Estimation of petrophysical and fluid properties using integral transforms in nuclear magnetic resonance
Graphical abstract
Highlights
▸ We propose an analysis method that directly computes linear functionals of the relaxation times. ▸ Based on integral transforms of the measured data. ▸ No need for regularization. ▸ Estimates are more accurate. ▸ Uncertainty of the estimates can be estimated as well.
Introduction
Our study has been guided by NMR applications in porous media such as rocks and well-logging. For example, NMR can provide certain properties of rocks like porosity, movable fluid porosity, permeability, and pore size distribution [1]. These applications use “inside-out” NMR apparatus for in situ evaluation of geological formations in boreholes at depths up to 10 km and temperatures up to 200 °C [1], [2], [3]. In these applications, the measured NMR magnetization data denoted by is a multi-exponential decay, with a continuum of relaxation times [2], [3],The relaxation time is the characteristic time corresponding to loss of coherence by protons in hydrocarbons or water present in pores of a rock or in the bulk fluid. The corresponding non-negative amplitude is the unknown distribution of relaxation times and denotes additive, white, Gaussian noise with known statistics. In low-field measurements (as it is assumed in this paper) the relaxation times are essentially due to the surface-to-volume ratio of the pores so that, physically, the distribution corresponds to a pore size distribution where longer times represent larger pores [4]. In addition to the pore size, the distribution is also a function of the type of fluids present in the sample. The function is assumed to be known and referred to as the polarization factor and determined either by the measurement physics (corresponding to different pulse sequences) and/or tool physics (various operational conditions). For example,where is a function of pre-polarization time and longitudinal relaxation . In downhole applications, the function is a complex function of the tool geometry (such as length of the magnet and design of the RF antenna) and other operational constraints. In these applications, it is often well-fit by a polynomial function.
Traditionally, assuming is known, an inversion algorithm is used to estimate the distribution of relaxation times in Eq. (1) from the measured data [5]. Next, linear functionals of the estimated are used to estimate the petro-physical or fluid properties. For example, the area under the distribution is interpreted as the porosity of the rock. Often, based on lithology, a threshold is chosen as the cut-off characteristic time separating fluid bound to the pore surface and fluid that is not bound to the pore surface and can flow more easily. For example, in sandstones, relaxation times smaller than 33 ms have been empirically related to bound fluid volume. The remaining relaxation times correspond to free fluid volume. The mean of the distribution, , is empirically related to either rock permeability and/or to hydrocarbon viscosity. The width of the distribution, , provides a qualitative range of the distribution of pore sizes in the rock. Moments of relaxation time or diffusion are often related to rock permeability and/or hydrocarbon viscosity [6], [7], [8], [9], [10], [11]. Similar properties can be obtained from linear functionals computed from two-dimensional diffusion-relaxation data or relaxation data [12], [13], [14].
In this paper, we describe a method based on integral transforms that allows the direct computation of linear functionals of from the measured magnetization data without involving an intermediate computation of the full distribution , which would require the inversion of an ill-conditioned, non-linear problem (see, e.g., [15]). Examples of linear functionals include areas and moments of relaxation time . More complicated linear functionals can also be obtained by combining simpler ones by means of a convolution as shown later in this paper. This allows, for example, to find the moments of a specified region of the distribution. (see [22] for a classical treatment of the estimation of linear functionals and its applications to resolution limits in inverse problems).
This paper is organized as follows. In Section 2, we describe the estimation of moments and areas from integral transforms. We start with a brief review in Section 2.1 of a previous work by the authors [16] where the Mellin transform was used to obtain the -th moment of from fully polarized magnetization data. Then we describe an analogous method to estimate tapered areas of the distribution. In Section 2.2 these techniques are then extended to partially polarized measurements which is important in certain applications where there is not enough time to polarize the samples completely. Section 2.3 proposes some techniques to estimate other linear functionals of the distribution that are of interest in petrophysical analysis. Finally, Section 3 demonstrates the performance of the method on simulated data for typically assessed petrophysical properties.
Section snippets
Parameter estimation using integral transforms
The integral transform of the data is denoted by , and is defined byFrom Eqs. (1), (2),where the functions and form a Laplace-transform pair, withFrom the right-hand side of Eq. (3), for a desired linear transformation in the domain, our objective is to construct a kernel in the time-domain satisfying Eq. (4). In turn, from Eq. (2), the scalar product of with the measured
Simulation results
As an illustration of the use of the integral transforms described in the previous sections, we consider the six models with distributions shown in Fig. 4. Models 5 and 6 were derived from the stretched exponential (Kohlrausch function),a common function used to represent magnetization decay [19]. Model 5 used and while model 6 used and .
In all cases the area under the distribution is 1. For each of these models we generated the magnetization decay
Experimental results
Finally, we tested the algorithm on experimental measurements. The difficulty with real measurements is the lack a ground truth to use as a reference in the calculation of the error (rmse). This is specially true for field measurements taken downhole in oil wells. Nevertheless, in order to get validation with real data we obtained laboratory results for a sample of doped water (water with salt). Ten experiments on the same sample were ran. This 10 magnetization measurements were combined to
Summary
We have described a novel method for computing linear functionals of the distribution function (e.g., moments and tapered areas) directly from the measured magnetization data without the intermediate step of first estimating the distribution of relaxation times. This method involves a linear transform of the measured data using integral transforms. Different linear functionals of the distribution function can be obtained by choosing appropriate kernels in the integral transforms. Two main
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