Mathematics in Computational Science and Engineering. Группа авторов
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case example of monitoring fluid progression in IOR projects is discussed in Narayan and Dusseault [28].
Figure 2.2 Schematic diagram of the new method for sampling and measuring potential field at the surface. Current dipole is fixed at depth closer to the anomalous zone of interest.
Before continuing further, another advantage of a fixed-electrode strategy must be noted. Methods such as “4-D” seismic are inherently expensive because the excitation source typically involves large surface equipment (vibroseis units). Therefore, surveys tend to be at long time intervals, months or years. In resistivity monitoring, excitation is easily achieved using an economical power source, therefore repeated surveys can be taken within a day or a week, examined to see if sufficient changes have occurred, and if so, analyzed in greater detail. Furthermore, we note that arrays of electrodes at the surface are undoubtedly cheaper than geophones or accelerometer arrays. If appropriate well designs are used, such as fiber-glass casings, electrode arrays can be directly combined with seismic receivers, as well as with temperature and pressure monitoring sensors. These advantages and their inherent flexibility imply that resistivity methods can be extremely economic to operate once the array is in place.
2.7 Mathematical Quantification of Resistivity Resolution and Detection
A good description of resolution, or resolvability, may be explained in terms of a minimum separation of two geological features or anomalies to be delineated [28]. Detectability is usually defined in terms of the size of the anomalous zone or electrical contrast of the anomalous zone that can produce a measurable potential field response [28]. On-site geological noise level must be considered in defining detectability. To delineate and quantify changes within the anomalous zone of interest, changes in geophysical parameters (e.g., electrical conductivity or resistivity) must occur over a distance. A detailed description of “resolvability” and “detectability” may be found in the geophysical literature. Greaves et al. [29] have discussed about the geometry and spatial distribution of current dipoles and potential dipoles giving a better resolution and detectability of the anomalous zone to be monitored.
Resolution may be further improved if the transmitting dipole or receiving dipole is placed into a borehole. Furthermore, it reduces the range of possible solutions and makes the mathematical analysis and interpretation more tractable and concrete. As indicated above in Figure 2.2, ∆-monitoring by inverting the differenced vector of potential field measurements may further enhance detectability if there are no systematic and non-random sources of noise. Narayan [3] has given a through description of theoretical sensitivity analysis and study for the resolvability and detectability of resistivity anomalies. Accordingly, a large amount of the current must flow through the anomalous region of interest. In this way, potential field measurements at the surface are detected effectively. It is important to focus on maximizing current flow through the anomalous zone. Current flow depends on several factors (e.g., spacing of current dipole, location of current dipole, dimension of anomalous region, electrical resistivity contrasts, etc.). It may be evaluated mathematically via theoretical model simulations and using other available data.
Figure 2.3 illustrates that the changes in potential field response as a function of electrical resistivity contrast [3, 28]. This also verifies equation 2.31 numerically. Also, the Figure shows that electrical potential field response varies exponentially with increasing resistivity contrast. A linear relationship has been found for small resistivity changes. The following important observations from this numerically computed result are summarized below:
The magnitude of the electrical potential field response increases linearly for small resistivity contrasts of the anomalous zone. From the mathematical formulation for sensitivity analysis, this result is expected here.
Figure 2.3 Computation of changes in the potential field response with increasing resistivity contrast.
An important observation here is that a linear relationship exists up to a change in electrical resistivity by a factor of four. This may be found extremely helpful in implementing a linear resistivity inversion in imaging in-situ processes using a difference of observed potential field data from time A to time B.
A linearized inversion will be valid only in situations where the resistivity perturbation in the target region is not more than a factor of four during two consecutive resistivity measurements.
This linearity for small electrical resistivity changes may also be found useful as a basis to introduce the concept of adaptive resistivity inversion.
Adaptive resistivity inversion should permit interpretation of difference potential field responses in terms of changes in resistivity at depth.
Practically, it is important and advised to maximize current dissipation in and around the region that is to be monitored so that electrical field perturbations are large enough to detect at the surface. It is usually difficult to achieve enough current flowing through an anomalous region if only surface electrodes are used. However, by placing current dipole electrodes at the right depth with an appropriate current dipole spacing, sufficient current flux through the target zone can be achieved that in turn causes substantial variations in surface potential field measurements over the monitoring periods. In addition to increasing current flux, detectability may be enhanced in surface measurements by placing the current dipole adjacent to or across the anomalous zone that is to be monitored.
2.8 Scheme of Resistivity Data Presentation
When monitoring in-situ processes, a number of essential questions arise. What is the minimum signal amplitude that can be detected? How should we quantify detectability? What is the minimum signal repeatability? These are vital questions from the monitoring point of view and must be dealt with quantitatively. We attempt to answer some of these questions using numerical model responses and by defining measurability and detectability consistently.
The issue of detectability of a signal associated with an anomalous zone is related to the depth of exploration. For a given target, this is usually defined as a maximum depth that may be detected with a given electrode configuration. Ward [30] noted that the depth of exploration is a function of several parameters such as sensitivity to inhomogeneity & bedrock topography, lateral effects, general topography, dip, etc., and signal-to-noise ratio. As these issues were not studied systematically for a wide range of electrical resistivity models, he did not define different electrode configurations in terms of detectability. Oldenburg [31] also noted that the without studying a wide range of models, there is no analytical basis for considering one electrode configuration over another for resolving model parameters. However, the approach herein is somewhat different, using a mathematical basis to evaluate placement of a current dipole at depth in the proximity of an anomalous zone to be monitored, which increases the amount of current flux in the zone, leading to a better detectability. Other issues affecting detectability must be studied using a forward model on a case-by-case basis to optimize the array characteristics.
Narayan [3] defined the term “measurability” in terms of percent difference of the measured signal with respect to the background. This gives us an idea as to the signal levels that must be achieved on top of the background signal in order for an anomaly to be measured. Available commercial