Three-dimensional Forward Modeling Research Articles

3-D forward modeling of shale gas fracturing dynamic monitoring using the borehole-to-ground transient electromagnetic method

The borehole-ground transient electromagnetic method enhances the detection range and resolution by placing the transmitter electrode within the borehole, allowing for close-proximity excitation of subsurface targets and compensating for the shortcomings of traditional ground-based methods. Utilizing an unstructured vector finite element method combined with a second-order backward Euler variable time-stepping difference scheme and the MUMPS solver, we have achieved three-dimensional forward modeling simulation of the transient electromagnetic field for a borehole-ground electric source. Based on the validation of its accuracy, we analyzed the effectiveness of this method for dynamic monitoring of shale gas reservoir fracturing. Through forward modeling, we examined the characteristics of the radial electric field response during shale gas reservoir fracturing monitoring and evaluated the effectiveness of the borehole-ground TEM method in dynamic monitoring of shale gas reservoirs. The research results indicate that this method can meet the requirements for reservoir fracturing dynamic monitoring and has a promising application prospect.

Three-dimensional simulation and characterization of airborne transient electromagnetic response in rough media

In fractal, porous, and cranny rough media, the diffusion of the electromagnetic (EM) field differs from that in piecewise smooth media. This phenomenon is known as anomalous diffusion. To study the influence of anomalous diffusion on EM signals, we use the fractional time derivative and derive a governing equation describing the subdiffusion process of electric fields in rough media. Three-dimensional forward modeling of the time-domain airborne EM is performed using the finite-difference method and frequency-time transformation. We verify the accuracy of our forward-modeling method by comparing it with semianalytical solutions for a layered earth model. The numerical results indicate that as the conductivity of rough media decreases with time, the amplitude of the EM field decreases during the early periods but increases during subsequent periods compared with nonrough media. The EM field does not diffuse in the classical mode of the “smoke ring,” but instead demonstrates a “trailing pattern,” meaning that it maintains its maximum value at positions close to the source and diffuses outward and attenuates until it vanishes.

Fictitious Point Technique Based on Finite-Difference Method for 2.5D Direct-Current Resistivity Forward Problem

With the widespread application of the direct-current resistivity method, searching for accurate and fast-forward algorithms has become the focus of research for geophysicists and engineers. Three-dimensional forward modeling can be the best way to identify geo-electrical anomalies but are hampered by computational limitations because of the large amount of data. A practical compromise, or even alternative, is represented by 2.5D modeling characterized using a 3D source in a 2D medium. Thus, we develop a 2.5D direct-current resistivity forward modeling algorithm. The algorithm incorporates the finite-difference approximation and fictitious point technique that can improve the efficiency and accuracy of numerical simulation. Firstly, from the boundary value problem of the electric potential generated by the point source, the discrete expressions of the governing equation are derived from the finite-difference approach. The numerical solutions of the discrete electric potential are calculated after the approximate treatment of the boundary conditions with a finite-difference method based on a fictitious point scheme. Secondly, through the simulation of a homogeneous half-space model and a one-dimensional model, and compared with the analytical results, the correctness and stability of the finite-difference forward algorithm are verified. Lastly, through the numerical simulation for a two-dimensional model, 2.5D direct-current sounding responses are summarized, which can provide a qualitative interpretation of field data.

Three-Dimensional Forward Modeling of Transient Electromagnetic Method Considering Induced Polarization Effect Based on Spectral Element Method

The transient electromagnetic method (TEM) is widely used in the exploration of mineral, petroleum, and geothermal resources due to its sensitivity to low-resistivity bodies, limited site constraints, and strong resistance to interference. In practical applications, the TEM often uses a long wire source instead of an idealized horizontal electric dipole (HED) source as the excitation source. This is due to the complex external conditions and the relatively large distance between the receiving zone and the transmitter source. Compared to the HED, the long wire source can provide a larger excitation current, generating stronger signals to meet the requirements of a higher signal-to-noise ratio or deeper exploration. It also produces longer-duration signals, thereby providing better resolution. Additionally, for the interpretation of TEM data, three-dimensional forward modeling plays a crucial role. However, the mature traditional TEM forward method is based on a simple, sometimes inappropriate model, as it is well established that the induced polarization (IP) effect is widely present in the deep earth, especially in oil and gas reservoirs. The presence of the IP effect results in negative responses in field data that do not conform to the traditional theoretical decay law of TEM, which can significantly impact data processing and inversion results. To address this issue, a TEM forward modeling method considering the IP effect based on the spectral element method (SEM) has been developed in this study. Firstly, starting from the Helmholtz equation satisfied by the time domain electric field, we introduce the Debye model with polarization information into the forward modeling by utilizing the differential form of Ohm’s law. As a result, we derive the boundary value problem for the time domain electric field that considers the induced polarization effect. Using Gauss–Lobatto–Legendre (GLL) polynomials as the basis functions, the SEM is employed to discretize the governing equations at each time step and obtain spectral element discretization equations. Then, temporal discretization equations are derived using the second-order backward Euler formula, and the linear system of equations is solved using the Pardiso direct solver. Finally, the electromagnetic responses at any time channel are obtained via SEM interpolation and numerical integration, thereby achieving three-dimensional TEM forward modeling considering the IP effect. The results indicate that this method can effectively reflect the spatial distribution of polarizable subsurface media. It provides valuable references for studying the polarization parameters of subsurface media and performing a three-dimensional inversion of TEM data considering the induced polarization effect.

Three-dimensional Forward Modeling of Semi-airborne Frequency-domain tipper Sounding Method with Complex Arbitrary-source Geometry

In this study, we propose a three-dimensional forward modeling algorithm for the Semi-airborne frequency-domain tipper sounding method (SAFTSM) based on the finite volume method to analyze the characteristics of SAFTSM responses with complex arbitary source geometry. We the used QMR algorithm preconditioned by the Jacobi method to solve the coupled symmetric equations of magnetic vector potential A and scalar potential ϕ under the Lorenz criterion. In these numerical experiments, we used grounded wire as a transmitting source. The accuracy of the algorithm is verified by comparing the three-dimensional results with the one dimensional results. We first calculate the low resistance model tipper response of a single source. After that, a reverse source is added on the basis of a single source, calculating the tipper response with an inverse parallel double source system. Finally, by comparing different emission source systems, we analyze the influence of the multiple sources on SAFTSM, so that we can provide a theoretical basis for practical detection.

3-D Forward Modeling of Transient EM Field in Rough Media Using Implicit Time-Domain Finite-Element Method

In a heterogeneous medium (usually called a rough medium) with fractured formations, the propagation of an electromagnetic (EM) field is a type of subdiffusion. Current mainstream geophysical EM data processing methods cannot be applied to data acquired on heterogeneous Earth, as they are not governed by the classic diffusion theory. To evaluate the influence of roughness on the transient EM (TEM) signal for a complex model and contribute to data inversion, we proposed a novel three-dimensional (3-D) forward modeling scheme for TEM in rough media. First, we derived the governing equation with a fractional-order time derivative for the subdiffusion of EM waves in rough media. Then, we proposed a novel time discretization using an unequal step length for the Caputo operator, which significantly reduces the total number of time steps. Finally, an implicit time-domain finite-element method using unstructured tetrahedron discretization was adopted to solve the 3-D forward problem. Furthermore, an efficient time segmentation strategy combined with parallel RHS construction was proposed to accelerate modeling. The numerical results prove that the proposed method is accurate and efficient, and will be a powerful numerical method for analyzing TEM wave propagation and processing TEM data in areas with multiscale fractures or porosity.

Three-Dimensional Forward Modeling of Ground Wire Source Transient Electromagnetic Data Using the Meshless Generalized Finite Difference Method

Three-Dimensional Forward Modeling of Ground Wire Source Transient Electromagnetic Data Using the Meshless Generalized Finite Difference Method

Real-world wintertime CO, N2O, and CO2 emissions of a central European village

Abstract. Although small rural settlements are only minor individual sources of greenhouse gases and air pollution, their high overall occurrence can significantly contribute to the total emissions of a region or country. Emissions from a rural lifestyle may be remarkably different than those of urban and industrialized regions, but nevertheless they have hardly been studied so far. Here, flux measurements at a tall-tower eddy covariance monitoring site and the footprint model FFP are used to determine the real-world wintertime CO, N2O, and CO2 emissions of a small village in western Hungary. The recorded emission densities, dominantly resulting from residential heating, are 3.5, 0.043, and 72 µg m−2 s−1 for CO, N2O, and CO2, respectively. While the measured CO and CO2 emissions are comparable to those calculated using the assumed energy consumption and applying the according emission factors, the nitrous oxide emissions exceed the expected value by a magnitude. This may indicate that the nitrous oxide emissions are significantly underestimated in the emission inventories, and modifications in the methodology of emission calculations are necessary. Using a three-dimensional forward transport model, we further show that, in contrast to the flux measurements, the concentration measurements at the regional background monitoring site are only insignificantly influenced by the emissions of the nearby village.

Channel trajectories control deep‐water stratigraphic architecture

AbstractInterpretation of deep‐water channel deposits is challenging because the spatial arrangement of their constituent lithologies is highly variable. This variability is often thought to be a signature of complex interactions between controlling boundary conditions and processes. A three‐dimensional forward stratigraphic model of a sinuous meandering channel is used to explore the production of channelised deep‐water stratigraphy. This model highlights three stages of stratigraphic evolution for channel belts: (1) an initial phase of rapid growth in mean belt width and variability in belt width driven by increasing channel sinuosity; (2) a subsequent phase of reduced belt‐width growth rate because of cutoff processes; and (3) a mature phase during which repeated bend lifecycles act to produce a statistically stable channel‐belt width. When a trajectory defining the vertical movement of a channel over time is added to the model, commonly recognised patterns of deep‐water channel‐belt stratigraphy are produced. These results demonstrate how forward stratigraphic models provide insights into processes governing the evolution of deep‐water stratigraphy that elude interpretations of static outcrops and seismic images of subsurface examples.

Crustal thickness and structural pattern evaluation of Sinai Peninsula using three-dimensional density modeling with aeromagnetic and earthquake data

The Sinai Peninsula is a subplate located between the African and Arabian plates. It is an important subplate in the world; however, this region is poorly understood because of the shortage of geophysical data. Thus, this study focuses on built a three-dimensional (3D) high-resolution forward model of the crustal thickness with a new tectonic model and structural evaluation of the Sinai Peninsula to understand this area tectonically and determine its lithospheric and crustal thicknesses. Qualitative and quantitative interpretations of Bouguer gravity, reduced-to-the-pole (RTP) aeromagnetic, and earthquake data are explained to achieve the goals of this study. Two-dimensional (2D) interactive sequential modeling of gravity data is performed along with some well-selected profiles with existing crustal layer depth points from previous works in the studied region. Seventeen 2D models are constructed and used to determine the basement, Conrad (lower crust boundary), and Moho (upper mantle boundary) depths and to build a 3D model. Accuracy of the 3D built model is evaluated by extracting observed, calculated, and residual gravity anomaly maps. The results show that the basement rocks appear on the surface in the South of Sinai and deepen (down to 4.5 km) toward the North. The Conrad surface varies approximately from 16 km (in the northeast and southwest of Sinai) to 22.8 km (in the southeast). The Moho surface depth (i.e., crustal thickness) varies approximately from 28.8 to 34.2 km, and it deepens toward the northeastern and southwestern parts of Sinai. A tentative basement structure map is constructed from the horizontal gradient (H-gradient) filter and the 3D Euler deconvolution of the Bouguer gravity and RTP aeromagnetic data. Correlation between the geological background of the study area and the obtained results suggests that the Sinai Peninsula is tectonically active, mainly in its southern portion due to the tectonic movements along the Gulf of Suez and the Gulf of Aqaba. The crustal thickness increases southward and thins northward.

Three-dimensional forward modeling and characterization of the responses of the ground-airborne frequency-domain electromagnetic method

Investigation of three-dimensional forward modeling for the ground-airborne frequency-domain electromagnetic (GAFEM) method and the characterization of its responses are presented. The electric field Helmholtz equation in the frequency domain is discretized by the Galerkin weighted residual method. The secondary electric field is the unknown to be solved for, discretized using edge-element basis functions. The edge-element basis functions are also used as the weight functions in the Galerkin approach. The solution of the discretized system is then obtained by a direct matrix-equation solver. To verify the effectiveness of the forward-modeling method and the accuracy of the corresponding code for computing the EM fields and reproducing their physical behavior, the numerical solutions computed using the presented method are compared against the equivalent solutions calculated from other techniques for a 3-D conductivity model. The good agreement between the various numerical methods and codes for this model demonstrates the accuracy of the responses computed by the method presented here. The method is then used to analyze the characteristics of GAFEM responses by comparing the magnetic fields and transfer functions (i.e., the magnetic field components Hx, Hy and Hz, and the transfer functions Tzx = Hz / Hx and Tzy = Hz / Hy) for different source-receiver configurations, altitudes of the observations and frequencies for several 3-D earth models. The aim is to improve the understanding of GAFEM responses and to determine the preferred measurement components and survey parameters. The responses at different frequencies of a complex model that includes topography are also investigated to study the characteristics of GAFEM responses of realistic earth models.

Three-dimensional forward modeling and response characteristics analysis of foundation pit leakage electric-field considering electrokinetic effect

Three-dimensional forward modeling and response characteristics analysis of foundation pit leakage electric-field considering electrokinetic effect

Borehole transient electromagnetic stereo imaging method based on horizontal component anomaly feature clustering

In the present study, according to the borehole transient electromagnetic method, the transceiver is placed into the borehole in front of the heading face for the purpose of detection. This can not only avoid the interference of metal support in the roadway, but also detect the hidden disaster-causing water body within a short distance. However, there is an issue of abnormal location in borehole transient electromagnetic method. In order to determine the location of transient electromagnetic anomalies in the borehole, the transient electromagnetic three-dimensional forward modeling is used to analyze the response characteristics of three components. According to the response characteristics of the horizontal component of the abnormal body in different quadrants, a method for calculating the rotation angle of the XOY plane of the abnormal center is proposed, based on the amplitude of the horizontal component and the abnormal quadrant. The resistivity of each depth calculated by the vertical component is regarded as an independent abnormal body, and the depths' mapping relationship with the sampling time is established. Next, based on K-means clustering, the corresponding characteristic value of horizontal component anomaly curve are binary classified, and the resistivity distribution quadrant of the entire data set is automatically divided. Finally, the borehole transient electromagnetic inversion resistivity stereo imaging method is obtained. The algorithm of this paper is used to calculate the stereo imaging of the three-dimensional numerical model, and the results show that the algorithm achieves good results for the small-scale and low resistivity abnormal body in the radial direction of the borehole. In summary, the method of borehole transient electromagnetic stereo imaging based on clustering is an organic combination of geophysics and machine learning, which can provide technical support for advanced detection and fine interpretation of hidden water damage in underground heading faces.

Three-dimensional forward modeling for the SBTEM method using an unstructured finite-element method

Three-dimensional forward modeling for the SBTEM method using an unstructured finite-element method

Morphology and possible origins of the Perm anomaly in the lowermost mantle of Earth

We have constrained a small-scale, dome-shaped low-velocity structure near the core-mantle boundary (CMB) of Earth beneath Perm (the Perm anomaly) using travel-time analysis and three-dimensional (3-D) forward waveform modeling of seismic data sampling of the mantle. The best-fitting dome-shaped model centers at 60.0°E, 50.5°N, and has a height of 400 km and a radius that increases from 200 km at the top to 450 km at the CMB. Its velocity reduction varies from 0% at the top to –3.0% at 240km above the CMB to –3.5% at the CMB. A surrounding 240-km-thick high-velocity D'' structure has also been detected. The Perm anomaly may represent a stable small-scale chemical pile in the lowermost mantle, although the hypothesis of a developing mantle plume cannot be ruled out.

Three-dimensional forward modelling of gravity field vector and its gradient tensor using the compact difference schemes

SUMMARY The traditional gravity forward modelling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate 3-D forward modelling algorithm based on a fourth-order compact difference scheme. First, a 19-point fourth-order compact difference scheme with general meshsizes in x-, y- and z-directions is adopted to discretize the governing 3-D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the pre-conditioned conjugate gradient algorithm. To obtain the first-order (i.e. the gravity field vector) and second-order derivatives (i.e. the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.

Considering fault interaction in estimates of absolute stress along faults in the San Gorgonio Pass region, southern California

Abstract Present-day shear tractions along faults of the San Gorgonio Pass region (southern California, USA) can be estimated from stressing rates provided by three-dimensional forward crustal deformation models. Due to fault interaction within the model, dextral shear stressing rates on the San Andreas and San Jacinto faults differ from rates resolved from the regional loading. In particular, fault patches with similar orientations and depths on the two faults show different stressing rates. We estimate the present-day, evolved fault tractions along faults of the San Gorgonio Pass region using the time since last earthquake, fault stressing rates (which account for fault interaction), and coseismic models of the impact of recent nearby earthquakes. The evolved tractions differ significantly from the resolved regional tractions, with the largest dextral traction located within the restraining bend comprising the pass, which has not had recent earthquakes, rather than outside of the bend, which is more preferentially oriented under tectonic loading. Evolved fault tractions can provide more accurate initial conditions for dynamic rupture models within regions of complex fault geometry, such as the San Gorgonio Pass region. An analysis of the time needed to accumulate shear tractions that exceed typical earthquake stress drops shows that present-day tractions already exceed 3 MPa along portions of the Banning, Garnet Hill, and Mission Creek strands of the San Andreas fault. This result highlights areas that may be near failure if accumulated tractions equivalent to typical earthquake stress drops precipitate failure.

Numerical modelling analysis of multi-source semi-airborne TEM systems using a TFEM

Abstract Traditional transient electromagnetic methods use single sources for excitation and extract the characteristics of underground media by improving interpretation technology. This study focused the improvements of transient electromagnetic interpretation using complex source technology. A time-domain vector finite element method (TFEM) was applied on three-dimensional forward modelling of semi-airborne transient electromagnetic (TEM) with multiple electrical sources, and it analysed the characteristics of fields with multiple sources. The study used a model of an isolated anomalous body in a homogeneous medium as an example. The effects of different combinations of excitation sources on the distributions of the magnetic field characteristics were analysed. Numerical results showed that the magnetic field components in a specific area could be strengthened by changing the layout of the sources, which was significant for future field data collections. By comparing the transient electromagnetic fields of the vertical array dipole sources with that of the loop source, the anomaly transient electromagnetic field of multi-source was more obvious than the field with a single source. Taking a complex orebody model as an example, a cross-electric source was used to calculate the magnetic field components of the semi-airborne TEM method. The resistivity distribution characteristics of the underground medium were obtained using an apparent resistivity interpretation method of the vertical magnetic field, which fully demonstrated that a multi-source transient electromagnetic system had the ability to determine abundant resistivity information of a complex medium.

3D frequency-domain airborne EM forward modelling using spectral element method with Gauss–Lobatto–Chebyshev polynomials

ABSTRACTThe spectral element method (SEM) based on high-order complete orthogonal polynomials is an accurate and efficient numerical method for electromagnetic modelling due to its spectral accuracy and exponential convergence. The SEM combines the flexibility of the finite-element method and the high accuracy of the spectral method. In this paper, we introduce SEM into three-dimensional frequency-domain airborne electromagnetic forward modelling. Starting from Maxwell's equations, we obtain a vector Helmholtz equation for the electric field. We use the Galerkin method to discretise the Helmholtz equation, in which the curl-conforming Gauss–Lobatto–Chebyshev polynomials are used as basis functions. The GLC polynomials help to derive the analytical expressions of entries in the system matrix and thus guarantee the modelling accuracy. Finally, we use the direct solver MUMPS to solve for the electric field and calculate the magnetic field by interpolation. For numerical experiments, we first compare our results with the semi-analytical solutions of a homogeneous half-space to verify the accuracy of our algorithm. We then analyse the characteristics of SEM by assuming different orders of interpolation polynomials and meshes. We also compare our method with the finite element method and SEM based on Gauss–Lobatto–Legendre polynomials. The results show that SEM is an efficient and effective method for electromagnetic modelling, it can deliver very accurate results and is less sensitive to mesh quality than the finite element method.

Three-dimensional forward modelling of gravity data using meshfree methods with radial basis functions and unstructured nodes

A study is presented using a mesh-free approach and a radial basis function generated finite difference (RBF-FD) method for numerically modelling 3-D gravity data. The gravity responses, that is, vertical gravity and gravity gradients, are obtained by solving the partial differential equation (PDE), that is, the Poisson’s equation for gravitational potential. The mesh-free approach discretizes PDEs using exclusively a cloud of unconnected nodes, instead of traditional tessellated meshes as used by mesh-based numerical methods such as finite difference, finite element and finite volume. Thus, the potentially computationally expensive and unstable creation and manipulation of 3-D meshes can be entirely avoided. A new type of finite-smoothness radial basis functions (RBFs), namely, the quintic-order polyharmonic spline (PHS) RBF, is proposed here in the RBF-FD frame for solving the gravity problem. Previous geophysical data modelling studies using RBF-FD have employed the infinitely smooth RBFs, such as the popular Gaussian (GA) RBFs. Here, both GA and PHS RBFs were tested with different numbers of nodes per mesh-free subdomain and with various shape parameter values (only GA RBFs have a shape parameter). The test results show that the PHS RBF-FD method is more computationally efficient than the GA RBF-FD counterpart. To achieve more efficiency, unstructured node distributions are proposed in discretizing the density models. For both quasi-uniform and unstructured node distributions, numerical results from the proposed PHS RBF-FD demonstrate that the computed vertical gravity and gravitational potential values agree well with analytical solutions with a reasonable number of degrees of freedom. A comparison study of modelling a complex density model with the PHS RBF-FD scheme and nodal finite-element method shows that the RBF-FD scheme generates sparse, asymmetric, linear systems of equations, supports unstructured nodal discretization and local refinement, and can have nonlinear h-convergence under refinement. Finally, the proposed RBF-FD method was applied to obtain the vertical gravity and gravity gradients over a real-world density model, where the benefits of mesh-free discretization are clearly illustrated.