole geometry. The distribution of fracture aperture evolved into a log-normal distribution with two peaks: one for the initial aperture and one for the developed fractures.
Figure 6.6 presents the third simulation with the initial fracture aperture set to a log-normal distribution with mean µ = log(4× 10−4) m and standard deviation σ = 0.8. The aperture was set randomly in the space.
Figure 6.7 shows results after 26, 000 yr of simulated dissolution in the network with initial aperture with log-normal distribution. Developed geometries resemble wormholes and the distribution aperture also has two peaks.
The last results show that simulating dissolution in a randomly spatially distributed
Figure 6.5: Final fracture aperture of network with initial exponential distribution at simu-lation time t = 19, 000 yr.
Figure 6.6: Initial fracture aperture with log-normal distribution
Figure 6.7: Final fracture aperture of network after simulation time t = 27, 000 yr. with initial log-normal distribution.
network does not produce geometries that resemble cave geometries as the real measured ones in figure 6.1. Two of the simulated networks had heterogeneity in fracture aperture. However
the connectivity of fractures was evenly distributed and this may be a factor contributing to wormhole geometry development.
Another alternative to generate artificial networks is applying percolation theory. Per-colation models were used by Hendrick and Renard (2016) to study the fractal properties of karst networks in the study area. Percolation models are pure geometric models to study connected clusters in disordered media. Percolation models have 1 parameter which is the probability p of a site being occupied. The size of connected clusters diverges for a cer-tain value which is the critical point. At the critical point there is a non-zero probability that a site in the model domain belongs to a cluster that connects the model boundaries.
Probability p is the order parameter of the system. A model in the disordered phase has a zero probability of having clusters that connect the system boundaries. Models in the ordered phase have a non-zero probability of connecting model boundaries with a cluster.
The critical point if the transition point between ordered and disordered phases.
To build a percolation model the first step is to build a mesh which is the substrate. Then a probability of being occupied is assigned to mesh edges. This generates several clusters.
When working with the critical probability, the largest cluster connects the substrate (mesh) boundaries. The largest cluster is selected among the many generated clusters to study flow across the mesh (Hendrick, 2016) .
A percolation model scaled to dimensions (1000x800 m) relevant for calcite dissolution was used as initial fracture geometry of a fourth simulation. This percolation models has an occupation probability at the critical point: the largest cluster connects all the model boundaries. Figure 6.8 shows the initial fracture aperture and flow boundary conditions which are 7 injection points with equal maximum hydraulic heads of 50 m and maximum flow rate of 10 m3d−1. Figure 6.8 also presents the initial concentration of solute before dissolution. Injection points in the bottom have larger plumes because they are closer to the outlet boundary. They have less resistance to flow, then advective transport is stronger compared to injection point farther from the outlet. As the first simulations, the dissolution rate accounts for both the 1st and 4th order dissolution kinetics. Reaction rate is assumed as homogeneous in the model domain. The number of simulated fractures is 79, 000.
Figure 6.9 shows the fracture network geometry and aperture distribution after all in-jection points are connected with the outlet boundary by enlarged fractures. The resulting
Figure 6.8: Percolation cluster representing initial conditions of fracture network.
network after simulation time of t = resembles the cave topology like the ones presented on figure 6.1. This results is influenced by the properties of the percolation cluster where there are preferential flow paths. Some fractures are dead ends with no flow, thus they do not enlarge. The distribution of aperture has also two peaks, the first one represents the initial aperture of 0.0004 m and the second one the cave. The fractures near to the outlet boundary are larger, meaning that they developed faster compared to fractures farther from the outlet.
Along each fracture,the aperture is distributed from a maximum at the injection point and a minimum at the outlet boundary or at the intersection with other enlarged fractures. Along a fracture the aperture is almost homogeneous, heterogeneity in aperture developed only at intersections of developing fractures and dead ends where diffusion is the only available mechanism for mass transport. Flow in a preferential path and homogeneous dissolution rate will tend to homogenize conduits cross sections along the preferential flow path. This behavior is not observed in real caves where large cross sections are contiguous to smaller ones.
A fifth simulation was done assuming heterogeneity in the dissolution kinetics. In the study area there is heterogeneity in limestone properties according to the the origin of sedi-ments. The spatial distribution of limestone properties is related to the depositional history described in Chapter 3. Reefal limestone has a different texture and thus dissolution rate compared to backreef lagoon sediments with finer grain size. Both limestones are near 100%
calcite, and have the same equilibrium concentration corresponding to the carbon dioxide
Figure 6.9: Fracture network geometry after dissolution time t = 1, 200 yr. and aperture distribution of simulation 4. Initial percolation cluster is shown in figure 6.8.
in solution. However, differences in grain size, meaning different reaction surface area will result in different dissolution rates. The fifth simulation assumes reaction rate spatially dis-tributed in 50 m thick highly reactive bands intercalated among 150 m less reactive bands.
The reaction rate is 10 times faster in the highly reactive bands.
Figure 6.11 presents the results of fracture evolution assuming heterogeneity in reaction rate. Flow and mass boundary conditions are equal to simulation 4. Heterogeneity in reaction rate produces heterogeneity in fracture aperture. Large apertures are followed downstream by smaller apertures and then larger apertures once again. This behavior is similar to real caves presented in figure 6.1. But, the most interesting feature of simulation 5 is that the geometry of the developed cave system is different.
Figure 6.12 present the differences between simulation 4 with homogeneous reaction rate and simulation 5 with heterogeneous reaction rate. In simulation 4, the fracture that developed was the closer to the outlet in terms of resistance to flow. Since initial fracture aperture was the same for all fracture in simulation 4, the path with less resistance to flow was the shortest one. However, in simulation 5 the fracture that developed is not the shortest one. The fracture that developed is located in the lower dissolution reaction rate.
The intuitive result is that faster dissolution rate is a preferential flow path for fractures.
This is true at the injection point where a higher dissolution rate creates larger fractures,
Figure 6.10: Assumed heterogeneity in reaction rate in simulation 5. 50 m thick highly reactive bands (red), intercalated among 150 m less reactive bands (purple).
Figure 6.11: Fracture network geometry after dissolution time t = 1, 650 yr. and aperture distribution of simulation 5. Heterogeneity in reaction rate in contrast to results presented in figure 6.9.
but only near the injection point. The fracture enlarged in simulation 5 has less resistance to flow because it developed farther away from the injection point because water less saturated could travel further into the fracture network. Thus, development of fractures is controlled
by the enlargement of the whole fracture from the injection point to the outlet. This can be explained by the use of Hagen-Poiseuille linear law for flow. As fractures grow their contribution to friction is less important and the largest fraction of resistance to flow is caused by the fractures that have the initial aperture. Thus, higher reaction yields larger fracture cross sections where reactive water enters the system, but lower reaction rates control the development of longer fractures that connect the injection point with the outlets.
Figure 6.12: Differences in developed network geometry between simulations 4 and 5.