Modelling of groundwater flow and solute transport PGI: Aug , 2002 Dr.ir. Gualbert Oude Essink 1. Netherlands Institute of Applied Geosciences 2. Faculty of Earth Sciences, Free University of Amsterdam Mathematical models in hydrogeology Summer School Polish Geological Institute

Delft University of Technology, Civil Engineering: till 1997 Ph.D.-thesis: Impact of sea level rise on groundwater flow regimes Utrecht University, Earth Sciences: till 2002 Lectures in groundwater modelling and transport processes Variable-density groundwater flow Salt water intrusion and heat transport Netherlands Institute of Applied Geosciences (NITG): till 2004 Salinisation processes in Dutch coastal aquifers Free University of Amsterdam, Earth Sciences: till 2004 EC-project CRYSTECHSALIN (o.a. with J. Bear) Crystallisation processes in porous media NITG: VU: Curriculum Vitae

Modelling protocol Discretisation Partial Differential Equation (PDE) Groundwater flow: MODFLOW Solute transport: MOC3D Modelling of groundwater flow and solute transport PGI: Aug , 2002 Programme (I)

Programme (II) PGI: Aug , 2002 Thursday Augustus 22, 2002: modelling protocol Darcy’s Law, steady state continuity equation Laplace-equation Taylor series development boundary conditions: Dirichlet, Neumann, Cauchy non-steady state: explicit, implicit, Crank-Nicolson Friday Augustus 23, 2002: groundwater flow with MODFLOW: mathematical description short introduction Graphical User Interface PMWIN Saturday Augustus 24, 2002: solute transport with MOC3D: mathematical description numerical dispersion

The Hydrological Circle

Ten steps of the Modelling Protocol 1.Problem definition 2.Purpose definition 3.Conceptualisation 4.Selection computer code 5.Model design 6.Calibration 7.Verification 8.Simulation 9.Presentation 10.Postaudit

Why numerical modelling? -: simplification of the reality only a tool, no purpose on itself garbage in=garbage out: (field)data important perfect fit measurement and simulation is suspicious Modelling protocol +: cheaper than scale models analysis of very complex systems is possible a model can be used as a database simulation of future scenarios

3. Conceptualisation (I) Model is only a schematisation of the reality Which hydro(geo)logical processes are relevant? Which processes can be neglected? Boundary conditions Variables and parameters: subsoil parameters fluxes in and out initial conditions geochemical data Mathematical equations Modelling protocol

3. Concept (II): example of salt water intrusion Modelling protocol Boundary conditions no flow at bottom flux in dune area constant head in polder area Relevant processes: groundwater flow in a heterogeneous porous medium solute transport variable density flow natural recharge extraction of groundwater Negligible processes: heat flow swelling of clayey aquitard non-steady groundwater flow The Netherlands

4. Selection computer code (II) Groundwater computer codes Hemker (1994)

4. Selection computer code There are numerous good groundwater computer codes available, so don’t make your own code! See the internet, e.g.: U.S. Geological Survey: water.usgs.gov/nrp/gwsoftware/ Scientific Software Group: Waterloo Hydrogeologic: Groundwater Digest:

5. Model design (I) Conditions: initial conditions boundary conditions: 1.Dirichlet: head 2.Neumann: flux, e.g., no flow 3.Cauchy/Robin: mixed boundary condition Modelling protocol Choice grid  x: depends on natural variation in the groundwater system concept model data collection Choice time step  t

5. Model design (II): example Geometry, subsoil parameters, boundary conditions Modelling protocol

5. Model design (III): choice time step  t Modelling protocol

6. Calibration (I) Fitting the groundwater model: is your model okay? trial and error automatic parameter estimation/inverse modelling (PEST, UCODE) Modelling protocol

6. Calibration (II): example Measured and computed freshwater heads Modelling protocol

6. Calibration (III): example Measured and computed freshwater heads 95 useful observation wells maximum measured head in area: m minimum measured head in area: m average (measured -computed) = m average absolute|measured -computed| = 0.34 m standard deviation = 0.46 m Modelling protocol

6. Calibration: errors during modelling protocol Wrong model concept resistance layer not considered heat transport on groundwater flow not considered (  not constant) Incomplete equations decay term of solute transport not considered Inaccurate parameters and variables Errors in computer code Numerical inaccuracies  t,  t, numerical dispersion Modelling protocol

7. Verification: testing the calibrated model ‘verification problem’: there is always a lack of data Modelling protocol 1D, 2D -->3D stationary dataset --> non-steady state simulation

9. Presentation Simulation of scenarios Computation time depends on: computer speed size model efficiency compiler output format 8. Simulation Modelling protocol

10. Postaudit Postaudit: analysing model results after a long time Anderson & Woessner(’92): four postaudits from the 1960’s Errors in model results are mainly caused by: wrong concept wrong scenarios Modelling protocol

Modelling scheme Modelling protocol Anderson & Woessner (1992)

I. Darcy’s law (1856) giveswhere K= hydraulic conductivity [L/T ] Discretisation PDE

II. Darcy’s Law Discretisation PDE In pressures: Definition piezometric head: q=Darcian specific discharge [L/T]  =piezometric head [L] p=pressure [M/LT 2 ]  =intrinsic permeability [L 2 ]  =dynamic viscosity [M/LT]  =density [M/L 3 ] g=gravitaty [L/T 2 ] k i =hydraulic conductivity [L/T]

III. Darcy’s Law Discretisation PDE if then: gives Gives:

Analogy physical processes Discretisation PDE

Continuity equation: steady state Mass balance [M T -1 ] Discretisation PDE  =density [M L -3 ]

Steady state groundwater flow equation (PDE=Partial Differential Equation) + Continuity equation If k i =constant and  =constant then: Laplace equation Discretisation PDE Flow equation (Darcy’s Law) = Groundwater flow equation

Taylor series development (I) - Discretisation PDE Best estimate of  i+1 is based on  i

Taylor series development (II) + Discretisation PDE

Laplace equation in 2D Discretisation in x-direction (i ): Discretisation in y-direction (j ): If  x =  y then: ‘Fivepoint operator’ Discretisation PDE

Fivepoint operator (I): constant head example Discretisation PDE

Fivepoint operator (II): no-flow example Nodes on the edges of an element Discretisation PDE

Fivepoint operator (III): example Hydraulic conductivity k=10 m/dag Thickness aquifer D=50m Characteristics: Groundwater recharge N=0 mm/dag BOUNDARY 1: no-flow near the mountains Boundary conditions BOUNDARY 2: linear from 100 m (near mountains) to 0 m (near sea) BOUNDARY 3: constant seawater level of 0 m BOUNDARY 4: no-flow Discretisation PDE

Fivepoint operator (IV): example On BOUNDARY 1: On BOUNDARY 4: Discretisation PDE EXCEL example!

Fivepoint operator (V): effect convergence criterion Discretisation PDE Solution I: Initial estimate head=50m Convergence-criterion=1m Number of iterations=13 Head 1=29,17m Head 2=25,11m Solution II: Initial estimate head=50m Convergence-criterion=0.01m Number of iterations=74 Head 1=20,67m Head 2=23.07m

Discretisation PDE Numerical schemes: iterative methods 1.Jacobi iteration: only old values Gauss-Siedel iteration: also two new values Overrelaxation:

Non-steady state groundwater flow equation Multiply with constant thickness D of the aquifer gives: S=elastic storage coefficient [-] T=kD= transmissivity [L 2 /T] Flow equation (Darcy’s Law) + Non-steady state continuity equation = S s =specific storativity [1/L] W’=source-term Groundwater flow equation non-steady state

Explicit numerical 1D solution (I) One-dimensional non-steady state groundwater flow equation: Properties: direct solution can be numerical instable Explicit (‘forwards in space’): non-steady state

Explicit numerical 1D solution (II): example Subsoil parameters: storage=1/3 T=kD=10 m 2 /day N=0.001 m/day Numerical parameters: two sets timestep  t=1 day &  t=10 days length step  x=10m N  t/  =0.003 m & 0.03 m T  t/  x 2 =0.3 & 3 I. II. Groundwater flow in aquifer between two ditches: ditch recharge=1mm/day phreatic storage=1mm/day  x=10 non-steady state EXCEL example!

Explicit numerical 1D solution (III): stability analysis non-steady state Suppose:

Implicit numerical solution Implicit (‘backwards in space’): One-dimensional non-steady groundwater flow equation (now no source-term W): Characteristics: not a direct solution solving with matrix A  =R numerical always stable needs more memory non-steady state

Crank-Nicolson numerical solution Crank-Nicolson (‘central in space’):  =0.5 One-dimensional non-steady state groundwater flow equation (now no source term W): Characteristics: solving as implicit stable with temporarily oscillations non-steady state

Numerical schemes non-steady state

MODFLOW a modular 3D finite-difference ground-water flow model (M.G. McDonald & A.W. Harbaugh, from 1983 on) USGS, ‘public domain’ non steady state heterogeneous porous medium anisotropy coupled to reactive solute transport MOC3 (Konikow et al, 1996) MT3D, MT3DMS (Zheng, 1990) RT3D easy to use due to numerous Graphical User Interfaces (GUI’s) PMWIN, GMS, Visual Modflow, Argus One, Groundwater Vistas, etc.

Nomenclature MODFLOW element MODFLOW

MODFLOW: start with water balance of one element [i,j,k] MODFLOW

Continuity equation (I) In - Out = Storage MODFLOW

Continuity equation (II) MODFLOW In = positive

Flow equation (Darcy’s Law) MODFLOW where is the conductance [L 2 /T]

Groundwater flow equation MODFLOW

Takes into account all external sources MODFLOW The term Rewriting the term:

Thé MODFLOW Groundwater flow equation MODFLOW and: gives:

Boundary conditions in MODFLOW (I) Numeric model MODFLOW Example of a system with three types of boundary conditions

Boundary conditions in MODFLOW (II) For a constant head condition: IBOUND<0 For a no flow condition: IBOUND=0 For a variable head:IBOUND>0 MODFLOW

Packages in MODFLOW MODFLOW 1.Well package 2.River package 3.Recharge package 4.Drain package 5.Evaporation package 6.General head package

1. Well package MODFLOW Example: an extraction of 10 m 3 per day should be inserted in an element as: (in = positive)

2. River package (I) MODFLOW M  riv  i,j,k element i,j,k M river gains water RBOT  riv river loses water river bottom

2. River package (II) MODFLOW Example: the river conductance C riv is 20 m 2 /day and the rivel level=3 m, than this package should be inserted in an element as:

2. River package (III) MODFLOW where is the conductance [L 2 /T] of the river Determine the conductance of the river in one element:

2. River package (IV) MODFLOW Leakage to the groundwater system M  riv  i,j,k element i,j,k RBOT Special case: if  i,j,k <RBOT, then

2. River package (V) MODFLOW

3. Recharge package MODFLOW

4. Drain package MODFLOW Special case: if  i,j,k < d than

5. Evapotranspiration package MODFLOW

6. General head boundary package MODFLOW

Time indication MODFLOW ITMUNI=1: seconde ITMUNI=2: minute ITMUNI=3: hour ITMUNI=4: day ITMUNI=5: year MODFLOW

MODFLOW: implementation of geology Vcont = vertical leakage [T -1 ]: vertical hydraulic conductivity / thickness element MODFLOW kD = transmissivity [L 2 T -1 ]: horizontal hydraulic conductivity * thickness element

Layertypes in MODFLOW (LAYCON): 4* I. LAYCON=0 Confined: T=kD=constant Storage in element: MODFLOW

II. LAYCON=1 MODFLOW Unconfined: T=variable Storage in element:

II. LAYCON=2 MODFLOW Confined or unconfined: T=kD=constant Storage in element:

II. LAYCON=3 MODFLOW Confined of unconfined: T=variable as with LAYCON=1 Storage in element as with LAYCON=2

Solving of the MODFLOW matrix MODFLOW LU-decomposition rewriting as

Files in MODFLOW: infile.nam file MODFLOW Additional information (good documentation): MODFLOW: MOC3D:

Files in MODFLOW: *.bas file MODFLOW

Files in MODFLOW: *.sip file Files in MODFLOW: *.bcf file MODFLOW Files in MODFLOW: *.wel file

Variable density models: 1. Hypothetical cases: interface between fresh and salt evolution of a freshwater lens Henry case Hydrocoin Elder (heat transport) Rayleigh 2. Real cases in the Netherlands: Island of Texel Water board of Rijnland Province of North-Holland Wieringermeerpolder Board 14 basin MODFLOW: examples of cases MODFLOW

Formula of Theis (1935) Solution: Non-steady state groundwater flow towards an extraction well PDE: If u is small, than W(u2)=2ln(0.75/u)

Formula of Theis: data Non-steady state groundwater flow towards an extraction well  one layer: NLAY=1  in the horizontal plane: 19*19 elements of 50*50 m 2  initial head is 0 m  hydraulic conductivity k is 4 m/d  porosity n e =0.25  T=kD= m 2 /s (or 200 m 2 /d), so D=50 m  non-steady state: specific storativity S s = m -1  elastic storagecoefficient S=  extraction in element [i,j,k]=[10,10,1] with Q=345.6 m 3 /d  timestep: 1 day divided in 20 steps  three observation points on x=175, 375 and 475 m (y=475 m)  top systeem is 0 m M.S.L, bottom is -50 m M.S.L.

Groundwater movement near salt domes: The Hydrocoin case (I)

Groundwater movement near salt domes: The Hydrocoin case (II) Salt fraction:

Schematisation Dutch coastal aquifer system

Present ground surface in the Netherlands and position polder Beemster Wormer Schermer Purmer Haarlemmermeer polder Wieringermeer polder ~1930 Flevo polders s

Depth saline-fresh interface (150 mg Cl-/l)

Development of the Dutch polder area

Past and future sea level rise in the Netherlands

Historic lowering of the ground surface in Holland

Salt water intrusion in the Netherlands

Salt water intrusion (3D) in Texel: geometry

Geometry of the groundwater system in Noord-Holland

Solute transport equation MOC3D Partial differential equation (PDE): D ij =hydrodynamic dispersion [L 2 T -1 ] R d =retardation factor [-] =decay-term [T -1 ] dispersion diffusion advectionsource/sinkdecay change in concentration

Solute transport equation: column test (I): MOC3D

Solute transport equation: column test (II): MOC3D

Solute transport equation: column test (III): MOC3D

Hydrodynamic dispersion MOC3D hydrodynamic dispersion = mechanical dispersion+ diffusion mechanical dispersion: tensor velocity dependant diffusion: molecular process solutes spread due to concentration differences

Mechanical dispersion MOC3D

Solute transport equation: diffusion MOC3D diffusion is a slow process: diffusion equation only 1D-diffusion means: R d =1, V i =0, =0 and W=0 similarity with non-steady state groundwater flow equation

Solute transport equation: diffusion MOC3D diffusion is a slow process: diffusion equation

Method of Characteristics (MOC) MOC3D Solve the advection-dispersion equation (ADE) with the Method of Characteristics Splitting up the advection part and the dispersion/source part: advection by means of a particle tracking technique dispersion/source by means of the finite difference method Lagrangian approach:

Advantage of the approach of MOC? MOC3D It is difficult to solve the whole advection-dispersion equation in one step, because the so-called Peclet-number is high in most groundwater flow/solute transport problems. (hyperbolic form of the equation is dominant) The Peclet number stands for the ratio between advection and dispersion

Procedure of MOC: advective transport by particle tracking MOC3D Average values of all particles in an element to one node value Place a number of particles in each element Move particles during one solute time step  t solute Determine the effective velocity of each particle by (bi)linear interpolation of the velocity field which is derived from MODFLOW Calculate the change in concentration in all nodes due to advective transport Add this result to dispersive/source changes of solute transport

Steps in MOC-procedure MOC3D 5. Determine concentration in element node after advective, dispersive/source transport on timestep k 4. Determine concentration gradients on new timestep k* 3. Concentration of particles to concentration in element node 1. Determine concentration gradients at old timestep k-1 2. Move particles to model advective transport

Causes of errors in MOC-procedure MOC3D 4. Empty elements 3. Concentration of sources/sinks to entire element 1. Concentration gradients 2. Average from particles to node element, and visa versa 5. No-flow boundary: reflection in boundary

Reflection in boundary MOC3D

Stability criteria (I) MOC3D Stability criteria are necessary because the ADE is solved explicitly 1. Neumann criterion:

Stability criteria (II) MOC3D Change in concentration in element is not allowed to be larger than the difference between the present concentration in the element and the concentration in the source 2. Mixing criterion

Stability criteria (III) MOC3D 3. Courant criterium 

Files in MOC3D: *.moc file MODFLOW

Files in MOC3D: *_moc.nam and *.obs files MODFLOW

Numerical dispersion and oscillation

Derivation of numerical dispersion: 1D (I) Discretisation : backwards in space backwards in time By means of Taylor series development:

Derivation of numerical dispersion: 1D (II) Now Taylor series development with truncation erros!:

Derivation of numerical dispersion: 1D (III) Neglect 3rd and 4th order terms:

Derivation of numerical dispersion: 1D (IV) Rewriting term:

Derivation of numerical dispersion: 1D (V) Rewrite all extra terms:

Derivation of numerical dispersion: 1D (VI) Remedy to reduce numerical dispersion: 1.  x en  t smaller 2. Use a different numeric scheme (Crank-Nicolson) 3. Use a smaller D real Numerical dispersion:

Conclusions Modelling protocol helps you in making a numerical model Conceptualisation is important Stepwise from hydrogeological process to model: -Concept -Darcy and Continuity gives groundwater flow equation -Taylor series development gives numerical model -Stability criterion MODFLOW packages in mathematical formulation Use General Head Boundary package for a change! MOC3D: particle tracking technique Check input and output files in case of errors See documentation of USGS

Modelling of groundwater flow and solute transport Additional information: Groundwater modelling: ftp://ftp.geo.uu.nl/pub/people/goe/gwm1/gwm1.pdfftp://ftp.geo.uu.nl/pub/people/goe/gwm1/gwm1.pdf Density dependent groundwater flow: ftp://ftp.geo.uu.nl/pub/people/goe/gwm2/gwm2.pdfftp://ftp.geo.uu.nl/pub/people/goe/gwm2/gwm2.pdf Gualbert Oude Essink Faculty of Earth, Free University (VU) Netherlands Institute of Applied Geosciences (NITG) PGI: Aug , 2002