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STOMP Runs for Calcite Mixing/Precipitation Problem

This page presents a series of STOMP simulations of the INL intermediate-scale flow cell experiments. These are preliminary simulations for use in design of upcoming experiments. Each experiment evaluates the potential impact of one or more simple heterogeneities within the flow cell. Note that in the images below the flow cell has been rotated ninety degrees; the actual flow cell has upward flow but to get the flow boundaries to work correctly in STOMP we had to rotate it onto its side. Flow in the images below is from left to right. Concentration-dependent density effects are not considered in these simulations.

Parameters that are constant for all simulations shown below:

Run 0: Homogeneous Base Simulation

In this case, the inclusion is a large rectangle. However, the hydraulic properties of the inclusion are the same as those of the background material, so effectively this is a homogeneous system.

Run 1: Large Inclusion with Moderately Low K

In this case, the inclusion is assigned lower hydraulic conductivity and higher porosity (leading to slower advection). The hydraulic conductivity of the inclusion is only moderately lower than the background and causes some flow spreading around the inclusion but still allows significant flow through the inclusion. The resulting calcite band penetrates through the inclusion but is dim within the inclusion and stronger upstream and downstream of the inclusion.

Run 2: Large Inclusion with Low K

In this case, the inclusion is assigned much lower hydraulic conductivity and higher porosity (again leading to slower advection). The hydraulic conductivity of the inclusion causes significant flow spreading around the inclusion. The resulting calcite band effectively is prohibited from forming in the inclusion, although over longer time frames the solutes will eventually penetrate. Flow converges back downstream of the inclusion rapidly and a stronger mineral band forms just downstream.

Run 3: Small Inclusion with Low K

This case is similar to Run 2 but with a smaller inclusion. The hydraulic conductivity of the inclusion is even lower yet, a full two orders of magnitude lower than the background material. The porosity of both materials is the same. Results are similar to Run 2 but it will likely take less time for the solutes to eventually penetrate the low K zone and form a precipitate band because of its smaller size.

Run 4: Long Rectangular Inclusion with Low K

This case is similar to Run 3 but with a long rectangular inclusion directly on the flow cell centerline. Again there is little or no precipitation in the low-K zone but precipitation does occur immediately downstream of the inclusion.

Run 5: Long Rectangular High-K Inclusion (Focused Flow)

In this case, the inclusion is a long rectangle with high hydraulic conductivity that focuses flow through the inclusion. Here the highest precipitation occurs within the inclusion where flow was focused. If coupling (feedback) between precipitation and permeability were included this would lead to lowering of K in the inclusion zone until perhaps flow were re-equalized.

Run 6: Slightly Offset Long Rectangular Inclusion with Low K

This case is similar to Run 4 but the low-K inclusion is offset slightly off the centerline of the flow cell such that the flow diversion is assymetrical. Results in this case are interesting in that, although the mixing interface is not completely diverted to the near edge of the inclusion, there is enough calcium transmitted to the lower side to form a strong precipitate nodule near the corner of the inclusion. This does not propagate strongly downgradient, although there is a faint precipitate near the bottom downstream edge of the inclusion. Presumably this case is representative of what could happen experimentally with a low-K zone on the centerline, as it would be difficult to get it exactly centered on the flow field. Also interesting is the possibility that with feedback included (if precipitation reduced permeability) then the system could be self-stabilizing. That is, the precipitate nodule near the bottom would block flow thereby re-equalizing flow around the low-K zone and minimizing mixing overall.

Run 7: Offset High-K Long Rectangular Inclusion (Flow Focusing)

This case is similar to Run 5 but the high-K inclusion is offset off the centerline by its own width.

Run 8: Homogeneous With Different Inflow Rate

This case is the same as Run 0, except that the inflow rate on the two halves of the flow cell (corresponding to the two solutes) is not uniform. Previous runs used a uniform inflow rate over the whole boundary of 0.38 cm/sec.

The results of this simulation raised a red flag in that the precipitate body appears to form not along the mixing streamline (as would have been expected) but rather crosses streamlines and forms in the Ca-rich region. Below is a figure with the calcite concentration overlain by the streamlines:

To ensure that this result was not caused by numerical dispersion, we performed simulations with a refined mesh with square elements (aspect ratio = 1.0), and also using the TVD transport solver (which is a higher-order solver). All tests gave the same results as the original run, indicating that the paradoxical result was not caused by numerical problems.

Since mass can only cross streamlines by diffusion / dispersion, it appeared that there was some preferential dispersion occurring in the simulation. This led us to the idea that the local contrast in advective velocity at the center point of the injection face could cause larger transverse dispersivity above the mixing streamline, thus perhaps causing the mixing zone to occur above the mixing streamline. We tested this concept in Run 13 (see below), and found that indeed a sharp contrast in advective velocity causes the precipitate to form away from the mixing streamline. However, it formed in the lower velocity region, not the higher velocity region. Also, this effect was not observable unless the transverse dispersivity was increased to 0.5 cm. Therefore, this hypothesis appeared to be invalid.

Finally, we performed a series of simulations in which we varied (and in some cases completely turned off) one or both of the dispersivity components (longitudinal and transverse). The results below are for Run 8h, in which the longitudinal dispersivity was set to zero and the transverse dispersivity was set to 0.005 cm.

It is evident in the figure above that the simulated calcite precipitate forms along the mixing streamline in the case where there is no longitudinal dispersion. Therefore, we believe that the way STOMP implements directional dispersion is causing an artificial flux in the x-axis direction when longitudinal dispersivity is significant. Vicky Freedman is checking the STOMP code to verify this conclusion.

Run 9: Low-K Inclusion With Different Inflow Rate

Here the inflow is non-uniform as in Run 8 but the inclusion is assigned a low hydraulic conductivity (two orders of magnitude lower than background). This forces the flow down around the bottom of the inclusion.

Run 10: Multiple Randomly-Placed Low-K Inclusions

This case is similar to several of the runs above in that it has low-K heterogeneity. However, in this case, the heterogeneity is in the form of several smaller rectangles randomly placed in the domain. The resulting precipitate band follows a slightly sinous path in order to avoid low-K zones near the centerline.

Run 11: Multiple Randomly-Placed Low-K Inclusions

This case is the same as Run 10 but with a larger number of smaller inclusions.

Run 12: Sequential Solute Injection ("Chaser")

This run uses homogeneous material properties (flow solution from Run 0). Instead of side-by-side simultaneous injection of the two solutes, the calcium is injected first (simulated by an initial condition with uniform Ca++ concentration of 0.05 molar) and followed by a flush of CO3-- (at 0.05 molar concentration).

Run 13: Sharp Contrast in Velocity

This run imposes a higher flow rate in the top half of the domain, as in Run 8. However, the permeability of the upper half is increased proportionally to the flow (relative to that of the lower half) such that the streamlines in the system are all parallel and vertical. However, this creates a sharp contrast in advective velocity at the centerline (where the solute mixing would nominally be expected to occur). Because the dispersion coefficient is proportional to velocity (with the proportionality constant being the dispersivity parameter), one would expect faster transverse dispersion in the upper half than in the lower half, and that this could impact the location of the mixing zone and thus the precipitated calcite body. To make the effect more obvious, we increased the transverse dispersivity in this run to 0.5 cm.

The precipitated calcite forms below the centerline, in the lower-velocity region. This can be explained as follows. At the inlet on the centerline, there is a sharp gradient in both solute concentrations. Because the transverse dispersion is larger in the high velocity region, the transverse dispersive flux of CO3 away from the centerline is larger than that of Ca. Therefore, the region where the gradient of CO3 concentration is large moves downward more rapidly than that of the Ca moves upward, creating asymmetry in the concentration profiles. This results in the location of the highest concentration product occuring below the centerline of flow, and thus the highest rate of calcite precipitation occurs below the centerline. (NOTE: To verify that this was not a density effect, we ran a similar simulation with the high flow and high K zone on the bottom, and the calcite precipitation occured above the centerline).

Run 15: Increased CO3 Concentration

This set of runs uses high carbonate concentrations under a variety of conditions. The objective of the runs is to see whether the calcite precipitate "drifts" to one side or the other when the concentrations of the two solutes are different.