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We propose an approach to inversion of induction LWD measurements based on calculation of the synthetic signals by artificial neural networks (ANN) specially trained on some database. The database for ANN training is generated by means of the proprietary 2D solver Pie2d. Validation of the proposed approach and estimation of computation time is performed for the problem of reconstruction of the three–layer model with a wall. Also, we make uncertainty analysis for the reconstructed model parameters for two tool configurations.

Multi-frequency and multi-component extra-deep azimuthal resistivity measurements with depth of investigation of a few tens of meters provide advanced possibilities for mapping of complex reservoir structures. Inversion of the induction measurements set becomes an important technical problem. We present a regularized Levenberg–Marquardt algorithm for inversion of resistivity measurements in a 2D environment model with pixel-based resistivity distribution. The cornerstone of the approach is an efficient parallel algorithm for computation of resistivity tool signals and its derivatives with respect to the pixel conductivities using volume integral equation method. Numerical tests of the suggested approach demonstrate its feasibility for near real time inversion.

Fast and accurate simulation of responses of electromagnetic logging-while-drilling tools in 2D and 3D formations is important for further development of the proactive geosteering service, which is currently based on the 1D layered-model inversion. In search of an optimal method of forward simulation, we have developed boundary integral equations in a 2D formation model with planar boundaries. A peculiarity of the method is that the number of grid nodes required for reaching a certain accuracy, and hence the computation time increases as the transmitter approaches the boundary. To retain efficiency in the case of transmitter-boundary proximity, we use the following modification of the method. The main term of the calculated electromagnetic field, namely, the solution for the two-layered model corresponding to the closest boundary, is calculated by the well-known explicit formulas, whereas the anomalous field is found from the system of integral equations. The modified method is implemented and tested on complex realistic 2D formation models as well as basic models to be used in 2D inversion. Numerical simulations of real propagation and azimuthal resistivity tools show that software based on the modified method can be successfully used for synthetic log simulation in complex formations. However, the proposed method has two shortcomings: poor efficiency when the transmitter is close to the boundary endpoints and limitation to isotropic formations. Both are topics for further research. Currently, the singularity near the endpoints is treated by simply refining the grid, which deteriorates efficiency. Generalization to the anisotropy case would include modification of the basic method and derivation of the closed-form solution for the layered model without the assumption of transverse isotropy.

Reliable estimates of the fluid pressure in the pore space of rocks are critical for different aspects of petroleum exploration and production including injection operations and scenarios of water flooding. Numerous approaches are available for formation pore pressure evaluation, however, these measurements become a challenge inside a cased borehole, and a list of possible options is short: either the casing is to be perforated, or the production tubing needs to be disconnected to perform the pressure tests. We present a method for through-casing evaluation of formation pore pressure without shutting down production. We suggest equipping an observation well with a borehole gravimeter and acquiring time variations of the vertical component of the gravity field. Changes in gravity occur during gas production and are related to time variations of formation pore pressure. Gravity changes obtained in the observation well are supposed to be inverted for time-dependent formation pore pressure variations beyond the casing. Our results and recommendations are based on numerical modeling of pore pressure spatial distribution during gas field exploitation and relevant changes in borehole gravity. Benchmark models were elaborated in order to consider a dynamic process of pressure changes in time and space under conditions similar to those in the Medvezhye gas field (Russia). Different modeling scenarios are considered for early and late stages of gas field exploitation. The sensitivity analysis was performed to estimate quantitatively a sensitivity of borehole temporal gravity changes to variations in formation pore pressure behind the casing. Based on resolution analysis we justify the possibility to extract the gravity measurements directly related to changes in pore pressure from the total changes in the gravity field due to reservoir exploitation. The impact of pore pressure on the gravity field measured in boreholes during the water flooding is also evaluated, and obtained results are discussed.

In most cases, interpretation of resistivity measurements is performed using 1D multilayered formation models that are used to fit data locally in real-time applications. While drilling high-angle or horizontal wells, more complex scenarios may occur, such as faults, pinch-outs, or unconformities. In these cases, resistivity logging data inversion should be performed using at least a 2D model, which is a more complex computational problem. This paper presents a neural networks approach for solving this problem exemplified by the application of a deep azimuthal resistivity tool for geosteering in the vicinity of a tectonic fault. The tool operational frequencies of 400 kHz and 2 MHz produce eight measurements with a coaxial arrangement of transmitters and receivers, and two azimuthally sensitive measurements with axial transmitters and transverse receivers. This paper considers a 2D model of a tectonic fault composed of three parallel layers on the one side of a displacement plane and the same three layers on the other side dislocated at a certain distance along the displacement plane. The model is described with nine independent parameters. The artificial neural networks (ANNs) were designed and trained to calculate the tool signals based on the model parameters. The training was carried out using a synthetic database of 4·105 elements containing the model parameters and corresponding tool signals. The database was calculated using distributed computations with in-house Pie2D software that used the boundary integral equation technique. To estimate the accuracy of the ANNs designed, the signals calculated with the networks were compared against the exact values obtained with Pie2D for an independent sample of 1.8·104 points. The comparison gave a good match for all 10 measurements, with the relative error comprising less than one standard tool measurement error for most points of the sample. Computation with the ANN required a few microseconds to calculate one signal, while the algorithm based on boundary integral equations required several minutes. The obtained acceleration of ~106 indicates many opportunities for modeling and inversion of logging-while-drilling data.

Extra-deep azimuthal resistivity measurements improve the depth of investigation up to 30 m from the wellbore. Interpretation of electromagnetic logging data in the neighbourhood of a well becomes an important technical problem. We present an efficient parallel method for computation of induction tool responses with multiple transmitter–receiver configurations in 2D pixel-based anisotropic model crossed by an arbitrary well trajectory. The cornerstone of the approach is volume integral equation method. We consider the conductivity distribution as a sum of background and anomalous conductivities. Background conductivity is 1D-layered. Anomalous conductivity has arbitrary 2D distribution. Electromagnetic fields are the superposition of background and anomalous fields. Background fields are calculated exactly using rigorous analytical solution for 1D-layered background model. With this approach, the 2D pixel-based model is treated as an extension of the 1D-layered model. The anomalous fields are required only in pixels with conductivity different from the conductivity of the 1D-layered model. Anomalous fields are calculated using convergent series of integral operators [1]. The approach takes into account conductivity anisotropy and allows obtaining both the exact solution and the fast approximate one. The convergence of approximate solution is investigated on some synthetic examples.

Summary We developed an OpenMP parallel version of the solver for numerical simulation of deep and extra-deep resistivity logging data. In this version, nested parallelization is implemented. Depending on the simulation task and CPU resources, we propose an algorithm for choosing the parallelization scheme that provides the fastest simulation.

Fast and accurate simulation of responses of logging-while-drilling (LWD) electromagnetic (EM) tools in complex 2D and 3D formations is very important for reconstruction of the resistivity distribution in proactive geosteering. Currently, real-time interpretation is based on the 1D parametric inversion. Advances in fast simulation beyond the 1D model would open a way for real-time 2D inversion. We developed, implemented numerically, and tested an efficient method for simulation of LWD EM tools in complex 2D formations with an arbitrary 3D position of the tool. The method is based on the boundary integral equations for the tangential components of the field and the Fourier transform, which reduces the problem to a series of 1D integral equations. Computations are carried out simultaneously for the whole set of measurement points with the same matrix, which provides a short computation time per point. We verified the method by comparing our results with those obtained by the well-known explicit 1D method and by commercial software in the 2D case. Numerical results justified that the method is accurate and time efficient. Also, as an example, we simulated the signals of real propagation and azimuthal resistivity tools for a complex 2D formation model with fault. The software based on our method can be useful in planning geosteering jobs in complex formations.

We introduce a numerical solution of the inverse light-scattering problem for a single non-absorbing spherical particle. The solution is implemented by global optimization at preliminary constructed database of light-scattering patterns. We propose an adaptive method for database construction, which aims both at providing satisfactory local accuracy and at avoiding large errors of the inverse map. Several databases were constructed varying the required accuracy of solution of the inverse problem and parameters used to characterize a sphere. We tested accuracy of the method on synthetic data for spheres with and without noise, on synthetic data for slightly prolate and oblate spheroids, and on experimental data of polystyrene microspheres measured with a scanning flow cytometer. The constructed databases have shown appropriate results in determination of the size and refractive index of a sphere from the angle-resolved light scattering with given accuracy.

We prove the unique continuation property for the hyperbolic integro-differential equation

Abstract Traditional gravimetry applications are mining/oil exploration surface gravity and formation bulk density borehole gravity logging. Large-scale reservoir saturation monitoring is a new gravimetry application. Substitution of oil or gas by water leads to density changes in large reservoir volumes, which causes time dependent subsurface and surface gravity field variations. This case study presents a complex multilayered reservoir time-lapse gravity data inversion problem. The customary bitmap approach requires many input parameters with a well-known inversion ambiguity. This ambiguity is in this work reduced by introducing a priori information obtained by biasing the inversion with history matched reservoir simulation output. A synthetic gravity data set was first generated by forward gravity modeling using the simulation saturation output from an onshore giant Middle Eastern oil field. By analyzing the simulation saturation data, the reservoir layer specific behavior of the water saturation and oil-water flood front was understood and used as a priori input in the optimized inversion algorithm used to fine-tune the predicted location of the oil-water flood front on the basis of the synthetic gravity data. Numerical examples with associated graphics demonstrate how inversion and accuracy estimates work for this data set. The proposed inversion technique will depict differences from the history matched simulation saturation and the gravity data; therefore, actual field gravity data will allow the enhancement of the reservoir simulation history match precision. The presented inversion of time-lapse gravity data demonstrates a substantial potential for the 4D microgravity. Since borehole gravity sensors are less affected by near surface changes, the borehole gravity data has significantly improved spatial resolution and much higher measured ? in the gravity signal resulting from the fluid substitution of hydrocarbons by water. In the presented inversion work, the maximum synthetic time-lapse ? gravity signal at the top reservoir was 18 times greater in magnitude compared to the predicted maximum surface time-lapse gravity signal amplitude. For microgravity to be a serious alternative for inter-well hydrocarbon saturation mapping for oil fields under waterflood, the borehole microgravity hardware development needs to be given significant R&D priority to reduce the diameter of the borehole gravity tool and improve gravity meter precision for both surface and borehole sensors. Introduction Geology and reservoir engineering describe the subsurface reservoir by utilizing sparse well data. As only a very small fraction of the subsurface can be observed through cores, well logs and well production data, the inter-well reservoir characterization is very challenging, especially in heterogeneous or fractured reservoirs, where simply interpolating wellbore data is not adequate to infer fluid distribution. At present, dynamic changes between wellbore control points is only understood via matching reservoir simulation models to reproduce the acquired well data. In the Journal of Petroleum Technology (JPT) May 2011 issue, the Society of Petroleum Engineers (SPE) Research & Development (R&D) Committee defined the five grand R&D challenges facing the oil and gas industry (Judzis et al. 2011), which included higher resolution subsurface imaging of hydrocarbons. In a follow-up JPT article, Neal and Krohn (2012) identified the most advanced technology options offering a solution to the inter-well hydrocarbon mapping problem as:3D/4D Seismic: surface, borehole and cross-well acquisition;Electromagnetic: borehole to surface and crosswell;Microgravity: borehole and surface acquisition; andNanotechnology.

Gravimetry is a physical method with a large depth of investigation. Traditional applications include surface gravity observations for mining and oil exploration and borehole gravity logging for investigating formation bulk density. A new application of gravimetry is large-scale reservoir saturation monitoring. Replacement of oil or gas by water leads to density changes in large volumes of the reservoir, which causes changes of the gravity field down hole as well as on the surface. Since borehole gravity sensors are closer to the reservoir than for surface acquired gravity data, borehole gravity data has better spatial resolution and are less affected by near surface changes. This paper focuses on the problems of inversion of time-lapse gravity data for complex multilayered reservoirs and estimation of the accuracy of the reconstructed oil-water flood front. The traditional bitmap approach (dividing the reservoir into blocks) requires a huge number of parameters and leads to the well-known inversion ambiguity. This ambiguity can be reduced by introducing a priori information. The basic idea of the presented approach is to obtain this a priori information by biasing the inversion with output from a history matched reservoir simulation data set. In this case, reservoir simulation saturation data from an onshore giant Middle Eastern oil field was used as input. By processing the simulation saturation data, it was possible to understand the behavior of the water saturation and oil-water flood front in the different layers of the reservoir. Using this knowledge, a 3D model of density changes was introduced. This model formed the basis of the optimization inversion algorithm used to fine-tune the actual location of the oil-water flood front on the basis of gravity data. Numerical examples demonstrate how inversion and accuracy estimates work for data obtained from a realistic reservoir simulation. The proposed inversion technique will depict any differences from the history matched reservoir simulation saturation output and the gravity data; thus, the gravity data will allow enhanced precision of the reservoir simulation history match.

We consider an inverse problem for the stationary elasticity system with constant Lame coefficients and a variable matrix coefficient depending on the spatial variables and frequency. The right-hand side contains a delta-function whose support (source) varies in some domain disjoint from the support of the variable coefficient. The inverse problem is to find the coefficient from the scattered wave measured at the same point at which the perturbation originates. A uniqueness theorem is proven. The proof bases on reduction of the inverse problem to a family of equations with the M. Riesz potential.

On the indicated class of functions we define the so-called (nonstationary) Dirichlet-to-Neumann operator H that assigns to g(x, t) the normal derivative a~u of a solution to (1), (2) on the boundary aft x R. In the present article, we consider the inverse problem consisting in reconstructing an unknown memory k(x, t) given the Dirichlet-to-Neumann operator H. We prove conditional stability of a solution to the problem; i.e., we show that for some class of functions k(z, t) we may guarantee closeness of two different memories, provided that the corresponding Dirichlet-to-Neumann operators are close. We suppose that the function k(z, t) belongs to the well-posedness set

We theoretically consider a new approach for measurement of the two-dimensional light-scattering patterns (2D LSP) of individual particles (for example, blood cells). Unlike the original optical scheme of the scanning flow cytometer that integrates scattering intensity over the azimuth angle, the new scheme allows us to measure the 2D LSP. The approach assumes measurement of the integral distribution of intensity on the fixed plane with subsequent reconstruction of the pattern via solving a first-kind integral equation. The last problem is ill-posed and we solve this equation by the standard regularization method. Error sources of the new approach are discussed from a comparison of the initial and reconstructed 2D LSPs for non-spherical particles.