Reservoir Petrofacies Predicted Using Logs Data: A Study of Shale Oil from Seven Members of the Upper Triassic Yanchang Formation, Ordos Basin, China

Abstract

The identification and prediction of petrofacies plays a crucial role in the study of shale oil and gas “sweet spots”. However, the petrofacies identified through core and core test data are not available for all wells. Therefore, it is essential to establish a petrofacies identification model using conventional well logging data. In this study, we determined the petrofacies of shale oil reservoirs in the Upper Triassic Yanchang Formation, Ordos Basin, China, based on scanning electron microscopy, core porosity and total organic carbon (TOC), and brittleness index calculations from X-ray diffraction (XRD) experiments conducted on seven members of the formation. Furthermore, we compared the interpreted logs with the raw well logs data clustered into electrofacies in order to assess their compliance with the petrofacies, using the Multi-Resolution Graph-Based Clustering (MRGC) method. Through an analysis of pore structure type, core porosity, TOC, and brittleness index, we identified four types of lithofacies with varying reservoir quality: PF A > PF B > PF C > PF D. The compliance of the clustered electrofacies with the petrofacies obtained from the interpreted logs was found to be 85.42%. However, the compliance between the clustered electrofacies and the petrofacies obtained from the raw well logs was only 47.92%. Hence, the interpreted logs exhibit a stronger correlation with petrofacies characterization, and their utilization as input data is more beneficial in accurately predicting petrofacies through machine learning algorithms.

1. Introduction

Lake-phase mudstone, shale, and oil shale, along with other fine-grained sedimentary rocks, are abundant in the Triassic Yanchang Formation of the Ordos Basin, offering significant potential for unconventional oil and gas resources such as shale oil and gas and oil shale [1]. The classification of reservoir petrofacies serves as the foundation and key to oil and gas exploration, with facies exerting a significant control on the distribution of shale oil and gas. Therefore, it holds immense importance to accurately characterize and predict reservoir petrofacies [2,3]. Petrofacies studies typically rely on core data, encompassing observations of core hand specimens, thin sections, scanning electron microscopy, X-ray diffraction (XRD), and physical property tests [4,5,6,7]. However, the availability of cores is often limited due to constraints in drilling time and cost [3,8]. Geophysical logging data, on the other hand, contain rich petrophysical information, enabling indirect petrofacies classification. Nevertheless, given the complexity and heterogeneity of the reservoir, there exists a considerable amount of redundancy among the log curves, necessitating careful selection of curve attributes and algorithm preferences for accurate petrofacies prediction [5,9,10,11].

The classification of shale petrofacies typically relies on indicators such as hydrocarbon-bearing fracturability, organic matter abundance, sedimentary structure, and mineralogical fractions, which are crucial in assessing their potential for hydrocarbon production [12,13,14]. Venieri et al. (2021) classified organic-rich shales into five lithofacies categories based on core observations, X-ray diffraction (XRD) analyses, and log response characterization [15]. Atchley et al. (2021) classified shale reservoirs into six petrofacies based on gamma ray measurements, density porosity cut-offs, and total organic carbon (TOC) values [14]. Consequently, petrofacies often exhibit distinct characteristics in logging curves, forming the basis for petrofacies prediction through logging data. Currently, many scholars have achieved satisfactory results in petrofacies identification using machine learning algorithms [11,16,17]. In machine learning, facies prediction is typically accomplished through clustering or classification algorithms [18,19]. Clustering algorithms are straightforward, as they do not require prior knowledge regarding the number and attributes of facies, nor do they require training [9]. Among these clustering algorithms, the Multi-Resolution Graph-Based Clustering (MRGC) algorithm has proven successful in log curve reconstruction [20], permeability prediction [21], flow zone unit determination [22], and petrofacies identification [5,23,24].

The shale of the seventh member of the Triassic Yanchang Formation is widely distributed, and it serves as a significant hydrocarbon source rock with substantial resources. At the Ordos Basin, the estimated resource amount reaches 33 × 10⁸ metric tons [25]. In this study, the petrofacies type of the shale oil interval was determined through various methods, including scanning electron microscopy (SEM), core porosity and total organic carbon (TOC) analysis, and X-ray diffraction (XRD) experiments. Furthermore, we assessed the consistency between the electrofacies and the petrofacies derived from interpreted logs and raw well logs, which were clustered and used as input data in conjunction with the MRGC algorithm. Our findings contribute to the development of petrofacies classification schemes for shale reservoirs and provide a viable approach for high-precision petrofacies prediction.

2. Geological Background

The Ordos Basin, located in the western part of the North China Platform, is the second largest sedimentary basin in China. It can be divided into six major tectonic units: the Yimeng Uplift, Weibei Uplift, Western Overthrust Belt, Tianhuan Depression, Yishan Slope, and Jinxi Fold-Fault Belt (Figure 1a). This basin originated on the stable crystalline basement of the Taikonian period and experienced tectonic sedimentary infill evolution from the Paleozoic to the Cenozoic, resulting in the formation of a cover layer with an average thickness of 4–5 km [26]. During the Late Triassic, the southern part of the basin underwent uplift due to the collision between the Yangzi Plate and the North China Plate, serving as the primary source of material for the southern region of the basin [27]. The North China Plate experienced extrusion from the south by the offset of the Yangzi Plate and from the north by the Xingmeng Plate, causing subsidence and forming a slope with angles of 3.5° to 5.5° in the south and 1.5° to 2.5° in the north of the Late Triassic Basin, respectively [26,28]. In the early Late Triassic, the basin’s basement rapidly subsided, leading to the intrusion of lakes and the transportation and accumulation of sediments from the Qinling and Liupanshan regions in the south and the Yinshan Mountains in the north, respectively [26,29].

Our study area is located in the southern part of the Ordos Basin (Figure 1a), and several drilled hydrocarbon wells in the region encounter Triassic formations (Figure 1b). The Upper Triassic Yanchang Formation is stratigraphically divided into 10 sublayers, ranging from YC10 to YC1 (Figure 2). The YC7 reservoir is identified as an unconventional shale reservoir, with a thickness between 100 and 120 m. The lithology is predominantly dark black mud shale interbedded with thin layers of silty mudstone and silty siltstone. During the deposition of the YC7, the lake basin experienced significant tectonic activity and underwent rapid expansion, ultimately becoming the largest lake in the Mesozoic era within the Ordos Basin [32]. The lake-phase mudstone and shale extend in a northwest–southeast direction across the region, corresponding to the semi-deep to deep lake zone. The thickness of these shale deposits ranges from 10 to 120 m, with some areas exhibiting shale layers exceeding 10 m in thickness over an extent of up to 3 × 10⁴ square kilometers [33].

3. Materials and Methods

3.1. Materials and Experiments

Conventional logging data and ECS logging data were collected from two wells in the study area. The locations of the two wells are displayed in Figure 1. The conventional logging data comprise CAL (Caliper), SP (Spontaneous Potential), GR (Gamma ray), PE (Photoelectric Absorption Cross-Section Index), DT (Compressional Slowness), NPHI (Neutron Porosity), DEN (Bulk Density), RD (Deep Resistivity), and RS (Shallow Resistivity).

A total of 48 depth points were sampled to collect core samples for experimental analysis. Porosity measurements were conducted using an ULTRAPORE-200A helium core porosimeter (Core Lab, USA) at a temperature of 25 °C and pressure of 1.025 bar. To determine the total organic carbon (TOC) content in the shale samples, pyrolysis experiments were performed using the Rock-Eval 6 analyzer. The experimental results for core porosity and TOC are summarized in Table 1. X-ray diffraction (XRD) experiments were conducted on the shale samples to analyze the mineral fractions. A MiniFlex 600 instrument equipped with Cu-Kα radiation was used, and the experimental samples were 300 mesh powders. The XRD analysis involved scanning the samples from 5° to 90° in 0.02° increments to obtain precise and accurate results [35]. The microstructure of the samples was examined using scanning electron microscopy, and the corresponding results are presented in Table 2.

3.2. The Multi-Resolution Graph-Based Clustering (MRGC) Algorithms

The MRGC algorithm is an unsupervised clustering algorithm that combines the advantages of the KNN (K-Nearest Neighbors) algorithm and graph theory algorithm [36]. Two important parameters in this algorithm are the neighbor index (NI) and the kernel representative index (KRI). In the context of considering the sample set as an attraction set with an attraction relationship, the NI represents the ability of a sample point to attract all other sample points within the attraction set. A higher NI value indicates that the point is closer to the core of a class [20]. On the other hand, the KRI reflects the ability of the current attraction set to act as a kernel for fusing other attraction sets. A higher KRI value signifies that the attraction set has a greater dominance in the fusion process [37].

Let’s assume that there are N sample points in the sample set S to be classified, denoted as S = {$x$₁, $x$₂, …, $x$_i, …, $x$_N}. Each sample point $x$ in the sample set S is represented as a vector with multiple attributes, and y represents the nth closest sample point to $x$ in the sample set S, where n ≤ N − 1. In order to determine the point at which no further attraction exists between sample points, a threshold K is set, indicating the Kth nearest neighbor. The attractiveness of a sample point x towards its nth sample point can be mathematically expressed as follows:

(1)${\delta }_n\left(x\right)=\left\{\begin{array}{c}{e}^{-\frac{m}{\alpha }},x is the mth nearest neighbor sample point of y,m\le K \\ 0 , x does not belong to the set of K nearest neighbors of y\end{array}\right.$

where α represents the smoothing factor greater than zero [23]. For each sample point in the set, the following calculations are performed:

(2)$S\left(x\right)=\sum _{i=1}^{N-1}{\delta }_n\left(x\right)$

The neighbor index of a sample point x is determined by the normalized value of the $S\left(x\right)$ function and can be mathematically expressed as follows:

(3)$NI\left(x\right)=\frac{S\left(x\right)-S_{min}}{S_{max}-S_{min}}$

Among them,

(4)$S_{min}=Min{\left\{S\left(x_i\right)\right\}}_{i=\mathrm{1,2},\dots ,N}$

(5)$S_{max}=Max{\left\{S\left(x_i\right)\right\}}_{i=\mathrm{1,2},\dots ,N}$

$S_{min}$ and $S_{max}$ are the minimum and maximum values of the $S\left(x\right)$ function, respectively; $NI\left(x\right)$ lies between 0 and 1.

The values $S_{min}$ and $S_{max}$ represent the minimum and maximum values of the $S\left(x\right)$ function, respectively. The neighbor index $NI\left(x\right)$ is a normalized value that falls within the range of 0 and 1.

The neighbor index (NI) is a localized index derived from the kernel representative index (KRI) that incorporates both the neighbor index of the current sample point and the neighboring relationships and spatial distances of the sample points. The KRI can be mathematically expressed as follows:

(6)$KRI\left(x\right)=M\left(x,z\right)\times NI\left(x\right)\times D\left(x,z\right)$

where z represents the nearest neighbor sample points to the sample point $x$ based on the proximity index. M($x$, z) denotes the neighborhood number of the sample point $x$ relative to the sample z. D($x$, z) represents the distance function, commonly calculated using the Euclidean distance formula.

The kernel representative index (KRI) is sorted in descending order to create a curve, which exhibits multiple inflection points that mark the transition from one smooth segment to another. Each inflection point indicates a change in the classification level, representing different levels of clustering results [24]. Based on specific facies analysis requirements, users have the flexibility to set parameters such as the maximum number of clusters, the minimum number of clusters, and the maximum number of optimal clustering schemes. Through analysis and calculation, the MRGC algorithm can automatically identify and compare several optimal clustering schemes, enabling users to efficiently delineate electrofacies [5,23].

4. Results

4.1. Petrofacies

The total organic carbon (TOC) content, reservoir properties, and fracturability play crucial roles in the development of shale oil and gas reservoirs. In this study, a total of 48 core samples were collected to analyze porosity, TOC content, and mineral fractions. In a related study by Kang et al. (2020), a novel mineralogical brittleness index was introduced, which considers the presence of brittle minerals in shale formations [38]. The mineralogical brittleness index can be mathematically defined as a function of the weights of brittle minerals:

(7)${BI}_{bm}=\left(W_Q+0.49\times W_F+0.51\times W_c+0.44\times W_D\right)/W_T$

where $W_Q$, $W_F$, $W_c,$ and $W_D$ represent the weights percentage of quartz, feldspar, calcite, and dolomite, respectively. Additionally, $W_T$ denotes the total mineral weight percentage (=100%). The brittleness index was calculated for the mineral fractions obtained through experimental analysis. The results of the brittleness index calculations are summarized in Table 1.

The “sweet spot” criteria for a shale reservoir typically involve high porosity, total organic carbon (TOC) content, and brittleness index [15,39,40]. In this study, we examined the correlation between core TOC, porosity (POR), and the mineralogical brittleness index (BI_bm). TOC and POR showed a positive correlation based on the distribution of most points (Figure 3a). TOC and BI_bm showed a negative correlation based on the distribution of most points (Figure 3c). There was little correlation between POR and BI_bm (Figure 3b). For the “sweet spot” of shale oil reservoirs, we prefer reservoirs with high POR, high TOC, and high brittleness index. Consequently, the petrofacies classification necessitates a trade-off between BIbm and TOC.

Based on SEM, core porosity, TOC, and brittleness index, we classified petrofacies into four types (Table 2).

Petrofacies A: High porosity (1.48–4.70%, average 2.17%), high TOC (2.29–15.19%, average 6.35%), and high BI_bm (25.77–38.95%, average 32.87%). Clay mineral interlayer pores, fractures, and interparticle pores are developed.

Petrofacies B: Median porosity (1.34–3.90%, average 2.01%), median TOC (2.89–8.03%, average 5.04%), and high BI_bm (23.33–38.17%, average 29.76%). Intraparticle pores and interparticle pores are developed.

Petrofacies C: Low porosity (0.80–2.62%, average 1.61%), median TOC (3.16–7.26%, average 5.10%), and low BI_bm (10.25–29.87%, average 24.78%). Clay mineral interlayer pores and fractures are developed; interparticle and intraparticle pores are sparsely developed.

Petrofacies D: Ultra-low porosity (0.50–1.44%, average 1.01%), low TOC (0.48–9.93%, average 4.17%), and high BI_bm (17.12–50.67%, average 30.29%). Clay mineral interlayer pores and fractures are developed.

4.2. Electrofacies

We performed electrofacies analysis by the MRGC algorithm using two types of logging data: one with interpreted logs (Model 1) and the other with raw well logs (Model 2). We compared the predictive effectiveness of both models.

4.2.1. Electrofacies from Interpreted Logs

POR and TOC were interpreted using conventional logging using the method proposed by Yu et al. (2017, 2018) [39,40]. The brittleness index was calculated using ECS logging, using the method proposed by Kang et al. (2020) [38]. Interpreted logs (POR, TOC, and BI) were used as input data for clustering to obtain electrofacies via the MRGC algorithm. Interpreted logs ultimately classify electrofacies into four types (Figure 4). EF A: high porosity, high TOC, and high brittleness index; EF B: median porosity, median TOC, and high brittleness index; EF C: median porosity, median TOC, and low brittleness index; EF D: tight porosity, low TOC, and high brittleness index. The electrofacies types obtained by clustering have more consistent characteristics with petrofacies types.

Electrofacies clustering results for tight oil reservoirs from two wells are presented in log plots (Figure 5). The 10th track shows petrofacies obtained by core SEM, core porosity, core TOC, and brittleness index calculated from XRD. The 11th track shows the classification results of electrofacies from the interpreted logs data using the MRGC algorithm (Model 1). The core-based petrofacies are in agreement with the model 1 electrofacies. The electrofacies obtained from model 1 clustering are 85.42% compatible with the core-based petrofacies.

4.2.2. Electrofacies from Raw Well Logs

Electrofacies are often clustered using conventional logging data [5,41]. The conventional log data selected for this study include AC, DEN, NPHI, GR, PE, RD, and RS. Raw well logs were used as input data for clustering to obtain electrofacies via the MRGC algorithm. Raw well logs ultimately classify electrofacies into four types (Figure 6). EF A: low AC, low DEN, low NPHI, high GR, high PE, high RD, and high RS; EF B: low AC, low DEN, high NPHI, high GR, high PE, median RD, and median RS; EF C: low AC, median DEN, median NPHI, low GR, high PE, low RD, and low RS; EF D: high AC, high DEN, high NPHI, low GR, low PE, low RD, and low RS. Raw well logs curves indirectly respond to the properties of the reservoir.

Electrofacies clustering results for tight oil reservoirs from two wells are presented in log plots (Figure 5). The 12th track shows the classification results of electrofacies from the raw well logs data using the MRGC algorithm (Model 2). Electrofacies are characterized by homogeneity at successive depths and show weak sensitivity to petrofacies changes. The core-based petrofacies are in poor agreement with the model 2 electrofacies. The electrofacies obtained from model 2 clustering are 47.92% compatible with the core-based petrofacies.

5. Discussion

This paper classifies petrofacies of shale oil reservoirs based on SEM, core porosity, core TOC, and mineralogy of brittleness index calculated from XRD experiments from seven members of the Upper Triassic Yanchang Formation, Ordos Basin, China. Petrofacies are initially defined as “intervals of rock with a similar average pore throat radius, thus having similar fluid flow characteristics” [42,43]. Many authors also define petrography based on a combination of core analysis characteristics, petrographic features (grain size, sorting, mineralogy, and pore type), and well logging characteristics [2,44,45]. Our petrofacies classification scheme focuses on four key criteria of shale reservoirs: porosity and pore types determine the reservoir’s storage performance and fluid endowment state [46,47], TOC determines the reservoir’s hydrocarbon potential [48,49], and brittleness index determines the reservoir’s remodeling ability [50,51,52]. Venier et al. identified five petrofacies through core hand specimens, mineralogical composition obtained from XRD experiments, TOC, and logging responses [15]. We consider the mineralogical brittleness index obtained from XRD experiments to be a more direct criterion for the classification of petrofacies.

We used interpreted logs and raw well logs as input data to obtain four types of electrofacies by clustering with the MRGC algorithm and established the mapping relationship with petrofacies. Further comparisons were made between the performance of the two types of data modeling for petrofacies prediction. MRGC is one of the few non-parametric methods that is well suited for the learning and clustering of analyzed data from logs and drilled cores [5]. Many scholars have modeled the identification of facies using logging data via the MRGC algorithm [5,20,53,54]. The properties of the interpreted logs are more closely related to the characteristics of the petrofacies, and the raw well logs contain redundant signals that can only indirectly reflect the characteristics of the petrofacies. A limited set of good input datasets is better than a larger set of attribute input datasets, as it reduces data redundancy and improves clustering efficiency [9]. Analyses of the available data confirmed this conclusion. The number of interpreted logs is low but more closely related to petrofacies. Therefore, model 1 based on interpreted logs as the input has a higher accuracy for the identification of petrofacies compared to model 2 based on raw well logs as the input (Figure 5).

6. Conclusions

Petrofacies are an important tool that represents different reservoir qualities [2] and can be modeled for logging identification [9]. In this study, we classified the shale oil reservoir into four petrofacies based on the shale’s pore structure, porosity, TOC, and brittleness index through experimental data from the seven members of the Upper Triassic Yanchang Formation, Ordos Basin, China. PF A is characterized by optimal reservoir properties and has good reservoir performance, hydrocarbon potential, and fracturability. PF B is inferior to PF A in terms of reservoir performance and hydrocarbon potential. PF C has a poor brittleness index, making it unsuitable for fracture modification. PF D cannot be used as an effective reservoir.

Interpreted logs and raw well logs data were used as input clustering to obtain electrofacies using the MRGC algorithm. The clustered electrofacies of the raw logging data showed poor sensitivity to changes in petrofacies. In contrast, the clustered electrofacies from the interpreted logs matched the petrofacies more closely. Therefore, extracting interpreted logs related to petrofacies from conventional logs is necessary to improve the accuracy of petrofacies prediction.

Author Contributions

Conceptualization, K.M. and H.Y.; methodology, K.M. and M.W; software, Y.J. and C.M.; validation, S.Z. and H.Y.; investigation, S.Z. and Y.J.; data curation, M.W.; writing—original draft preparation, K.M. and P.X.; writing—review and editing, M.W., J.Z. and H.Y.; visualization, K.M. and P.X.; supervision, H.Y., J.Z. and M.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to restrictions of privacy.

Acknowledgments

The authors thank the reviewers for their valuable comments and are grateful to the editor for careful editing.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. (a) Major structural units of the Ordos Basin and the location of the study area [30]; (b) cross-section (A to A′) in (a) of the Ordos Basin, showing the various tectonic units and strata (Triassic rocks in yellow) [31].

Figure 2. Stratigraphic subdivision of the Yanchang Formation in the research area [2,34].

Figure 3. (a) The cross-plot of core TOC and porosity; (b) the cross-plot of core BI and porosity; and (c) the cross-plot of core BI and TOC.

Figure 4. Final classified characterization of interpreted logs for electrofacies model 1, using MRGC algorithm (The green line is the Gaussian curve of the frequency histogram; the black line is the cumulative percentage of the frequency histogram).

Figure 5. The clustered electrofacies results from the two types of models via the MRGC algorithm. (a) YY22; (b) YY28.

Figure 6. Final classified characterization of raw well logs for electrofacies model 2, using MRGC algorithm (The green line is the Gaussian curve of the frequency histogram; the black line is the cumulative percentage of the frequency histogram).

Table 1

The petrophysical features of the four types of petrofacies. Porosity and TOC results were obtained from core experiments, and brittleness index calculation results were obtained from XRD experiments.

NO.	POR(%)	TOC(%)	BIbm(%)	PF	NO.	POR(%)	TOC(%)	BIbm(%)	PF
1	1.25	1.44	36.28	PF D	25	1.86	3.59	29.43	PF D
2	1.77	3.21	36.67	PF D	26	2.33	3.16	29.87	PF D
3	0.68	3.45	28.81	PF D	27	1.79	3.87	24.82	PF D
4	0.70	2.84	29.29	PF D	28	1.40	2.68	24.33	PF D
5	0.96	1.48	43.85	PF D	29	0.39	2.12	20.89	PF D
6	2.03	3.75	26.75	PF B	30	1.15	0.88	50.67	PF D
7	1.47	4.12	24.21	PF B	31	1.15	0.88	32.503	PF C
8	1.87	4.77	25.21	PF D	32	1.40	4.89	33.917	PF C
9	2.08	4.43	34.12	PF B	33	1.80	6.11	26.619	PF C
10	2.58	2.89	34.38	PF B	34	1.10	5.22	22.703	PF B
11	1.61	4.93	27.19	PF C	35	1.70	5.16	30.889	PF B
12	2.12	4.55	25.73	PF B	36	1.90	7.26	24.224	PF C
13	1.61	4.18	32.19	PF B	37	0.90	7.93	21.579	PF B
14	1.05	3.34	30.78	PF D	38	1.90	3.11	38.167	PF D
15	2.24	2.9	25.77	PF A	39	1.40	6.19	10.254	PF C
16	1.82	3.23	30.66	PF A	40	3.70	9.19	38.946	PF B
17	1.91	4.49	27.82	PF A	41	1.40	7.87	23.333	PF B
18	1.56	5.61	31.22	PF B	42	2.10	6.38	28.549	PF B
19	2.12	5.19	27.84	PF C	43	1.50	5.2	29.125	PF D
20	1.87	3.74	25.5	PF C	44	2.00	6.7	22.759	PF C
21	2.19	4.92	21.33	PF C	45	1.60	6.62	17.123	PF B
22	1.48	5.38	28.31	PF B	46	2.60	7.03	26.764	PF B
23	1.56	5.97	37.2	PF C	47	1.50	6.13	33.026	PF D
24	1.95	2.29	36.84	PF C	48	1.70	3.2	26.964	PF D

Table 2

Characterization of petrofacies types for the shale oil reservoirs from the 7 members of the Upper Triassic Yanchang Formation, Ordos Basin, China.

Petrofacies	Characterization	SEM	XRD
PF A	High porosity, high TOC, and high brittleness index; clay mineral interlayer pores and fractures, and interparticle pores.	[Image omitted. Please see PDF.]	[Image omitted. Please see PDF.]
PF B	Median porosity, median TOC, and high brittleness index; interparticle pores and intraparticle pores.	[Image omitted. Please see PDF.]	[Image omitted. Please see PDF.]
PF C	Low porosity, median TOC, and low brittleness index; clay mineral interlayer pores and fractures, and sparsely developed interparticle and intraparticle pores.	[Image omitted. Please see PDF.]	[Image omitted. Please see PDF.]
PF D	Ultra-low porosity, low TOC, and high brittleness index; clay mineral interlayer pores and fractures.	[Image omitted. Please see PDF.]	[Image omitted. Please see PDF.]

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