Direct Current Resistivity Modeling Research Articles

A new method of 3D direct current resistivity modelling using a long electrode source for forward probing in tunnels

ABSTRACTThe direct current resistivity method has been widely used for the prediction of water‐bearing structures in subway tunnels and mine tunnels. The traditional direct current resistivity uses point electrode sources in tunnels to excite the electric field; potential electrodes are arranged on the back of the tunnel face to measure the position of the anomaly using the apparent resistivity sounding curve. Due to the dimension limitations of the tunnel cavity, the maximum distance of potential electrodes is often restricted. Thus, the exploration depth often cannot cover the target appropriately. Additionally, owing to the inaccessibility of point electrode sources on the tunnel face, point electrode sources can only be set near the tunnel face, so the electric signal excited by point electrode sources cannot be coupled with the water‐bearing anomaly at a large depth. As such the traditional point electrode method for forward probing in tunnels has a shortcoming of small detection depth. To overcome this problem by increasing the detection depth, a horizontal drilling copper casing is first used as a new long electrode source array, which (as far as the authors are aware) has never been reported in forward probing in tunnels. We developed a new method of 3D direct current resistivity modelling with a long electrode source for forward probing in tunnels using the finite‐element method. The long electrode current source was used to replace the conventional point electrode current source, and the observed potential electrodes were respectively arranged in the tunnel floor, tunnel face and horizontal drilling. Theoretical formulas for the long electrode source for the whole‐space direct current method are first applied in forward probing in tunnels. An analytical test for a whole‐space model with the long electrode source has been used to validate the accuracy of the 3D algorithm. Three synthetic geological models for the long electrode source were used to examine the applicability by investigating the detection capability to predict the anomaly. The observed signal of apparent resistivity excited by the long electrode source is stronger than that of the point electrode source because the long electrode source is closer to the anomaly. This new method using the long electrode source greatly improves the resolution of the anomaly, which is of great significance for the safety of the tunnel construction.

3-D DC resistivity modeling and inversion using multi-resolution framework

We discuss the implementation of multi-resolution framework to 3-D Direct Current (DC) problem. Commonly used staggered (SG) grid fixes the horizontal grid resolution for all depths. Thus, employin ...

2.5D electric resistivity forward modeling with element-free Galerkin method

An element-free Galerkin method (EFGM) is proposed to perform the direct current (DC) resistivity modeling problems. The advantage of this method is the absence of elements, which makes nodes free from the elemental restraint. In this method, the moving least square (MLS) approximation is employed to construct the shape function. And the cubic spline function with high continuity is used as a weight function. Then, the boundary value problem and the corresponding variation problem of DC resistivity modeling problems are discussed. The discrete equations of two and a half dimensional (2.5D) DC resistivity modeling problems are obtained by using the Galerkin weak form formulation. Thereby, a 1D multilayered geoelectric model is adopted to verify the validity of the proposed EFGM. And the further tests of complicated geoelectric models illustrate that the correctness, good adaptability and flexibility of EFGM compared with the finite element method (FEM). The EFGM can conveniently simulate the undulating terrain and improve the simulation accuracy by refining nodes in local domain. In addition, the discretization of test models shows that EFGM is easy to handle the complicated geometrical geoelectric models by using arbitrary and irregular nodes. All these tests demonstrate that the EFGM is suitable to perform forward modeling for complicated geoelectric models.

A hybrid finite difference–finite element method to incorporate topography for 2D direct current (DC) resistivity modeling

Finite difference (FD) and finite element (FE) approximations are two commonly used methods for solving Poisson's equation from a known electrical resistivity model. FD discretizes the model with rectangular blocks, while FE discretizes it with right triangular elements. The advantage of FD is that it requires less computational time and memory storage, while the advantage of FE is its ability to handle topography. Here we introduce a hybrid FD–FE algorithm which requires less computational resources than a typical FE algorithm but can still handle topography. The model is first discretized as FD rectangular blocks. Some of these FD rectangular blocks beneath topographical features are replaced with FE right triangular elements. A system of equations for the hybrid method is then formed according to the types of discretization used. The system is a linear combination of the FD and FE equations. All three methods are applied to several models with and without topography. Comparisons in terms of computational time, memory and accuracy show that, as expected, FD is a better choice when modeling without topography, whereas FE is more practical when modeling with topography. However, our developed hybrid FD–FE method is efficient in both circumstances. Without topography, it behaves exactly like FD. With topography, it is as accurate as FE but its speed and memory usage are slightly larger than FD. In addition, we describe a scheme for automatic mesh generation. This scheme has led to another variant method, a mixed grid hybrid method. Our hybrid scheme and the automatic grid generation can be used for slopes of up to 75°. This hybrid method can be further extended and implemented for 3D DC resistivity modeling for more efficient computation.

3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method

A new adaptive finite-element method for solving 3D direct-current resistivity modeling problems is presented. The method begins with an initial coarse mesh, which is then adaptively refined wherever a gradient-recovery-based a posteriori error estimator indicates that refinement is necessary. Then the problem is solved again on the new grid. The alternating solution and refinement steps continue until a given error criterion is satisfied. The method is demonstrated on two synthetic resistivity models with known analytical solutions, so the errors can be quantified. The applicability of the numerical method is illustrated on a 2D homogeneous model with a topographic valley. Numerical results show that this method is efficient and accurate for geometrically complex situations.

2.5-D/3-D resistivity modelling in anisotropic media using Gaussian quadrature grids

SUMMARY We present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the ‘Gaussian quadrature grid’ (GQG) method, cooperatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method but does not employ the constant elements or require the mesh generator to match the Earth's surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. We show it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system that can be solved by an iterative or matrix inversion method. Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissa number) and the subdomain size. The higher the order or the smaller the subdomain size that is employed, the more accurate are the results obtained. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography, are presented and found to yield results comparable to finite element solutions involving a dense mesh.

DC resistivity modelling with a Gaussian quadrature grid

We present a new numerical scheme for 2D/3D direct current resistivity modelling. This method co-operatively combines the solution of the variational principle of the partial differential equation, Gaussian global quadrature abscissae and local cardinal functions so that it has the main advantages of the finite element method and the spectral method. The formulation shows that the method is close to the spectral element method, but it does not require the element mesh or the element integrations, and it makes it much easier to deal with geological models having a 2D/3D complex topography than the traditional numerical methods. It can achieve a similar convergence rate to the spectral element method. We show it transforms the 2D/3D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.

Finite element resistivity modelling for three-dimensional structures with arbitrary anisotropy

We present a three-dimensional finite element algorithm for direct current resistivity modelling. The standard Fortran code allows for nearly arbitrary conductivity structures including general anisotropy. The problem is formulated in terms of secondary potentials where mixed boundary conditions are incorporated. Also in case of anisotropy, this type of boundary condition is superior to the Dirichlet type. We have verified the finite element method using an anisotropic two-layered earth, whose analytical solutions are available. For this simple model, the algorithm achieves high accuracy. The relative deviation between numerical and analytical solution is less than 1.2%. Due to the lack of further analytical references, the responses of three representative model types serve as a cross-check for plausibility and prove the operativeness of the code.

Three-dimensional DC resistivity forward modelling using finite elements in comparison with finite-difference solutions

SUMMARY A 3-D finite-element scheme for direct current resistivity modelling is presented. The singularity is removed by formulating the problem in terms of the secondary potential, which improves the accuracy considerably. The resulting system of linear equations is solved using the conjugate gradient method. The incomplete Cholesky preconditioner with a scaled matrix has been proved to be faster than the symmetric successive overrelaxation preconditioner. A compact storage scheme fully utilizes the sparsity and symmetry of the system matrix. The finite-element (FE) and a previously developed finite-difference (FD) scheme are compared in detail. Generally, both schemes show good agreement, the relative error in apparent resistivity for a vertical dike model presented in this paper is less than 0.5 per cent overall. The FD scheme produces larger errors near the conductivity contrast, whereas the FE scheme requires approximately 3.4 times as much storage as the FD scheme and is less robust with respect to coarse grids. As an improvement to the forward modelling scheme, a modified singularity removal technique is presented. A horizontally layered earth or a vertical contact is regarded as the normal structure, the solution of which is the primary potential. The effect of this technique is demonstrated by two examples: a cube in two-layered earth and a cube near a vertical contact.

Finite element three-dimensional direct current resistivity modelling: accuracy and efficiency considerations

The finite element method is a powerful tool for 3 D DC resistivity modelling and inversion. The solution accuracy and computational efficiency are critical factors in using the method in 3 D resistivity imaging. This paper investigates the solution accuracy and the computational efficiency of two common element type schemes: trilinear interpolation within a regular 8 node solid parallelepiped, and linear interpolations within six tetrahedral bricks within the same 8 node solid block. Four iterative solvers based on the pre conditioned conjugate gradient method (SCG, TRIDCG, SORCG and ICCG), and one elimination solver called the banded Choleski factorization are employed for the solutions. The comparisons of the element schemes and solvers were made by means of numerical experiments using three synthetic models. The results show that the tetrahedron element scheme is far superior to the parallelepiped element scheme, both in accuracy and computational efficiency. The tetrahedron element scheme may save 43 per cent storage for an iterative solver, and achieve an accuracy of the maximum relative error of < 1 per cent with an appropriate element size. The two iterative solvers, SORCG and ICCG, are suitable options for 3 D resistivity computations on a PC, and both perform comparably in terms of convergence speed in the two element schemes. ICCG achieves the best convergence rate, but nearly doubles the total storage size of the computation. Simple programming codes for the two iterative solvers are presented. We also show that a fine grid, which doubles the density of a coarse grid, will require at least 27 = 128 times as much computing time when using the banded Choleski factorization. Such an increase, especially for 3 D resistivity inversion, should be compared with SORCG and ICCG solvers in order to find the computationally most efficient method when dealing with a large number of electrodes.