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in Vasco et al. (2008), the onset time can be related to the travel time of the pressure front initiated by the start of injection. Specifically, for an elastic medium and a sharp injection profile resembling a step function, the onset of the peak rate of volume change, T _peak , is related to the phase σ of the propagating pressure front according to (Vasco et al., 2000). For propagation governed by the diffusion equation, the phase is given by the solution of an eikonal equation,

(2.18)

      where γ (x) is the inverse of the hydraulic diffusivity (Vasco et al., 2000; Vasco & Datta‐Gupta. 2016, p. 138). The nonlinear, first‐order eikonal equation is equivalent to the system of ordinary differential equations,

(2.19

      and

(2.20)

      where x(s) is the flow path in the porous medium and s is the distance along the path. These expressions may be used to find the hydraulic diffusivity, and consequently the effective permeability, within the reservoir. As was shown by Rucci et al. (2010), permeability estimates based upon diffusive travel times are primarily sensitive to the kinematics of the pressure propagation and not sensitive to the coupling between the magnitudes of reservoir pressure and volume change.

2.4. FIELD APPLICATIONS

      In this section, we describe three large‐scale CO₂ storage projects that made, or are making, use of geodetic monitoring. All three projects incorporated InSAR observations into their monitoring work flow.

      2.4.1. In Salah, Algeria

At In Salah, Algeria, excess carbon dioxide from the production at three gas fields was removed, processed, and reinjected into the flanks of an anticline defining one of the fields. The target reservoir for long‐term storage was a sandstone layer roughly 20 m thick, overlain by almost 1 km of shales and siltstones and an additional kilometer of interbedded sandstones and shales. Initially, it was supposed that the carbon dioxide would simply reside in the reservoir. Since the start of injection in 2004, roughly 3.8 million tonnes of carbon dioxide have been sequestered. Initial predictions of surface uplift, obtained from coupled hydrological‐geomechanical simulations, varied from a few millimeters to about one centimeter (Bissell et al., 2011). InSAR data acquisition was part of an extensive monitoring effort devised by the partners of a Joint Industry research and development program (Mathieson et al., 2010; Ringrose et al., 2013). Several of the monitoring activities, including microseismic and InSAR data collection, were in collaboration with Lawrence Berkeley National Laboratory. Fortunately, the surface conditions at the In Salah site are quite favorable to SAR monitoring and the permanent scatterer approach, with a boulder‐strewn, hard‐packed surface, and a restricted supply of mobile sand. The project benefited by both large‐scale (Bissell et al., 2011) and detailed coupled modeling of the flow and related geomechanics.

       Envisat Range Change Observations

For the first phase of this work, Lawrence Berkeley National Laboratory contracted TRE to process existing data in the European Space Agency Envisat archive from 12 July 2003 to 19 March 2007. Two satellite tracks covered the region containing the three injection wells (Tracks 65 and 294). During this initial study period, the respective tracks contained 26 and 18 satellite images, with one or more months between each image. The data were processed using the permanent scatterer algorithm described above (section 2.2.3). The range change data for both tracks provided an estimate of ground displacement along the line of sight of the satellite, that is, along the look direction of the satellite. The analysis of the Envisat data revealed observable surface deformation associated with the injection of carbon dioxide. Peak velocities of over 5 mm/year were found for both tracks, exceeding the estimated errors of 1 mm/year. Elongated patterns of range decrease were imaged, suggesting uplift over the three injection wells KB‐501, KB‐502, and KB‐503 (Fig. 2.3).

Initially, the range decreases in the region overlying the well KB‐501 were assumed to be the result of injection‐related volume change within the reservoir (Vasco et al., 2008). Following the procedure described above (see equation 2.11) the reservoir volume surrounding the injector was mapped into a grid of cells. A regularized least‐squares approach was used to estimate the fractional volume changes within each grid block. A mapping of the sequence of range change into reservoir‐volume change in the region surrounding well KB‐501 indicated preferential migration to the northwest of the injection well. Using the diffusive imaging technique described by equations (2.18, 2.19, and 2.20). the propagation times of the volume changes were used to calculate permeability variations within the reservoir layer (Vasco et al., 2008).

Figure 2.3 Range changes above the carbon storage site at In Salah, Algeria, 1,261 days after the start of injection. The black lines are the traces of the injection wells within the target formation. The open circles denote the wellhead locations of the gas producers.

Specifically, the time series of volume changes for each grid block were used to define the onset of the peak rate of change T _peak . This time is related to the phase of the propagating front σ , according to (Vasco et al., 2000). In Figure 2.4, the variations in the phase are plotted for those regions where the volume changes were significant. The trajectories are found by solving equation (2.19). The solution is quite simple because the right‐hand side is the known phase field σ . So the trajectories are found by marching down the gradient from each grid block center back to the well. Given the trajectories and the spatial distribution of σ , one can estimate the diffusivity by solving equation (2.20) for κ . This is a tomographic problem of a slightly different nature than that of cross‐well imaging. For the problem at hand, there are many trajectories traveling along a similar flow path but for varying distances. The density of trajectories sampling the flow path depends upon the temporal sampling provided by the geodetic data. Thus, one can incrementally build up the diffusivity distribution along each flow path, based upon the set of trajectories sampling that path. The resulting permeability

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