Mathematics in Computational Science and Engineering. Группа авторов
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2.1), equation (2.31) becomes
The current densities for each block can be computing using a forward resistivity modelling scheme, so that the equation (2.32) reduces to
The above relationship may further be simplified for 2-D and 1-D models. The current density in Y-coordinate direction of 2-D geologic model is negligible and the average current density in the strike direction is zero, in which case, equation (2.33) becomes
(2.34)
for 2-D geologic models, or
(2.35)
for 1-D geologic models.
2.6 Mathematical Concept for Application to Geoengineering Problems
Resistivity inversion methods have been implemented successfully for a variety of applications. However, the method has not been tested fully in various possible applications, such as for monitoring in-situ processes for improved oil recovery (IOR), environmental and geotechnical aspects of landfills and similar retainment structures. This may be because field surveys conducted until recently were done manually. Manual execution involves direct human activity to set up current and potential electrodes, electrode connections, and to take measurements of the induced potential field arising from current injection into the ground; this tends to make long-term investigations uneconomical or impractical. Another reason may be that field data are sometimes difficult to interpret in terms of a geologic model, owing to a lack of an appropriate interpretive tool (inversion model), poor resolution, poor quality data, or poor data coverage. The advent of the personal computer has led to dramatically increased efficiency in data collection. It is now possible to measure and interpret field data with a far better resolution and coverage than could be obtained with manual data collection, particularly if a fixed-electrode strategy is used. This in turn enhances the possibility of obtaining unambiguous geological interpretations of the field data because incomplete or varying locations for data sets over a time interval can be difficult to interpret. Mathematical tool discussed herein believes that the possible applications of direct-current resistivity methods are now limited mainly by our lack of imagination or opportunity, and it is likely that many more applications will be attempted in the future.
Whenever a sufficient resistivity change over a region or at a front is generated as a result of a dynamic process such as groundwater contamination or IOR processes, the induced electrical-field response to that process can be modeled with an appropriate mathematical tool, and an optimum monitoring strategy determined. This monitoring capability can be achieved with currently available technology at relatively low expense, and it may be highly complementary to other monitoring methods (e.g., seismic response, geochemistry changes, surface displacement data, and pressure-volume-temperature (PVT) data in the case of IOR projects).
It is important to note that electrical resistivity methods highly used for pore fluid delineations. Particularly if fluid electrical conductivity is changing but the saturation is not. Seismic properties are effectively insensitive to fluctuations in fluid ionic properties occurring as a result of IOR processes. However, pore pressure changes without resistivity changes clearly affect wave velocities and attenuation, but have less effect on electrical properties. Seismic response differences over time (“4-D” seismic) have been successfully used to map IOR processes. Pore-fluid type, salinity, temperature, and saturation, as well as lithology and to a lesser degree stress and pressure, all influence resistivity. There are rock parameters that affect either electrical response, seismic response, or both.
In this article, applications of direct-current resistivity methods for monitoring in-situ processes are investigated and emphasized based on a solid mathematical basis. Attempt is made herein to explain the mathematical concept that can be used for monitoring in-situ processes (e.g., processes associated with geotechnical problems, processes of geo-environmental problems and processes of IOR projects involving water flooding and steam flooding).
Referring to a mathematical equation (2.31), J’(x, y, z) is a current density in the region of resistivity change after applying reciprocity between the receiver and transmitter. J’(x, y, z) may also be viewed mathematically as a Green’s function. The symbol τ in the above equation is the volume of an anomalous region where resistivity has been perturbed. Using equation (2.31), measurement sensitivity of the surface potential field to electrical resistivity changes can be expressed by the inner dot product of current densities in the anomalous zone to be monitored. This is mathematically a fundamental principle and concept that form the basis to introduce new techniques and strategies for resistivity measurements and tools for data interpretation. The physical insight derived from this analysis is that surface measurement sensitivity of the potential field is proportionally related to the amount of power dissipated (current density) around the zone of interest. This theoretical concept used in the derivation of equation (2.31) is commonly known as an adjoint solution or an adjoint state technique in the geophysical literature. This technique involves the transformation of the differential equation (in the case of resistivity, it describes the potential field due to a direct current point source) to yield a Green’s function. The Green’s function approach is quite common for implementing inversion of geophysical data. After deriving the theoretical formulation, an attempt has been made to interpret it physically. The physical insight derived from the relation has been used to guide sensitivity analyses as well as to introduce a new technique for resistivity measurements [3].
Current density is a function of several parameters such as depth of zone to be monitored, electrode spacing, electrode orientation, resistivity of the sounding medium, and the amount of current injected. To delineate and investigate a deeper anomalous zone, larger currents and sampling electrode separations are required. It is necessary to maximize current dissipated in and around the anomaly at depth to cause perturbations in the electrical-field large enough to detect reliably at the surface [3]. If only a surface configuration is used, it is difficult to have enough current flowing through the plume or anomalous zone of interest even at a moderate depth. However, if the current electrodes are placed at the right depth, a sufficient current flux through the anomalous region can be achieved to cause substantial variations in surface potential field measurements over time intervals.
Narayan [3] has showed that detectability of resistivity changes at depth could be maximized in surface measurements by increasing the amount of power dissipation in the zone of changing resistivity; this is achieved by placing the transmitting dipole adjacent to it or across it. This clearly is a basic aspect of practical recommendations for any subsurface-to-surface resistivity method. The transmitting dipole location, its spacing, and its proximity with respect to the zone control this power dissipation, therefore a parametric study of these factors has been conducted numerically, and these results provide a basis for a new method of resistivity measurements (Figure 2.2). Based on the theory and method of resistivity measurements discussed