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# On the influence of the correlation path in incremental multi-well stratigraphic correlation

Guillaume Caumon, Christophe Antoine

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Guillaume Caumon, Christophe Antoine. On the influence of the correlation path in incremental multi-well stratigraphic correlation. International Geostatistics Congress 2024, Sep 2024, Ponta Delgada Açores, Portugal. pp.45-56, 10.1007/978-3-031-92870-3_5. hal-05066225

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On the influence of the correlation path in incremental
multi-well stratigraphic correlation

Guillaume Caumon<fnref n="1" />[0000-0002-7828-4600] and Christophe Antoine<fnref n="2" />

**Abstract.** Stratigraphic correlation between several one-dimensional borehole sections is generally achieved by expert-based interpretation and sedimentological reasoning to produce a best-case scenario. To gain productivity and make the process reproducible, automation methods defined since the 1980's have mainly translated the correlation (or alignment) problem as an optimization task. Therefore, stratigraphic layering, which determines rock unit volumes, connectivity and ways to measure distances in geomodels, is most often deterministic. However, the stratigraphic correlation problem is highly non-unique, because of sparse sampling, lateral variability, complex interactions between tectonic and sedimentary processes and because the types of data and the stratigraphic principles are many. In this work, we discuss some of the computational strategies to sample this uncertainty. N-best solutions are a possible option for pairwise correlations but become computationally intractable when the number of stratigraphic sections increases. Moreover, they explore the space of solutions only locally around the optimum. In multi-well correlation, finding the global optimum itself is highly challenging when the number of sections and samples is large. We use a hierarchical and incremental correlation which guarantees the consistency of the solution and naturally prevents crossings. However, the correlation path has an impact on the solution, so we propose strategies to randomize this path based on well distances. This is achieved either using geographic distances between boreholes or distances coming from local pairwise correlations. We test these strategies on a synthetic benchmark data set generated from a forward stratigraphic model, and we evaluate the effect of the randomization strategies on the produced correlation outcomes. Finally, we discuss the implications of this work both for finding the optimal correlation and for the generation of a diverse but acceptable set of scenarios.

**Keywords:** Multiple sequence alignment, Uncertainty, Dynamic Time Warping.

<footnote n="1">¹ Université de Lorraine, CNRS, GeoRessources, ENSG, F-54000 Nancy, France</footnote>

<footnote n="2">² Institut universitaire de France</footnote>

<footnote n="3">³ ASGA (Association Scientifique pour la Géologie et ses Applications, F-54000 Nancy,
France</footnote>

<footnote>Guillaume.Caumon@univ-lorraine.fr</footnote>
# 1 Introduction

Well correlation is an important interpretation task to characterize and quantify stratigraphic systems. Indeed, correlations affect the geometry and connectivity of stratigraphic units. Because correlations also define relative geological time, they also define the coordinate system in which geostatistical inference and modeling are performed [1]. Traditionally, well correlation is performed by sedimentologists who instill a significant amount of expert knowledge and interpretive concepts in well data analysis and interpretation. This approach is well suited to the complexity of the task, but it is time consuming and it does not capture interpretation uncertainties in a reproducible way.

Since the 1980's, Dynamic Time Warping (DTW) has emerged as a method of choice to automate the correlation between two wells [2–5]. DTW finds the best association between all the samples of two sequences (or wells) $W^1$ and $W^2$ using dynamic programming. The idea of DTW is to stretch or squeeze signals to obtain the best match. Consistently with stratigraphic superposition principles, the method guarantees that correlation lines never cross, so it should only be applied to wells in which relative geological time is known to increase with depth. DTW relies on sample association costs which can be visualized in a cost matrix, as well as gap costs corresponding to the insertion of a stratigraphic hiatus in either sequence. Correlating two sequences of $n_1$ and $n_2$ samples is performed in $O(n_1 \times n_2)$ complexity with DTW, by iteratively selecting the best out of three possible transitions in the cost matrix [6]. Whereas this approach can be extended to multiple wells with multi-dimensional cost matrices, it has been shown to be NP-hard [7]. Therefore, most approaches proceed by sequential applications of pairwise DTW to reduce the complexity, but this may lead to sub-optimal correlations, and may even generate inconsistent results if loops exist in the correlation path [4]. In this case, incremental application of DTW can be applied iteratively to ensure the consistency of the solution [3, 4]. Recently, Wheeler and Hale [8] proposed to consider the results of multiple pairwise correlations followed by a least-squares computation of the associated shifts. However, this process may also lead to crossings (or negative shifts), which call for correcting the solution, e.g., by replacing reversed sequences by gaps [5].

Incremental correlation naturally avoids such inconsistencies by aggregating sets of wells along a correlation path [9–11], which ensures the absence of crossing when the path does not include loops. In this paper (Section 2), we describe an incremental correlation method (WeCo) inspired by progressive approaches for multiple sequence alignment in bioinformatics [12–15]. This method hierarchically correlates two groups of correlated wells. As all incremental approaches, this method ensures that no sequence reversal occurs on the solution, but it does not guarantee the optimality of the solution.

Another difficult question in well correlation, be it manual or automatic, relates to the management of uncertainties. Indeed, wells provide sparse and ambiguous views on stratigraphic formations: well logs and core analysis only offer an indirect view on rocks and their depositional history, and lateral facies variations may occur within a stratigraphic unit. In automatic correlation, two main approaches have been proposed to address uncertainty by generating several possible sets of correlations between two
wells. Randomized DTW [1, 4] randomly samples the transition at each step of the DTW algorithm instead of selecting the best transition. N-best DTW [16, 17] uses a more systematic approach and tracks the N optimal correlation paths during the algorithm. In this paper, we use N-best DTW in incremental multi-well correlation to propagate a set of possible solutions (Section 2). Additionally, we use the order by which wells are integrated into the process to produce a diverse set of solutions (Section 3).

Section 4 presents some results on a synthetic benchmark model generated from a forward stratigraphic simulation, together with some visualization tools to aggregate and analyze the various solutions.

# 2 Incremental multi-well correlation: the WeCo approach

## 2.1 Multi-well correlation: problem statement

While the correlation of two wells has been widely studied and can be achieved very efficiently with dynamic programming [2, 6], the correlation of multiple wells remains a difficult computational challenge.

Consider a set of M subvertical wells $\Omega = \{W^1, W^2, ..., W^M\}$, each well $W^m$ consisting of an ordered sequence of samples $S^m = [s_1^m, s_2^m, ..., s_{N_m}^m]^T$ and a geographic position $x_m = (x_m, y_m)$. Each sample $s_i^m$ may represent one or several well logs values or interpretations derived from interpretations of the well data, and is generally associated by a measured depth (or a depth range) along the well path. For simplicity, we will not consider depths at this stage, and only assume that the samples follow the stratigraphic superposition principles, i.e., that deeper samples are older. In relative chronostatigraphic terms, the goal of multi-well correlation is to find, for all possible relative geological times $\tau$, all the subsets $C^\Omega(\tau) = \{s_{f_1(\tau)}^1, s_{f_2(\tau)}^2, ..., s_{f_M(\tau)}^M\}$ of age $\tau$.

Multiple correlation (or multiple alignment) aims at finding the optimal functions $f^m(\tau)$ that give the index of the sample of age $\tau$ in well $W^m$, under the constraint that $f^m(\tau_1) \ge f^m(\tau_2)$ for any two ages $\tau_1 > \tau_2$ (note that the case where $f^m(\tau_1) = f^m(\tau_2)$ may occur when the sampling interval is too large, or when stratigraphic gaps are present). In practice, as the number of samples is finite, the deterministic outcome of multi-well correlation can be stored as a sequence of sample subsets $C^\Omega = [C^\Omega(\tau_{min}), ..., C^\Omega(\tau_{max})]^T$. Traditionally, a correlation outcome is visualized using lines connecting the samples of $C^\Omega(\tau_i)$ between several wells.

Because the shift functions $f^m(\tau_i)$ at each well $W^m$ are non-unique, multiple alignment seeks functions that minimize the global cost function

$$ \Phi^{\Omega} = \sum_{\tau_i=\tau_{min}}^{\tau_{max}} \varphi(C^{\Omega}(\tau_i)) + \sum_{\tau_i=\tau_{min+1}}^{\tau_{max}} \psi(C^{\Omega}(\tau_{i-1}), C^{\Omega}(\tau_i)), \quad (1) $$

where $\varphi(C^{\Omega}(\tau_i))$ is a lateral association cost function penalizing unlikely horizontal correlations, and $\psi(C^{\Omega}(\tau_{i-1}), C^{\Omega}(\tau_i))$ is a transition cost function penalizing unlikely vertical successions. In this paper, we simply consider $\varphi$ as the minimal variance for
the samples in $C^\Omega(\tau_i)$, and use a small constant value for $\psi$, but more elaborate examples of cost functions can be considered [11, 17]. Note that in the absence of absolute datings (e.g., isotopes), these cost functions are generally moderately informative, in the sense that they leave room for many acceptable correlation scenarios.

<figure><img src="image_4.png" /><figcaption>**Fig 1:** Incremental correlation and representation of multiple correlation results with a directed acyclic graph in WeCo. By default, the wells are placed in a binary search tree (Kd-tree) (b) based on their geographic location (a). Each non-terminal node of the tree corresponds to a correlation task $T_{\{C^L-C^R\}}$, where $C^L$ and $C^R$ refer either to wells or outcomes of previous correlation tasks. Each outcome of a correlation task is stored as a directed acyclic graph $C^{LR}$ (c).</figcaption></figure>

## 2.2 Multiple correlation with WeCo

The multiple alignment problem has been much addressed in bioinformatics, where ‘progressive’ approaches have been introduced [12–15]. For instance, CLUSTAL [12] starts by computing pairwise correlation costs between *M* sequences. The resulting $\binom{N}{2}$ distances between the sequences are analyzed to establish a phylogenetic tree (orrogram) that clusters all pairs hierarchically. This tree is then traversed to incrementally align larger and larger groups of sequences that are more and more dissimilar. WeCo uses the same idea to incrementally correlate groups of correlated wells. In the default implementation, a binary subdivision tree (Kd-tree) is generated based on the well positions (Fig. 1a-b). The root of the tree corresponds to the whole set of wells $\Omega$. Branches recursively split the well set of the parent node into two balanced subsets. At each step of the tree construction, the *x* or *y* axis is selected based on the longest edge of the bounding box of the current well subset, and the threshold is selected so that children branches have approximately the same size. In the end, the wells $W^m$, $m = \{1, 2, ..., M\}$ form the leaves of this binary tree. Each branch of the tree corresponds to
a pair correlation task $T_{\{C^L-C^R\}}$, where $C^L$ and $C^R$ denote the left and right solutions to correlate. Some of these pair correlations can be run in parallel (for example, in Fig. 1b, task $T_{4-5}$ may be executed concurrently with tasks $T_{12}$ and $T_{12-3}$), but the binary tree uniquely defines the order of dependent correlation tasks.

Each deterministic correlation yields a sequence $C^{LR}$ whose nodes aggregate samples of the right and left subsets connected by a line (or, equivalently, having the same geological age). In the presence of gaps, some nodes are duplicated along one well. An example of such a sequence is illustrated by black arrows on the right-hand side of Fig. 1c, corresponding to dashed black correlation lines.

However, considering only the best solution for each of these binary correlation tasks may introduce a strong dependency of the global multi-well solution. To alleviate this influence, WeCo considers the set of *N*-best solutions at each stage of incremental correlation. Each correlation task $T_{\{C^L-C^R\}}$ then consists in correlating the set of right and left solutions obtained during previous tasks. For this, WeCo starts by representing the set of solutions $C^{LR}$ obtained at each step as a directed acyclic graph (DAG), which share the same initial and terminal nodes. In each DAG, the common parts of the paths (corresponding to the same correlation lines) are merged. Conversely, branches occurring in the graph correspond to differences in correlations. By construction, the DAG representing sets of solutions do not have cycles.

Figure 1c illustrates the DAG creation on a simple example with two wells, considering two solutions represented by the dashed black and red lines. Note that the mapping between a set of *N* correlations solutions and a DAG is not bijective in general, as *K* ≥ *N* solutions can be generated from the DAG. This is not a problem in practice because larger sets of solutions are preferable to avoid local minima. If needed (for instance to spare computer memory), the *N* best solutions can still be recovered by ranking all possible solutions in the DAG by their total correlation cost.

The last component of WeCo is a generalized DTW algorithm that correlates not just sequences but either wells, deterministic correlation outcomes or sets or correlation outcomes. All these three options can be represented as a DAG, so the generalized DTW solves the correlation task $T_{\{C^L-C^R\}}$ that creates the $C^{LR}$ correlation outcome from the input DAGs $C^L$ and $C^R$. This generalized DTW is achieved by using the same dynamic programming principles as in the original DTW algorithm [2, 6]. The main difference is that there may be more than three ways to transition from one node to the next.

### 3 Considering alternative correlation sequence orders

The incremental correlation described in Section 2 guarantees the absence of reversals by building a unique and sensible order for correlation tasks. However, the correlation outcomes generally depend on this order (or path). This problem has received significant attention in the area of bioinformatics [13–15]. In geosciences, linear incremental correlation paths have been proposed [9, 10]. For example, Wu et al. [10] look for an optimal correlation path by mixing well length (to start preferentially from complete
stratigraphic sections), and geometric distances distorted using discontinuities identi-
fied on seismic data. WeCo can, in principle, reduce the effect of correlation order by
considering a large enough set of N of optimal correlations outcomes at each step of
the incremental method. In spite of the DAG optimization, this strategy, however, is
costly in terms of RAM and CPU costs. Moreover, in practice, even a relatively large
number of scenarios (e.g., N = 1000) only capture a small fraction of all the possible
solutions. In our experience, it appears that the N-best approach tends to generate solu-
tions which only vary locally, hence tend to under-estimate the diversity of acceptable
correlation scenarios.

In this section, we rather consider this dependence on the correlation path to generate
a diverse set of solutions. To allow for more path variations than allowed by the default
WeCo binary tree, we start by defining a graph connecting nearby wells to define alter-
native incremental correlation tasks. As in [5], this graph can be generated by running
a Delaunay triangulation on the well positions **x**<sub>m</sub>, m = {1,2,...,M}, or by forming the
K-nearest neighbors of each well, with an optional maximum distance threshold.

<figure><img src="image_5.png" /><figcaption>**Fig. 2:** From an initial graph (e.g., a Delaunay triangulation, **a**) and the associated inter-well distances (numbers on graph edges), disjoint pairs of wells are ranked by increasing distances (**b**). Then, correlation tasks are run and the process is repeated by computing distances between previous correlation outcomes (**c**) or remaining isolated wells until a global correlation is obtained. The overall correlation order can be displayed as a binary tree (**d**).</figcaption></figure>
Then, distances are associated to the edges of this graph to determine the incremental
correlation sequence order, starting with the shortest isolated pairs of wells: first, the
list of graph edges is sorted by increasing distance, and disjoint shortest edges are se-
lected to create the corresponding correlation tasks (Fig. 2a and first line of Fig. 2d).
Then, the graph is updated by considering the average distance of the already correlated
wells, and the process is repeated until all pairs of wells groups have been processed
(Fig. 2 b-c). Using different distances before applying this algorithm can generates al-
ternative incremental correlation paths.

**4 First results and discussion**

**4.1 Reference data set: bathymetry data in a forward stratigraphic model**

A general difficulty in stratigraphic correlation is the absence of high-quality reference
3D data sets. Therefore, we consider a data set made of 8 vertical wells drilled virtually
in the 3D model generated with the diffusion-based forward stratigraphic modeling
software DionisosFlow [18], as described by Edwards et al. [11]. Figure 3 shows the
reference depositional bathymetry model and the location of virtual wells. The grid
layering provides the reference geological time corresponding to the forward simula-
tion time steps.

**4.2 Experiments using bathymetry to estimate isochron correlation lines**

For the correlation, we use the bathymetry log taken from the stratigraphic model. Us-
ing this information is a challenging stress test for the method, because of lateral vari-
ations of bathymetry along E/W lines. Therefore, we consider the *vertical derivative* of
the bathymetry along the wells as sample data **S** instead of the bathymetry itself. This
aims at capturing eustatic level changes as a proxy for depositional age, under the as-
sumption that the sedimentation rate is homogeneous laterally.

<figure><img src="image_6.png" /><figcaption>**Fig. 3:** 3D stratigraphic grid produced by a forward stratigraphic modeling software (a) and well location map with bathymetry map for the uppermost horizon (b)</figcaption></figure>
<table><caption>**Table 1.** Metrics and orderings for some of the produced correlation outcomes.</caption><thead><tr><th>Graph, distance method</th><th>Task Ordering</th><th>Global correlation cost Φ<sup>Ω</sup></th><th>Score Δ<sup>Ω</sup></th></tr></thead><tbody><tr><td>Kd-Tree</td><td>[[[56] [[23]4]] [[78][01]]]</td><td>47.9</td><td>10.8</td></tr><tr><td>Delaunay, L<sub>2</sub> distance</td><td>[[[08] [[12][34]] [[56]7]]</td><td>51.4</td><td>14.9</td></tr><tr><td>Delaunay, Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[67][08]] [35]] [12]] 4]</td><td>48.8</td><td>13.7</td></tr><tr><td>Delaunay, (1 + r<sub>1</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[45][67]] [[08]3]] [12]]</td><td>49.5</td><td>13.8</td></tr><tr><td>Delaunay, (1 + r<sub>2</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[67]4] [12]] [35]] [08]]</td><td>48.6</td><td>13.5</td></tr><tr><td>Delaunay, (1 + r<sub>3</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[67][25]] [18]] [03]] 4]</td><td>49.5</td><td>12.7</td></tr><tr><td>Delaunay, (1 + r<sub>4</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[45][67]] [[03][12]] 8]]</td><td>51.9</td><td>12.8</td></tr><tr><td>Delaunay, (1 + r<sub>5</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[[67][18]] [35]] 4] 0] 2]</td><td>45.7</td><td>13.3</td></tr><tr><td>Delaunay, (1 + r<sub>6</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[67][08]] [[13][25]]] 4]</td><td>50.6</td><td>13</td></tr><tr><td>Delaunay, (1 + r<sub>7</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[03] [[67][45]]] [[12] 8]]</td><td>48.7</td><td>12.2</td></tr><tr><td>Delaunay, (1 + r<sub>8</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[08] [[67][45]]] [[13] 2]]</td><td>47.2</td><td>12.8</td></tr><tr><td>Delaunay, (1 + r<sub>9</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[25] [[67][08]]] [[13] 4]]</td><td>51.2</td><td>12.7</td></tr><tr><td>Delaunay, (1 + r<sub>10</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[[67] 8] [45]] [03]] [12]]</td><td>49.1</td><td>12.7</td></tr><tr><td>Delaunay, (1 + r<sub>11</sub>)Φ<sup>W<sup>i</sup>W<sup>j</sup></sup></td><td>[[[[45][67]] [[01]8]] [23]]</td><td>47.3</td><td>10.3</td></tr></tbody></table>

The multi-well correlation is run using several ordering methods, using only the five best correlations during each incremental task: a base correlation outcome obtained with the spatial subdivision tree; two outcomes produced with Delaunay triangulation using the geometric distance and pairwise DTW costs ($\Phi^{W^iW^j}$ on each Delaunay edge connecting wells $W^i$ and $W^j$, Fig. 4c); 20 additional correlation outcomes obtained by adding to the pairwise DTW cost $\Phi^{W^iW^j}$ a uniform scaling $r_i \sim U[-0.8, 0.8]$. Table 1 presents the corresponding ordering tasks and the obtained global correlation costs for the best correlation outcome of each run. The score $\Delta^\Omega$ for each solution measures to what extent the best correlation outcome differs from the reference geological time $\tau_i^{\text{ref}}$:

$$ \Delta^{\Omega} = E\{\text{StdDev}(\tau_i^{\text{ref}})\}, \qquad (2) $$

where the expectation $E\{\cdot\}$ is computed over all correlation lines and the standard deviation $\text{StdDev}(\cdot)$ is computed over all the wells.

## 4.3 On cost function bias, visualization and uncertainty assessment

These results show that the optimal multi-well solution in terms of global cost function is not necessarily the most accurate one. This stems from the inappropriate choice of correlation cost functions, which minimize the variations of the bathymetry gradient. We posit that errors probably come from the absence of consideration for erosions,
which introduce noise in the estimated bathymetry gradient, and inappropriate assumption of a steady sedimentation rate. Another possible source of error may also come from the equal weight given to all samples in the selected cost function. Weighting the samples by increasing distances as in [10] could possibly address this problem. Nonetheless, log-based correlation remains daunting in general when lateral facies variations are present or suspected, so this clearly deserves further investigation to better integrate geological concepts and diverse types of data to reduce uncertainty [4, 11, 17, 19].

In terms of uncertainty assessment, the proposed randomization strategy achieves its objective by producing a relatively large set of scenarios to explore correlation uncertainties consistently with the chosen cost function. Figure 4 propose two types of visualizations for the various correlation outcomes on a SW-NE transect. Figure 4a simply overlays two outcomes obtained with the default solution using the Kd-Tree and the solution obtained with Delaunay triangulation and pairwise cost functions, corresponding to the first and third lines of Table 1. The bathymetry derivative and the reference geological age are shown for reference in the left and right well track, respectively. Figure 4b displays all the correlation lines corresponding to lines of the solution matrices $C^{\Omega}$ colored by their frequency of apparition.

On the considered example, results suggest that correlation uncertainties are mild between wells 1 and 2. This relative stability is observed despite different correlations paths which do not necessarily include these two wells one after the other. Interestingly, these two wells also have a relatively small score with regard to the reference time lines. This is consistent with the location of these two wells which have the same distality with regard to the depositional system, which makes the log-based correlation relevant because little lateral facies variations are expected. Conversely, areas of large variability from one solution to the next might suggest the inadequacy of the correlation principles embedded into the method.

# 5 Conclusions

Well correlation is a difficult challenge prone to many uncertainties, for which computational methods can be of significant value to help interpreters quantify uncertainty and increase the degree of reproducibility as compared to manual correlations. We have presented a new incremental multi-well correlation method that aims to find the right balance between optimization and exploration of the correlation search space. The order of correlations appears as a powerful way to generate a diverse set of correlations consistent with the correlation rules. The visualization of results can provide interesting insights about where correlation uncertainties are important for a given rule set. This approach opens the door to more research on more realistic correlation rules able to better integrate the available information, and ultimately increase the predictability of subsurface models.
<figure><img src="image_7.png" /><figcaption>**Fig. 4:** Correlation uncertainty visualization. Correlation outcomes can be compared using two colors for two solutions (a). For a larger set of solutions (here, the 20 solutions of the randomized pairwise DTW distances), the correlation outcomes are aggregated, and the correlation lines are colored by frequency, highlighting the stable areas and the more uncertain areas.</figcaption></figure>

**Acknowledgments.** We acknowledge the support for the RING Consortium managed by ASGA (www.ring-team.org/Consortium) for supporting the WeCo development. We also acknowledge Beipic for academic licenses of DionisosFlow and AspenTech for academic licenses of SKUA-GOCAD. Finally, we thank Florent Lallier, Jonathan Edwards, Paul Baville, Xinming Wu and Dave Hale for earlier discussions on well correlations algorithms.

**Disclosure of Interests.** The authors declare no competing interest.
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