Training slides for the groundwater modelling software FEFLOW for non-commercial use.

1. Slide 1 4-Oct-22
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format.
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FEFLOW TRAINING MATERIAL
Introduction to Flow Modelling
© AQUASOIL Ingenieure und Geologen GmbH 2022

2. Slide 2 4-Oct-22
AQUASOIL Ingenieure und Geologen GmbH

3. Slide 3 4-Oct-22
Offices
• Westheim (Headquarters)
• Berlin, Stuttgart, Passau
Fields of work
• groundwater and surface water modelling
• geotechnics
• contaminated sites
• water management, hydrogeology and hydrology
• soil and foundation engineering
• road and underground construction planning
AQUASOIL INGENIEURE UND GEOLOGEN GMBH

4. Slide 4 4-Oct-22
AQUASOIL GROUNDWATER MODELLING TEAM
Dr. Frank
Wenderoth
Dr. Jörn
Bartels
Peter Schätzl
Dr. Junfeng
Luo
Hannes
Pommer
Bastian Rau
Sven Seifert
• Experts for groundwater and surface water
• Infrastructure development (rail and road tunnels, dewatering)
• Geothermics (deep and near-surface)
• Groundwater management and climate change
• Groundwater in mining

5. Slide 5 4-Oct-22
• Construction-site dewatering / infiltration ponds / siphon dimensioning / …:
• Underground train station and railway tunnel construction (Stuttgart 21, 2. SBSS München,
Berlin City-S-Bahn)
• Geothermics:
• BHE systems, energy piles, open-loop geothermal systems, deep geothermics, ATES
• Berlin Airport, Bibliothek 21 Stuttgart, BMW Dingolfing, Geothermie Pullach
• Integrated water resources management:
• Coupled groundwater-surface water model for the Zayandeh Rud Basin, Iran
• Climate-change prediction / MAR:
• Groundwater model for the Poverty Bay Flats, Gisborne, New Zealand
• Drinking water abstraction:
• Groundwater model Friedrichshagen waterworks (Berlin)
• Contamination:
• Plume migration due to geothermal system Staatstheater Karlsruhe
AQUASOIL GROUNDWATER MODELS - EXAMPLES

6. Slide 6 4-Oct-22
POROUS / FRACTURED / KARSTIC AQUIFERS
Porous aquifer
Water fills pores between sand
grains – mostly in sedimentary
deposits and sandstones
Fractured aquifer
Water fills fissures and fractures in
hard rock
Karstic aquifer
Water fills caves and openings in
limestone
Typical groundwater models apply Darcy‘s equation, describing flow in porous aquifers. Its application to fractured
rocks and Karst systems may be feasible under certain conditions (depending on the questions asked to the model,
but additional care has to be taken with respect to the parameterisation and result interpretation of such models.

7. Slide 7 4-Oct-22
CONFINED VS. UNCONFINED AQUIFERS
Aquitard
Aquifer
Aquifer
Unconfined Aquifer
Aquifer is partially filled with water. Water level
changes lead to filling or draining of the aquifer
medium.
Confined Aquifer
Aquifer is completely filled with water, the piezometric
head is higher than the aquifer top elevation.
Pressure changes lead to a compression or
expansion of the aquifer medium.

8. Slide 8 4-Oct-22
HYDRAULIC HEAD
ℎ =
𝑝
𝜌 ⋅ 𝑔
+ 𝑧
• p: pressure
• 𝜌: density
• g: acceleration of gravity
• z: elevation
Hydraulic Head is also called Piezometric Head and corresponds to the water level observed in a piezometer.
h

9. Slide 9 4-Oct-22
• q: volume flux vector
• 𝐾: hydraulic conductivity tensor
• h: hydraulic head
DARCY‘S EQUATION FOR GROUNDWATER FLOW
𝑞 = −𝐾 ⋅ ∇ℎ
𝑣 =
𝑞
𝜀
• v: effective flow velocity
• q: volume flux vector
• 𝜀: effective porosity
Darcy‘s law is applicable to laminar
flow in porous media. For very
coarse-grained sediments and/or
high flow velocities, Darcy‘s law
may no longer be valid as turbulent
flow may occur!

10. Slide 10 4-Oct-22
Numerical Model
Mathematical Model
Conceptual/Hydrogeological
Model
Complex Reality
GROUNDWATER MODELING – MODEL BUILDING AND APPLICATION
Simplification
/
Model
Building
Model
Application
The process of model building
includes a step-wise simplification of
reality. This model-building process is
highly dependent on the expected
model applications. A numerical model
can typically not be safely used for
other than the intended purposes
without adaptations, hereby re-
building additional/alternative apects
of reality into the model.
Model results can only be applied to
reality (for decision-making) if the
numerical model is fit-for-purpose,
covering the aspects relevant for the
decision to make.

11. Slide 11 4-Oct-22
Model
setup
• Mesh / geometrical setup
• Parametrization
Model
Calibration
• Comparison to observed data
• Adaptation of parameters so that the model matches observations
Model
Verification
• Comparison to observed data for another system status (other time
period / other stresses to the system)
Model
Application
• Calculation of scenarios
• Predictive model runs
MODELING WORKFLOW

12. Slide 12 4-Oct-22
3D Mesh Setup
Supermesh design
MODEL SETUP
Mesh generation
Parameterization

13. Slide 13 4-Oct-22
Model
setup
Steady-
state
Calibration
Transient
Calibration
Verification
Model
Application
Classical workflow
MODEL CALIBRATION
Calibrated
Parameters
• Hydraulic
Conductivity
• Groundwater
Recharge
• River-bed
Conductance
Calibrated
Parameters
• Storage
Parameters
(Sy, Ss)
• River-bed
Conductance
Calibrated
Parameters
• Ideally none
Iterations required –
Potentially even on model setup!

14. Slide 14 4-Oct-22
Model
setup
Steady-
state
Calibration
Transient
Calibration
Verification
Model
Application
Alternative workflow
MODEL CALIBRATION
Potential advantage:
Shorter iteration cycles (cross-check
steady-state calibration regularly with
transient runs – not only at the end)

15. Slide 15 4-Oct-22
• Manual calibration: by modifying parameters in the GUI
• Parameter estimation: by modifying parameters through inverse modeling
(PEST/FePEST)
• Common pitfalls:
• Overfitting: finer adaptation of parameters than suggested/supported by measurements
• Surrogate parameters: A calibrated parameter (most times hydraulic conductivity) is
modified to unrealistic values, filling in for a flaw in the model that is not addressed (e. g.,
processes not covered by the mathematical model, structural deficiencies of the conceptual
model)
• Non-uniqueness of parameter sets: There are usually multiple possible parameter sets
providing equally well calibrated models
MODEL CALIBRATION
While overfitting and using surrogate parameters lead to an apparently better calibration quality (less deviation
between observed and calculated values), they in fact result in poorer predictive ability, even more for model
stresses not covered in the calibration / verification datasets.

16. Slide 16 4-Oct-22
How to avoid overfitting, surrogate roles of parameters, non-uniqueness and non-
conservative parametrization:
• Keep model parameters within physically expected ranges
• When calibration is not possible with parameters in expected ranges, think
about potentially missing features/processes and add them – or make a clear
and well-grounded decision about accepting model deficiencies
• When having to deal with uncertainty, prefer parameter sets that are
conservative for the expected model applications
• Use known parameter zonation (e. g., hydrogeological units)
• Limit parameter variation within zones to the bare minimum (only where clearly
supported by observations)
• Consider observation noise and error – finish calibration before going below
expected noise/error for the deviation between calculated and observed values.
MODEL CALIBRATION

17. Slide 17 4-Oct-22
• Run the model for a situation not covered by calibration
• other/extended time period for transient calibration (preferred)
• other conditions (e. g., low / high groundwater levels rather than medium) for steady-state
calibration
• If the model does not perform sufficiently well for the verification case, re-iterate
calibration, including the verification dataset.
MODEL VERIFICATION

18. Slide 18 4-Oct-22
• Make sure model stresses for the application/prediction cases are sufficiently
covered in calibration/verification datasets, e.g.,
• don‘t do temporal predictions with a model only calibrated in steady state
• don‘t predict for time periods much longer than calibration/validation period
• Consider applying uncertainty bands for model parameters (manually or
through PEST/FePEST)
• Explain (and possibly quantify) uncertainties to everyone supposed to be
making decisions based on model results
• Make sure decisions are based on a known risk level, depending on their
tolerance for failure (uncertainty might be tolerated when dealing with
environmental flows to a river, but a strictly conservative approach may be
needed when dewatering behind a pit wall)
MODEL APPLICATION

19. Slide 19 4-Oct-22
Introduction to FEFLOW

20. Slide 20 4-Oct-22
• (Analytical Equations)
• for simple cases only
• Analytical Elements (AEM)
• AnAqSim, SLAEM, MLAEM/2, MLAEM
• Finite Differences (FDM)
• MODFLOW
• Finite Volumes (FVM)
• MODFLOW-USG, TOUGH2
• Finite Elements (FEM)
• FEFLOW, Aqua3D, MicroFEM, OpenGeoSys, HydroGeoSphere
GROUNDWATER MODELLING SOFTWARE

21. Slide 21 4-Oct-22
• Flexible meshing
• Broad application range
• Convenient pre and post processing
• Programming interfaces
• Reliability
• Consistent further development
• World-wide acceptance
WHY DO WE USE FEFLOW?

22. Slide 22 4-Oct-22
• a predecessor developed by Prof. Hans-Jörg Diersch since 1973 at the
Academy of Sciences in the former GDR („FINEL“)
• since 1979 called „FEFLOW“, run on mainframes IBM 370, EC 1055, BESM-6
with punch-card input
• since 1987: graphical user interface (for ATARI ST and UNIX workstations first)
• since 1990: commercial product of WASY GmbH, Berlin
• since 2007: commercial product of DHI Group (acquisition of WASY GmbH)
• current release: FEFLOW 7.5 as part of MIKE 2022
FEFLOW HISTORY

23. Slide 23 4-Oct-22
• Groundwater flow
• Contaminant transport (non-reactive and reactive, density-dependent)
• Heat transport
• Saturated and unsaturated flow and transport
• Porous systems and discrete fractures
• via external coupling / plug-ins:
• Surface-water flow
• Freeze-thaw processes
• Subsidence
• Chemical reactions (incl. equilibrium chemistry)
PROCESSES COVERED

24. Slide 24 4-Oct-22
• FEFLOW Help
http://www.feflow.info/html/help75/feflow/01_Introduction/intro.htm
• FEFLOW Book
https://link.springer.com/book/10.1007/978-3-642-38739-5
Similar content to the book (but older and partially less detailed):
• FEFLOW Reference Manual
https://www.yumpu.com/en/document/view/20781782/reference-manual-feflow
• FEFLOW White Papers
http://www.feflow.info/uploads/media/white_papers_vol1_01.pdf
http://www.feflow.info/uploads/media/white_papers_vol2_01.pdf
http://www.feflow.info/uploads/media/white_papers_vol3_01.pdf
http://www.feflow.info/uploads/media/white_papers_vol4.pdf
FEFLOW DOCUMENTATION

25. Slide 25 4-Oct-22
SOFTWARE STRUCTURE
FePEST Graphical
User Interface
FEFLOW
Graphical User
Interface
FEFLOW
Command-line
Mode
FEFLOW Python
Interface
Plug-ins
Python
scripts
FEFLOW
Kernel

26. Slide 26 4-Oct-22
FILE STRUCTURE
FEFLOW
Map data
(.shp, .dxf,
…)
Selections,
settings, …
Model File
.fem
Results File
.dac *
Plots,
curves, …
Supermesh
File
.smhx
* .dac includes all
information in .fem
Results File
.dar

27. Slide 27 4-Oct-22
DEFAULT FOLDER STRUCTURE
Project directory
femdata (.fem)
results (.dac, .dar)
import+export (.shp, .dat, .dxf, …)
plot (.shp, .dxf, …)
Supermsh (.smh, .smhx)

28. Slide 28 4-Oct-22
Supermesh and Finite-Element Mesh

29. Slide 29 4-Oct-22
Classical workflow for 3D models
MODEL SETUP AND SUPERMESH
Supermesh
Polygons, lines, points
2D Finite-element mesh
Elements, nodes
3D Finite-element mesh
Elements, nodes
Layers, slices
1 2 3

30. Slide 30 4-Oct-22
USER INTERFACE COMPONENTS
View Windows
Panels
Toolbars
Menu
Status Bar
The GUI can be customized – here the default layout is shown.

31. Slide 31 4-Oct-22
• Data:
• Access to all model parameters
• Maps:
• Load background maps
• Link map data for import
• View Components:
• Customize view contents
• Entities:
• Access to layers, slices and other geometrical entities
• Selections:
• Stored or current selections of nodes, elements, etc.
THE MOST IMPORTANT PANELS

32. Slide 32 4-Oct-22
A supermesh
• provides the framework for mesh generation
• must contain at least one polygon
• can have a number of polygons, lines, and points
• must not have any overlaps of polygons
• should not have vertices too close to each other
SUPERMESH

33. Slide 33 4-Oct-22
Why different polygons?
• Geological zones (can be used later on in the model)
• Natural/technical structures (river banks, tunnels, basements)
• Local mesh refinement
SUPERMESH
Geological zone
boundary
River banks
Local mesh
refinement

34. Slide 34 4-Oct-22
Why lines?
• Natural/technical structures (rivers, faults, …)
• Local mesh refinement
• Creation of parallel lines
Why points?
• Natural/technical structures
(wells, springs)
• Local mesh refinement
SUPERMESH
Well location
Drainage
channel

35. Slide 35 4-Oct-22
Mesh nodes
• Calculation of results (hydraulic head, concentration, temperature)
• Elevation (3D models)
• Initial conditions
• Boundary conditions
Mesh elements
• Material properties
(hydraulic conductivity, …)
FINITE-ELEMENT MESH
Mesh element
Mesh node

36. Slide 36 4-Oct-22
Inner angles of triangles:
• ideal: 60°
• avoid: very obtuse angles
• check with:
Auxiliary Data – Max. interior angle of triangles
Absolute triangle size
• ideal: depends on model objective
• avoid: strong local contrasts in size
• check with:
Auxiliary Data – Elemental diameters
MESH QUALITY MEASURES

37. Slide 37 4-Oct-22
Delauney Criterion
• not crucial for FEFLOW simulation
• ideal: no violations
• check with:
Auxiliary Data – Delauney criterion violations
MESH QUALITY MEASURES
fulfilled
violated

38. Slide 38 4-Oct-22
Hypothetical example - Well with fixed head in distance
• Variation of element size
around the well node
IDEAL ELEMENT SIZE

39. Slide 39 4-Oct-22
Hydraulic head calculated at a well node depends on spatial
discretization (can be calculated too high or too low), so either
• do not use head at well nodes or
• use ideal element size around well
IDEAL ELEMENT SIZE
regional hydraulic head
independent from local
element size
local hydraulic head
depends on element size
Screen
mesh too fine:
head to low
mesh too coarse:
head to high
ideal size:
head equal to analytical

40. Slide 40 4-Oct-22
IES is derived from 2D horizontal case (confined) by setting (linearized) FEM
solution equal to analytical solution and solving for element size.
𝐼𝐸𝑆 = 𝑤𝑒𝑙𝑙 𝑟𝑎𝑑𝑖𝑢𝑠 ∗ 𝑒
2∗π
𝑛𝑒∗tan(
π
𝑛𝑒)
ne: number of elements around well node, often 6 elements are used
Practical application:
• add points to supermesh with distance IES from well point
• create mesh so that only one element is created in between
For point generation:
• Excel table, Mesh Node Createor or Python script
• available on request
IDEAL ELEMENT SIZE

41. Slide 41 4-Oct-22
2D MESH GENERATORS
Mesh Generator Advantages Disadvantages
Advancing Front • very regular triangles • no lines and points considered
Triangle • extremely fast
• very complex supermeshes
possible
• local refinement at lines, points,
polygon boundaries
• triangles less regular – smoothing
recommended
• may fail under certain
circumstances
Gridbuilder • good gradation from fine to
coarse
• produces parallel lines of nodes
• can fail under certain circumstances
Transport Mapping • produces quad meshes • only for simple geometries (all
supermesh polygons quadrilateral)

42. Slide 42 4-Oct-22
• Typically right after mesh generation
• Select all nodes that may be moved:
• Select nodes to be kept, usually by using the supermesh geometries in the Maps panel for
selection
• Invert the selection
• Smooth mesh
MESH SMOOTHING

43. Slide 43 4-Oct-22
• May be numerically more stable, especially for transport models
• Reduces the effort for matrix assembly
MIXED QUAD AND TRIANGULAR MESH
Multiple smoothing steps
recommended after converting mesh
Do zone by zone using elemental selections in case
internal geometry needs to be retained in the mesh!

44. Slide 44 4-Oct-22
Management of layers and slices
DIALOGS – 3D LAYER CONFIGURATOR

45. Slide 45 4-Oct-22
• Vertical subdivision of aquitards needed for transport simulation and particle
tracking
• Reason: calculation of Darcy vectors (basis for transport simulation) at mesh
nodes
• Often done with thin buffer layers on top and bottom of each aquitard
BUFFER LAYERS
Nodes in aquitard show
low aquitard velocities.
Nodes at aquifer/aquitard
boundary show mixed
velocities.

46. Slide 46 4-Oct-22
Model Settings

47. Slide 47 4-Oct-22
Basic settings for simulation model
DIALOGS – PROBLEM SETTINGS

48. Slide 48 4-Oct-22
• Horizontal
• Simulate one aquifer in plan view
• Assumption: vertically averaged conditions
• Vertical, planar
• Cross-sectional model
• Assumption: all flow in cross-sectional plane (no flow perpendicular to the cross section)
• Vertical, axisymmetric
• Cross-sectional model with radial symmetry, e. g., cross section through a well cone
• Assumption: all flow with radial symmetry (e. g. aquifer layering identical all around the well)
MODEL PROJECTIONS 2D

49. Slide 49 4-Oct-22
• Typical:
• X-Y plane in the horizontal
• Triangular mesh in the horizontal, layers in the vertical
• Vertical layers
• X-Y plane in the vertical
• Starts from 2D cross-sectional model
• Extending the cross section by using a number of cross-sectional layers
• Unstructured mesh in 3D
• Typically X-Y plane in the horizontal (though rotation is possible)
• Tetrahedral meshing in 3D
• No layering
MODEL PROJECTIONS 3D

50. Slide 50 4-Oct-22
Confined:
Aquifer completely filled with water.
Unconfined:
Aquifer partially filled with water,
assuming a groundwater table
Unsaturated/variably saturated:
Aquifer partially filled with water,
considering transition between partially
saturated and saturated
CONFINED / UNCONFINED / UNSATURATED

51. Slide 51 4-Oct-22
• in 2D horizontal models:
• input parameter Transmissivity
• T = kf * t
• anisotropy in x/y via angle and factor
• in 3D models:
• input parameter hydraulic conductivity kf
• anisotropy possible (conductivities kxx, kyy, kzz or rotated anisotropy)
CONFINED CONDITIONS
t
kf
Confined conditions:
Storage using only
- Specific storage (compressibility)

52. Slide 52 4-Oct-22
• in 2D horizontal models:
• input parameters conductivity and top/bottom
elevation (as material properties)
• anisotropy in x/y via angle and factor
• transmissivity internally calculated from
conductivity, water level and bottom elevation
• confined when hydraulic head > top elevation
UNCONFINED CONDITIONS 2D
top
kf
bottom
Unconfined conditions:
Storage using
- Specific storage (compressibility)
- Drain-/fillable porosity

53. Slide 53 4-Oct-22
• in 3D models (horizontal layers):
• input parameters conductivity and elevations
of nodes
• transmissivity internally calculated from
conductivity, water level and nodal elevations
• confined when hydraulic head > top elevation
UNCONFINED CONDITIONS 3D
kf
Unconfined conditions:
Storage using
- Specific storage (compressibility)
- Drain-/fillable porosity

54. Slide 54 4-Oct-22
Two basic options
UNCONFINED CONDITIONS 3D
kf
Free: moving slice elevations so that water level is equal
to top of the model
Phreatic: no slice movement, reduced transmissivity for
(partially) dry layers
Advantage: no partially saturated/unsaturated elements,
no reduction of hydraulic conductivity
Disadvantage: Interpolation of material properties where
the water level cuts through the initial slice stratigraphy
Advantage: no movement of slices, no additional effort
for fully unsaturated calculation
Disadvantage: sharp contrasts in hydraulic conductivity,
difficult handling of groundwater recharge

55. Slide 55 4-Oct-22
• moving top surface with hydraulic head / water level
• lower slices automatically moved with top if not set to ‘fixed’
• potentially problematic interpolation of material properties where the water level
cuts through slices of the original stratigraphy, especially in layered aquifer-
aquitard systems
UNCONFINED: FREE

56. Slide 56 4-Oct-22
• linear reduction of hydraulic conductivity based on water depth (equals average
saturation) in each element
• residual water depth (RWD) controls minimum saturation / minimum hydraulic
conductivity for dry conditions:
• groundwater recharge applied to (potentially dry) model top – increase RDW in
case of instabilities (though leading to larger residual conductivity)
• options for behavior on model top (confined/unconfined/free outflow) when
head > top) and bottom (constrained or unconstrained water level)
UNCONFINED: PHREATIC
water level < bottom:
kf,red = RWD / t * kf,sat
water level >= top:
kf,red = kf,sat
bottom < water level < top:
kf,red = WD / t * kf,sat
t: element thickness

57. Slide 57 4-Oct-22
Layer-by-layer settings
• Free: Slice may move (only top slice)
• Phreatic: Use phreatic setting for layer below (linear relative conductivity)
• Dependent: Behavior according to slice above
• Confined: Layer below slice is always confined
• Use ‚confined‘ for the uppermost layer that is always fully saturated when using
‚free‘ in order to avoid slice movement in lower layers
• Combinations as shown may be used if more than one phreatic surface is to be
simulated for the entire model domain
(very rarely applicable!). A ‚confined‘ layer
needs to be between any two ‚free‘ or ‚phreatic‘
layers.
UNCONFINED CONDITIONS 3D

58. Slide 58 4-Oct-22
• Basic assumption:
pore pressure = 0 at ‚groundwater table‘ (boundary of fully saturated soil)
above / at lower saturation level: negative pore pressure (suction)
below: positive pore pressure
• Hydraulic head ℎ =
𝑝
𝜌∗𝑔
+ 𝑧
Thus hydraulic head is less than elevation at partially saturated conditions,
equal or higher than elevation at saturated conditions.
UNSATURATED FLOW
z
p
0

59. Slide 59 4-Oct-22
• Saturation/water content depends on suction pressure and vice versa (water
retention curve)
• Hydraulic conductivity is a function of saturation/water content
• The dependencies are highly dependent on the soil type.
• The curves are hysteretic, which is often neglected for simplicity.
UNSATURATED FLOW – CONSTITUTIVE RELATIONSHIPS
Sand
Silt
Loam
Clay
Rel k!

60. Slide 60 4-Oct-22
FEFLOW provides a number of constitutive relationships to choose from:
• Spline models based on data
• Empirical models:
• Van Genuchten
• Modified van Genuchten (free m and exponential relationship for krel)
• Brooks & Corey
• Haverkamp
• Exponential
• Linear
UNSATURATED FLOW – CONSTITUTIVE RELATIONSHIPS
FEFLOW provides the respective equations used for the
relationships as tooltips in the Data Panel.
Different constitutive relationships can be used in a
model on an element-by-element basis.

61. Slide 61 4-Oct-22
Head-based formulation:
UNSATURATED FLOW – RICHARDS‘ EQUATION
Richards‘ equation is fairly difficult to solve. Depending on the constitutive relationship applied, very fine spatial
and/or temporal resolution might be required to obtain a stable solution.
𝜕𝜃
𝜕ℎ
⋅
𝜕ℎ
𝜕𝑡
= ∇ 𝐾 ℎ ∇𝐻
• h: matric head
• t: time
• 𝜃: volumetric water content
•
𝜕𝜃
𝜕ℎ
: specific moisture capacity
• 𝐾: hydraulic conductivity tensor, dependent on matric potential
• H: hydraulic head

62. Slide 62 4-Oct-22
• Steady-state:
• as if identical conditions for an unlimited time
• leads to single result (equilibrium conditions)
• no storage (inflow = outflow)
• fast solution, low data input requirements
• Transient
• simulation for a defined time period
• water can be stored (rising gw level or pressure) or
released from storage (sinking gw level or pressure)
• requires time stepping (discretization in time)
• more data input requirements (storage properties, time series
for BCs, etc.)
STEADY-STATE VS. TRANSIENT
time
time
Often a steady-state simulation is done before a
transient simulation to obtain initial conditions and/or do
a rough calibration before refining in transient.

63. Slide 63 4-Oct-22
• FEM results in a large equation system that needs to be solved in every step
• Direct solvers provide a perfect solution (without residual error), but are
memory-intensive and potentially slow.
• PARDISO – Parallel Direct Solver by O. Schenk and K. Gärtner
• Iterative solvers have a residual error, but require less memory and may be
faster.
Standard iterative – classic methods, good for transient models
• PCG – preconditioned conjugate-gradient method (not parallelized!)
• PETSc Krylov-subspace solvers (from FEFLOW 7.5, no experience yet)
Algebraic multigrid – especially good for steady-state models
• SAMG – Algebraic Multigrid by K. Stüben (decides internally on using AMG or PCG)
• PETsc AMG (from FEFLOW 7.5, no experience yet)
EQUATION-SYSTEM SOLVERS

64. Slide 64 4-Oct-22
Phreatic groundwater surfaces or unsaturated conditions lead to non-linear
equation system, e. g. transmissivity (input parameter) changes with a changing
hydraulic head (result)
ITERATIVE SOLUTION AND ERROR CRITERIA
(Iterative) equation system solver
Inner iteration
convergence criteria of the
solver
Outer iteration
convergence criteria in
Numerical Parameters
next time step
or steady-state
solution
last time step
or initial
conditions

65. Slide 65 4-Oct-22
• Relative error criterion
𝐸 =
𝑖=0
𝑛
𝑒
𝑛
ℎ𝑚𝑎𝑥
𝑒: 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 ℎ𝑦𝑑𝑟𝑎𝑢𝑙𝑖𝑐 ℎ𝑒𝑎𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝑛: 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑑𝑒𝑠
ℎ𝑚𝑎𝑥: 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 ℎ𝑦𝑑𝑟𝑎𝑢𝑙𝑖𝑐 ℎ𝑒𝑎𝑑 𝑖𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑟 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠
• Default criterion: 1e-3
• Different averaging norms:
• Maximum error norm
• Absolute L1 error norm (shown in above equation)
• Euclidean L2 (RMS) norm (default)
ERROR TOLERANCE
The relative formulation of the error criterion leads to a dependency of the outer iteration convergence on absolute
elevation: the default criterion of 1e-3 means an average error of 1 cm when maximum head in the model is 10 m,
but 1 m when maximum head in the model is 1000 m. Decrease the Error tolerance when at higher altitudes!

66. Slide 66 4-Oct-22
Boundary Conditions

67. Slide 67 4-Oct-22
• Types:
• Hydraulic-head BC (Dirichlet BC)
• Fixed hydraulic head on a node
• Examples: outer model boundaries, well connected rivers, springs with known level
• Fluid-flux BC (Neumann BC)
• Fixed flux (in or out) at the nodes covering an element face
• Examples: outer model boundaries, areal infiltration
• Fluid-transfer BC (Cauchy BC)
• Fixed hydraulic head linked to the model via an additional conductance parameter
• Examples: rivers, outer model boundary as general head BC
• Well BC
• Fixed inflow or outflow at a node
• Examples: pumping wells, infiltration wells, springs with known discharge
• Multilayer Well
• Fixed inflow or outflow along a line
• Examples: Pumping wells, infiltration wells
BOUNDARY CONDITIONS

68. Slide 68 4-Oct-22
• Fixed hydraulic head at a node
• Default unit: [m]
• Often used to:
• delimit model area at a head isoline
• simulate rivers or lakes that are well connected to
the groundwater system
HYDRAULIC-HEAD BC
Hydraulic-head BC fix the groundwater level, no matter at what
inflow or outflow rates. Make sure to check the water budget to
ensure that flow rates are realistic!
Check whether the use of Fluid-flux BCs or Fluid-transfer BCs
may be more realistic. Fixed head

69. Slide 69 4-Oct-22
• Fixed flux (in or out) at the nodes covering
an element face
• Default unit: [m/d] in 3D and 2D unconfined
or cross-sectional models,
[m2/d] in 2D confined models
• Internally multiplied with face area
• Often used to:
• define lateral inflow, e. g. from hard rock areas
• simulate areal infiltration with known infiltration
rate
FLUID-FLUX BC
Fluid-flux BC fix the inflow or outflow rates, no matter at what
hydraulic head levels. Make sure to check the resulting hydraulic
head in order to see whether the specified rates and/or hydraulic
conductivity in the model are realistic!
in
-
out
+

70. Slide 70 4-Oct-22
• Reference water level at the nodes covering
an element face
• Default unit: [m]
• Additional material properties:
• In-transfer rate, Out-transfer rate
• Often used to:
• simulate rivers or lakes
• specify a general-head BC, i.e., apply a fixed
hydraulic head in a distance from the model
boundary
FLUID-TRANSFER BC
Fluid-transfer BC lead to inflow or outflow rates based on the
difference in head between BC and groundwater – without
feedback on the river/lake level. Make sure to check the resulting
flow rates in order to see whether they are realistic in comparison
to water flow in the river or water volume in the lake!

71. Slide 71 4-Oct-22
Calculation of inflow / outflow from head differential:
𝑄𝑖𝑛 = 𝐴 ∗ 𝑇𝑅 ∗ ℎ𝑟𝑒𝑓 − ℎ𝑔𝑤
A: Area of boundary condition [m2]
TR: transfer rate [1/d]
href: Reference water level (boundary condition value) [m]
hgw: Hydraulic head in groundwater [m]
• Transfer rate TR is used as In-transfer rate in case of flow into the model (href > hgw) and
Out-transfer rate in case of outflow from the model (hgw > href)
• Practically In-transfer rate may be smaller than out-transfer rate as suspended material from
river water tends to clog the pores in the river bottom
• Transfer rate may be estimated from properties (conductivity, thickness) of the clogging
layer TR =
𝑘𝑓
𝑡
FLUID-TRANSFER BC

72. Slide 72 4-Oct-22
• Fixed inflow or outflow at a node
• Default unit: [m3/d]
• Often used to:
• simulate pumping wells or
infiltration wells
• consider springs with known discharge
WELL BC
in
-
out
+
The hydraulic head calculated for a specific discharge or recharge
in a well depends on spatial model discretization. Use the ideal
element size in case the hydraulic head at the well node is a
relevant model result! In no case well-internal processes, such as
a skin effect, are considered in FEFLOW.
mesh too fine:
head to low
mesh too coarse:
head to high
ideal size:
head equal to analytical

73. Slide 73 4-Oct-22
• Fixed inflow or outflow along a line
• Default unit: [m3/d] plus add.
parameters (radius [m], etc.)
• Often used for:
• Pumping wells, infiltration wells screened
in multiple model layers
• Horizontal wells (Ranney wells)
• Well BC with a conductive well pipe
connecting the nodes
• Well pipe simulated as 1D discrete
feature with laminar flow (Hagen-
Poiseuille law)
• Definition based on element edges
rather than nodes
MULTILAYER WELL
in
-
out
+

74. Slide 74 4-Oct-22
• Used mainly as time series for
boundary conditions (BC) and
constraints (BCC)
• Can also be used in
expressions (e. g., expression-
based material properties)
• Defined in Time-series Editor
(<Ctrl>-T or Edit – Time Series)
TIME SERIES
Time steps in time series used for boundary
conditions are met by the automatic time
stepping procedure. A fine temporal resolution
can therefore lead to long simulation time. Time
steps in different series should be aligned for
the same reason.

75. Slide 75 4-Oct-22
• Interpolated time series
indicated by i in the view
• Time-series IDs for interpolated
time series start at 10,000
• Name for interpolated time
series is Internal
• Time series in use cannot be
deleted
TIME SERIES - IDS

76. Slide 76 4-Oct-22
Curve Type
• determines
interpolation
between given
time stages
TIME SERIES – CURVE TYPE
linear
constant
Akima 1
Akima 2
Constant steps will cause
the automatic time
stepping procedure to
calculate on step before
and one after every
change of value and may
thus lead to more time
steps simulated than the
other options.

77. Slide 77 4-Oct-22
• Linear: use first/last value before/after defined period
• Cyclic: repeat defined period endlessly
TIME SERIES – TIME MODE
linear cyclic
The cyclic option requires the first and last values defined for the series to be identical. The last step corresponds to
the first step of the next cycle.

78. Slide 78 4-Oct-22
• Gaps can be used to
remove the corresponding
BC for some time period
• Gaps start right after the
previous time stage and
end at the next defined
stage
• Gaps can also be used at
the beginning and end of a
time series
TIME SERIES - GAPS
gap

79. Slide 79 4-Oct-22
• File extension is *.pow
• Time series ID indicated by #
• Time series name and
(optionally) settings in lines
indicated by !
• Time/value separated by
blank(s) or tab
• Gap represented by keyword
GAP
• Single time series closed by
END
• Entire file closed by another
END
TIME SERIES – FILE FORMAT
When exported from FEFLOW, pow files use scientific notation
(see first series in example). For importing, however, any
number format can be used (second series).

80. Slide 80 4-Oct-22
Boundary Condition Constraints

81. Slide 81 4-Oct-22
Boundary Condition Constraints (short: BCC)
• Limit the application of a boundary condition to certain conditions
• Work (typically) by internally (temporarily) swapping boundary condition and
constraint value
• Added to FEFLOW GUI by using context menu BC (Add parameter)
• Examples:
• Well with minimum water level
• Head-BC with maximum inflow or outflow
• Fluid-flux BC with maximum water level
• Fluid-transfer BC with river bed elevation
BOUNDARY CONDITION CONSTRAINTS
in
+
out
-
For BCCs, inflow is considered as
positive, outflow as negative, similar to
budgeting in FEFLOW.
This is opposite to the definition of Fluid-
Flux and Well Boundary conditions!
BCCs require a conditional checking of simulation results
with potentially following changes in BC. This may cause
an oscillatory behavior which cannot be easily detected in
steady-state results. Use BCCs with care in steady-state
models or avoid their usage alltogether!

82. Slide 82 4-Oct-22
• BC: Hydraulic Head [m]
• BCC: Minimum and/or maximum flow [m3/d]
• Example: Spring with free outflow when groundwater
level is higher than ground elevation at spring
• BC: Hydraulic-head BC at ground elevation
• BCC: Maximum flow rate of 0 m3/d
• Hydraulic head > BC: Outflow from model (negative),
not exceeding maximum – no change in BC
• Hydraulic head < BC: Inflow to model (positive), exceeding
maximum – internal swap to Well BC with value 0 m3/d
CONSTRAINT ON HYDRAULIC-HEAD BC
in
+
out
-
This combination is also called a Seepage-face BC which is
readily available in the FEFLOW Editor toolbar, so that BC and
BCC do not have to be set separately.

83. Slide 83 4-Oct-22
• BC: Fluid flux [m/d]
• BCC: Minimum and/or maximum hydraulic head [m]
• Example: Lateral inflow where groundwater level
cannot exceed ground surface
• BC: Fluid-flux BC
• BCC: Maximum hydraulic head at elevation of ground surface
• Hydraulic head < BCC: constraint condition not violated, no change in BC
• Hydraulic head > BCC: constraint condition violated, BC swap – application of hydraulic-
head BC (value: ground surface elevation)
CONSTRAINT ON FLUID-FLUX BC
Fluid-flux BC
BCC: hmax
Hydraulic-
head BC:
hmax
1 2
1
2

84. Slide 84 4-Oct-22
• BC: Fluid transfer [m/d]
• BCC: Minimum and/or maximum flow [m3/d]
• Rarely used
• Example: River with maximum infiltration to groundwater
• BC: Fluid-transfer BC
• BCC: Maximum infiltration per river node (positive)
• Flow < BCC: Inflow to model (positive) not exceeding maximum or outflow from model
(negative) – no change in BC
• Flow > BCC: Inflow to model (positive), exceeding
maximum – internal swap to Well BC with value of BCC
CONSTRAINT ON FLUID-TRANSFER BC - I
in
+
out
-

85. Slide 85 4-Oct-22
• BC: Fluid transfer [m/d]
• BCC: Minimum and/or maximum level [m]
• Example: River with river bed elevation
• BC: Fluid-transfer BC [m]
• BCC: minimum hydraulic head hbed [m]
• Hydraulic head > BCC: 𝑄𝑖𝑛 = 𝐴 ∗ 𝑇𝑅 ∗ ℎ𝑟𝑒𝑓 − ℎ𝑔𝑤
– no change in BC
• Hydraulic head < BCC: 𝑄𝑖𝑛 = 𝐴 ∗ 𝑇𝑅 ∗ ℎ𝑟𝑒𝑓 − ℎ𝑏𝑒𝑑
– limited infiltration when groundwater level lower than
river bed elevation (unsaturated conditions below river)
• BC < BCC: 𝑄𝑖𝑛 = 0
– no flow when rivel level lower than river bed elevation
(used in transient simulations to avoid inflow from dry
parts of the river bed)
CONSTRAINT ON FLUID-TRANSFER BC – II
hgw hbed
href
hgw
hbed
href
hgw
hbed
href
1
2
1
2
3
3

86. Slide 86 4-Oct-22
• BC: Well [m3/d]
• BCC: Minimum and/or maximum hydraulic head [m]
• Example: Well with minimum level
• BC: Well BC [m]
• BCC: minimum hydraulic head hmin [m]
• Hydraulic head > hmin: no change in BC
• Hydraulic head < hmin: swap BC to hydraulic-head BC with h = hmin
CONSTRAINT ON WELL BC
When the groundwater level falls below hmin around the well because of
influences external to the well (e. g., a neighboring well), the well will NOT be
turned off, but it will start infiltrating water through the hydraulic-head BC. See next
slide for a better alternative setup.
hmin
Q
hgw
1 2
2
hgw
This only works well whenever the Well BC is the only source for drawdown
in the area. The resulting flow is then less than the Q defined for the Well BC.
1

87. Slide 87 4-Oct-22
• BC: Hydraulic head at hmin
• BCC:
• Minimum flow-rate at -Q (negative of the pumping rate)
• Maximum flow-rate at 0 m3/s to avoid inflow
• Example: Well with minimum level
• hgw > hmin and Q < Qmin:
swap BC for Q = Qmin, regular pumping
• hgw > hmin and Q > Qmin:
no change in BC, reduced pumping
• hgw < hmin :
swap BC for Q = Qmax = 0 m3/d, no pumping, no infiltration
WELL BC WITH LIMITED WATER LEVEL – BETTER SOLUTION
hmin
Q
hgw
1
2
hgw
hgw
3
x
1
2
3

88. Slide 88 4-Oct-22
User Data

89. Slide 89 4-Oct-22
• User-defined parameters in Data panel
• Nodal or elemental
• Value-based or expression-based, or predefined ($)
• Stored in FEFLOW model
• Edited in the same way parameters are edited
• Used for
• results evaluation (expression-based derived values from primary results)
• calculating expression-based parameters (value-based User Data can be used in other
expressions)
• provide parameters for FEFLOW plug-ins and Python scripts
• pure visualization (add a value distribution to the view that is not used as model data)
• monitoring spatial distribution of model error of outer iteration (see model settings) during
simulation (pre-defined distributions)
USER DATA

90. Slide 90 4-Oct-22
MAP HANDLING

91. Slide 91 4-Oct-22
• Support for many map formats
• Folders by default organized by file type,
but arbitrary folders possible
• Map icon shows type (point, line,
polygon)
• Map links for data import (see section
Editing Tools)
• Map layer holds visualization properties
and can be added to views (by double
click)
• By default one layer („Default“), more
can be added
MAPS PANEL
Folder
Map
Map link
Map layer

92. Slide 92 4-Oct-22
Classification
• Opened via context menu of the map
layer
• Different options for classification based
on attributes
• Options for visualization styles,
annotation, …
MAP PROPERTIES
Select row(s)
for modifying
styles
Check for 3D
drawing in 3D
views
Marker and
annotation
properties

93. Slide 93 4-Oct-22
• Available from context menu on map
• Add additional maps/tables (by using
) and join them via an attribute (by
drag-and-drop of the corresponding
attribute)
• Use SQL-style selection statements to
filter map features by a selection based
on the map attributes
MAPS – JOIN AND SELECT
Map attributes
Joined map
Selection
statement

94. Slide 94 4-Oct-22
Editing Tools

95. Slide 95 4-Oct-22
• Quick Import
• used when exporting from one
FEFLOW model and importing
to another
• Automatic detection of
FEFLOW parameter
• assignment based on
node/element number or
location
• Assignment to current selection
or all nodes/elements
• Link To Parameter
• See following slides
IMPORTING DATA FROM MAPS

96. Slide 96 4-Oct-22
• Set a parameter based on map data and using a regionalisation method
• Link a point map to a FEFLOW parameter
• Create the link via context menu on point map in Maps panel
LINK TO PARAMETER

97. Slide 97 4-Oct-22
1. Define link with all interpolation settings
2. Double-click link in order to set active parameter and assignment method
3. Select target geometry (e. g., nodes or elements) if selection is not not already
associated to the link
4. Execute parameter assignment in the Editor toolbar
LINK TO PARAMETER
Active parameter Assignment type: Maps Execute

98. Slide 98 4-Oct-22
• Link a source-data attribute to
be used as parameter value to a
FEFLOW parameter by
selecting the source data
attribute on the left and double-
clicking a parameter on the
right.
• Use context menu on parameter
to switch between different
types of transient data
assignment:
REGIONALISATION OPTIONS I
The context menu on the input map attributes
can be used to join source-data tables, use
SQL-style data selection or to re-calculate the
source data using an expression.

99. Slide 99 4-Oct-22
• Link Type: Link to time-constant
or time-dependent data in
FEFLOW
• Source Data Unit: Unit conversion
by choosing the source-data unit
• Topology Processing: Choose
map attribute in source data
relating to FEFLOW node or
element ID (to be used instead of
map point location)
• Default Link Selection: Choose a
map to select target
nodes/elements rather than using
a selection in FEFLOW
REGIONALISATION OPTIONS II

100. Slide 100 4-Oct-22
• Neighborhood Relationship
• Inverse Distance
• Kriging
• Akima
• 1D linear interpolation along lines
REGIONALISATION METHODS
When not sure what interpolation method
might be best for a purpose, try different ones
and check the outcome to see whether it fits to
your expert knowledge about the system.

101. Slide 101 4-Oct-22
Assign values based on the distance
between input data point in the map
and FEFLOW mesh node or element.
• Snap Distance
Consider data points with a distance up to the snap distance from the target
node or element center.
• Single Target
When checked, the value of each input data point is assigned to the closest
target geometry only, even if the snap distance would include more.
• Aggregation
In case more than one source data point is close to the same target geometry,
different aggregation strategies can be used (average, sum, or nearest)
• Logarithmic
Do interpolation based on logarithmized values rather than original ones.
REGIONALISATION METHODS – NEIGHBORHOOD RELATIONSHIP

102. Slide 102 4-Oct-22
Assign values based on an inverse
distance algorithm.
• Neighbors
Number of neighboring source data points to consider for the interpolation for
each target geometry.
• Exponent
Exponent of distance for weighting.
• Logarithmic
Do interpolation based on logarithmized values rather than original ones when
checked (e. g., for interpolating hydraulic conductivity).
REGIONALISATION METHODS – INVERSE DISTANCE

103. Slide 103 4-Oct-22
Assign values based on the simple or
ordinary Kriging algorithm.
• Neighbors
Number of neighboring source data points to consider for the interpolation for
each target geometry.
• Logarithmic
Do interpolation based on logarithmized values rather than original ones when
checked (e. g., for interpolating hydraulic conductivity).
• Type
Choose Simple or Ordinary Kriging.
REGIONALISATION METHODS – KRIGING

104. Slide 104 4-Oct-22
Assign values based on a linear
interpolation along a line map provided
under Default Link Selection.
• Input Data Point Distance
Maximum distance of source points
to be used from the line geometries.
• Neighborhood Relation / Neighborhood Distance
Set a maximum distance (target geometries from source geometries) where
source values are used rather than interpolated values.
REGIONALISATION METHODS – 1D LINEAR INTERPOLATION I
For branching river networks, it may be important that the interpolation is done from downstream branches (main
river) to upstream branches, so that for the latter the water level elevation of the receiving stream can be taken into
account by using “Incorporate existing data”. In this case the map should contain a field that is used as “Selection
Order Field” that holds the intended interpolation order in either ascending or descending order.

105. Slide 105 4-Oct-22
• Temporal Interpolation
for transient interpolation only: Do spatial interpolation based on values for
each time in the source data, or do a spatio-temporal interpolation, interpolating
also along time axis (for example, to reflect progression of a flood wave in a
river over time).
• Extrapolation
Choose behavior of regionalization along a line beyond the input data points.
• Incorporate existing data
Incorporate exisiting data in the model (e. g., already defined boundary
conditions) as additional source data values for the interpolation.
• Logarithmic
Do interpolation based on logarithmized values rather than original ones.
REGIONALISATION METHODS – 1D LINEAR INTERPOLATION II
1D linear interpolation can also be done directly, without using a point map as a data source. For this, choose the
Linear 1D Interpolation tool in the editor toolbar.

106. Slide 106 4-Oct-22
• Calculations for many purposes in FEFLOW
• Automatically checked
for syntax
• Use of additional
variables possible
• Unit handling for some
applications
EXPRESSION EDITOR
Pick any parameter in
FEFLOW to be used in the
expression
Both nodal and elemental
properties can be used in
expressions, regardless of what
the target geometry of the
expression is. Note that this
requires a conversion to be
done in the background.
Expression
Store/load
expressions
Pick pre-defined
expression elements
Note the warning regarding equality in expressions.

107. Slide 107 4-Oct-22
Results Evaluation

108. Slide 108 4-Oct-22
• Water budget of steady-state models or at a
single step of a transient model
• For the entire domain or a selection (nodal or
elemental)
• Storage changes only
in transient models
• Internal Transfer only
in case of an elemental
selection as DOI
RATE BUDGET PANEL
in
+
out
-
Flows at boundary
conditions
Recharge and
areal/volumetric
sources/sinks
Water to (-) /
from (+) storage
Balance error
Water flow into (+)
and out of (-) the
element selection
(via porous
medium and
discrete features)
Right click for unit change

109. Slide 109 4-Oct-22
• Water budget of steady-state models
or at a single step of a transient model
• For evaluating flows within the
model (e. g., across a section face)
• Defined as an interface between
a domain of interest (DOI) and a
masking domain (MD)
• DOI and MD defined as elemental
selections
• Positive flow is into the DOI
SUBDOMAIN RATE BUDGET PANEL
in
+
out
-
Domain of Interest Masking Domain Boundary masked
define DOI and
MD from stored
selections
BCs fully on the
interface
flow through the
interface
total flow through
the interface
net flow through
the interface
Right click for unit change

110. Slide 110 4-Oct-22
• Streamlines are tangent to the flow velocity vectors
• Streamlines reflect the current system status (steady-state or single time step
of a transient flow model)
• Pathlines follow single fluid particles over time and thus reflect the transient
nature of the velocity vector field
• Streamlines and Pathlines assume purely advective flow, i. e., they ignore
dispersive and diffusive processes
• With the random-walk method, streamlines and pathlines can be added a
random component, on average representing the additional dispersive and
diffusive components in the movement of the fluid and dissolved chemical
components
STREAMLINES, PATHLINES, AND THE RANDOM-WALK METHOD

111. Slide 111 4-Oct-22
1. Select a starting geometry:
• a node selection in the Selections panel (Current Node
Selection or named stored selection) or
• A 3D Point Set, 3D Loop, 3D Point Group, 3D Sphere
Object or 3D Cylinder Object in the Entities panel
2. Double click Streamlines – Forward or
Streamlines – Backward (Pathlines, Random-
Walks resp.) in the Data panel
3. Double click the new entry in the View
Components panel
4. Set the properties such as number of seed points
etc. in the Properties panel
5. Activate a visualization style in the View
Components panel
STREAMLINES, PATHLINES, AND THE RANDOM-WALK METHOD
1 2
3
4
5

112. Slide 112 4-Oct-22
Properties:
• min. / max. Travel time (in days), by default Autorange (all values)
• Radius (for starting particles at a circle around node selections or
3D Point Groups), one starting particle when radius is 0 m
• Seeds per Node: number of particles started on a circle or on a
3D Sphere / 3D Cylinder surface
• Flux-weighted: When unchecked, equal distance for starting
points, when checked distance depends on groundwater flux
along the circle/3D line or on the sphere surface (more starting points where
more flux, i. e. each particle representing the same flow rate across
line/surface)
• Surface only: checked: starting points on circular line or shpere/cylinder
surface, unchecked: starting points distributed within circle/sphere/cylinder
STREAMLINES, PATHLINES, AND THE RANDOM-WALK METHOD

113. Slide 113 4-Oct-22
CONTACT YOUR TRAINERS
Peter
Schätzl
p.schaetzl@aquasoil.de
+49 30 684007600
Bastian
Rau
Sven
Seifer
t
s.seifert@aquasoil.de
b.rau@aquasoil.de