Saturated/Unsaturated Interface Simulations

In many engineering problems of industrial interest, if the saturated/unsatured interface is not accurately resolved the associated fluid flow can not be modeled accurately, and this translates into errors of considerable magnitude when the fluid flow is needed to compute quantities of interest. For example, when the fluid flow below the water table is used for simulations of remediation, decontamination, etc.

ReSolution is capable of resolving saturation interfaces without artificial diffusion, even when the fluid interfaces are arbitrarily located on the underlying grid/mesh. To achieve simulation results of comparable quality with the classical Finite Element/Volume/Difference Methods (FEM/FVM/FDM) would require simulation grids with extremely small elements. Such meshes would be very difficult to generate and would be very time consuming. If classical adaptive refinement is used, many levels of local refinement would be required to capture the saturation interface, and this would also translate into a very large number of degrees of freedom and discretization errors associated with mesh gradation.

The following validation cases are very simple to highlight the quality of results that ReSolution can deliver.

Contents:

Single-well drainage - water table simulation
Comparison of ReSolution with classical simulation codes
Conclusions

Single-well drainage - water table simulation

To show qualitatively the accuracy of meshless solutions to water table problems, we include a simple example consisting of a rectangular region containing a dewatering well at the center and fixed pressure boundary condition on the sides.

Using a coarse structured cartesian grid:

Drainage area is 100 ft x100 ft. Depth: 30 ft. Well rate: 100 barrels per day.

Side view of the base simulation grid (meshless components not shown)
Animation of the saturated-unsaturated interface

This animation shows how well the time dependent saturated/unsaturated interface is resolved. Note that the base grid used for numerical integration is very coarse, but the interface is accurately resolved without the smearing typical of static grids. The meshless approach works wells regardless of the thickness of each layer.

Using a coarse tetrahedral grid:

Drainage area is 100 ft x100 ft. Depth: 25 ft. Well rate: 100 barrels per day. (Note: this is not exactly the same case as above, not only the thickness is different, also the rock permeability and the well perforation length)

Side view of the base simulation grid (meshless components not shown)
Animation of the saturated-unsaturated interface

Only the vertical and horizontal grid lines are displayed for reference.

Animation of the water table with depth magnification (x3)

Only the vertical and horizontal grid lines are displayed for reference.

Animation of pressure isosurfaces with depth magnification

Only the vertical and horizontal grid lines are displayed for reference.

The examples presented above illustrate how accurate the GFEM technique is for the simulation of saturated/unsaturated flow problems. Other engineering applications of this technique are: mold filling, resin transfer molding, etc.

Comparison of ReSolution with classical simulation codes

A simple quasi one-dimensional example is used to illustrate quantitatively and qualitatively how accurate ReSolution is for the simulation of flows in porous media. This simulation shows a sharp saturation interface moving across the blocks of the grid with no artificial diffusion when ReSolution is used. A comparison with classical solutions is presented.

A rock core 100 ft high is subject to 100 psig of fluid pressure at the bottom. Initially only the lower 10% of the rock core is saturated with fluid. Specific gravity of the fluid 0.5 psi/ft. See full description of the problem and analytic solution.

Mesh consisting of only 2 elements. Technique: ReSolution with dynamic adaptation. Initial saturation and pressure. Saturation and pressure at the end of the simulation time.

Animation of saturation and pressure fields.
Almost exact solution compared with the analytical solution to this problem. See close up view of saturation at the top.
The total CPU time of this simulation is 0.5 sec (PII 400 Mhz).

Mesh consisting of 20 elements. Technique: classical FEM with static mesh. Initial saturation and pressure. Saturation and pressure at the end of the simulation time.

Animation of saturation and pressure fields.
The solution exhibits a "LOT" of smearing of the saturation front and the associated pore pressure.
The total CPU time of this simulation is 3.3 sec (PII 400 Mhz).

Mesh consisting of 100 elements. Technique: classical FEM with static mesh. Initial saturation and pressure. Saturation and pressure at the end of the simulation time.

Animation of saturation and pressure fields.
The smearing of the saturation front is small but considerable in comparison with ReSolution's approach. Also there is a small lagging error in the front location.
The total CPU time of this simulation is 15.0 sec (PII 400 Mhz). Thirty times the CPU of the dynamically adapted GFEM approach.

This simple comparison shows that for this class of problems ReSolution can deliver very accurate solutions with much less CPU time and hardware requirements than the classical FE/FV/FD techniques with static meshes.

Conclusions

A large number of engineering applications require accurate modeling of fluid saturation interfaces. Comparisons between the classical FEM, FVM, FDM, and ReSolution's approach indicate that ReSolution's is by far the best approach to simulate problems involving fluid saturation interfaces.

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