Aspects of Conditional Simulation and estimation of hydraulic conductivity in coastal aquifers" Luit Jan Slooten

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Aspects of Conditional Simulation and estimation
of hydraulic conductivity in coastal aquifers"
Luit Jan Slooten
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Contents

• Introduction
• Review of Inverse problem
• Review of conditional simulation
• A method to perform conditional simulation for
  saltwater intrusions
• Use of tidal wave data
• Incorporation of tidal wave data into
  conditioning of saltwater intrusion data

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Saltwater intrusion
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Saltwater intrusion in heterogeneous media

• Small and medim scale
• heterogeneity effective
• equations/parameters
• Large scale heterogeneity inverse problem

From E. Abarca
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Inverse problem (I)

• Find parameters that produce best model fit
• Model fit expressed in objective function
  weighted squared residual norm with
  regularization term

• Model performance can be judged in terms of
  weighted residual bias and variance ideally,
  bias is zero and variance is small

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Inverse problem (II)

• Methods of minimizing F with respect to p
• Global search simulated annealing, swarm
  methods, monte carlo
• extensive sampling of parameter space
• Attempt to find global minimum
• Feasibly only for small amount of parameters to
  estimate
• Local search steepest gradient,
  Marquardt-Levenberg
• Need far less objective function evaluations
• Need gradient of objective function and sometimes
  approximation of second derivatives
• Attempt to find local minimum ? result depends
  on starting parameters
• Can deal withlarge numbers of parameters

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Derivatives of objective function
Solve each timestep n1 times
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Derivatives of objective function
For each timestep, solve nonlinear problem, and
as many linear systems as there are parameters
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Parameterization methods

• Express discrete model parameter vector in
  function of estimable parameters

Example Zones of constant parameter value
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Conditional Estimation

• Prior information in pilot points is obtained by
  kriging from observations
• Covariance matrix of parameters is estimation
  covariance of this kriging system (as such it is
  conditional to observations)

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Conditional Simulation

• Simulated K is random but respects measurements
• Prior information in pilot points is value of
  simulation field value obtained by kriging from
  observations
• Covariance matrix is 2 times the covariance
  computed from the kriging from the observations

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Comparison of methods
Conditional Estimation
Conditional simulation
TRUE
(64 pilot points, 10 observations of log10K k,
160 of hydraulic head)
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Conditional Estimation and Simulation

• Initially, field is conditioned to parameter
  measurements
• Using an inverse problem approach the pilot
  points are estimated to condition the resulting
  fields to observations of state variables.
• Simulation represents local heterogeneity better
  but requieres many realizations of drift field
• Both work best with large number of pilot points
  (regularization term stabilises to allow this)

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Inverse problem for saltwater intrusion

• Simplifications are common to reduce running time
  
• Constant density model
• Homogeneous model or zones

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Inverse problem for saltwater intrusion

• Approach using steady state approximation of
  system dynamics
• For SWI, steady state is computed by a long
  transient simulation with constant boundary
  conditions.

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Derivatives of objective function
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Derivatives of objective function
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Computation time
Solving variable density flow and transport
problem at a mesh of 861 nodes. Estimating 16
parameters. Transient simulation simulates
36000s divided over 492 timesteps
Time for 1 inverse problem iteration Derive transient Derive steady state only
Parameter perturbation 34 min 2min 50 s
Automatic derivation 15 min 2 min 10 s
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Test

• Generated random field
• Extracted observations and prior information (172
  hydraulic head, 172 concentration, 10 log K)
• Added gaussian noise
• Used this dataset to optimize 64 pilot points

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Test case
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Example result
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Tidal wave propagation
Propagation inland in homogeneous confined
aquifer perpendicular to coastline produces
DAMPING and PHASE SHIFT
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Tidal wave propagation (homogeneous case,
constant denstity)
S 0.01 m-1 K 1.58 m/d
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Effect of tidal wave on velocity(homogeneous
confined aquifer)
Difference between steady state velocity and
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Do we need density dependent flow to model tidal
wave propagation?
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Heterogeneous case
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Effect of tidal wave on velocity(heterogeneous
confined aquifer)
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Tidal wave propagation (heterogeneous case)
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Modeling tidal wave propagation amplitude

• Only flow is sufficient
• Mesh must be sufficiently long
• Mesh may be coarser than the one needed for
  correct simulation of saltwater intrusion
• Problem is linear

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Compute transient simulation until reaching
steady state (NO derivatives)
Compute steady state simulation using transient
solution as initial guess, and derivatives
simultaneously
Solve tidal fluctuation amplitude using much
larger and coarser mesh, ONLY Flow, Transient,
and compute derivatives simultaneously
Compute new parameter set
Converged?
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Use of tidal wave propagation for aquifer
characterization

• Carr and Van Der Kamp (1969) Determining
  Aquifer Characteristics by the Tidal Method
• 80s , 90s many analytical solutions for
  different aquifer geometries
• Alcolea et al Inverse Modeling of Coastal
  Aquifers Using Tidal Response and Hydraulic
  Tests (uses constant density model for
  horizontal 2d plane)

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To do

• Evaluate whether weighted resiudal variance and
  bias is reduced if transmissivity fields are
  conditioned not only to head and concentration,
  but also to tidal wave amplitude date

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Conclusions

• A method was presented to perform conditional
  simulation and estimation of saltwater intrusions
  in steady state in an efficient way
• It was shown that tidal wave amplitude data can
  be modeled with the flow equation only
• An algorithm was presented to condition random
  fields to tidal wave data, head and concentration
  without loosing the steady state approximation of
  the coupled problem.