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

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

Saltwater intrusion

Saltwater intrusion in heterogeneous media Small and medim scale heterogeneity: effective equations/parameters Large scale heterogeneity: inverse problem From E. Abarca

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

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

Derivatives of objective function Compute objective function at p_i Perturbate all parameters 1 time objective function for each Compute gradient and new parameter set Converged done Solve each timestep n+1 times

Derivatives of objective function Compute objective function at p_i and gradient simulatneously Compute new parameter set Converged done For each timestep, solve nonlinear problem, and as many linear systems as there are parameters

Parameterization methods Express discrete model parameter vector in function of estimable parameters Example: Zones of constant parameter value

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)

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

TRUE Conditional Estimation Conditional simulation Comparison of methods (64 pilot points, 10 observations of log 10 K k, 160 of hydraulic head)

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)

Inverse problem for saltwater intrusion Simplifications are common to reduce running time –Constant density model –Homogeneous model or zones

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.

Derivatives of objective function Compute transient simulation until reaching steady state (NO derivatives) Compute steady state simulation using transient solution as initial guess, and derivatives simultaneously Compute new parameter set Converged?

Derivatives of objective function Compute transient simulation until reaching steady state (NO derivatives) Compute steady state simulation using transient solution as initial guess. The do the same with each parameter perturbed Compute new parameter set Converged?

Computation time Time for 1 inverse problem iteration Derive transientDerive steady state only Parameter perturbation 34 min2min 50 s Automatic derivation 15 min2 min 10 s Solving variable density flow and transport problem at a mesh of 861 nodes. Estimating 16 parameters. Transient simulation simulates 36000s divided over 492 timesteps

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

Test case

Example result

Tidal wave propagation Propagation inland in homogeneous confined aquifer perpendicular to coastline produces DAMPING and PHASE SHIFT

Tidal wave propagation (homogeneous case, constant denstity) S =0.01 m -1 K = 1.58 m/d

Effect of tidal wave on velocity (homogeneous confined aquifer) Difference between steady state velocity and

Do we need density dependent flow to model tidal wave propagation?

Heterogeneous case

Effect of tidal wave on velocity (heterogeneous confined aquifer)

Tidal wave propagation (heterogeneous case)

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

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

Use of tidal wave propagation for aquifer characterization Carr and Van Der Kamp (1969): “Determining Aquifer Characteristics by the Tidal Method” ´80s, 90´s : 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)

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

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.