Inversion of nuclear well‐logging data using neural networks

Abstract

A B S T R A C TThis work looks at the application of neural networks in geophysical well-logging problems and specifically their utilization for inversion of nuclear downhole data. Simulated neutron and γ -ray fluxes at a given detector location within a neutron logging tool were inverted to obtain formation properties such as porosity, salinity and oil/water saturation. To achieve this, the forward particle-radiation transport problem was first solved for different energy groups (47 neutron groups and 20 γ -ray groups) using the multigroup code EVENT. A neural network for each of the neutron and γ -ray energy groups was trained to re-produce the detector fluxes using the forward modelling results from 504 scenarios. The networks were subsequently tested on unseen data sets and the unseen input parameters (formation properties) were then predicted using a global search procedure. The results obtained are very encouraging with formation properties being predicted to within 10% average relative error. The examples presented show that neural networks can be applied successfully to nuclear well-logging problems. This enables the implementation of a fast inversion procedure, yielding quick and reliable values for unknown subsurface properties such as porosity, salinity and oil saturation. I N T R O D U C T I O NIn many geophysical problems, the aim is to apply a technique which will enable the determination of subsurface properties (e.g. lithology, porosity, density, hydraulic conductivity, resistivity, salinity and water/oil saturation) through the use of either surface or borehole measurements. This constitutes what is known as a geophysical inverse problem in which a mathematical model is used to relate the measured/observed data to the subsurface model parameters. In order to recover correctly the unknown parameters in the mathematical model an error function, otherwise known as the objective function, is set up. This function measures the discrepancy between the observations and predictions from a forward-modelling calculation. Minimizing this error function leads to the recovery of the unknown parameters, yielding optimal solutions. *

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“…In BP algorithms, the learning rate, H, is a small number (0.1 < H < 1.0) (Aristodemou et al, 2005) that controls the amount of error that will be negatively added to the interconnection weights for the next iteration (Cranganu, 2007). If the learning rate is large, then large changes are allowed in the weight changes, and no learning occurs.…”

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“…2) (i) a set of nodes (artificial neurons) where the nodes perform simple computations, (ii) a set of interconnections or synapses linking pair of nodes, and (iii) a set of labels known as weights, associated with each interconnection and identifying some property of interconnection. These weights correspond to the synaptic efficiency of the biological neurons (ARISTODEMOU et al, 2005).…”

“…2), whereas a corresponding output response is presented to the last layer of processing elements. All connections between the input layer, the ''hidden'' layers, and the output layer are then adjusted using an objective function (or ''cost function'') such as the mean-squared error (MSE) (ARBOGAST et al, 2000), the sum-squared error (SME) (ARISTODEMOU et al, 2005) or the global error E (LUHTI and BRYANT, 1997). For example, the global error E is computed as the sum of the squared differences between the computed output y k and the desired output d k (i.e., the local errors).…”

“…The addition of momentum is credited with smoothing out wild changes of weights, and also with helping the network converge faster when error is successively changing in the correct direction. Typical values for momentum fall between 0 and 1.0 (ARISTODEMOU et al, 2005).…”

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“…For example: (1) for seismic event classification (Dystart and Pulli, 1990), (2) well log analysis (Aristodemou et al, 2005;Maiti et al, 2007;Maiti and Tiwari, 2007, 2010b, (3) first arrival picking (Murat and Rudman, 1992), (4) earthquake prediction (Feng et al, 1997), (5) inversion (Raiche, 1991;Devilee et al, 1999), (6) parameter estimation in geophysics (Macias et al, 2000), (7) prediction of aquifer water level (Coppola Jr. et al, 2005), (8) magneto-telluric data inversion (Spichak and Popova, 2000), (9) magnetic interpretations (Bescoby et al, 2006), (10) signal discrimination (Maiti and Tiwari, 2010a), (11) modeling (Sri Lakshmi and Tiwari, 2009), (12) DC resistivity inversion (Qady and Ushijima, 2001;Lampinen and Vehtari, 2001;Singh et al, 2005Singh et al, , 2006Singh et al, , 2010.…”