Construction of a Robust 3-D Particle Tracking Tool for...
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Construction of a Robust 3-D Particle Tracking Tool for Visualizing Transient Groundwater Flow Fields Generated by Finite-Element Numerical Models Shawn Leppert—Leppert Associates Inc. Golden, Colorado Spawned from need to analyze transient groundwater capture zones for large scale, public supply extraction wells, the authors wrote a computer algorithm to track the movement of non-reactive particles within a transient Darcy velocity field generated by numerical groundwater flow models. Advanced particle tracking capabilities were required to analyze results from finite-element models which represent the complex site hydrogeologic concep- tual model being considered. The numerical models simulated an environment where the magnitude and direction of groundwater flow varies dramatically throughout the seasons. Although MODPATH (USGS) can provide advanced particle tracking capabilities for results from finite-difference numerical models, there is a lack of publicly available particle track- ing tools equipped to precisely track the simulated movement of particles within a wildly transient finite-element velocity field. The particle tracking scheme constructed employs techniques that enable very precise tracking within a prescribed velocity field, even where flow direction can change by 180 degrees within a few days or less. Interpolation tech- niques were used that accurately estimate the Darcy velocity within the finite-element cells. Unlike other available particle tracking schemes, the tracker demonstrates its ability to appropriately represent extraction well capture zones, even when the finite-element mesh is relatively course. The algorithm was written to be flexible. It incorporates both techniques drawn from the scientific literature and methods developed by the authors. Adaptive time stepping and coordination with temporal changes in the velocity field were incorporated to enhance the precision of the particle tracking. 3-D capture zones and flow nets are provided to demonstrate the tool’s ability to visualize complex groundwater flow model results, particularly for transient flow fields. ABSTRACT PROBLEM APPLICATION SELECT REFERENCES TECHNIQUE The particle tracking computer code developed was suc- cessfully used to analyze transient, 3-D groundwater flow fields generated by the finite-element model. In fact, the tracker proved to be less sensitive to mesh re- finement than particle trackers which use velocities cal- culated at the finite element nodes, such as what is pro- vided by Wasy in the FEFLOW simulator. As part of the validation process, the tracker’s sensitivity to model mesh size was compared to that of the FEFLOW particle tracker. Particle defined capture zones were used to make the comparisons. When applying FEFLOW’s parti- cle tracking method, the size of a extraction well’s cap- ture zone exhibits a dependency on the level of mesh re- finement in the area surrounding the extraction node. The simulated capture appears to increase in size as the mesh is refined. The particle tracking algorithm de- scribed here does not show the same dependency. The adjacent figures illustrate the particle tracks defining a simulated extraction well’s capture zone. Three differ- ent cell refinements were simulated and their flow line- defined capture zones are plotted using different colors; green having the coarsest discretization and red the fin- est. In addition to the analyses the tracker was initially de- veloped for, it has been used to quantify other numeri- cally generated transient groundwater velocity fields. The particle tracker was used to simulate transient cap- ture zones in the design of a sophisticated groundwater injection/extraction remediation system. The site’s groundwater flow field was sensitive to precipitation de- rived recharge as well as intermittent groundwater ex- traction used to supply local canneries. The tracker was used, in concert with a well-calibrated finite- element groundwater flow model, to provide real-time information on the current effectiveness of the hydrau- lic controls. In another application, the particle tracking algorithm was used to evaluate the effects from mine dewatering on local shallow groundwater supply wells. At the site, past mine dewatering operations effectively eliminated the use of groundwater resources for local rural water supplies. Mining has ceased and the water table is slowly rebounding. A saturated/unsaturated finite- element model was constructed to evaluate the flow field. Discrete hydraulic features were used to emulate the remaining mine tunnels. The tracker was being used to analyze the transient groundwater recovery. Our particle tracker was constructed in a modular fashion with the intent to advance it through time. From its inception, a number of additional capabilities were planned to be incorporated. The computer algorithm was programmed with specific hooks to cou- ple four specific advancements. Soon these advancements will be realized. Schafer-Perini, A.L. and J.L. Wilson, 1991. Efficient and Accurate Front Tracking for Two- Dimensional Groundwater Flow Models. Water Resour. Res., 27(7), 1471-1485. Leppert, S.C., 1990. Capture Zone in Transient Flow Fields: Simulations and Analyses. Masters Thesis for New Mexico Institute of Mining and Technology, 111 pp. Wasy, 2001. FEFLOW Finite Element Subsurface Flow and Transport Simulation System. Version 4.8. In 2001, the authors scoured the scientific literature and queried software venders searching for a particle tracking algorithm capable of tracking within simu- lated 3-D groundwater flow fields described by a tran- sient finite-element model. We had been tasked with evaluating a public drinking water resource supplied by shallow aquifers where the magnitude and direc- tion of the groundwater velocities vary radically through time. This highly transient flow field repre- sented an environment where large-scale seasonal changes in the local hydrologic cycle heavily influ- ence the groundwater flow within the aquifers. The 3-D finite-element simulator FEFLOW was selected to construct the groundwater flow models. At the time, FEFLOW’s (WASY, 2001) particle tracking capa- bilities were limited by the assumption that the groundwater flow field is at steady-state and can be described by a single velocity distribution realiza- tion. FUTURE ADVANCEMENTS • Measured Potentiometric Surface Tracker — pro- vides the ability to input the hydraulic heads meas- ured in monitoring wells, generate a single layer fi- nite-element mesh, assign discrete porous media, and use the tracker to generate flow lines. • Adaptive Time Step Control — is a technique that adapts the tracking time step to account for ground- water flow divergence which can introduce errors (Leppert,1990). • Dynamic Front Tracking — a front tracking method that allows for the precise tracking of groundwater flow fronts. It backfills particles once adjacent par- ticles in a front exceed a prescribed distance (Schafer-Perini and Wilson, 1991). • Fast, efficient particle tracking using direct ac- cess files to store the massive amounts of infor- mation potentially generated by a transient finite- element model. • Particle tracking input is provided through a basic user interface. A specific interface with FEFLOW has been constructed but the model is designed for input of common finite-element model results. • The algorithm was constructed in a modular fash- ion using Fortran 90. It was equipped with dy- namic memory allocation to accommodate differ- ent sized models. The interface was written in Fortran 90 as well. • Specific tools are provided to configure particle tracking for common analyses including: capture zones, quantitative flow nets, and discrete particle placement. • This particle tracker assimilates results from sophisticated models that may incorporate deformable grids, discrete hydraulic features (including line-element wells), and satu- rated/unsaturated fluid flow. Jeff Weaver—GEOMATRIX Denver, Colorado Bill Linderfelt—GEOMEGA Albuquerque, New Mexico Capture zone sensitivity to grid discreti- zation using the particle tracker. Capture zone sensitivity to grid discreti- zation using FEFLOW Construction of the particle tracking algorithm drew from the basic principals of tracing the move- ment of a conservative, non-reactive particle through a prescribed transient, 3-D velocity field. The velocity field was derived from the transient, 3-D finite-element nodal hydraulic heads calculated by the numerical model. An Euler-type numerical time stepping scheme was used to march the parti- cles through the velocity field. In the finite-element solution, hydraulic heads are calculated for each time step at the six nodes de- fining each element prism. Using linear basis func- tions, two cell velocities were calculated for each finite-element cell within the model domain. These basis functions describe the slope of a plane fit through the three hydraulic head elevations calcu- lated for both the upper and lower cell-halves. The magnitude and direction of the hydraulic gradient coupled with the cell’s hydraulic conductivity and porosity were used to calculate the cell velocities. This approach results in the prescription of a discontinuous velocity distribution over the model domain. A discontinuous velocity field is acceptable provided that its spatially vary- ing nature remains small with respect to the element dimension. Movement of particles through the finite-element cells is accom- plished by setting a small time step, ∆t, and calculating ∆x, ∆y, and ∆z values using ∆(x, y, or z ) = v x, y or z * ∆t. In 3-D, the veloc- ity can vary vertically such that the upper and lower faces of a six-sided prism will have different head distributions and there- fore, different velocities. Half-cell velocities were assigned to be constant between the midpoint and the upper and lower cell sur- faces. The new particle positions are determined from (x, y, or z) new = (x, y, or z) old + ∆(x, y, or z). The spatiotemporal location of the particle is checked after each time step to determine if it has passed into a new element or time-dependent velocity field. When developing a transient, numerical model particle tracking algorithm, managing the data is the most difficult task. Signifi- cant effort was made to integrate the velocity-field change intro- duced during time step changes. The computer code was built to account for the cell-by-cell fluid mass in an effort to provide in- sight on mass balance related issues. The tracker also accounts for prescribed model boundary conditions. Temporal and spatial variations in groundwater flow – note that different colors represent differ- ent depths within the flow field. Separately and together, the authors have extensive experience in developing particle track- ing algorithms. Although a bit drastic, we undertook the development of a particle tracking computer program for use in analyzing transient finite-element model results representing the site. A robust but flexible algorithm was needed to thoroughly analyze the model re- sults. The requirements of the particle tracking algorithm were dictated by the complexi- ties inherent in the site’s hydrogeologic conceptual model. First, the tracker was required to easily read the large amounts of data generated by a finely discretized, transient, 3-D nu- merical model. FEFLOW is capable of emulating very intricate hydrogeologic environments using sophisticated hydrologic and physical representations including: saturated/ unsaturated flow; deformable grids; discrete hydraulic features; and complex boundary con- ditions. The particle tracker was built to accommodate these possible complexities. And fi- nally, the tracker was designed to visualize the particle tracking results using a number of different quantitative formats such as extraction well capture zones and precise flow nets. FEATURES Diagram of the finite-element cells, their representative hydraulic head facets and associated velocity vectors. Verification of particle tracker results; velocity field, forward tracks, and backward tracks. • Spatial Probability Distribution Analysis — a tool that will generate a probability distribu- tion for the likelihood that groundwater contained within a finite-element cell will be cap- tured by an extraction well simulated in a transient groundwater environment. A precise flow net generated from the numeri- cal model prescription of inflow into the model domain. Each of the 132 stream tubes has a flow rate = 129.9 m 3 /day. 30 35 40 45 E l e v a t i on ( m e t e r s a ms l ) Bedrock-2 UG Bedrock-1 UG Bedrock-15 Bedrock-13 UG Bedrock-11 UG Bedrock-12 UG Bedrock-10 UG Bedrock-9 UG Bedrock-8 UG Bedrock-7 UG Bedrock-6 UG Bedrock-5 UG Bedrock-4 UG Bedrock-3 UG Injection-2 Bedrock-16 DG Bedrock-14 DG Injection-4 Injection-1 Injection-3 Bedrock Aquifer Flownet Legend N Vertical Exaggeration = 50x m Scale: 225 0 Injection Trench Particle Trace Injection Well Extraction Well 3-D capture tube of a simulated pumping well extraction groundwater from a tran- sient groundwater flow field.
Transcript of Construction of a Robust 3-D Particle Tracking Tool for...
Construction of a Robust 3-D Particle Tracking Tool for Visualizing Transient Groundwater Flow Fields Generated by Finite-Element Numerical Models
Shawn Leppert—Leppert Associates Inc. Golden, Colorado
Spawned from need to analyze transient groundwater capture zones for large scale, public supply extraction wells, the authors wrote a computer algorithm to track the movement of non-reactive particles within a transient Darcy velocity field generated by numerical groundwater flow models. Advanced particle tracking capabilities were required to analyze results from finite-element models which represent the complex site hydrogeologic concep-tual model being considered. The numerical models simulated an environment where the magnitude and direction of groundwater flow varies dramatically throughout the seasons. Although MODPATH (USGS) can provide advanced particle tracking capabilities for results from finite-difference numerical models, there is a lack of publicly available particle track-ing tools equipped to precisely track the simulated movement of particles within a wildly transient finite-element velocity field. The particle tracking scheme constructed employs techniques that enable very precise tracking within a prescribed velocity field, even where flow direction can change by 180 degrees within a few days or less. Interpolation tech-niques were used that accurately estimate the Darcy velocity within the finite-element cells. Unlike other available particle tracking schemes, the tracker demonstrates its ability to appropriately represent extraction well capture zones, even when the finite-element mesh is relatively course. The algorithm was written to be flexible. It incorporates both techniques drawn from the scientific literature and methods developed by the authors. Adaptive time stepping and coordination with temporal changes in the velocity field were incorporated to enhance the precision of the particle tracking. 3-D capture zones and flow nets are provided to demonstrate the tool’s ability to visualize complex groundwater flow model results, particularly for transient flow fields.
ABSTRACT
PROBLEM
APPLICATION
SELECT REFERENCES
TECHNIQUE The particle tracking computer code developed was suc-cessfully used to analyze transient, 3-D groundwater flow fields generated by the finite-element model. In fact, the tracker proved to be less sensitive to mesh re-finement than particle trackers which use velocities cal-culated at the finite element nodes, such as what is pro-vided by Wasy in the FEFLOW simulator. As part of the validation process, the tracker’s sensitivity to model mesh size was compared to that of the FEFLOW particle tracker. Particle defined capture zones were used to make the comparisons. When applying FEFLOW’s parti-cle tracking method, the size of a extraction well’s cap-ture zone exhibits a dependency on the level of mesh re-finement in the area surrounding the extraction node. The simulated capture appears to increase in size as the mesh is refined. The particle tracking algorithm de-scribed here does not show the same dependency. The adjacent figures illustrate the particle tracks defining a simulated extraction well’s capture zone. Three differ-ent cell refinements were simulated and their flow line-defined capture zones are plotted using different colors; green having the coarsest discretization and red the fin-est. In addition to the analyses the tracker was initially de-veloped for, it has been used to quantify other numeri-cally generated transient groundwater velocity fields. The particle tracker was used to simulate transient cap-ture zones in the design of a sophisticated groundwater injection/extraction remediation system. The site’s groundwater flow field was sensitive to precipitation de-rived recharge as well as intermittent groundwater ex-traction used to supply local canneries. The tracker was used, in concert with a well-calibrated finite-element groundwater flow model, to provide real-time information on the current effectiveness of the hydrau-lic controls. In another application, the particle tracking algorithm was used to evaluate the effects from mine dewatering on local shallow groundwater supply wells. At the site, past mine dewatering operations effectively eliminated the use of groundwater resources for local rural water supplies. Mining has ceased and the water table is slowly rebounding. A saturated/unsaturated finite-element model was constructed to evaluate the flow field. Discrete hydraulic features were used to emulate the remaining mine tunnels. The tracker was being used to analyze the transient groundwater recovery.
Our particle tracker was constructed in a modular fashion with the intent to advance it through time. From its inception, a number of additional capabilities were planned to be incorporated. The computer algorithm was programmed with specific hooks to cou-ple four specific advancements. Soon these advancements will be realized.
Schafer-Perini, A.L. and J.L. Wilson, 1991. Efficient and Accurate Front Tracking for Two-Dimensional Groundwater Flow Models. Water Resour. Res., 27(7), 1471-1485.
Leppert, S.C., 1990. Capture Zone in Transient Flow Fields: Simulations and Analyses. Masters Thesis for New Mexico Institute of Mining and Technology, 111 pp.
Wasy, 2001. FEFLOW Finite Element Subsurface Flow and Transport Simulation System. Version 4.8.
In 2001, the authors scoured the scientific literature and queried software venders searching for a particle tracking algorithm capable of tracking within simu-lated 3-D groundwater flow fields described by a tran-sient finite-element model. We had been tasked with evaluating a public drinking water resource supplied by shallow aquifers where the magnitude and direc-tion of the groundwater velocities vary radically through time. This highly transient flow field repre-sented an environment where large-scale seasonal changes in the local hydrologic cycle heavily influ-ence the groundwater flow within the aquifers. The 3-D finite-element simulator FEFLOW was selected to construct the groundwater flow models. At the time, FEFLOW’s (WASY, 2001) particle tracking capa-bilities were limited by the assumption that the groundwater flow field is at steady-state and can be described by a single velocity distribution realiza-tion.
FUTURE ADVANCEMENTS
• Measured Potentiometric Surface Tracker — pro-vides the ability to input the hydraulic heads meas-ured in monitoring wells, generate a single layer fi-nite-element mesh, assign discrete porous media, and use the tracker to generate flow lines.
• Adaptive Time Step Control — is a technique that adapts the tracking time step to account for ground-water flow divergence which can introduce errors (Leppert,1990).
• Dynamic Front Tracking — a front tracking method that allows for the precise tracking of groundwater flow fronts. It backfills particles once adjacent par-ticles in a front exceed a prescribed distance (Schafer-Perini and Wilson, 1991).
• Fast, efficient particle tracking using direct ac-cess files to store the massive amounts of infor-mation potentially generated by a transient finite-element model.
• Particle tracking input is provided through a basic user interface. A specific interface with FEFLOW has been constructed but the model is designed for input of common finite-element model results.
• The algorithm was constructed in a modular fash-ion using Fortran 90. It was equipped with dy-namic memory allocation to accommodate differ-ent sized models. The interface was written in Fortran 90 as well.
• Specific tools are provided to configure particle tracking for common analyses including: capture zones, quantitative flow nets, and discrete particle placement.
• This particle tracker assimilates results from sophisticated models that may incorporate deformable grids, discrete hydraulic features (including line-element wells), and satu-rated/unsaturated fluid flow.
Jeff Weaver—GEOMATRIX Denver, Colorado
Bill Linderfelt—GEOMEGA Albuquerque, New Mexico
Capture zone sensitivity to grid discreti-zation using the particle tracker.
Capture zone sensitivity to grid discreti-zation using FEFLOW
Construction of the particle tracking algorithm drew from the basic principals of tracing the move-ment of a conservative, non-reactive particle through a prescribed transient, 3-D velocity field. The velocity field was derived from the transient, 3-D finite-element nodal hydraulic heads calculated by the numerical model. An Euler-type numerical time stepping scheme was used to march the parti-cles through the velocity field. In the finite-element solution, hydraulic heads are calculated for each time step at the six nodes de-fining each element prism. Using linear basis func-tions, two cell velocities were calculated for each finite-element cell within the model domain. These basis functions describe the slope of a plane fit through the three hydraulic head elevations calcu-lated for both the upper and lower cell-halves. The magnitude and direction of the hydraulic gradient coupled with the cell’s hydraulic conductivity and porosity were used to calculate the cell velocities.
This approach results in the prescription of a discontinuous velocity distribution over the model domain. A discontinuous velocity field is acceptable provided that its spatially vary-ing nature remains small with respect to the element dimension.
Movement of particles through the finite-element cells is accom-plished by setting a small time step, ∆t, and calculating ∆x, ∆y, and ∆z values using ∆(x, y, or z ) = vx, y or z * ∆t. In 3-D, the veloc-ity can vary vertically such that the upper and lower faces of a six-sided prism will have different head distributions and there-fore, different velocities. Half-cell velocities were assigned to be constant between the midpoint and the upper and lower cell sur-faces. The new particle positions are determined from (x, y, or z)new = (x, y, or z)old + ∆(x, y, or z). The spatiotemporal location of the particle is checked after each time step to determine if it has passed into a new element or time-dependent velocity field. When developing a transient, numerical model particle tracking algorithm, managing the data is the most difficult task. Signifi-cant effort was made to integrate the velocity-field change intro-duced during time step changes. The computer code was built to account for the cell-by-cell fluid mass in an effort to provide in-sight on mass balance related issues. The tracker also accounts for prescribed model boundary conditions.
Temporal and spatial variations in groundwater flow – note that different colors represent differ-ent depths within the flow field.
Separately and together, the authors have extensive experience in developing particle track-ing algorithms. Although a bit drastic, we undertook the development of a particle tracking computer program for use in analyzing transient finite-element model results representing the site. A robust but flexible algorithm was needed to thoroughly analyze the model re-sults. The requirements of the particle tracking algorithm were dictated by the complexi-ties inherent in the site’s hydrogeologic conceptual model. First, the tracker was required to easily read the large amounts of data generated by a finely discretized, transient, 3-D nu-merical model. FEFLOW is capable of emulating very intricate hydrogeologic environments using sophisticated hydrologic and physical representations including: saturated/unsaturated flow; deformable grids; discrete hydraulic features; and complex boundary con-ditions. The particle tracker was built to accommodate these possible complexities. And fi-nally, the tracker was designed to visualize the particle tracking results using a number of different quantitative formats such as extraction well capture zones and precise flow nets.
FEATURES
Diagram of the finite-element cells, their representative hydraulic head facets and associated velocity vectors.
Verification of particle tracker results; velocity field, forward tracks, and backward tracks.
• Spatial Probability Distribution Analysis — a tool that will generate a probability distribu-tion for the likelihood that groundwater contained within a finite-element cell will be cap-tured by an extraction well simulated in a transient groundwater environment.
A precise flow net generated from the numeri-cal model prescription of inflow into the model domain. Each of the 132 stream tubes has a flow rate = 129.9 m3/day.
30
35
40
45
Elevat ion
(meters
amsl)
Bedrock-2UG
Bedrock-1UG
Bedrock-15
Bedrock-13UG
Bedrock-11UG
Bedrock-12UG
Bedrock-10UG
Bedrock-9UG
Bedrock-8UG
Bedrock-7UG
Bedrock-6UG
Bedrock-5UG
Bedrock-4UG
Bedrock-3UG
Injection-2
Bedrock-16DG
Bedrock-14DG
Injection-4
Injection-1
Injection-3
Bedrock AquiferFlownet
Legend
N
Vertical Exaggeration = 50x
mScale:
2250
Injection Trench
ParticleTraceInjection Well
Extraction Well
3-D capture tube of a simulated pumping well extraction groundwater from a tran-sient groundwater flow field.
