Improving Aquifer Imaging and Long-term Monitoring with Oscillating Signals: Oscillatory Hydraulic Tomography

Michael Cardiff (cardiff@wisc.edu) and YaoQuan Zhou (zhou86@wisc.edu) University of Wisconsin-Madison, Department of Geoscience

Poster H33I-1440

1. Motivation

References

6. Long-term Monitoring

3. Modeling Approach Hydraulic tomography (HT) is a “next generation” aquifer characterization method in which pressure changes (measured at several locations) from a series of pumping tests are jointly analyzed to image aquifer heterogeneity. Most often, hydraulic tomography is performed using traditional, constant rate pumping tests (Cardiff and Barrash 2011) We present an improved modified approach, Multi-Frequency Oscillatory Hydraulic Tomography (M-OHT), in which multiple oscillating pumping tests of several frequencies are used as the aquifer stimulation. Practical Advantages: •  No net injection or extraction of water •  Little movement of existing contamination •  Multiple tests can be performed without moving / re-arranging

equipment Computational Advantages: •  Minimal impact of model boundary conditions (due to conservative

pumping) •  Oscillatory signals easily extractable from noisy data, even with low

SNR (signal-to-noise ratio) and drift •  Fast inversion by using a “steady-periodic” numerical model •  Reduced ill-posedness of the inverse problem by using multiple

frequencies each with different sensitivity structures

Cardiff, M., T. Bakhos, P. K. Kitanidis and W. Barrash (2012-in review). "Oscillatory Hydraulic Tomography: A new concept for aquifer imaging and long-term monitoring with periodic signals." Water Resources Research.

Cardiff, M. and W. Barrash (2011). "3-D transient hydraulic tomography in unconfined aquifers with fast drainage response." Water Resources Research 47: W12518.

The key concept of the approach that allows fast analysis of oscillatory signals is to ignore early-time transient data and focus on the long-term response through the use of a Steady-Periodic Numerical Approach. For each stimulation frequency, steady-periodic model runs were performed in which the “wave phasor” is solved for. The phasor is a complex-valued field variable that compactly stores the wave’s properties (e.g., amplitude and phase) at each point in the domain. For this model, with the same discretization and convergence criteria, the time-domain model required 10 minutes of runtime, whereas the phasor domain required less than 10 seconds.

Diagrammatic explanation of how a transient (head-based) formulation with a given input-frequency source term can be converted to a steady-

state (phasor-based) formulation.

DEPARTMENT  OF  GEOSCIENCE

By applying an adjoint state sensitivity approach to the phasor-based governing equations, we have derived the sensitivity of signal properties (phase offset and amplitude for a given frequency) to spatially-distributed aquifer properties. Once calculated, the sensitivity maps are used in highly-parameterized inverse modeling. Example sensitivity maps for a single pumping well and single observation well are shown to the right. These maps indicate how measurements of different frequency stimulations average over different aquifer volumes Since the sensitivity structure at each frequency is different, the sensitivity maps presented suggest that additional information about aquifer heterogeneity can be obtained through multi-frequency aquifer stimulation.

Using the test model, sinusoidal pumping is simulated at periods of 5, 10, 20, 100, 200, and 300 seconds. Pumping took place at the central well (maximum flowrate 1.2 liters/second) and was measured at 4 surrounding wells. For each frequency, 1 hour of synthetic data was generated and noise was added to each temporal record. Wave properties were then extracted through signal processing and inverted using sensitivity calculations from 6 steady-periodic model runs (one steady model run per pumping frequency). Total inversion time in all cases was less than 1 hour on a standard desktop PC.

5. Single- and Multi-frequency Imaging Results

Problem geometry showing heterogeneous synthetic aquifer and well arrangement.

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2. Synthetic Experiment Setup A synthetic 2-D heterogeneous aquifer used to test the M-OHT concept is shown below. Oscillatory pumping was performed only at the central (red) well, with observations at surrounding (white) wells. Both transient and steady-periodic physics were simulated using the COMSOL Multiphysics finite element model.

4. Sensitivity Analysis

Validation comparison of results from transient (solid) and steady-periodic (dotted) modeling for the test

problem.

Since oscillating signals do not result in net injection or extraction of water, M-OHT may represent a valuable strategy for long-term monitoring of aquifer processes. In this example, a simple model demonstrates how the propagation of oscillatory signals is changed, over time, by an infiltrating high-viscosity fluid

iωSs x( )Φω = ∇⋅ K x( )∇Φω( ) +Qmax x( ) ∀x∈Ω

Φω = 0 ∀x∈Γdirichlet

∇Φ⋅n = 0 ∀x∈Γneumann

Ss x( ) ∂h∂t

= ∇⋅ K x( )∇h( ) + q ∀t ≥ 0,x∈Ω

h = 0 ∀t ≥ 0,x∈Γdirichlet

∇h ⋅n = 0 ∀t ≥ 0,x∈Γneumann

q =Qmax x( )cos ωt( ) q = Re Qmax x( )exp iωt( )⎡⎣ ⎤⎦

Euler’s Formula

h = Re Φω x( )exp iωt( )⎡⎣ ⎤⎦

Expected long-term Response

Time Domain Phasor Domain

Sensitivity of measured sinusoid properties to aquifer transmissivity at periods of 10 seconds and 50 seconds. Background color represents

positive (white) / negative (grey) sensitivity. Contours represent absolute magnitude. Blue represents observation location, and red oscillating

pumping well

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Signal log(Amplitude) Signal phase

Imaging results obtained for progressively higher number of oscillatory

testing frequencies.

True Log10(K) within central 25m x 25m

region of the synthetic model

Research Supported by NSF Grant EAR-1215746

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Above: COMSOL Model Geometry of sandbox experiment. Infiltration of high-viscosity, high-density fluid is simulated by changing effective K within “plume” region. Rectangle above pumping well represents low-K inclusion. Oscillating pumping takes place at central (red) point, with observations shown for two locations (blue and green) points. Right: Changes in oscillatory signal response before infiltration (dashed) vs. after infiltration (solid). Pumping stimulations are 1 second period (top) and 10 second period (bottom). Note that changes in phase and amplitude are dependent on pumping oscillation period.

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