Forward Solver Research Articles

AN ADAPTIVE STRATEGY FOR SEQUENTIAL DESIGNS OF MULTILEVEL COMPUTER EXPERIMENTS

Investigating uncertainties in computer simulations can be prohibitive in terms of computational costs, since the simulator needs to be run over a large number of input values. Building an emulator, i.e., a statistical surrogate model of the simulator constructed using a design of experiments made of a comparatively small number of evaluations of the forward solver, greatly alleviates the computational burden to carry out such investigations. Nevertheless, this can still be above the computational budget for many studies. Two major approaches have been used to reduce the budget needed to build the emulator: efficient design of experiments, such as sequential designs, and combining training data of different degrees of sophistication in a so-called multifidelity method, or multilevel in case these fidelities are ordered typically for increasing resolutions. We present here a novel method that combines both approaches, the multilevel adaptive sequential design of computer experiments in the framework of Gaussian process (GP) emulators. We make use of reproducing kernel Hilbert spaces as a tool for our GP approximations of the differences between two consecutive levels. This dual strategy allows us to allocate efficiently limited computational resources over simulations of different levels of fidelity and build the GP emulator. The allocation of computational resources is shown to be the solution of a simple optimization problem in a special case where we theoretically prove the validity of our approach. Our proposed method is compared to other existing models of multifidelity Gaussian process emulation. Gains in orders of magnitudes in accuracy or computing budgets are demonstrated in some numerical examples for some settings.

Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation

<p style='text-indent:20px;'>Bayesian inverse problems for inferring the unknown forcing and initial condition of Navier-Stokes equation play important roles in many practical areas. The computation cost of sampling the posterior probability measure can be exceedingly high. We develop the Finite Element Multilevel Markov Chain Monte Carlo (FE-MLMCMC) sampling method for approximating expectation with respect to the posterior probability measure of quantities of interest for a model problem of Navier-Stokes equation in the two dimensional periodic torus. We first consider the case where the forcing and the initial condition are bounded for all the realizations and depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. We establish the essentially optimal convergence rate of the method and verify it numerically. The method follows from that developed in V. H. Hoang, Ch. Schwab and A. M. Stuart, Inverse problems, vol. 29, 2013 for inferring the coefficients of linear elliptic forward equations under the uniform prior probability measure. In the case of the Gaussian prior probability measure, numerical results, using the MLMCMC method developed for the Gaussian prior in V. H. Hoang, J. H. Quek and Ch. Schwab, Inverse problems, vol. 36, 2020, indicate the essentially optimal convergence rates. However, a rigorous theory for the MLMCMC sampling procedure is not available, due to the non-integrability with respect to the Gaussian prior of the theoretical finite element errors of the forward solvers that are available in the literature.</p>

Numerical Determination of Thermal Conductivity in Heat Equation under Nonlocal Boundary Conditions and Integral as Over specified Condition

In this article, an inverse problem of finding timewise-dependent thermal conductivity has been investigated numerically. Numerical solution of forward (direct) problem has been solved by finite-difference method (FDM). Whilst, the inverse (indirect) problem solved iteratively using Lsqnonlin routine from MATLAB. Initial guess for unknown coefficient expressed by explicit relation based on nonlocal overdetermination conditions and intial input data .The obtained numrical results are presented and discussed in several figures and tables. These results are accurate and stable even in the presense of noisy data.

Azimuthal amplitude difference inversion constrained by azimuth velocity anisotropy

SUMMARYAmplitude variation with incident angle and azimuth (AVAZ) inversion is usually used for estimating subsurface medium properties from azimuthal stacked seismic data for fracture prediction. And weak anisotropy parameters play an important role in the fracture prediction of shale reservoirs. However, the low contribution of the anisotropic parameters to the reflection coefficient and too many condition numbers of forward solver lead to insufficient inversion stability of the anisotropic parameters. Therefore, a new azimuthal-amplitude-difference inversion method with the azimuth-velocity-anisotropic constraints is proposed to improve the stability of inversion and obtain a reasonable estimation of anisotropic parameters. The azimuthal amplitude difference AVAZ forward solver in the anisotropic medium is initially established from a rewritten approximate reflectivity equation and variable substitution. Sensitivity analysis of estimated parameters of the rewritten approximate reflectivity indicates that our approach is sufficiently suitable for seismic inversion of anisotropic parameters. Furthermore, an azimuthal amplitude difference Bayesian AVAZ inversion approach with azimuth-velocity-anisotropic constraint is developed to improve the stability of the inversion. We add Gaussian noise to the synthetic seismic records and assume the Cauchy distributions as a prior constraint on the model parameters. The anisotropic information obtained from the azimuth-velocity-anisotropic inversion is utilized as a constraint for improving the stability of inversion. Synthetic data tests reveal that the anisotropic parameters can be estimated stably even with moderate noise. Field data tests illustrate the feasibility and reliability of the presented inversion method for estimating anisotropic parameters in a shale reservoir with vertical or near-vertical fractures.

Learning-Based Subsurface Quantitative Imaging via Near-Field Scanning Microwave Microscopy

Accurate characterization of sample dielectric property is essential for scientists and engineers to analyze and evaluate the performance of the invested sample. This article proposes a learning-based (LB) method for quantitative subsurface imaging via coaxial-resonator-based near-field scanning microwave microscopy (CR-NFSMM) in a nondestructive way. To efficiently generate the database, a well-designed forward solver is critical since many samples will be included in the database, each sample needs a scanning procedure, and each scanning point involves solving the forward problem once. A fast forward problem solver is used to evaluate the tip–sample interaction in CR-NFSMM that avoids repeated meshing during the scanning process and it can apply to an arbitrary tip shape. Once the neural network is trained, it generates the reconstructed image within 1 s. Numerical and experimental results show that the proposed LB method could improve the resolution of the image and recover the dielectric properties of the subsurface perturbation pixel-by-pixel. Moreover, the proposed method outperforms the traditional objective function approach in terms of image resolution and time cost. The proposed method is promising to realize a nondestructive and real-time local dielectric evaluation of the subsurface object.

Ultrasonic characterization of frozen soils using a multiphase poromechanical approach

The ability to quantitatively and non-invasively characterize complex multiphase geomaterials is still a major challenge to the engineering, construction, and geophysical fields. In the context of accelerating climate change, construction on permafrost requires remedial measures and an appropriate characterization of permafrost (e.g., ice content, unfrozen water content, porosity, ice lenses, soil type, and mechanical properties). Current techniques are insufficient for efficient characterization of permafrost samples. Here, we propose an ultrasonic sensing technique and a signal interpretation method based on an inverse multiphase poromechanical approach to overcome critical gaps in permafrost characterization. This technique allows us to interpret the measured electrical signal using a theoretical transfer function obtained through an efficient multiphase poromechanical forward solver. Our study demonstrates the potential of the ultrasonic sensing technique for the rapid characterization of permafrost samples in terms of both physical and mechanical properties.

Parameter identification and state estimation for nuclear reactor operation digital twin

Reactor Operation Digital Twin (RODT) is now receiving increasing attention and investment in nuclear engineering domain. A prototype of a RODT was first brought out by Gong et al. at Nuclear Power Institute of China. The RODT contains a forward solver for online real-time simulation and an inverse problem solver for parameter identification and state estimation. To further improve the efficiency and accuracy of RODT and promote the practical deployment of RODT (i) we first propose an advanced differential evolution algorithm to upgrade the inverse solver; (ii) then we bring out a systematical uncertainty quantification of RODT considering assimilation noisy observations. The accuracy and validity of the proposed RODT are tested along one cycle operating stage of HPR1000, at various operating temperatures, control rod steps, general power level and the inlet temperature of a practical nuclear reactor core. Numerous numerical results confirm its potential for practical engineering applications for on-line parameter identification and state estimation.

Efficient Model Assisted Probability of Detection Estimations in Eddy Current NDT with ACA-SVD Based Forward Solver.

Model assisted probability of detection (MAPoD) is crucial for quantifying the inspection capability of a nondestructive testing (NDT) system which uses the coil or probe to sense the size and location of the cracks. Unfortunately, it may be computationally intensive for the simulation models. To improve the efficiency of the MAPoD, in this article, an efficient 3D eddy current nondestructive evaluation (ECNDE) forward solver is proposed to make estimations for PoD study. It is the first time that singular value decomposition (SVD) is used as the recompression technique to improve the overall performance of the adaptive cross approximation (ACA) algorithm-based boundary element method (BEM) ECNDE forward solver for implementation of PoD. Both the robustness and efficiency of the proposed solver are demonstrated and testified by comparing the predicted impedance variations of the coil with analytical, semi-analytical and experimental benchmarks. Calculation of PoD curves assisted by the proposed simulation model is performed on a finite thickness plate with a rectangular surface flaw. The features, which are the maximum impedance variations of the coil for various flaw lengths, are obtained entirely by the proposed model with selection of the liftoff distance as the uncertain parameter in a Gaussian distribution. The results show that the proposed ACA-SVD based BEM fast ECNDE forward solver is an excellent simulation model to make estimations for MAPoD study.

How to lighten parametric inverse computations if the misfit is non-convex and the forward solver needs stabilization

Solving parametric inverse problems (IPs) for partial differential equations (PDEs) plays an important role in medical diagnosis, resource investigation, nondestructive testing and many other human activities. Real-world IPs formulated as misfit functional minimization tasks are frequently ill-conditioned. The origins of this ill-posedness are the misfit multimodality and insensitivity as well as the instability of the utilized numerical PDE solver. We propose a complex multi-population memetic strategy HMS combined with the Petrov–Galerkin method stabilized by the Demkowicz operator to overcome these obstacles. The paper delivers a rigorous mathematical formulation of the strategy as well as a theoretical motivation for common inverse/forward error scaling, which significantly reduces the computational cost of the strategy. The presented theory is illustrated with two examples. The first one shows the analytical construction of the Demkowicz operator. The second one is an application of HMS in solving an IP utilizing the approximate stabilization with a forward solver that uses the isogeometric residual minimization (iGRM) method.

Accounting for Modeling Errors in Linear Inversion of Crosshole Ground‐Penetrating Radar Amplitude Data: Detecting Sand in Clayey Till

AbstractMapping high permeability sand occurrences in clayey till is fundamental for protecting the underlying drinking water resources. Crosshole ground penetrating radar (GPR) amplitude data have the potential to differentiate between sand and clay, and can provide 2D subsurface models with a decimeter‐scale resolution. We develop a probabilistic straight‐ray‐based inversion scheme, where we account for the forward modeling error arising from choosing a straight‐ray forward solver. The forward modeling error is described by a Gaussian probability distribution and included in the total noise model by addition of covariance models. Due to the linear formulation, we are able to decouple the inversion of traveltime and amplitude data and obtain results fast. We evaluate the approach through a synthetic study, where synthetic traveltime and amplitude data are inverted to obtain slowness and attenuation tomograms using several noise model scenarios. We find that accounting for the forward modeling error is fundamental to successfully obtain tomograms without artifacts. This is especially the case for inversion of amplitude data since the structure of the noise model for the forward modeling error is significantly different from the other data error models. Overall, inversion of field data confirms the results from the synthetic study; however, amplitude inversion performs slightly better than traveltime inversion. We are able to characterize a 0.4–0.6 m thick sand layer as well as internal variations in the clayey till matching observed geological information from borehole logs and excavation.

Patient-specific solution of the electrocorticography forward problem in deforming brain

Invasive intracranial electroencephalography (iEEG), or electrocorticography (ECoG), measures electric potential directly on the surface of the brain and can be used to inform treatment planning for epilepsy surgery. Combined with numerical modeling it can further improve accuracy of epilepsy surgery planning. Accurate solution of the iEEG forward problem, which is a crucial prerequisite for solving the iEEG inverse problem in epilepsy seizure onset zone localization, requires accurate representation of the patient’s brain geometry and tissue electrical conductivity after implantation of electrodes. However, implantation of subdural grid electrodes causes the brain to deform, which invalidates preoperatively acquired image data. Moreover, postoperative magnetic resonance imaging (MRI) is incompatible with implanted electrodes and computed tomography (CT) has insufficient range of soft tissue contrast, which precludes both MRI and CT from being used to obtain the deformed postoperative geometry. In this paper, we present a biomechanics-based image warping procedure using preoperative MRI for tissue classification and postoperative CT for locating implanted electrodes to perform non-rigid registration of the preoperative image data to the postoperative configuration. We solve the iEEG forward problem on the predicted postoperative geometry using the finite element method (FEM) which accounts for patient-specific inhomogeneity and anisotropy of tissue conductivity. Results for the simulation of a current source in the brain show large differences in electric potential predicted by the models based on the original images and the deformed images corresponding to the brain geometry deformed by placement of invasive electrodes. Computation of the lead field matrix (useful for solution of the iEEG inverse problem) also showed significant differences between the different models. The results suggest that rapid and accurate solution of the forward problem in a deformed brain for a given patient is achievable.

A Multi‐Resolution Finite‐Element Approach for Global Electromagnetic Induction Modeling With Application to Southeast China Coastal Geomagnetic Observatory Studies

AbstractWe present a multi‐resolution finite‐element approach for three‐dimensional (3D) electromagnetic (EM) induction modeling in spherical Earth. First, the secondary electric field approach is employed so that both magnetospheric and ionospheric current sources are naturally considered. Second, the multi‐resolution tetrahedral grids are used to approximate the heterogeneous crust and mantle, so that the local ocean effects at coastal and island observatories can be accurately simulated. Furthermore, a parallel goal‐oriented hp‐adaptive finite‐element method with Nédélec vector elements is employed to guarantee the accuracy of solutions for arbitrary 3D conductivity distributions. Finally, two synthetic models are used to verify the accuracy and efficiency of our newly developed forward modeling solver. Results show that accurate solutions can be obtained for problems with several million to hundreds of millions of unknowns in a few minutes using 128 cores on a cluster. We apply this approach to correct the near‐surface ocean effects for several unused Chinese coastal observatories by performing multi‐resolution 3D modeling. The corrected data are inverted for the subsurface layered mantle conductivity structures. The conductivity model beneath southeast China is more resistive than that beneath northeast China by more than half an order of magnitude. By comparing the inverse models with the latest laboratory conductivity‐depth profiles, the estimated transition zone water content is less than 0.01 wt% beneath southeast China irrespective of which laboratory data is used. Considering the low‐velocity anomalies in this region, which suggest high‐temperature structures, less water is expected. We, therefore, infer that the mantle transition zone beneath southeast China is dry.

A Bayesian Optimization Approach to the Simultaneous Extraction of Intrinsic Physical Parameters fromT1 andT2 Relaxation Responses

SummaryNuclear magnetic resonance (NMR) relaxation responses in porous media provide a sensitive probe of the microstructure and yet are influenced by a number of factors which are not easily detangled. Low-field T2 transverse relaxation measurements can be carried out quickly and are frequently used as pore size distributions, while adding T1 longitudinal relaxation measurements provides additional insights into surface properties and fluid content. Here we present an inverse solution workflow extracting related intrinsic physical parameters of the system by fitting experiment and numerical simulation(s). An efficient NMR forward solver for the simultaneous calculation of T1 and T2 responses is introduced, which honors existing inequality relationships between T1 and T2 parameters. We propose a Bayesian optimization approach that jointly identifies T1- and T2-related properties satisfying physical constraints by simultaneously fitting T1 and T2 experiments to simulations. This dual-task inverse solution workflow (DT-ISW) identifies the solution by minimizing the sum of the L2 norm of the fitting residuals of both T1 and T2 distributions into a single objective and jointly models the two highly correlated objectives with high efficiency using the vector-valued Gaussian process (GP) kernel for transfer learning. A multimodal search strategy is used to identify nonunique solution sets of the problem. The workflow is demonstrated on Bentheimer sandstone, identifying five intrinsic physical parameters. The performance of the joint DT-ISW (DT-ISW-J) is compared to a sequential DT-ISW (DT-ISW-S) approach as well as an independent single-task ISW (ST-ISW) of the T1 and T2 responses. Both dual-task versions converge more than two times faster than the single-task version. DT-ISW-J equally minimizes the L2 norm of T1 and T2 fitting residuals whereas DT-ISW-S only preferentially minimizes the objective assigned higher importance. A Pareto optimal solution (POS) is provided to allow operators to subjectively balance the preference of T1 and T2 data fits for the slightly conflicting objectives. The ability to extract five intrinsic physical parameters simultaneously provides new techniques for tracking wettability alteration and assessing the influence of clay amount and distribution on petrophysical property estimates.

Extending a low-order inhomogeneous adjoint equations model to a higher-order model with verification on integral applications

A higher-order nodal mathematical inhomogeneous adjoint model conjugate to the NEM-M2B2 nodal diffusion forward model is developed and introduced in this research. Verification of the developed model is presented through applications in perturbation analysis and the IAEA-3D benchmark including adjusted forms of it. This paper’s objective is to explore ways of extending and optimizing a mathematical adjoint capability suitable for use in an industrial reactor code, such that it becomes not merely an approximate but rather the exact adjoint counterpart to the typically used higher-order nodal forward solvers used in mature industrial reactor codes. Specifically, it is investigated how to upgrade an already available lower-order nodal mathematical adjoint solver towards higher-order accuracy. An example of the latter is the lower-order nodal adjoint solver used in the ARTEMIS reactor code, in the technical context of stabilization and acceleration of embedded control rod search mechanisms. Though the latter adjoint solver proved suitable for the needed preconditioning purposes, while also enabling the benefit of computationally very lean adjoint iterations, several future developments could benefit from having a higher-order adjoint nodal solver available as well. By using a preconditioned form of the base NEM-M2B2 nodal diffusion forward model and by using variational analysis, we have obtained a higher-order nodal mathematical adjoint that can have a physical interpretation associated with it as a Lagrangian multiplier. The nodal mathematical adjoint is then developed for the Axial Offset (AO) as a Response of Interest (RoI) which leads to an inhomogeneous adjoint system of equations. A solution verification of the adjoint developed is done through analyzing the effects coming from perturbations in the absorption and the scattering cross-sections. The applications investigated include axially and radially traveling perturbations along the reactor’s core. Several locations for the traveling perturbations are chosen to represent important locations in the core. Comparison between the low-order and the higher-order adjoint models is conducted. The forward model is set to the NEM-M2B2 nodal diffusion equation for both adjoints during the comparison. The higher-order adjoint model developed show consistent results in comparison to its lower-order sibling, suggesting the preference of using the developed higher-order model for adjoint computations.

Machine learning enhanced electrical impedance tomography for 2D materials

Electrical impedance tomography (EIT) is a non-invasive imaging technique that reconstructs the interior conductivity distribution of samples from a set of voltage measurements performed on the sample boundary. EIT reconstruction is a non-linear and ill-posed inverse problem. Consequently, the non-linearity results in a high computational cost of solution, while regularisation and the most informative measurements must be used to overcome ill-posedness. To build the foundation of future research into EIT applications for 2D materials, such as graphene, we designed and implemented a novel approach to measurement optimisation via a machine learning adaptive electrode selection algorithm (A-ESA). Furthermore, we modified the forward solver of a python-based EIT simulation software, pyEIT, to include the complete electrode model (CEM) and employed it on 2D square samples (Liu B et al 2018 SoftwareX 7 304–8; Somersalo E et al 1992 SIAM J. Appl. Math. 52 1023–40). In addition, the deep D-Bar U-Net convolutional neural network architecture was applied to post-process conductivity map reconstructions from the GREIT algorithm (Hamilton and Hauptmann 2018 IEEE Trans. Med. Imaging 37 2367–77; Adler et al 2009 Physiol. Meas. 30 S35). The A-ESA offered around 20% lower reconstruction losses in fewer measurements than the standard opposite–adjacent electrode selection algorithm, on both simulated data and when applied to a real graphene-based device. The CEM enhanced forward solver achieved a 3% lower loss compared to the original pyEIT forward model. Finally, an experimental evaluation was performed on a graphene laminate film. Overall, this work demonstrates how EIT could be applied to 2D materials and highlights the utility of machine learning in both the experimental and analytical aspects of EIT.

Bayesian uncertainty quantification of local volatility model

Local volatility is an important quantity in option pricing, portfolio hedging, and risk management. It is not directly observable from the market; hence calibrations of local volatility models are necessary using observed market data. Unlike most existing point-estimate methods, we cast the large-scale nonlinear inverse problem into the Bayesian framework, yielding a posterior distribution of the local volatility, which naturally quantifies its uncertainty. This extra uncertainty information enables traders and risk managers to make better decisions. To alleviate the computational cost, we apply Karhunen–Lòeve expansion to reduce the dimensionality of the Gaussian Process prior for local volatility. A modified two-stage adaptive Metropolis algorithm is applied to sample the posterior probability distribution, which further reduces computational burdens caused by repetitive numerical forward option pricing model solver and time of heuristic tuning. We demonstrate our methodology with both synthetic and market data.

Bayesian tomography with prior-knowledge-based parametrization and surrogate modelling

SUMMARYWe present a Bayesian tomography framework operating with prior-knowledge-based parametrization that is accelerated by surrogate models. Standard high-fidelity forward solvers (e.g. finite-difference time-domain schemes) solve wave equations with natural spatial parametrizations based on fine discretization. Similar parametrizations, typically involving tens of thousand of variables, are usually employed to parametrize the subsurface in tomography applications. When the data do not allow to resolve details at such finely parametrized scales, it is often beneficial to instead rely on a prior-knowledge-based parametrization defined on a lower dimension domain (or manifold). Due to the increased identifiability in the reduced domain, the concomitant inversion is better constrained and generally faster. We illustrate the potential of a prior-knowledge-based approach by considering ground penetrating radar (GPR) traveltime tomography in a crosshole configuration with synthetic data. An effective parametrization of the input (i.e. the permittivity distributions determining the slowness field) and compression in the output (i.e. the traveltime gathers) spaces are achieved via data-driven principal component decomposition based on random realizations of the prior Gaussian-process model with a truncation determined by the performances of the standard solver on the full and reduced model domains. To accelerate the inversion process, we employ a high-fidelity polynomial chaos expansion (PCE) surrogate model. We investigate the impact of the size of the training set on the performance of the PCE and show that a few hundreds design data sets is sufficient to provide reliable Markov chain Monte Carlo inversion at a fraction of the cost associated with a standard approach involving a fine discretization and physics-based forward solvers. Appropriate uncertainty quantification is achieved by reintroducing the truncated higher order principle components in the original model space after inversion on the manifold and by adapting a likelihood function that accounts for the fact that the truncated higher order components are not completely located in the null space.

A Systematic Material Characterization Method via Near-Field Scanning Microwave Microscopy

Accurate measurement of material dielectric property is essential for engineers to synthesize microwave devices. This article proposes a systematic material characterization method via the coaxial resonator (CR)-based near-field scanning microwave microscopy (NFSMM) with an arbitrary tip shape, from the modeling method to the inversion algorithm. The effective interaction region in the CR-based NFSMM is investigated and determined, which contributes most to the contrast capacitance caused by the tip-sample interaction. Then the Huygens’ principle is applied in the forward solver of NFSMM when the samples under test are homogeneous. The relative error of our model is less than 2.1% in numerical validations. Based on the proposed forward solver, a dielectric characterization method via the conjugate gradient (CG) algorithm is proposed and verified with experimental results. The retrieved permittivity of unknown samples matches well with that measured by the transmission line method, and the discrepancy between the two results is less than 3%. The proposed systematic method is promising to provide a new pathway to quantitatively determine the dielectric property of homogeneous samples.

Interactive software for processing and modeling oil well resistivity logs as part of a two-dimensional interpretation approach

Resistivity well logging is widely used to identify reservoir rocks and evaluate their oil potential. However, the integral nature of the rock response to the field excited by the logging tool leads to the fact that the tool signal opposite the target formation is influenced by a well filled with drilling fluid, overlying sediments, drilling fluid filtrate invasion into the formation, etc. Today's widespread methods of resistivity logs interpretation are based on excessive simplifications, which in complex conditions leads to errors in determining the oil saturation coefficient of the target reservoir, costs of testing water-saturated intervals and missing the potentially promising reservoirs. The paper presents the results of developing new interactive software for working with well logging data and geoelectric model of the environment, focused on a 2D approach to interpreting resistivity logs. The use of a 2D axisymmetric geoelectric model of the medium as a base allows one to correctly take into account the above factors when interpreting practical data. Original realization of user interaction with geoelectric model of near-wellbore space takes into account the features of 2D approach to resistivity logs interpretation and provides convenience of work with multilayer models. The computing server AlondraCore integrated with the software supports resistivity logs simulation using two types of forward problem solvers: high-precision finite-element and fast neural network-based ones. The finite-element algorithms provide high log simulation accuracy with the ability to adjust the tool parameters. On the other hand, the neural network-based algorithms provide fast solution of forward problems for common logging tools.

2D inversion of ultradeep electromagnetic logging measurements for look-ahead applications

Knowing formation ahead of the drill bit can help with hydrocarbon exploration and production, especially for low-angle and vertical wells. The advent of electromagnetic look-ahead technology makes it possible to interrogate formations tens of feet ahead of the drill bit. The formation can be very complex, such as fault and unconformity. However, existing approaches to recover the formation rely primarily on the 1D forward solver, which presumes that the formation is layered and transversely isotropic. For complicated formation, it will give rise to incorrect resistivity distribution. As a consequence, we have investigated and implemented a 2D pixel-based inversion algorithm to interpret the measurements for complex scenarios. We conduct a sensitivity study to illustrate the differences between look-ahead applications and look-around applications. To improve the look-ahead ability, we propose an effective technique to incorporate the prior information. In addition, we develop several examples to demonstrate the performance of the 2D inversion algorithm. It is found that using prior information as a reference model in the objective function enhances inversion performance significantly, and the proposed method can reasonably reconstruct some complex structures.