While proactive geosteering, special inversion algorithms are used to process the readings of logging-while-drilling resistivity tools in real-time and provide oil field operators with formation models to make informed steering decisions. Currently, there is no industry standard for inversion deliverables and corresponding quality indicators because major tool vendors develop their own device-specific algorithms and use them internally. This paper presents the first implementation of vendor-neutral inversion approach applicable for any induction resistivity tool and enabling operators to standardize the efficiency of various geosteering services. The necessity of such universal inversion approach was inspired by the activity of LWD Deep Azimuthal Resistivity Services Standardization Workgroup initiated by SPWLA Resistivity Special Interest Group in 2016. Proposed inversion algorithm utilizes a 1D layer-cake formation model and is performed interval-by-interval. The following model parameters can be determined: horizontal and vertical resistivities of each layer, positions of layer boundaries, and formation dip. The inversion can support arbitrary deep azimuthal induction resistivity tool with coaxial, tilted, or orthogonal transmitting and receiving antennas. The inversion is purely data-driven; it works in automatic mode and provides fully unbiased results obtained from tool readings only. The algorithm is based on statistical reversible-jump Markov chain Monte Carlo method that does not require any predefined assumptions about the formation structure and enables searching of models explaining the data even if the number of layers in the model is unknown. To globalize search, the algorithm runs several Markov chains capable of exchanging their states between one another to move from the vicinity of local minimum to more perspective domain of model parameter space. While execution, the inversion keeps all models it is dealing with to estimate the resolution accuracy of formation parameters and generate several quality indicators. Eventually, these indicators are delivered together with recovered resistivity models to help operators with the evaluation of inversion results reliability. To ensure high performance of the inversion, a fast and accurate semi-analytical forward solver is employed to compute required responses of a tool with specific geometry and their derivatives with respect to any parameter of multi-layered model. Moreover, the reliance on the simultaneous evolution of multiple Markov chains makes the algorithm suitable for parallel execution that significantly decreases the computational time. Application of the proposed inversion is shown on a series of synthetic examples and field case studies such as navigating the well along the reservoir roof or near the oil-water-contact in oil sands. Inversion results for all scenarios confirm that the proposed algorithm can successfully evaluate formation model complexity, recover model parameters, and quantify their uncertainty within a reasonable computational time. Presented vendor-neutral stochastic approach to data processing leads to the standardization of the inversion output including the resistivity model and its quality indicators that helps operators to better understand capabilities of tools from different vendors and eventually make more confident geosteering decisions.
A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization