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ScienceDirect Russian Geology and Geophysics 56 (2015) 487–492 www.elsevier.com/locate/rgg

Correlation of well logs as a multidimensional optimization problem V.V. Lapkovsky a,b,*, A.V. Istomin a, V.A. Kontorovich a,b, V.A. Berdov a a

A.A. Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 3, Novosibirsk, 630090, Russia b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia Received 23 April 2013; accepted 10 July 2013

Abstract We consider a variant of automatic correlation between well sections based on construction of multidimensional functions of differences between fragments of logs. The solution is obtained either by successive correlation between the boundaries based on the projective model of Haites and definition of the boundary position at the minimum of the difference function or by construction of the lines of optimal trajectories on a 2D Zhekovskii’s plot. These algorithms are implemented as a program package for original interpretation of well data and as a plug-in for the automatic correlation module of the Schlumberger Petrel software using the Ocean development tools. © 2015, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: automatic correlation; well log interpretation; optimal correlation; comparison of wells

Introduction Attempts to develop automatic systems for correlating horizons in well sections have been made with varying degrees of success since the 1960s. The need for such tools has become particularly urgent in studies of closed areas, where information on sedimentary sequences is obtained mainly by well logging and seismic surveys (Kirichkova et al., 2007). The publication of the collection of papers (Payton, 1977) gave impetus to the development of methods for correlating seismic horizons with stratigraphic boundaries, as well as the structural features of wave fields with the structure of sedimentary rocks under different facies conditions. New areas of stratigraphic research have emerged based on a continuous survey of the geological environment—seismic stratigraphy (Payton, 1977) and sequence stratigraphy (Margulis, 2008). For these areas, initially based on the use of digital geophysical data, there is a wide range of computer technologies built into integrated systems of geological modeling. However, most of the implemented technological solutions give the interpreter a set of tools for analyzing, editing, and visualizing well data. Selecting the correlation variant and proper identification of horizons and boundaries in different wells is the prerogative of the specialist and depends on its installations, experience,

* Corresponding author. E-mail address: [email protected] (V.V. Lapkovsky)

and the “art of reading” well logs. For a field with hundreds or thousands of wells, the construction of a detailed correlation model, even in a technologically advanced instrumental environment is a very time-consuming task requiring long efforts of an experienced professional. In this connection, the development of effective computer making-decision technologies for stratigraphic identification is of special importance. Most attempts in this direction were unsuccessful, which has even led to a comparison of the development of automatic correlation systems with the problem of perpetual motion (Kashik et al., 2010). Nevertheless, there are now two recognized, technologically very advanced solutions that allow correlations to be entrusted to a computer. One of them is implemented in the DV Geo system of the Central Geophysical Expedition (Kovalevskii et al., 2007), and the other (AutoCorr) is developed in the State University of Oil and Gas (Gutman et al., 2006, 2010). Despite the fact that these solutions are substantially different from each other, they are based on a common approach—correlation comparison of a pair of wells located on the same edge of the graph obtained by triangulation of the position of the wells in the plane. Pairwise comparisons give rise to discrepancies which have to be eliminated. These discrepancies are due to the fact that in moving along the edges of the graph from one well to the other and determining stratigraphic analogs in them, one can select different routes, which, sometimes, lead to substantially different solutions. The DV Geo system offers a complex algorithm for local correc-

1068-7971/$ - see front matter D 201 5, V.S. So bolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.rgg.2015.02.009

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V.V. Lapkovsky et al. / Russian Geology and Geophysics 56 (2015) 487–492

tion of correlation variants to match pairwise comparisons of wells into a consistent correlation model. In the event of significant discrepancies in the AutoCorr software, the corresponding edges of the graph are color-coded on the plot and the expert is offered to manually resolve the conflicts of pairwise comparisons. In this paper, we propose a solution that prevents the occurrence of discrepancies—the use of multidimensional measures of differences in comparing wells. The essence of this approach is that, using originally constructed pairwise measures of differences between well log fragments being compared, one constructs a multidimensional difference function which is used to compare each well simultaneously with the totality of its surrounding wells and optimize the correlation solution by minimizing this difference.

2. All functions g (x) ∈ C (R 1) used for well log correlation are normalized in the range from 0 to 1, i.e., ||g (x)|| ≤ 1 over the entire interval studied. This makes it possible to use different measures of difference, in particular those including weighted combinations of measurement results from several normalized logs satisfying these conditions. 3. For each of the compared logs for constructing a Zhekovskii’s plot, one of two methods for calculating distances defined as follows. Suppose that x1 is the depth of the central point of the interval in the first of the wells compared, x2 is the that in the first of the wells compared in the second, A is the length of the interval, s is the variable along which the integration from –A/2 to A/2 is performed, g is the log given in both wells and for which the distance is calculated, and w(s) is the weight function; then 1/2

Principle of correlation of horizons for a pair and group of wells The approach is based on three basic constructs. 1. The perspective (projective) correlation model proposed by T.B. Haites (Haites, 1963; Salin, 1979). 2. The pairwise optimal correlation comparison of the proposed by B. Zhekovskii (1963). This work was published as a short article in an issue of temporary storage of library, and is currently not accessible. The Zhekovskii approach was further developed by domestic resrarchers (Grishkevich, 1984; Guberman and Ovchinnikova, 1972; Gutman et al., 2006, 1010; Kashik et al., 2010). This solution was also investigated in (Lineman et al., 1987) and implemented in the form of a software product that performs pairwise correlation (Mirowski et al., 2005). 3. Similarity-difference measures for comparing well log fragments. Modified measures proposed for interval comparison of logs are used (Guberman and Ovchinnikova, 1972; Vistelius and Romanova, 1962).

Correlation for a pair of wells A plot (or a two-dimensional difference function) is constructed for one or several well logs. This is done using the following assumptions and constraints: 1. Let X = (x1, x2, ..., xN) be the region of Euclidean space EN of the depths of all wells limited by the correlation interval. As well-log curves we will consider the bounded real function g (x) ∈ C (R 1), where x ∈ R 1 is any of the types of depth used to represent well data. These data can be obtained from the results of instrumental measurements directly in the well (on the cable or casing). In addition, arbitrary transformations of the source log are applicable (e.g., v (g (x)) ∈ C (R 1), where x ∈ R 1), which allows a comparison of wells based on logs free from the effect of the degree of formation saturation, technological factors such as the type of mud, wellbore design, measurement time.

⎫ ⎧ A/2 ⎪ 2 ⎪ ⎪ ∫ w (s) [g1 (x1 + s) − g2 (x2 + s)] ds ⎪ ⎪ −A / 2 ⎪ fd (x1, x2) = ⎨ ⎬ A/2 ⎪ ⎪ ⎪ ⎪ w (s) ds ∫ ⎪ ⎪ −A / 2 ⎭ ⎩

.

(1)

As the weighted function we used the Gaussian func2 2 tion w (s) = e−s / 2 σ with the specified external parameter σ, i.e., w (s) is also bounded and continuous in the interval of integration. Another type of distance function is defined through the weighted moving correlation coefficients: fc (x1, x2) =

1− K (g1 (x1), g2 (x2)) 2

.

(2)

where K (g1 (x1), g2 (x2)) is the correlation coefficient between the logs g1 (x1) and g2 (x2) describing the same property in different wells, for example, the spontaneous-potential (SP) curve; the coefficient is calculated in the interval from –A/2 to A/2 of the neighborhood of the points compared. 4. Calculations of difference functions from several types of logs reduce to weighted averaging of the difference functions obtained for each curve separately. For example, in constructing the plot, one can take into account the dissimilarity of log fragments based on SP and gamma-ray (GR) logs simultaneously. A plot constructed on the basis of these two types of log curves is shown in Fig. 1. Figure 1 illustrates the principle of constructing a correlation solution on the example of part of the Nizhnyaya Kheta and Sukhaya Duda Formations for two wells of the Suzunskaya area of the northeast of West Siberia. The map in Fig. 1a corresponds to the value of the function of the measure of the difference between the well fragments compared f (x1, x2). Here x1 and x2 are the depths of the central points of these fragments. The black diagonal line is the optimal correlation variant. This line, denoted by L, is the solution of the optimization problem

Fig. 1. Correlation model for comparing sections of two wells. a, In the form of a Zhekovskii’s plot, relative depths of the Suzunskaya 12 well are plotted on the x axis, and those of the Suzunskaya 3 well on the y axis, the black line shows the optimal correlation; b, correlated well sections, the depth levels connected in different wells correspond to individual points of the optimal line (see part a).

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∫ f (x1, x2) dL → min ,

L

(3)

L

which has a solution because of the properties of functions (1) or (2). In the central part of the model, one of the stratigraphic boundaries—the roof of the Nizhnehetskaya Formation—is specified by an expert. Thus, the admissible solutions are limited and the corresponding stratigraphic boundary (or several boundaries) separates areas where correlation comparisons are possible and the difference coefficients can be calculated from those where such correlations are not possible (see Fig. 1a, white filling). The well log correlation model is obtained by constructing the optimal trajectory in the field of the function f (x1, x2) and solving problem (3). The optimal line was constructed using an ant colony algorithm (Dorigo, 1992) and a wave algorithm (Lee, 1961). The points of this line are the values of pairs of depths of the wells being compared that are minimally different from each other according to the set of log curves used. Therefore, each of the points can be represented as a segment connecting two depths in the wells compared (see Fig. 1b). An advantage of this method is that the optimization is performed over the entire interval in which the solution is constructed, i.e., the horizons are compared simultaneously, rather than separately, to find their analogs. The problem for several wells is solved using multidimensional difference function. As mentioned above, this eliminates correlation discrepancies, which are inevitable in sequential pairwise comparison of wells. Multidimensional difference characterizes the general closeness of the sections of all wells being compared for some depth vector— x = (x1, x2, ..., xn). An example of a multidimensional measure of difference is variance. A difficulty in constructing this measure for log curves is that we do not have mean values which are typically used in the calculation of variances. There is no curve of mean values with which the results of downhole measurements can be compared, and it can only be constructed using the correlation results. At the same time, there are a large number of estimates of differences for all pairs of wells compared. They can be used as a basis in the construction of multidimensional difference functions in which the argument is the vector of arbitrary depths of the group of neighboring wells. The validity of estimating the statistical parameters of spatial data based on pairwise differences of individual samples is grounded in geostatistics. It is in this way that variograms for estimating the variances and covariance of spatial data are constructed. The possibility of direct calculation of the variance, covariance, and correlation from a set of pairwise differences without calculating the means is shown in (Chechulin, 2011). In particular, for the variance, he proves the formula D2 (X) =

1 Σ ni = 1 Σ nj = 1 (xi − xj)2. 2n2

where n is the number of samples and xi and xj are the values of some random variable measured on these samples.

In the implemented algorithms of automatic correlation, we used two types of difference measures. The first is the Euclidean distance: Σ [f (xi, xj) wij⎯]2 . ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ F (x1, x2, ..., xn) = √

(4)

Here f(xi, xj) are all calculated functions of pairwise difference measures and wij are the individual weights of each of them. The other measure is of the geometric mean types and is given by the expression F (x1, x2, ..., xn) =

Σ wij

Π wij ⋅ f (xi, x⎯j) . ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ √

(5)

The weights wij in (4) and (5) are assumed to be different from zero for wells where the correlation solution has already been obtained, as well as for the well in which it is sought at the moment. In all other wells, zero weight is specified. The solution of the optimization problem for the whole set of wells can be represented as

∫ FX (x1, x2, ..., xn) dL → min.

L

L

However, since its solution is computationally very expensive because the dimension of the space EN and the number of discrete measurements in the correlation interval can reach several thousand, a serial method was proposed to solve this problem. In each step, the following optimization problem is solved: _ (6) ∫ FXk + 1 (x k, xi) dL k + 1 → min. k+1

L

L

_______ Here k = 1, N − 1 is the dimension of the subspace, which _ coincides with the step number, x k is the vector of the wells recorded in the previous step, xi is a free term, Lk+1 is the curve in the space Ek + 1 which is the optimal correlation line (multidimensional correlation line according to (Grishkevich, 1984)). Process of constructing correlation solutions for a group of wells The problem of matching the horizon boundaries for several wells is solved in one of the following two processes (or their sequence). 1. Tracing some of the most expressive boundaries from well to well using the Haites model as an initial approximation, after which the position of the boundary is refined and it is tied to the local minimum of the multidimensional difference function with consideration of the admissible deviation level and some predetermined linearity parameter. 2. Obtaining the solution in the entire range of depths by constructing an optimal correlation on a Zhekovskii’s plot. In the first step, two wells for which there is a calculated pairwise difference measure are combined into a model. Here the correlation line is the ordered set of vectors of the two depth


491

Fig. 2. Example of correlation of part of the Sukhodudinskaya and Nizhnekhetskaya Formations for a group of wells in the Suzunskaya area, according to the well-log data, in the GISWell environment. 1, resulting correlation lines; 2, correlation line taken as a new stratigraphic boundary; 3, originally defined stratigraphic boundaries (roofs of the Sukhodudinskaya and Nizhnekhetskaya Formations).

values (of the first and second wells). Next, this line is straightened (parameterized by its length) and becomes one of the axes of the plot in the next step; i.e., on this axis, we plot dependent pairs of values rather than depth values of an individual well. On the other axis, we measure depths in the well that, in this step, is added to the existing model. As a result, we have the optimal correlation line in the form of the ordered set of vectors of three depth values. The process is repeated until all wells are added to the general solution. A feature of this approach is that, in each step, the optimal correlation between the well and its neighbors is found over the entire range of predetermined depths. This is the fundamental difference between this method and the situation where the best correlation is sought between different horizons separately. In stratigraphic interpretation, the horizon boundaries can be arranged automatically using one of the previously implemented criteria for fragmentation of well sections (Berdov et al., 2012).

An example of automatic correlation of part of the Sukhaya Duda and Nizhnyaya Kheta Formations on the Suzunskaya area of the northeast of West Siberia is shown in Fig. 2. In the construction of this model, the solution was constructed not on the profile by comparing only adjacent wells, but the relationships of each well to its surroundings in the deposit area were considered. The resulting correlation models can be used to directly calculate a number of geometric parameters of the stratified formation and construct structural maps for any correlation levels and maps of thicknesses and mean well-log values on any stratigraphic range. The experience of using automatic technologies of well correlation shows that sometimes insufficiently well-founded comparisons arise (especially when using inconsistent, incomplete or ill-conditioned data). Quality control of constructions can be efficiently carried out by analysis of correlation models, both in the form of sections and also on calculated maps of hypsometry of boundaries, thicknesses, and mean well-log values in controlled stratigraphic ranges.

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Conclusions At present, this approach has two software implementations—as a software package in the GISWell product (IPGG SB RAS) and as a plug-in for the Schlumberger Petrel software. In each implementation, the problem can be solved using the processes described above, and, in addition, the model can be corrected manually in the intermediate and final stages of model building. Further development of the technologies related to the proposed approach involves the use of the results of stratigraphic interpretation of seismic data to minimize uncertainty in correlation constructions.

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Editorial responsibility: A.E. Kontorovich