Atc dimensions

Karaoulis, A. Revil, D. Werkema, B. Minsley, W.

Fuse (automotive)

Karaoulis, A. Revil, D. Werkema, B. Minsley, W. Woodruff, A. Induced polarization more precisely the magnitude and phase of impedance of the subsurface is measured using a network of electrodes located at the ground surface or in boreholes. This method yields important information related to the distribution of permeability and contaminants in the shallow subsurface.

We propose a new time-lapse 3-D modelling and inversion algorithm to image the evolution of complex conductivity over time. We discretize the subsurface using hexahedron cells. Each cell is assigned a complex resistivity or conductivity value. Using the finite-element approach, we model the in-phase and out-of-phase quadrature electrical potentials on the 3-D grid, which are then transformed into apparent complex resistivity.

Inhomogeneous Dirichlet boundary conditions are used at the boundary of the domain. The calculation of the Jacobian matrix is based on the principles of reciprocity. The goal of time-lapse inversion is to determine the change in the complex resistivity of each cell of the spatial grid as a function of time.

This approach can be simplified into an inverse problem looking for the optimum of several reference space models using the approximation that the material properties vary linearly in time between two subsequent reference models.

Regularizations in both space domain and time domain reduce inversion artefacts and improve the stability of the inversion problem. In addition, the use of the time-lapse equations allows the simultaneous inversion of data obtained at different times in just one inversion step 4-D inversion. The advantages of this new inversion algorithm are demonstrated on synthetic time-lapse data resulting from the simulation of a salt tracer test in a heterogeneous random material described by an anisotropic semi-variogram.

Electrical resistivity is sensitive to salinity, porosity, saturation, pore shape, temperature, clay content and biological activity e. Variability in any of these parameters can have an influence on resistivity and can be monitored by time-lapse electrical resistivity tomography TL-ERT.

In the recent literature, TL-ERT has started to be a popular method to monitor dynamic processes occurring in the shallow subsurface typically the first hundred metres, see Legaz et. The TL-ERT imaging, often involving permanent electrode installations, has proven to provide information complementary to in situ geochemical measurements.

Daily et. Slater et al. Nguyen et. In an effort to extract more information about the subsurface geology e. Such a geophysical method is called complex resistivity, complex conductivity, time- or frequency-domain induced polarization or low-frequency dielectric spectroscopy in the literature.

In frequency-domain induced polarization, an alternating current is injected and retrieved into the ground using two electrodes A and B. Both the resulting magnitude and phase of the voltage between two potential electrodes M and N are measured and used to define impedance, which once corrected for the position of the electrode is used to define an apparent complex resistivity.

This method was originally developed for the exploration of ore bodies Pelton et. The sensitivity enhancement of modern equipment has increased the measurement resolution of the phase lag between the current and the voltage typically 0. Olhoeft, personal communication and Vaudelet et. Recently, Revil et al. Leroy et. However, all these approaches do not include a description of membrane polarization and a unified model including this contribution has still to be done.

The approach described in Leroy et. The introduction of time into the inversion of geophysical data sets can be achieved with the use of time-lapse algorithms. In this case, several strategies are possible.

A standard approach is to independently invert the measured data acquired at each monitoring step and to reconstruct time-lapse images e. As suggested by several researchers, the independent time-lapse inversion images may be strongly contaminated with inversion artefacts due to the presence of noise in the measurements and independent inversion errors.

In this work, we describe a new induced polarization time-lapse tomography algorithm. Forward modelling is presented in Section 2. In Section 3, we present a new 4-D algorithm for induced polarization based on an active time constrained ATC approach. Our work extends the recent work of Karaoulis et. Time-lapse time-domain IP data could be treated the same way. In our approach, the subsurface is defined as a space—time model and the regularization over time is active where it allows variability between different time steps depending on the degree of spatial complex resistivity changes occurring among different monitoring stages time steps.

As a result, the 4-D-ATC algorithm can help in focusing on the 3-D spatio-temporal changes of the complex resistivity. We will present the results for a single-frequency application of the algorithm; however, the extension of the algorithm to multifrequency time-lapse data can be done with the successive application of the algorithm to a set of data taken at distinct frequencies.

Along the same lines, the approach of Kemna et. Using spectral-induced polarization data, a relaxation model such as the Cole—Cole model can be fitted for each cell and the evolution of the Cole—Cole parameters can be followed over time. Kemna In this work, we used mixed boundary conditions, which can be implemented in the complex case analogous to the dc case Kemna In Eq.

The active time Lagrangian, expressed with the matrix , controls the time-related changes. Effectively, such a scheme should vary the time normalization between the parameters of different time steps proportionally to the spatial resistivity changes occurring amongst different monitoring locations.

The determination of the time regularization parameter may depend on the spatio-temporal characteristics of the process, which is controlling the changes in complex resistivity. Ideally, matrix entries associated with areas of significant property changes must be assigned low time regularization values and vice versa.

Two methods are proposed to assign the appropriate values to the time regularization parameter: one based on a fast pre-estimation of the first independent inversion iteration and one, more accurate, after a full inversion Karaoulis et. In this work, we used the accurate calculation of the time Lagrange matrix.

The creation of the matrix is similar to the dc real values problem with one exception. In the induced polarization case, two models must be considered, one for time-lapse changes in the amplitude and one for the time-lapse changes of the phase. Note that the resistivity and the phase can change over time independently from each other see Vaudelet et.

To perform this task, we follow the following steps: 1 we generate a time-related distribution of values for the Lagrangian parameter as a function of the difference in amplitude between two sequential time steps, 2 we generate a time-related distribution of values for the Lagrangian parameter from the difference in phase between two sequential time steps and 3 we combined these two time-related Lagrangian value distributions in one scheme e. Trial-and-error testing showed that for our numerical examples the two time-related Lagrangian values must be between 0.

The 4-D-ATC algorithm is going to be tested with synthetic data and compared to the prediction of using independent inversion tomographies performed independently at each time step. In the case of field data, it is expected that the artefacts associated with the presence of noise in the data is significant and independent inversion must, therefore, be avoided. For the comparison between the two approaches to be objective, all algorithms were based on the same 3-D finite element forward modelling and inversion platform, the principles of this platform having already been discussed in Section 2.

Note that the same homogeneous half-space was used as the starting model for all the tested techniques and that all the synthetic data are considered as measured simultaneously for each time step.

In this paper, the phase and amplitude are shown it is implicit that the phases have negative values. The data misfit was smaller than 5 per cent for the two examples discussed later. Modelled data obtained for five different time steps representing a hypothetical time-lapse induced polarization change are depicted in Figs 1 and 2. The pseudosection comprises a total of measurements for each time step. In this specific example, the synthetic data are taken noise-free.

The 4-D induced polarization model used in this work showing the changes in amplitude through time five time steps. The grey cubes denote the synthetic model used in the previous time step. The red cubes show the change in that time step with respect to the previous time steps. Same as Fig. The background model has a phase of —5 mrad. Figs 1 and 2 show the modelled evolution of both the amplitude of the resistivity and the amplitude of the phase.

The grey cubes show the changes in both amplitude and phase that remain stable through time. Red cubes reveal the modelled changes in both the amplitude of the resistivity and that of the phase between two sequential time steps. For instance, the red cube shown in time step 1 in amplitude, remains stable from time step 2 on so it is denoted as grey in all later time steps , where a new red cube is introduced, which shows the modelled change between those two time steps.

As discussed in Section 4, the 4-D-ATC technique requires a priori information on the expected time related changes, so the matrix could be formulated.

The matrix must consider time related changes in both amplitude and phase, to adjust appropriate weight. Cold colours, that is, low values on the time-related Lagrangian, indicate areas with expected changes in both amplitude and phase and hot colours large time-related Lagrangian values indicate areas with no time changes.

Therefore, Fig. The relation between low time-related Lagrangian values with the actual changes is quite good, even considering the fact that the estimation seems to be spread.

Note that the matrix is just a pre-estimation of where the expected change is located between two time steps. The matrix was calculated using the full independent inversion of each data set. In Figs 4 and 5 , the first series of images upper part shows the difference in amplitude between two sequential time steps; in Figs 6 and 7 , the first series of images upper part denotes the difference in the phase.

A combination of the amplitude and phase time-related changes is then used to create the matrix Fig. The distribution of Lagrange parameters based on the independent inversion as a prior information used in the ATC approach.

The cold colours indicate areas with significant changes. These areas are characterized by low values of the Lagrange parameters. The hot colours indicate areas with no changes, that is, areas characterized by high values of the Lagrange regularization parameters. The grey cubes show the position of the true changes in the synthetic model. Difference images for the synthetic model of resistivity presented in Figs 1 and 2.

The 4-D-ATC lower row and independent inversion upper row difference amplitude images are shown for time steps 2—1 left side and 3—2 right side , respectively. The grey cube shows the position of the true change according to the synthetic model.

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