This paper presents a logical approach to ensure successful nuclear magnetic resonance (NMR) logging for fluid-characterization purposes. Numerous examples are shown in a three-stage process consisting ofplanning,processing and interpretation, andvalidation and integration. First, modeling is performed to assess the NMR contrast, using any available information concerning anticipated fluid types. Knowledge of downhole conditions also allows the estimation of the NMR signal-to-noise ratio (SNR) from which the sensitivity and statistical error can be evaluated. The outcome of the planning process is a suite of NMR pulse sequences that optimizes resolution of the different fluids and minimizes error with the shortest possible acquisition time. Interpreting fluid-characterization NMR logs is relatively straightforward when the fluids have large NMR contrast and obey the standard correlations. However, dissolved gas, wettability variations, internal field gradients, and restricted diffusion can cause deviations from model behavior and have significant effects on NMR relaxation data. These effects need to be recognized and accounted for. To gain insights into the NMR-based fluid evaluation process, we have developed a model-independent technique that gives various sets of D-T1-T2 maps. The maps are analogous to the crossplots commonly used in log interpretation practices and are indispensable in the interpretation of the data. We also explain quantitative interpretation techniques from D-T2 maps. Integrating NMR fluid-characterization results with other information and openhole logs constitutes the final step. As with any technique, results need to be crosschecked with other log data for consistency. The answer gives a water saturation that can be verified with other tools and techniques such as the dielectric log, resistivity logs, and the density magnetic resonance technique method. Introduction Current methods to analyze fluids using suites of NMR measurements employ model-based inversions. Two examples of the forward-modeling approach are the MACNMR1 and Magnetic Resonance Fluid (MRF) characterization methods.2 The MRF technique is based on physical laws that are calibrated empirically to account for the downhole fluid NMR responses. By using realistic fluid models, MRF aims to minimize the number of adjustable parameters to be compatible with the information content of typical NMR log data. Since the model parameters are by design related to the individual fluid volumes and properties, determination of the parameter values (i.e., data-fitting) leads directly to estimates of the fluids petrophysical quantities. Any forward-model approach relies on the validity of the fluid models employed. In "non-ideal" situations where the fluid NMR response deviates from the model behavior (such as internal gradient, oil-wet rocks, restricted diffusion etc.), these techniques may lead to erroneous answers. In some circumstances, "non-ideal" responses may be identified by a poor fit-quality of the echo data, in which case the fluid models can be adjusted by modifying the appropriate model parameters. However, it may not be obvious which element of the fluid model should be modified and what modification is needed to get the desired answer. The maximum entropy principle (MEP) method is a model-independent inversion that provides a simple graphical representation of NMR data for fluid analysis in all environments. The graphical representations (i.e., multi-dimensional distributions) can themselves be used directly for interpretation or, alternatively, they may be used to guide the selection of parameters for model-based processing such as MRF. It is important to recognize that the MEP technique as well as the methods to interpret D-T2 maps are applicable to both CPMG (Carr, Purcell, Meiboom, and Gill) and DE (diffusion editing)3 measurements.
Abstract. We investigate positive solutions (x, y) of the Diophantine equation x 2 − (k 2 + 1)y 2 = k 2 that satisfy y < k − 1, where k ≥ 2. It has been conjectured that there is at most one such solution for a given k.
Logging While Drilling (LWD)* tools are formation evaluation sensors engineered into drill collars. Using LWD in horizontal wells offers significant advantages over other logging systems for formation evaluation. When LWD is provided in real-time via mud pulse telemetry the system becomes a geological steering tool which can be used to accurately place the horizontal section with reference to formation features and/or fluid contacts. The combined benefits of formation evaluation and real-time geological steering make LWD an essential tool for many horizontal wells. We will describe the benefits of LWD measurements and the improvements in both drilling and logging efficiency which LWD offers. These features are illustrated with practical examples of LWD from a number of horizontal wells in the North Sea. Although these examples are from horizontal wells in particular, the same benefits can be realized in directional wells in general.
Abstract. We derive midpoint criteria for solving Pell's equation x 2 − Dy 2 = ±1, using the nearest square continued fraction expansion of √ D. The period of the expansion is on average 70% that of the regular continued fraction. We derive similar criteria for the diophantine equation y 2 = ±1, where D ≡ 1 (mod 4). We also present some numerical results and conclude with a comparison of the computational performance of the regular, nearest square and nearest integer continued fraction algorithms.
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