resume - jhmer

A new approach for the investigation of the local regularity of borehole
wire-line logs
Saïd Gaci * and Naïma Zaourar
Département Géophysique - FSTGAT, Université des Sciences et de la
Technologie Houari Boumediene (USTHB), BP 32 El Alia, 16111, Algiers,
Algeria
Received: 21 April 2010 Accepted after revision: 28 September
2010 Published online: 8 October 2010 Abstract: In previous researches, borehole wire-line logs were described as
fractional Brownian motions (fBms) characterized by Hurst (or Hölder)
exponents which measure their global regularity degrees. Since theses
monofractals are everywhere regular with the same Hölder exponent, they do
not reflect the depth-evolution of the local regularity of the logs. For
this purpose, we suggest a general framework, multifractional Brownian
motion (mBm), to describe well logs and propose an algorithm based on the
generalized quadratic variations (GQV) to estimate the local Hölder
exponent function. First, synthetic logs data simulated by the successive
random additions (SRA) method are used to assess the potential of this
algorithm; it is observed that the estimated Hölder functions (or
regularity profiles) are very close to the theoretical Hölder functions.
Second, this analysis is extended to sonic logs data recorded at the KTB
pilot borehole. The obtained regularity profiles allow to perform a
lithological segmentation and to identify fault contacts on the geological
layers crossed by the well. A strong correlation between the variations of
the Hölder exponent value and the lithological changes is also noted.
Keywords: Well logs, Hölder exponent, fractal, multifractional [pic] Une nouvelle approche pour l'examen de régularité locale des
enregistrements de diagraphie Résumé : Dans les recherches précédentes, les enregistrements diagraphiques
ont été décrits comme des mouvements Browniens fractionnaires (fBms)
caractérisés par des exposants de Hurst (ou de Hölder) qui mesurent leurs
degrés de régularité globale. Comme ces mono-fractals sont partout
réguliers avec le même exposant de Hölder, ils ne reflètent pas l'évolution
en profondeur de la régularité locale des logs. Nous suggérons alors un
cadre plus général, le mouvement Brownien multifractionnaire (mBm), pour
décrire les logs, et proposons un algorithme basé sur les variations
quadratiques généralisées pour estimer la fonction de l'exposant de Hölder
local. Dans un premier temps, des données de logs synthétiques, simulées
par la méthode des sommations aléatoires successives, ont été utilisées
pour apprécier le potentiel de cet algorithme; il a été montré que les
fonctions de Hölder (ou les profils de régularité) estimé(e)s sont très
proches des fonctions de Hölder théoriques. Ensuite, cette analyse a été
étendue aux données de logs sonic enregistrées dans le puits pilote du KTB.
Les profils de régularité obtenus ont permis d'effectuer une segmentation
lithologique et d'identifier des contacts de failles sur les couches
géologiques traversées par le puits. Une forte corrélation entre les
variations de la valeur de l'exposant de Hölder et les changements
lithologiques a été également notée.
Mots clés : logs de puits, exposant de Hölder, fractal, multifractionnaire [pic]
1. Introduction Borehole measurements are an essential complement to exploration
activities (seismic, drilling, etc.) because they provide additional
information from the borehole that can not be derived from other subsurface
investigations. The analysis of these recorded data may bring further
information about the earth's heterogeneities.
Numerous studies have shown that borehole wire-line logs may be described
by non-stationary fractional Brownian motions (fBms). These monofractal
processes are characterized by a fractal [pic]-power spectrum model where k
is the wavenumber and H is the Hurst exponent (Todoeschuck et al., 1990;
Pilkington and Todoeschuck, 1991; Bean, 1996; Dolan and Bean, 1997; Dolan
et al., 1998). The Hurst parameter H gives an indication about the self-
similarity degree and long-range dependence of the well log. When[pic], the
process is reduced to the ordinary Brownian motion (namely the process has
no memory); for [pic], it is characterized by a persistence (positive or
negative)- namely the process shows a clear trend; and finally, for [pic],
it exhibits an anti-persistent behavior. From a regularity point of view,
the pointwise Hölder exponent is almost surely equal to H for all the
samples of the fBm; the higher H, the more regular the fBm. The estimation
of the H parameter can be carried out using several methods (Dolan and
Bean, 1997; Dolan et al., 1998, Leonardi and Kümpel, 1998, 1999; Malamud
and Turcotte, 1999; Arizabalo et al., 2004, 2006; Chamoli et al., 2007).
Indeed, fBms are everywhere singular with the same Hölder exponent, and
thus do not suit to study well logs whose local regularity varies rapidly
with depth. The introduction of a multifractal analysis is inappropriate,
since it yields a global map of the statistical distribution of regularity,
and all depth-dependent information is lost on the multifractal spectrum.
Recently, developed stochastic models and associated regularity estimation
methods allow to investigate the depth-evolution of the local regularity
even on extremely erratic data. Specifically, in this study, we will
consider the multifractional Brownian motion (mBm) as a model for borehole
logs and use algorithm based on the generalized quadratic variations (GQV)
for estimating their regularity profiles. This analysis attempts to
complete the approaches suggested in our previous researches (Gaci and
Zaourar, in press; Gaci et al., 2010, in press).
The present paper is organized as follows. First, we present a short
description of the mBm model and the GQV algorithm. Then, we show the
results obtained by this technique on synthetic logs data simulated by the
successive random additions (SRA) algorithm. Finally, the GQV algorithm is
implemented on P- and S-wave sonic measurements recorded at the KTB pilot
borehole. 2. Theory
Multifractional Brownian motion One of the models which can be used to describe the behavior of the
borehole wire-line logs is the mBm which generalizes the fBm. Then its
introduction requires the recalling of the main features of its famous
special case (fBm).
Defined by Mandelbrot and Van Ness (1968), the fBm is characterized by a
slowly decaying autocorrelation function depending on the parameter[pic],
named Hurst exponent, and admits the following moving average
representation:
[pic] (1) and [pic] where dB stands for the ordinary Brownian motion
process motion and G represents a strictly-positive scaling parameter:
[pic] with C as a positive constant. If G = 1, the motion is said
standard.
The process[pic] is self-similar of parameter H and has stationary
increments. Its covariance function is expressed as:
[pic] (2)
The fBm can be generalized by replacing the constant Hurst parameter H by a
function H(t). This extension leads to define the mBm which has the
following representation (Peltier and Lévy-Véhel, 1995; Lévy -Véhel, 1995;
Ayache and Lévy- Véhel, 2000):
[pic] (3)
where [pic] is required to be a Hölder function of order [pic] to ensure
the continuity of the motion. In the case of the Hurst function [pic] is
constant, [pic] is reduced to a simple fBm. The mBm's increments are in
general neither independent nor stationary. It can be shown that they
display long range dependence for all admissible non-constant regularity
functions [pic] (Ayache et al., 2000). Contrarily to fBm, the pointwise
Hölder exponent of [pic],[pic], may depend on the location. It equals with
probability one to [pic] for each t (Peltier and Lévy-Véhel, 1995; Benassi
et al., 1997; Ayache and Taqqu, 2005; Ayache et al., 2007):
[pic] (4)
Thus, mBm allows to describe and to model phenomena whose regularity varies
in time/space. Recall that the pointwise Hölder exponent of a stochastic
process [pic]at [pic]is defined by:
[pic] (5)
Another important property of mBm is that it does not remain self-similar
but is locally asymptotically self-similar (in short LASS). That means that
for each t, there is a fBm of Hurst parameter H(t), which is tangent to mBm
(Benassi et al., 1997; Falconer, 2002; 2003). Local Regularity Estimation Ayache and Lévy-Véhel (2004) suggested a parametrical method to identify
the Hölder exponent of mBm processes. This method is based on the
computation of the so-called "Generalized Quadratic Variations" (GQV).
For a trajectory of the process[pic], discretized at times[pic], [pic] with
[pic], the GQV are defined by:
[pic] (6)
where [pic] can be considered as "the neighborhood" of t.
The authors demonstrated that under some conditions imposed on the
constants, [pic]and[pic], the following relation is satisfied with
probability one for any mBm [pic].
[pic] (7)
Consequently, an estimation algorithm of the local Hölder exponent of a mBm
[pic], called here GQV algorithm, is resulted:
[pic] (8) 3. Application to simulated sonic logs data In this section, the GQV algorithm is tested on synthetic sonic logs data.
These logs are assumed to be mBms and modeled using the approach suggest