Inversion problem in regional electromagnetic induction studies

For completeness of the theory proposed by Oni (1972), the inversion problem is examined. It is shown that parameters which constrain distinct solutions of the inverse problem can be determined in the application of the theory. The fundamental basis of the inversion and the strategy to be adopted are discussed.     

Generalised relative and cumulative response functions for electromagnetic induction conductivity meters operating at low induction numbers

Relative and cumulative analytical response functions have been widely used as a powerful tool for forward modelling and interpretation of measurements obtained by electromagnetic induction conductivity meters operating at low induction numbers for one‐dimensional layered earth models. These well‐known functions were derived and should be used for the instruments laid on the surface of the earth. In this paper, we extended the response functions and obtained new generalised analytical expressions, which can be used for instruments carried at any height from the surface. The proposed new equations were compared with numerically constructed functions, obtained using the full solution of Maxwell's equations, and proved to be in very good agreement at low induction numbers. Quantitative analyses of the behaviour of the relative response and the depth of investigation of electromagnetic induction instruments, when raised from the ground, could also be done using the generalised functions.     

Low induction number, ground conductivity meters: A correction procedure in the absence of magnetic effects

Ground conductivity meters, comprising a variety of coil–coil configurations, are intended to operate within the limits provided by a low induction number (LIN), electromagnetic condition. They are now routinely used across a wide range of application areas and the measured apparent conductivity data may be spatially assembled and examined/correlated alongside information obtained from many other earth science, environmental, soil and land use assessments. The theoretical behaviour of the common systems is examined in relation to both the prevailing level of subsurface conductivity and the instrument elevation. It is demonstrated that, given the inherent high level of accuracy of modern instruments, the prevailing LIN condition may require operation in environments restricted to very low (< 12 mS/m) conductivities. Beyond this limit, non-linear departures from the apparent conductivity that would be associated with a LIN condition occur and are a function of the coil configuration, the instrument height and the prevailing conductivity. Using both theory and experimental data, it is demonstrated that this has the potential to provide biased and spatially distorted measurements. A simple correction procedure that can be applied to the measured data obtained from any of the LIN instruments is developed. The correction procedure would, in the limit of a uniform subsurface, return the same (correct) conductivity, irrespective of the ground conductivity meter used, the prevailing conductivity or the measurement height.     

Evaluation of streambed hydraulic conductivity heterogeneity in an urban watershed

It has long been understood that streambed hydraulic conductivity plays an important role in surface-subsurface solute exchange. Using a portable falling head permeameter in situ, we estimated the horizontal hydraulic conductivity, K, of the near-surface streambed sediments at a total of 85 locations encompassing two depth intervals: 7.5–10 and 10–12.5 cm. The measurements were conducted in an 80 m reach of Indian Creek, a small urban stream in Philadelphia, PA, USA. We found that the ln K data within each sediment layer were Gaussian, but the combined data set was not. The results indicated that while the mean hydraulic conductivity decreased with depth, horizontal heterogeneity (e.g. the variance) increased with depth. This strong contrast between layers suggests that they should be treated as separated entities in modeling studies. Variogram analyses across the stream suggested symmetry with respect to the thalweg in the upper layer and fractality in the lower layer. The variograms along the streams suggested that the K data are random.     

Non-linear inversion of controlled source multi-receiver electromagnetic induction data for unexploded ordnance using a continuation method

This work adopts a continuation approach, based on path tracking in model space, to solve the non-linear least-squares problem for discrimination of unexploded ordnance (UXO) using multi-receiver electromagnetic induction (EMI) data. The forward model corresponds to a stretched-exponential decay of eddy currents induced in a magnetic spheroid. We formulate an over-determined, or under-parameterized, inverse problem. An example using synthetic multi-receiver EMI responses illustrates the efficiency of the method. The fast inversion of actual field multi-receiver EMI responses of inert, buried ordnances is also shown. Software based on the continuation method could be installed within a multi-receiver EMI sensor and used for near-real-time UXO decision-making purposes without the need for a highly-trained operator.     

Finite element computation of shape factors for Kirkham's piezometer tube method for hydraulic conductivity of saturated soil

The Kirkham piezometer tube method of measuring hydraulic conductivity of saturated soil uses a numerical constant (shape factor) to account for the relative geometrical parameters of the cavity and the soil domain. Published shape factors have mostly been determined using an electrical analogue. In this paper the application of the finite element method to shape factor determination is considered. Convergence is studied and methods of extrapolation evaluated. A determination is made of the size of finite region required to represent adequately the radially unbounded domain assumed for the piezometer method. Computed shape factors were found to be in good agreement with analogue data.     

Substitute conductors for electromagnetic response estimates

Various concepts exist to define substitute conductors for empirical response estimates at singular frequencies: Chapman's shell-core model, the Cagniard-Tikhonov apparent resistivity, the Niblett-Bostick and Molochnov transformation, thep ^* —z ^* transformation. They are all interrelated and assign comparable resistivities to the substitute conductor at a given frequency. Applications to synthetic response data of plane and spherical conductors show under which conditions these substitutions come closest to the model and which influence of source dimensions and Earth's sphericity can be expected.p ^* —z ^* transformed global response data forS andDst variations demonstrate how substitute conductors may serve as useful guides in inverse procedures.     

用地质雷达数据资料反演二维地下介质的方法

从二维麦克斯韦方程组出发推导出反演介电常数和电导率等二维介质物性参数的反演公式.反演的步骤是: 建立初始猜测模型,利用电磁波时间域有限差分法模拟正演数据,用正演数据与观测数据之间的数据残差建立目标函数,通过引入一个由麦克斯韦方程计算的伴随场,将目标函数对介质参数的导数表示成显式形式,应用最优化理论得出对初始猜测模型的修改,用共轭梯度法迭代,最终得到反演结果.用合成数据反演具有粗糙地表的非导电介质的介电常数,用实验数据同时反演介电常数和电导率,并比较了麦克斯韦方程反演结果与声波方程反演结果、波动方程偏移剖面的差异.     

Appreciation of spherically symmetric models of electrical conductivity

A procedure is suggested of a more effective and faster computation of the impedance, the transfer function and amplitudes of the induced field in a spherically symmetric model of the electrical conductivity. The existing induction data have been supplemented by about 80 new values derived from the analysis of daily means. The fit of the existing 1-D models of the electrical conductivity of the mantle to the set of induction data is investigated. The characteristic equation for the free electromagnetic oscillations of a radially inhomogeneous Earth is derived and its possible importance in solving the inverse problem of electric conductivity is pointed out.     

A concise parameterization of the hydraulic conductivity of unsaturated soils

The parameter n in the well-known expression for hydraulic conductivity K=K₀S_eⁿ (where K₀ is its value at satiation and S_e the effective saturation) is determined as a function of the exponent in the power form of the soil–water retention relationship. The result is validated with an extensive experimental database comprising some 43 soils, collected by Mualem.     

基于电流密度连续性条件的直流电阻率各向异性问题自适应有限元模拟

直流电阻率法被广泛应用在工程和环境及水文地球物理、野外采矿、地热探测等领域.地下岩石常具有层理面和裂缝等具有方向依赖性的结构,岩石电导率常常呈现各向异性特征,因此研究复杂直流电阻率各向异性问题的高精度正演算法具有迫切的理论和学术需求.本文利用面向目标的自适应有限元算法和非结构化网格相结合的方式,解决了带地形任意复杂直流电阻率各向异性问题的高精度正演这一难题.有别于前人的研究成果,本文提出了一种特别的二次虚拟场算法实现带源的任意起伏地形问题模拟;另外,本文第一次基于电流密度连续性条件构建适合直流电阻率各向异性问题的后验误差估计算法,有效地驱动面向目标有限元网格自适应加密过程.最后,通过三组电阻率各向异性模型验证本文提出算法的正确性和适应性,测试结果表明：对于任意复杂直流电阻率各向异性问题,本文提出的算法具有精度高、适应性强等特点;另外,我们还发现电流密度连续性条件可用于设计直流电阻率问题的有效后验误差估计算法.

     

Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media

The behavior of the mean equivalent hydraulic conductivity normal and parallel to stratification (K₁, and K₂, respectively) is studied here through Monte Carlo simulations of three-dimensional, steady-state flow in statistically anisotropic, bounded, and heterogeneous media. For water flow normal to stratification in strongly heterogeneous porous media (²_Y=3) the value of K₁ is not unique; it ranges from an arithmetic to a geometric, and finally, to a harmonic mean behavior depending on field dimensions, and medium anisotropy. For a fixed anisotropy ratio and variance of Y = ln K, the larger the distance, in the direction perpendicular to stratification, over which water flow takes place, the faster the rate at which, K_H, behavior is approached. However, even for large anisotropy ratios, harmonic mean behavior appears to be a good approximation only for aquifer thickness L₁ that is large enough to allow stratified flow to occur. For small aquifer thickness (L₁/₁<8, where ₁ is the integral scale normal to stratification) the limiting behavior, for large anisotropy ratios, appears to be, instead, that of two-dimensional flow, i.e., water flows primarily parallel to the planes of stratification. When the aquifer thickness is very small compared to the horizontal dimensions (and with relative similar integral scales in the three directions) a behavior resembling arithmetic mean conditions is exhibited, i.e., water flow takes place through heterogeneous, vertical, soil volumes. The geostatistical expressions of Desbarats (1992a) for upscaling hydraulic conductivity values were utilized and closed form empirical relations were developed for the main components of the upscaled hydraulic conductivity tensor.     

用相位感应测井数据反演地层电阻率和介电常数

基于相位感应测井的两个接收线圈获取的总场相位差和幅度比曲线,在二维轴对称非均匀介质中同时反演地层电阻率和介电常数. 通过相位差和幅度比测井曲线提取地层纵向边界位置的初始值,利用测井响应方程和变分原理建立反演方程,然后采用共轭梯度(CG)法进行求解. 对多层地层的电阻率、介电常数和纵向边界位置进行整体反演,模拟计算结果表明了方法的可行性.     

Application of space analysis of electromagnetic fields to investigation of the geoelectrical structure of the Earth

Lateral composition inhomogeneities of the Earth's deep geoelectric structure require special consideration for any conductivity evaluation of a region. This paper presents a review of some theoretical techniques for determining both the vertical and horizontal conductivity profiles of a region using a spatial distribution of observed electromagnetic fields at the Earth's surface. Effects of shallow positioned anomalies upon a deep conductivity determination are also considered. An application of the procedure is illustrated by a conductivity study in the Soviet Carpathians.     

Effect of exchangeable Mg on saturated hydraulic conductivity, disaggregation and clay dispersion of disturbed soils

Different opinions exist regarding the specific effect of Mg on soil physical and chemical properties. We hypothesized that Mg²⁺, compared with Ca²⁺, reduces saturated hydraulic conductivity (K_s) via promoting clay swelling, disaggregation, and clay dispersion. Two soils (mixed, mesic Typic Hapludalfs) in packed soil columns were leached with either Ca- or Mg-containing solutions at the successive concentrations of 250, 10, 2, 0.5, and 0 mM. Critical flocculation concentration (CFC) in either Ca or Mg systems was determined with flocculation series tests. Aggregate stability and mean weight diameter (MWD) were assessed by wet-sieving. The CFCs were higher in Mg than in Ca for both soils, indicating that Mg is more dispersive than Ca. The MWDs measured using 1–2 mm aggregates of both soils were significantly larger for Ca-soils than for Mg-soils (P=0.05). The K_sr (normalized with initial K_s) started to decline at higher concentrations for Mg than for Ca, and the reduction was much greater in Mg than in Ca above 0.5 mM. The K_sr and percent transmittance (inversely related to turbidity) of leachate at a given eluted pore volume following ‘steady state’ were higher in Ca than in Mg for both soils (P=0.1), indicating lower permeability and more clay dispersion with the Mg treatment. Swelling and disaggregation, which reduced large pores, appeared to be the dominant process causing the rapid initial decline of K_sr. Clay dispersion and subsequent pore plugging became progressively important when electrolyte concentration was reduced to below CFCs.     

金属套管对多分量电磁感应测井信号的影响

研究金属套管对低频电磁场各分量透射信号和反射信号的影响规律对采用过套管电磁感应测井技术监测生产过程中的流体移动具有重要价值.采用柱状成层各向异性介质中的并矢Green函数模拟当发射源在套管内、接收器分别在套管内和套管外时多分量电磁感应的响应,并系统分析了金属套管对透射磁场和反射磁场各主分量的影响规律.计算结果表明,无论是透射场还是反射场,由于所采用的频率较低,金属套管对每个分量信号的影响可以与地层对该分量信号的影响分开.在低频段,磁性金属套管对磁场各分量仍存在稳定的反射,但不同分量被反射的程度不同,受套管磁屏蔽作用大的分量（如ρρ分量）被反射的程度高于受套管磁屏蔽作用小的分量（如zz分量）.随着频率的升高,反射磁场的强度逐渐增大并最终达到饱和,信号被全反射.磁性金属套管的存在导致透射磁场各分量出现不同程度的幅度衰减和相位延迟,受套管磁屏蔽作用大的分量的幅度衰减程度大.在固定套管参数和频率的情况下,磁性套管对透射磁场不同分量的影响程度虽然不同但是恒定,其所导致的各分量的幅度衰减和相位延迟与线圈系位置和地层电参数无关.在低频段,套管外透射场的各分量处于准静态,其幅度和相位均不随频率而改变,受套管磁屏蔽作用大的分量的幅度随套管磁导率和厚度的不同而存在明显变化.当频率超过某一临界值后,套管的影响逐渐从以静态屏蔽为主转到以趋肤效应为主,导致透射场幅度的衰减和相位的延迟,套管的厚度越大、套管的电导率和相对磁导率值越大,所对应的频率临界值越小.

     

三维MCSEM利用电磁场分解消除空气波效应

对于浅水域可控源电磁勘探,在资料处理和解释过程中需要消除空气波影响,电磁场上下场分解方法是消除这种影响的一种有效手段.为了快速有效实现电磁场分解,假定电磁场沿水平方向变化缓慢,基于麦克斯韦方程组,本文推导了一种仅利用电磁场水平梯度实现电磁场分解新方法.数值试验表明利用这种分解方法可以得到稳定可靠的分解结果,同时论证了这种方法是一种海洋可控源电磁法（MCSEM）数据快速定性解释手段.

     

Influence of bedding orientation on the anisotropy of hydraulic conductivity in a well-sorted fluvial sediment

The paper describes a permeameter test method for determination of the hydraulic conductivity(AT) along multi-directions in fluvial sediments with cross beddings.Unlike existing in-situ permeameter methods that determine hydraulic conductivity for submerged streambeds,our method was intended to measure hydraulic conductivity of exposed streambeds or fluvial sediments.The method was applied to the Wei River,Shaanxi Province,Central China for characterization of the anisotropy of K in a well-sorted fluvial sediment.The results illustrated that even in well-sorted sediments,cross-bedding and sediment fabrication(or texture) can lead to varied K values along different measurement directions.The K value was the largest along the dip direction(or the major direction) that is parallel to the orientation of cross bedding and the smallest in the direction perpendicular to the bedding(or the minor direction). The K value in a given direction between the major and minor direction often fell in the range bounded by the K values in the major and minor directions.The anisotropy ratio of K(the ratio of K value between the major and minor directions) in two trenches for this well-sorted fluvial sediment was up to 1.14 to 1.23,respectively.Our results also demonstrated that even for well-sorted sediments,the K values between two sampling points only about 10 cm apart can differ.It is clear that the K distribution strongly correlates to the bedding orientation.     

A fast integral equation solver for 3D induction well logging in formations with large conductivity contrasts

Simulation of induction logging responses in formations with large conductivity contrasts is an important but challenging problem due to the singularity of a linear system caused by large contrasts. Also, three‐dimensional (3D) analysis of complex geophysical structures usually encounters high computational demands. In this paper, a pre‐corrected fast Fourier transform (pFFT)‐accelerated integral equation method is applied to overcome these difficulties. In the approach, the entire formation is included in the solution domain. The volume integral equation is set up in the region based on the fact that the total field is the summation of the excitation field and the secondary field. The emitted field by the transmitter coil (treated as a magnetic dipole) is regarded as the excitation of the system. Then the method of moments (MoM) is used to solve the integral equation. To reduce the high computational requirements of the MoM, the pFFT method is used to speed up the solution of the matrix equation and reduce the memory requirement as well. The resultant method is capable of computing induction logging problems involving large and complex formations. For problems with high conductivity contrasts, the solution of the matrix equation usually converges very slow or even fails to converge due to the large condition number of the coefficient matrix. To overcome this difficulty, an incomplete LU pre‐conditioner is used to significantly speed up the convergence of the matrix equation, thus further reducing the computation time. Numerical results show that the present method is efficient and flexible for 3D simulation of induction logging and is specifically superior for problems with high conductivity contrasts.     

Upscaling hydraulic conductivity based on the topology of the sub-scale structure

The hydraulic conductivity of heterogeneous porous media depends on the distribution function and the geometry of local conductivities at the smaller scale. There are various approaches to estimate the effective conductivity K_eff at the larger scale based on information about the small scale heterogeneity. A critical geometric property in this ‘upscaling’ procedure is the spatial connectivity of the small-scale conductivities. We present an approach based on the Euler-number to quantify the topological properties of heterogeneous conductivity fields, and we derive two key parameters which are used to estimate K_eff. The required coefficients for the upscaling formula are obtained by regression based on numerical simulations of various heterogeneous fields. They are found to be generally valid for various different isotropic structures. The effective unsaturated conductivity function K_eff (ψ_m) could be predicted satisfactorily. We compare our approach with an alternative based on percolation theory and critical path analysis which yield the same type of topological parameters. An advantage of using the Euler-number in comparison to percolation theory is the fact that it can be obtained from local measurements without the need to analyze the entire structure. We found that for the heterogeneous field used in this study both methods are equivalent.