Summary
A numerical method is proposed for solving an inverse problem for a
pulsed-neutron gamma inelastic log. The objective is to determine a composition
of the in-pore fluid and of the rock skeleton by gamma ray measurement data in
spectral windows of four elements: C, O, Ca, and Si. Inversion of the data is
based on an iteration procedure, on each step of which the corresponding direct
problem is solved for transport equations of neutrons and gamma quanta. Results
of numerical experiments are presented that confirm a convergence to exact
solution for most of the unknowns. The proposed numerical method is the
development of an approach for solving inverse problems in nuclear geophysics
(Khisamutdinov and Blankov 1989; Khisamutdinov and Minbaev 1995; Khisamutdinov
1999; Khisamutdinov and Phedorin 2003; Khisamutdinov 2005).
Introduction
Inversion methods are an important component of nuclear-geophysical
technologies. The methods commonly used now are approximate. In some methods,
the transport equations are replaced with diffusion equations; in others,
measured gamma ray spectra are approximated with a linear combination of given
standard monoelemental spectra, using laboratory measurements for further
evaluating the elemental content of a formation. We do not review inversion
methods here, but we provide a few references (Hertzog 1980; Herron 1988;
Roscoe et al. 1991; Gilchrist et al. 1999). In this paper, we develop a
rigorous method, contrary to approximate ones.
The problem of pulsed-neutron gamma logging with measurements of inelastic
scattering gamma-quanta (PNG-I) is discussed. This type of logging is based on
irradiating the rock by monoenergetic neutrons emitted by a pulsed generator
located in the borehole and subsequently registering the gamma ray energy
spectra. This gamma radiation is caused by inelastic scattering of neutrons on
the atoms of the medium. Measurements are taken during the pulse duration of
the generator when most neutrons have not been slowed down yet. This enables us
to distinguish the gamma ray spectrum of inelastic scattering from the capture
gamma ray spectrum. Gamma ray energy that a nucleus irradiates at inelastic
scattering of a neutron is uniquely characteristic for the element (the
isotope) where the scattering has occurred. Modern PNG-I tools mainly are aimed
at measuring spectral yields of four elements, C, O, Ca, and Si, which is
related to high cross sections of inelastic scattering at these elements or to
their high abundance in rocks and fluids.
Quantitative interpretation of experimental data is difficult for several
reasons. Both the neutron field and the scattered gamma-quanta field depend not
only on concentrations of the previously mentioned four elements in the
formation but also on concentrations of other elements and on wellbore
properties (including the fluid that fills it). In this paper, the numerical
method is constructed and one particular problem of interpreting pulsed-neutron
gamma-log data is solved. We treat the interpretation problem as an inverse
problem of evaluating parameters for the transport equation. This approach to
inversion of measured nuclear-geophysical data was developed in Khisamutdinov
and Blankov 1989, Khisamutdinov and Minbaev 1995, Khisamutdinov 1999,
Khisamutdinov and Phedorin 2003, and Khisamutdinov 2005; and it can be summed
up as follows:
- Qualitative elemental composition of the formation being investigated is
chosen properly, the set of unknown parameters is fixed, and constraint
equations are written out.
- Equations for measurements are defined; in combination with constraint
equations, they form the system of nonlinear equations for unknown
parameters.
- Taking into account the first and second items, a numerical method is
constructed to solve the system of equations for the inverse problem, and this
method should be convergent to the exact solution.
Results of numerical experiments shown in this paper seem to confirm the
convergence of this method to the exact solution for oil/water saturation and
for most mineral components of the formation. Note that unknown parameters are
linearly contained in the coefficients of transport equations (i.e., in the
cross sections of interactions between particles and the medium).
The present work relates to both geophysics and applied mathematics. We hold
the following method: A realistic problem containing all key points is chosen
to be solved, and we construct an inversion method for this problem. It is
important that the method can be generalized to more-complicated and integrated
problems. At first, a model of medium and measured values should be formulated
as key objects. The method can be generalized, in particular, to a case of
joint inversion of gamma spectra from inelastic scattering and radiative
capture of neutrons.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
17 September 2006
- Meeting paper published:
3 October 2006
- Revised manuscript received:
5 May 2008
- Manuscript approved:
21 August 2008
- Published online:
16 March 2009
- Version of record:
1 March 2009
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