International Journal of Geophysics

Volume 2018 (2018), Article ID 8592054, 11 pages

https://doi.org/10.1155/2018/8592054

## QVOA Techniques for Estimation of Fracture Directions

Correspondence should be addressed to Vladimir Sabinin

Received 11 August 2017; Revised 8 November 2017; Accepted 14 December 2017; Published 16 January 2018

Academic Editor: Alexey Stovas

Copyright © 2018 Vladimir Sabinin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Some new computational techniques are suggested for estimating symmetry axis azimuth of fractures in the viscoelastic anisotropic target layer in the framework of QVOA analysis (Quality factor Versus Offset and Azimuth). The different QVOA techniques are compared using synthetic viscoelastic surface reflected data with and without noise. I calculated errors for these techniques which depend on different sets of azimuths and intervals of offsets. Superiority of the high-order “enhanced general” and “cubic” techniques is shown. The high-quality QVOA techniques are compared with one of the high-quality AVOA techniques (Amplitude Versus Offset and Azimuth) in the synthetic data with noise and attenuation. Results are comparable.

#### 1. Introduction

Explicit numerical techniques of AVOA analysis (Amplitude Versus Offset and Azimuth) are well known for determining the direction of fractures (see [1]). They are based on the equation for approximating the reflection coefficient at the boundary between elastic anisotropic layers [2–4]. Implicit numerical AVOA techniques leading to the solution of nonlinear systems of equations have been proposed in [5], and their effectiveness has been shown therein.

However, the AVOA methods do not take into consideration the presence of additional viscous attenuation of seismic waves in target oil layers. In this context, the interest is considering for oil layers the function of quality factor , which characterizes the magnitude of this viscous attenuation. The equation for dependence of factor on the direction of fractures was proposed by Chichinina et al. in works [6, 7] and serves as the basis for QVOA techniques (Quality factor Versus Offset and Azimuth).

Generally speaking, the approximate analytical QVOA equation (see (1) below) is similar to the analogous AVOA equation. However, numerical techniques for its solutions can have their own characteristics. Therefore, to make a conclusion about the adequacy of QVOA methodology to determine the direction of the fractures, one should consider different numerical techniques and test them in a strict numerical experiment.

A number of efficient numerical QVOA techniques were proposed in [8]. The present paper proposes several additional techniques. The qualitative and quantitative properties of the techniques are investigated using numerical experiments. Also, I compare the results with the best AVOA technique from [5] with the use of synthetic seismograms with attenuation and noise.

#### 2. Models of 3D Seismic Data

In the QVOA techniques (as in AVOA), one uses 3D seismic data obtained from the receivers located on the ground surface as if at the points of a rectangular grid. The symmetry axis azimuth of the target layer fractures is evaluated for a rectangle surrounding the point of the grid (for a bin), taking into account only those traces for which the midpoint (MP) is inside the bin. If there are not enough points for the accuracy of the calculations, the adjacent bins are combined into a superbin and calculations are carried out for it. Therefore, the preliminary stage of evaluation is extracting from the 3D data all traces whose MPs lay inside the superbin. QVOA techniques are applied to seismic traces selected for a single superbin.

In the numerical experiment below, the location of the receivers on the surface is more ideal: at the points of the polar grid. I have made this to apply the 2D numerical model for generating 3D seismograms by rotating it around the source.

#### 3. Overview of QVOA Techniques

If fractured rocks with a preferential orientation of fractures are saturated with a fluid, then the fluid flows may lead to azimuthally varying attenuation of seismic waves. Let us define the attenuation factor (where is a complex velocity [7]) that corresponds to the definition of the quality factor , where is the wave energy and is the energy lost per cycle due to attenuation [9]. If we set , it is possible to derive the approximate equation [7] for the attenuation factor in the following form:where is the azimuth and the incidence angle is the angle relative to the normal vector in the target layer.

This approximation was made under the assumption that the term with is negligible.

According to [7], for HTI media,where is the symmetry axis azimuth of the fracture-strike direction.

Five numerical techniques for estimating the symmetry axis azimuth using QVOA were proposed in [8]. Three of them, the general technique (G), the truncated technique (T), and the sectored technique (S), are based on (1)-(2).

In technique G, the equation that is obtained by substituting (2) into (1) is solved by the least-squares method:

In technique T, a truncated version of (3) without the last term is solved.

Technique S solves in turn (1) and the first equation of (2). For this technique, the seismic data of the superbin should be separated by sectors. Equation (1) is solved for each sector independently to obtain the function , and it is assumed that in each sector —the middle azimuth of the sector.

The least-squares method is used to obtain in these three techniques, but it gives nonlinear systems of equations that have no analytical solution. To deal with the analytical solutions, the first equation of (2) can be replaced by the following approximate equation in techniques T and S:

As a result of this replacement, two additional techniques arise, approximate truncated technique (AT) and approximate sectored technique (AS), which are similar to the linear and sector AVOA methods (see [5]). They have the analytical solutions.

#### 4. Generalized and Enhanced QVOA Techniques

The practice of applying the approximate techniques (AT and AS) showed that they provided more preferred results than corresponding T and S techniques, based on (2), for certain seismic data. Perhaps, this is due to the fact that (4) has one parameter more () than (2) and therefore a more general form than (2).

Let us try to replace (2) by the following generalized equations:

After formal substitution of (5) into (1), generalized equation is obtained instead of (3):where and .

Equation (6) is the same as the equation for the general method of AVOA [5], except for the replacement of the amplitude function in it by attenuation factor . Let us call this new QVOA technique based on (6) the generalized technique (Gd). The solution algorithm for it is given in Appendix.

To account for nonlinearity in the sectored technique, one can base it on (1), function , and the second equation of (5). Here, the least-squares solution gives a system of four nonlinear equations, which is simpler than technique Gd. Let us call this new technique SC—sectored on C.

As shown in Figure 2, the dependence of on can differ from the quadratic dependence prescribed by (6). There are two ways to take this discrepancy into account. The first is to formally add a term in (6). This leads to solving a nonlinear system of 11 equations in the least-squares method. The solution of this system is not given here due to its bulkiness. It can be derived by analogy with the Appendix. This new technique will be called cubic (C).

The second way is to improve the quality of the approximate equation (1). It was obtained in [7] from a more complex equation of the following form:

Substitution of generalized equations of the form (5) into (7) gives

Application of the least-squares method to (8) gives a nonlinear system of 12 equations, which is solved similarly to Gd (see Appendix). The improved technique thus obtained (which is an enhanced version of the generalized technique) will be called EGd—enhanced Gd.

It is also worth considering a truncated version of (8) for simplicity:

This leads to solving a system of 9 nonlinear equations in the least-squares method. The technique, based on (9), will be called the truncated enhanced Gd technique (TEGd).

I do not cite here the systems of equations for EGd and TEGd techniques, because they can be derived by analogy with the Appendix.

Finally, to complete the view, it is necessary to consider the substitution of theoretical equations (2) into (7). This gives two more techniques that will be called by analogy: the enhanced general technique (EG) and the truncated enhanced general technique (TEG).

All 12 numerical QVOA techniques for evaluation of the symmetry axis azimuth are listed in Table 1 for reference.