Inversion of time domain three-dimensional electromagnetic data

We present a general formulation for inverting time domain electromagnetic data to recover a 3-D distribution of electrical conductivity. The forward problem is solved using finite volume methods in the spatial domain and an implicit method (Backward Euler) in the time domain. A modified Gauss–Newton strategy is employed to solve the inverse problem. The modifications include the use of a quasi-Newton method to generate a pre-conditioner for the perturbed system, and implementing an iterative Tikhonov approach in the solution to the inverse problem. In addition, we show how the size of the inverse problem can be reduced through a corrective source procedure. The same procedure can correct for discretization errors that inevidably arise. We also show how the inverse problem can be efficiently carried out even when the decay time for the conductor is significantly larger than the repetition time of the transmitter wave form. This requires a second processor to carry an additional forward modelling. Our inversion algorithm is general and is applicable for any electromagnetic field  ( E , H , d B / dt )  measured in the air, on the ground, or in boreholes, and from an arbitrary grounded or ungrounded source. Three synthetic examples illustrate the basic functionality of the algorithm, and a result from a field example shows applicability in a larger-scale field example.     

TEM study of the geoelectrical structure and groundwater salinity of the Nahal Hever sinkhole site, Dead Sea shore, Israel

Since the 1990s a large number of sinkholes have appeared in the Dead Sea (DS) coastal area. Sinkhole development was triggered by the lowering of the DS level. In the literature the relationship between the sinkholes and the DS level is explained by intrusion of relatively fresh water into the aquifer thereby dramatically accelerating the salt dissolution with creation of subsurface caverns, which in turn cause sinkholes. The main goal of our project was detection and localization of relatively fresh groundwater. During our study we used the transient electromagnetic method (TEM) to measure the electrical resistivity of the subsurface. As a test site we selected Nahal Hever South which is typical for the DS coast. Our results show that resistivity of the shallow subsurface reflects its vertical and lateral structure, e.g., its main hydrogeological elements explain the inter-relations between geology, hydrogeology, and sinkholes. The TEM method has allowed detailed differentiation of layers (clay, salt, etc.) in the subsurface based on their bulk resistivity. The 10 m-thick salt layer composed of idiomorphic crystals of halite deposited during the earlier Holocene period was extrapolated from borehole HS-2 through the study area. It was found that in Nahal Hever the typical value of the bulk resistivity of clay saturated with the DS brine varies between 0.2 and 0.3 Ωm, whereas saturated gravel and sandy sediments are characterized by resistivity between 0.4 and 0.6 Ωm. The high water salinity of the aquifer (enveloping the salt layer) expressed in terms of resistivity is also an important characterization of the sinkhole development mechanism. The electrical resistivity of the aquifer in the vicinity of the salt unit and its western border did not exceed 1 Ωm (in most cases aquifer resistivity was 0.2-0.6 Ωm) proving that, in accordance with existing criteria, the pores of the alluvial sediments are filled with highly mineralized DS brine. However, we suggest that the criterion of the aquifer resistivity responsible for the salt dissolution should be decreased from 1 Ωm to 0.6 Ωm corresponding to the chloride concentration of approximately 100 g/l (the chloride saturation condition reaches 224 g/l in the northern DS basin and 280 g/l in the southern one).Based on TEM results we can reliably conclude that in 2005, when most of sinkholes had appeared at the surface, salt was located within a very low resistivity environment inside sediments saturated with DS brine. Intrusion of relatively fresh groundwater into the aquifer through the 600 × 600 m²area affected by sinkholes has not been observed.     

Numerical Study of A Generic Tidal Turbine Using BEM Optimization Methods

Three blade-geometry optimization models derived along with assumptions from the blade element momentum(BEM) approach are studied by using a steady BEM code to improve a small horizontal-axis rotor of three blades that has been previously used in experiments. The base rotor blade has linear-radially varying chord length and pitch angle, while the other three models noted as Burton, Implicit and Hansen due to their references and characteristics yield blades of non-linearly varying chord length and pitch angle. The aim is to compare these rapid models and study how assumptions embedded in them affect performance and induction factors. It is found that the model that has the least assumptions(Hansen) and which considers the blade-profile drag in its optimization procedure yields the highest power coefficient, C_P, at the optimal tip speed ratio(TSR), about 7% higher than the base one and also higher C_P at high TSR. It produces an axial induction factor distribution along the blade that is closest to the 1 D optimal value of 1/3. All optimized tangential induction-factor distributions along the blade closely vary as inverse to the square of the radial distance, while being mildly higher than the base distribution. It shows that sufficient swirl is necessary to increase power but at a level causing not too much energy loss in unnecessary swirl of the wake. At high TSR, all optimized rotors adversely produce higher thrust than the base one, but the one with most embedded assumptions(Burton) produces the highest thrust. Details of all three optimization models are given along with the distributions of the power, thrust, blade hydrodynamic efficiency and induction factors.     

An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain

In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique.     

A GCV based method for nonlinear ill-posed problems

This paper discusses the inversion of nonlinear ill-posed problems. Such problems are solved through regularization and iteration and a major computational problem arises because the regularization parameter is not known a priori. In this paper we show that the regularization should be made up of two parts. A global regularization parameter is required to deal with the measurement noise, and a local regularization is needed to deal with the nonlinearity. We suggest the generalized cross validation (GCV) as a method to estimate the global regularization parameter and the damped Gauss-Newton to impose local regularization. Our algorithm is tested on the magnetotelluric problem. In the second part of this paper we develop a methodology to implement our algorithm on large-scale problems. We show that hybrid regularization methods can successfully estimate the global regularization parameter. Our algorithm is tested on a large gravimetric problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

Model Fusion and Joint Inversion

Inverse problems are inherently non-unique, and regularization is needed to obtain stable and reasonable solutions. The regularization adds information to the problem and determines which solution, out of the infinitely many, is obtained. In this paper, we review and discuss the case when a priori information exists in the form of either known structure or in the form of another inverse problem for a different property. The challenge is to include such information in the inversion process. To use existing known structure, we review the concept of model fusion, where we build a regularization functional that fuses the inverted model to a known one. The fusion is achieved by four different techniques. Joint inversion of two data sets is achieved by using iterative data fusion. The paper discusses four different methods for joint inversion. We discuss the use of correspondence maps or the petrophysics of the rocks, as well as structure. In particular, we suggest to further stabilize the well-known gradient cross product and suggest a new technique, Joint Total Variation, to solve the problem. The Joint Total Variation is a convex functional for joint inversion and, as such, has favorable optimization properties. We experiment with the techniques on the DC resistivity problem and the borehole tomography and show how model fusion and joint inversion can significantly improve over existing techniques.     

Scandium-bearing carbonates of the Tarr albitite complex,southeast Sinai

Unusually high Sc-concentrations (30 to 150 ppm) were found in the carbonate minerals (dolomite, breunnerite, calcite) of the Tarr albitite complex. Scandium is probably part of the carbonate crystal structure substituting for (Mg, Fe²⁺). Scandium is considered to have originated from a source of mantle-affinity, thus supporting an earlier suggestion that the Tarr albitite complex is related to an ophiolite at depth.     

Hydro‐frac monitoring using ground time‐domain electromagnetics

As motivation for considering new electromagnetic techniques for hydraulic fracture monitoring, we develop a simple financial model for the net present value offered by geophysical characterization to reduce the error in stimulated reservoir volume calculations. This model shows that even a 5% improvement in stimulated reservoir volume for a 1 billion barrel (bbl.) field results in over 1 billion U.S. dollars (US$) in net present value over 24 years for US$100/bbl. oil and US$0.5 billion for US$50/bbl. oil. The application of conductivity upscaling, often used in electromagnetic modeling to reduce mesh size and thus simulation runtimes, is shown to be inaccurate for the high electrical contrasts needed to represent steel‐cased wells in the earth. Fine‐scale finite‐difference modeling with 12.22‐mm cells to capture the steel casing and fractures shows that the steel casing provides a direct current pathway to a created fracture that significantly enhances the response compared with neglecting the steel casing. We consider conductively enhanced proppant, such as coke‐breeze‐coated sand, and a highly saline brine solution to produce electrically conductive fractures. For a relatively small frac job at a depth of 3 km, involving 5,000 bbl. of slurry and a source midpoint to receiver separation of 50 m, the models show that the conductively enhanced proppant produces a 15% increase in the electric field strength (in‐line with the transmitter) in a 10‐Ωm background. In a 100‐Ωm background, the response due to the proppant increases to 213%. Replacing the conductive proppant by brine with a concentration of 100,000‐ppm NaCl, the field strength is increased by 23% in the 100‐Ωm background and by 2.3% in the 10‐Ωm background. All but the 100,000‐ppm NaCl brine in a 10‐Ωm background produce calculated fracture‐induced electric field increases that are significantly above 2%, a value that has been demonstrated to be observable in field measurements.     

Signal detectability of marine electromagnetic methods in the exploration of resistive targets

We compare selected marine electromagnetic methods for sensitivity to the presence of relatively thin resistive targets (e.g., hydrocarbons, gas hydrates, fresh groundwater, etc.). The study includes the conventional controlled‐source electromagnetic method, the recently introduced transient electromagnetic prospecting with vertical electric lines method, and the novel marine circular electric dipole method, which is still in the stage of theoretical development. The comparison is based on general physical considerations, analytical (mainly asymptotic) analysis, and rigorous one‐dimensional and multidimensional forward modelling. It is shown that transient electromagnetic prospecting with vertical electric lines and marine circular electric dipole methods represent an alternative to the conventional controlled‐source electromagnetic method at shallow sea, where the latter becomes less efficient due to the air‐wave phenomenon. Since both former methods are essentially short‐offset time‐domain techniques, they exhibit a much better lateral resolution than the controlled‐source electromagnetic method in both shallow sea and deep sea. The greatest shortcoming of the transient electromagnetic prospecting with vertical electric lines and marine circular electric dipole methods comes from the difficulties in accurately assembling the transmitter antenna within the marine environment. This makes these methods significantly less practical than the controlled‐source electromagnetic method. Consequently, the controlled‐source electromagnetic method remains the leading marine electromagnetic technique in the exploration of large resistive targets in deep sea. However, exploring laterally small targets in deep sea and both small and large targets in shallow sea might require the use of the less practical transient electromagnetic prospecting with vertical electric lines and/or marine circular electric dipole method as a desirable alternative to the controlled‐source electromagnetic method.