A Sand-to-Shale Ratio Prediction Method for Variogram Optimization by Constructing Virtual Wells Using Seismic Attributes

A Sand-to-Shale Ratio Prediction Method for Variogram Optimization by Constructing Virtual Wells Using Seismic Attributes | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Sand-to-Shale Ratio Prediction Method for Variogram Optimization by Constructing Virtual Wells Using Seismic Attributes Qinghui Xie, Chuanjin Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8736617/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Aiming at the low accuracy of variograms caused by sparse and uneven drilling data in traditional sand-to-shale ratio prediction, this paper proposes a sand-to-shale ratio prediction method that integrates geostatistics, seismic geophysical technologies, and the concept of virtual well modeling. With geostatistics as the core, this method achieves accurate prediction of reservoir sand-to-shale ratio through a comprehensive workflow, including basic data preparation, seismic attribute extraction and optimization, virtual well construction, variogram optimization, and Gaussian random function simulation. Taking the upper member of the Pinghu Formation in the study area as a case study, the results indicate that the RMS attribute exhibits the strongest correlation with the sand-to-shale ratio. The variogram optimized using virtual well data exhibits a spherical model, where the major range and minor range are in good agreement with the distribution characteristics of geological bodies within the study area.), Gaussian random function simulation was performed with actual drilling data as hard constraints and seismic attributes as soft constraints. The generated planar distribution map of sand-to-ground ratio aligns well with the sedimentary facies pattern of "higher in the northwest and lower in the southeast" and demonstrates superior detail accuracy compared to traditional methods.. The detailed accuracy of this map outperforms that of traditional methods, providing reliable support for reservoir evaluation and development scheme design. Sand-to-shale ratio prediction Virtual well Variogram Seismic attribute Gaussian random function simulation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 0 Introduction The sand-to-shale ratio, defined as the ratio of sandstone thickness to total stratigraphic thickness within a given interval, is a critical indicator of sand body development in reservoirs. It reflects the abundance of reservoir sand bodies and the scale of effective pore space. A higher sand-to-shale ratio generally suggests greater sandstone content, better pore development, and more favorable reservoir flow properties; conversely, a lower ratio indicates a higher proportion of non-reservoir rocks, which diminishes exploration and development potential. Predicting the sand-to-shale ratio is significant for hydrocarbon exploration and development. It serves as a key parameter for reservoir evaluation, such as delineating reservoir boundaries and assessing quality, and guides decision-making in exploration and development, thereby helping to reduce risks and optimize investments (Jia, 2010 ). Conventional sand-to-shale ratio prediction typically involves defining sandstone-mudstone cut-offs from actual well data, calculating sandstone and total thicknesses for each interval, and subsequently deriving the ratio statistically. This ratio is then interpolated between wells under geological and seismic constraints to generate a spatial distribution map. However, the spatial distribution of wells is often sparse and uneven, primarily due to practical limitations in drilling placement and operations. Wells tend to be clustered in prioritized areas such as structural highs and early exploration success zones (Zhou, et al., 2010 ; Feng, et al., 2024 ; Wei & Sun, 2021 ), leaving structural lows, basin margins, and geologically complex areas with very low drilling density or even creating "data gaps." This sampling bias directly compromises the reliability of variogram parameters calibrated from well data. Sparse or clustered well points provide insufficient spatial point pairs for robust variogram analysis, particularly in data-gap regions or areas with large well spacing. Moreover, the uneven distribution increases the variogram's sensitivity to local anomalies, preventing it from accurately capturing the true regional spatial structure. Previous research on variogram optimization has primarily focused on three aspects: (1) enhancing the calculation algorithms to improve accuracy (Wang & Hu, 1988 ; Sun, 1990 ); (2) refining model-fitting methods to ensure they effectively capture reservoir spatial architecture (Zhang & Tan, 2010 ); and (3) exploring the application of variograms in reservoir modeling (Wang, et al., 2010 ; Liu, et al., 2021 ). In recent years, attention has shifted toward utilizing seismic data to constrain horizontal variograms. For instance, Liu et al. and Chen et al. proposed methods to optimize variograms by constructing virtual wells from seismic impedance data. However, in the process of virtual well construction, existing studies are hard to deeply integrate the advantage of lateral continuity of seismic data with the supplementary spatial sampling capability of virtual wells. To bridge this gap, this paper proposes a sand-to-shale ratio prediction method based on variogram optimization via seismic-attribute-derived virtual wells. The method is grounded in geostatistical principles and leverages the complementary strengths of two data types: the extensive lateral continuity of seismic data and the enhanced spatial sampling provided by virtual well modeling. The workflow consists of three key steps: (1) establishing a mapping relationship between seismic attributes and the sand-to-shale ratio through attribute extraction and sensitivity analysis; (2) optimizing the variogram model using the denser sample set from virtual wells; and (3) performing Gaussian random function simulation, constrained by hard data from actual wells and soft data from the optimized seismic attribute, to predict the sand-to-shale ratio across the entire study area. 1 Theoretical Methodology The variogram (also referred to as the semi-variogram or structural function) is defined as half the variance difference between the values of a regionalized variable \(\:\text{Z}\text{(}\text{x}\text{)}\) at two spatial locations \(\:\text{x}\) and \(\:\text{x}\text{+h}\) separated by a lag distance \(\:\text{h}\) . It serves as a fundamental geostatistical tool for quantifying spatial variability and characterizing the underlying spatial structure of regionally distributed random variables (Sun, 1990 ). Its mathematical expression is: $$\:\text{γ}\text{(h)=}\frac{\text{1}}{\text{2}}{\text{E}\text{[}\text{Z}\text{(}\text{x}\text{)−}\text{Z}\text{(}\text{x}\text{+h)]}}^{\text{2}}$$ 1 The experimental (or sample) variogram is calculated as: $$\:{\text{γ}}^{\text{∗}}\text{(h)=}\frac{\text{1}}{\text{2}\text{N}\text{（h）}}\sum\:_{\text{k}\text{=1}}^{\text{N}\text{(h)}}{\text{[}\text{Z}\text{(}{\text{x}}_{\text{k}}\text{)−}\text{Z}\text{(}{\text{x}}_{\text{k}}\text{+h)]}}^{\text{2}}$$ 2 Where \(\:\text{N}\text{(h)}\) is the number of data pairs separated by lag distance \(\:\text{h}\) , and \(\:{\text{γ}}^{\text{∗}}\text{(h)}\) is the value of the experimental variogram. Based on known reservoir parameter values at the well locations, a series of experimental variogram values γ∗( hi ​) can be calculated for different lag distances hi ​ ( i = 1, 2, …, n) along a given direction. Plotting h on the abscissa and γ∗( hi​ ) on the ordinate yields a set of points ( h , γ∗( hi​ )), known as a variogram plot (Fig. 1 ). The variogram model is typically characterized by several key parameters: the range a , the nugget effect \(\:\text{c}\) 0 ​, the sill c , and the partial sill C (also referred to as the structural variance). Range ( a ) defines the spatial correlation distance of the regionalized variable. Data points separated by a distance less than aa are spatially correlated, whereas those beyond aa are not. Nugget effect ( c 0 ​ ) represents microscale variability and measurement error, manifesting as a discontinuity at the origin. A higher nugget indicates poorer short-range continuity and a greater purely random component. Sill ( c ) denotes the total variance of the data, representing the maximum variability reached at or beyond the range. A larger sill corresponds to greater overall variability in the parameter. Partial sill ( c c ) is the sill minus the nugget ( c c =c−c 0 ​ ). It quantifies the spatially structured variance observable at the scale of measurement. When the nugget is zero, the sill equals the partial sill. Commonly used variogram models include the exponential, spherical, and Gaussian models (Zhang & Tan, 2010 ; Wang, et al., 2010 ). The selection of an appropriate model depends on the underlying geological continuity: The exponential model is often applied to fluvial or channelized deposits, reflecting a relatively high degree of local randomness in spatial variation. The spherical model is suitable for representing spatial structures in environments like large-scale channel belts or relatively stable deltaic settings, characterized by moderate randomness. The Gaussian model typically describes phenomena with strong, smooth spatial continuity, commonly found in stable, widespread depositional environments such as marine or lacustrine settings. The mathematical expressions for these models are: $$\:\text{γ}\text{(h)=}\left\{\begin{array}{c}\text{0}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{h}\text{=}\text{0}\\\:{\text{c}}_{\text{0}}\text{+}\text{c}\text{(}\frac{\text{3h}}{\text{2}\text{a}}\text{−}\frac{\text{3}{\text{h}}^{\text{3}}}{\text{2}{\text{a}}^{\text{3}}}\text{)}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{0}\text{≤}\text{h}\text{≤}\text{a}\\\:{\text{c}}_{\text{0}}\text{+}\text{c}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{h}\text{>}\text{a}\end{array}\right.$$ 3 The primary goal of calculating a variogram is to characterize the spatial correlation structure of geological variables using the principal, secondary, and vertical range parameters. Core calculation settings include the search radius, bandwidth, lag distance (or step size), and angular tolerance (Liu, et al., 2021 ). The search radius should not exceed the maximum distance between data points within the study area to maintain the validity of spatial analysis. Typical default values are often set as follows: the bandwidth is approximately twice the average well spacing; the lag tolerance is half the average well spacing; and the angular tolerance is suggested to be π/8. The nugget effect quantifies short-range variability and heterogeneity. The range reflects the spatial scale of continuity, with the principal range representing the maximum correlation distance along the dominant direction of anisotropy. Determining the principal direction requires a synthesis of geological understanding, such as sedimentary facies trends, and analysis of attribute continuity to ensure consistency with the depositional model. 2 Application Example 2.1 Regional Geological Setting The study area (Area P) is situated within the Xihu Sag, part of the eastern depression zone of the East China Sea Shelf Basin. The stratigraphic sequence, from oldest to youngest, comprises the Paleocene, the Eocene Baoshi and Pinghu formations, the Oligocene Huagang Formation, the Miocene Longjing–Liulang formations, the Pliocene Santan Formation, and the Quaternary Donghai Group. The target interval, the Eocene Pinghu Formation, was deposited in a restricted marine environment during the late syn-rift stage. Influenced by Pacific Plate compression coupled with fluvial and tidal processes, a tide-dominated delta system developed extensively. In the upper member of the Pinghu Formation, water shallowing reduced tidal influence, leading to a gradual transition to a fluvial-dominated delta system. The target interval is characterized predominantly by braided-channel deposits, with a principal sediment source from the northeast. Consequently, sand body distribution generally trends from northeast to southwest. A total of 25 wells have been drilled in the area, with an average spacing of about 1 km and a maximum spacing reaching 35 km. These wells are clustered primarily on structural highs in the western part of the study area, resulting in uneven spatial coverage, sparse data in the eastern sector, and localized data gaps. 2.2 Method Workflow This study proposes a workflow centered on geostatistics, integrating seismic geophysical interpretation with virtual well modeling, to address the common challenge of low variogram reliability caused by sparse and unevenly distributed well data in conventional sand-to-shale ratio prediction, thereby enabling more accurate reservoir prediction (Fig. 2). 1) Data Preparation For the study area, a combination of well logs was used as the discriminant criterion. Typical log responses include low gamma ray, high density, and low acoustic velocity for sandstone, and high gamma ray, low density, and high acoustic velocity for mudstone. A threshold was determined from a gamma-ray-density cross-plot. Sandstone and mudstone thicknesses were then calculated for each stratigraphic interval, yielding the sand-to-shale ratio at each well location. 2) Seismic Attribute Extraction A multi-dimensional suite of seismic attributes was extracted from the 3D seismic volume using specialized geophysical software. The extracted attributes encompassed kinematic, dynamic, geometric, and instantaneous types to capture comprehensive subsurface information (Chen, et al., 2019; Liu, et al., 2015; Ou, et al., 2018). 3) Sensitive Attribute Optimization A three-step methodology including spatial matching, quantitative modeling, and correlation analysis was employed to identify the most sensitive attribute. First, seismic attribute values were spatially matched and paired with the sand-to-shale ratio at corresponding well locations, visualized via cross-plots. A linear regression was then performed to establish a quantitative relationship between the sand-to-shale ratio and each attribute, from which the coefficient of determination (R²) was calculated. Finally, the attribute with the highest R² value was selected as the optimal predictor, with this statistical result being validated against the known sedimentary facies distribution. 4) Virtual Well Construction and Sampling A regular grid of virtual wells with varying spacing was superimposed on the study area. The optimized seismic attribute value was extracted and assigned to each virtual well location. This process generated a combined dataset of actual wells and dense virtual wells, effectively mitigating the original sample sparsity issue. 5) Variogram Optimization To enhance the reliability of variogram parameter estimation, key calculation parameters were carefully designed. The search radius was constrained to the maximum inter-well distance within the study area to capture the full range of spatial correlation. Other parameters, including lag distance and bandwidth, were set to ensure a sufficient number of data pairs for stable statistical calculation. Variograms were then computed and plotted for the combined well dataset (actual and virtual) across multiple scales. Finally, the key variogram parameters (nugget, sill, range) were derived from these plots for subsequent simulation. 6) Gaussian Random Function Simulation The simulation was performed under a dual-constraint framework. The sand-to-shale ratio values from the actual wells served as hard data, ensuring the simulation honored the measured data points. The optimized seismic attribute, calibrated to the sand-to-shale ratio, was used as a soft data trend to guide the spatial distribution between wells, ensuring geological realism. Using the optimized variogram model parameters, sequential Gaussian simulation was run multiple times to generate a set of equiprobable realizations of the sand-to-shale ratio distribution. The ensemble of realizations was analyzed, and any statistically anomalous outcomes were filtered out to produce a final, robust prediction map. 2.3 Sand-to-Shale Ratio Data Preparation The lithology discrimination for the study area was established using a combination of well logs. Characteristic log responses were defined as follows: sandstone exhibits low gamma ray, high density, and low acoustic velocity, whereas mudstone exhibits high gamma ray, low density, and high acoustic velocity. The lithologic cut-off was determined from a gamma-ray versus density cross-plot. Sandstone and mudstone thicknesses were then calculated for each stratigraphic interval, resulting in the sand-to-shale ratio for all 25 wells (Table 1). These well-derived ratios provide the hard data constraints for the subsequent geostatistical simulation, ensuring the results are anchored to reliable geological observations. Table 1 Sand-to-Shale Ratios of Target Interval Well name Sand-to-Shale Ratio (%) Well name Sand-to-Shale Ratio (%) Well name Sand-to-Shale Ratio (%) B6S 34.48 P10 66.06 S8 24.63 B3 66.31 P11 32.74 T1 32.85 B4 68.52 P113 54.96 T2 72.33 F1 39 S3 13.4 T3 76.18 F1S 31.15 S4 43.01 T4 51.31 P1 13.43 P5 11.02 T5 40.27 P2 68.26 P6 74.3 TT6 54.78 P3 32.88 P8 60.85 - - P4 46.94 P9 18.48 - - 2.4 Seismic Attribute Extraction and Optimization First, guided by the geological context of the fluvial-deltaic system in the upper Pinghu Formation, a suite of seismic attributes indicative of sandstone reservoir characteristics was extracted. These included attributes such as RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration. The spatial distributions of these attributes are displayed as attribute maps (Fig. 3). The attributes displayed include RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration. Subsequently, to quantify the relationship between seismic response and reservoir quality, cross-plots were constructed pairing the sand-to-shale ratio at each well with the corresponding seismic attribute value at that location. Linear regression was applied to each attribute to establish a quantitative relationship and calculate the coefficient of determination (R²) (Fig. 4). The analysis revealed a strong negative correlation between RMS amplitude and the sand-to-shale ratio (R² > 0.8), identifying it as the optimal predictor. In contrast, maximum duration showed the weakest correlation (R² < 0.2), rendering it unsuitable for sandstone characterization. The remaining attributes exhibited moderate correlations, with R² values ranging from 0.2 to 0.6. Based on this quantitative analysis, RMS amplitude was selected as the primary soft data constraint for subsequent modeling. The analyzed attributes include RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration. 2.5 Variogram Calculation Variogram parameters were estimated at three distinct scales of data support. The primary scale was derived directly from the sand-to-shale ratio values at the actual well locations. The secondary scale utilized the spatial trend captured by the optimized seismic attribute map. A tertiary scale was introduced through the construction of a dense grid of virtual wells. 1) Variogram from Actual Well Data For the calculation based solely on the 25 actual wells, key parameters were set as follows (Table 2): a search radius of 35 km (constrained by the maximum inter-well distance in the study area), a lag distance equal to the average well spacing (~ 1 km), a bandwidth of twice the average spacing, a lag tolerance of half the average spacing, and an angular tolerance of π/8. These settings aimed to balance the capture of spatial structure with the generation of a statistically sufficient number of point pairs. After statistical analysis and iterative adjustment of the principal anisotropy direction, the following best-fit model parameters were obtained for the target interval: a major range of 2700 m, a minor range of 1800 m, a sill of 0.98, a nugget of 0.28, a major direction azimuth of 26°, and a minor direction azimuth of 154° (Table 3, Fig. 5). Table 2 Variogram parameter design Well Spacing /m Direction Search Radius/km Basic Lag /m Distance Tolerance /m Angular Tolerance Bandwidth /m 1000 Major Direction 35 1000 500 π/8 2000 800 Minor Direction 20 800 400 π/8 1600 Table 3 Fitting parameter table of sand-to-shale ratio variogram Average Well Spacing /m Major Range /m Minor Range /m Sill Nugget Value 1000 2700 1800 0.98 0.26 (a: major range; b: minor range) 2) Variogram from the Seismic Attribute Map A variogram was also calculated directly from the spatial distribution of the optimized RMS amplitude attribute map. This analysis provided estimates for the structural ranges and orientations: a major range of 5680 m, a minor range of 4250 m, with azimuths of 25° and 125°, respectively (Fig. 6). However, as the attribute map represents a continuous, interpreted trend rather than direct point measurements of the target variable, it does not yield estimates for parameters such as the nugget effect or sill. 3) Variogram from Virtual Well Construction Using the optimized RMS amplitude attribute, a virtual well dataset was generated by sampling the attribute grid at four different spacing intervals: 500 m, 1000 m, 1500 m, and 2000 m. Prior to variogram estimation, key parameters were defined (Table 4): a search radius of 35 km, a lag distance equal to the tested spacing, a bandwidth of twice the spacing, a lag tolerance of half the spacing, and an angular tolerance of π/8. Table 4 Variogram parameter design Well Spacing /m Direction Search Radius /km Basic Lag /m Distance Tolerance /m Angular Tolerance Bandwidth /m 500 Major Direction 35 500 250 π/8 1000 1000 Major Direction 35 1000 500 π/8 2000 1500 Major Direction 35 1500 750 π/8 3000 2000 Major Direction 35 2000 1000 π/8 4000 Fitting variograms to the data from each virtual well set (Fig. 7) yielded the following key model parameters (Table 5): an average major range of 3425 m (ranging from 3250 to 3600 m), an average minor range of 2537 m (ranging from 2400 to 2650 m), and a principal direction azimuth of 25°, which aligns with the known northeast sediment provenance. The average nugget value calculated from the dense virtual well sets was 0.11; however, a value of 0 was applied during the final simulation to enhance the spatial continuity of the model. Table 5 Fitting variogram parameter table of virtual well seismic attribute Virtual Well Spacing /m Major Range /m Minor Range /m Sill Nugget Value 500 3250 2400 0.98 0.28 1000 3500 2500 0.95 0.15 1500 3600 2650 1.00 0 2000 3350 2600 0.98 0 Analysis of the results leads to three main conclusions regarding sample spacing and variogram reliability: 1. Impact of Well Spacing on Parameter Accuracy: At smaller well spacings (e.g., Fig. 7a and 7b), the experimental variograms are defined by a sufficient number of point pairs, enabling robust parameter estimation. As spacing increases, the number of point pairs decreases significantly. At larger spacings, the scarcity of point pairs at short lag distances (near the origin) critically compromises the accuracy of range estimation, as these short-distance pairs are most sensitive for defining the variogram structure. 2. Nugget Effect from Dense Sampling: The dense virtual well grids (500 m and 1000 m spacing) reveal a measurable nugget effect, with an average value of 0.11, indicating inherent short-scale variability. 3. Consistency of Anisotropy Structure: Despite variations in well spacing, the derived spatial structure remains consistent: the major range varies between 3250–3600 m (avg. 3425 m), the minor range between 2400–2650 m (avg. 2537 m), and the principal direction is stable at 25° (with a perpendicular minor direction at ~ 155°), corroborating the northeast-southwest depositional trend (Table 6). Table 6 Fitting variogram parameters from sand-to-shale ratio of different data sources Data Source Major Range /m Minor Range /m Sill Nugget Value Principal Direction Actual Wells 2700 1800 0.98 0.26 26° Seismic Attribute Map 5680 4250 / / 25° Virtual Wells 3425 2537 0.98 0.11 25° 4) Comparative Analysis and Final Parameter Selection A comparative analysis of the variograms derived from the three data sources (actual wells, seismic attribute map, and virtual wells) reveals key insights: Anisotropy Direction The principal anisotropy directions estimated by all three methods are in close agreement and align with the known northeast-southwest sediment provenance direction. Anisotropy Ratio : The major range consistently exceeds the minor range across all methods, with an average ratio of approximately 3:2. This confirms a well-defined, elongated spatial continuity of the sand bodies along the principal direction. Scale Discrepancy The major range estimated from the sparse actual well data (~ 2700 m) is substantially shorter than that derived from the continuous seismic attribute map (~ 5680 m). Given the limitations of the actual well data (sparse, uneven distribution) and the overly smoothed, non-point-support nature of the attribute map, the variogram model calibrated from the dense virtual well dataset was selected for the final simulation. This model provides an optimal balance, mitigating the undersampling bias of the wells while preserving more realistic spatial structure than the attribute map. For the subsequent sequential Gaussian simulation, the final model parameters were set as follows: a major range of 3425 m, a minor range of 2537 m, a principal direction of 25°, and a sill derived from the well data variance. To achieve the dual objectives of (1) strictly honoring the hard well data and (2) generating a spatially continuous model, the nugget value was set to 0. These parameters were used to generate the final sand-to-shale ratio distribution map, providing a critical input for subsequent reservoir evaluation. 3 Application Results For the fluvial-deltaic system of the upper Pinghu Formation, the spherical variogram model was selected due to its suitability for representing spatial structures in large-scale channel and stable deltaic settings. Using the optimized variogram parameters (major range: 3425 m, minor range: 2537 m, azimuth: 25°), a sequential Gaussian simulation was performed under a dual-constraint framework: hard data from the actual well sand-to-shale ratios and a soft data trend from the RMS amplitude attribute (R² > 0.8). This process generated a high-resolution realization of the sand-to-shale ratio distribution (Fig. 8 ). The resulting model demonstrates significant improvements over conventional approaches: Compared to a simulation using only well data (no seismic trend), the new map more accurately reflects the known regional "higher in the west, lower in the east" geological trend. Compared to a simple interpolation of the well data, the new map captures significantly richer geological detail, clearly resolving smaller-scale sedimentary heterogeneities while remaining consistent with the depositional facies model. The final sand-to-shale ratio distribution conforms to the expected geological framework: it shows a "higher in the northwest, lower in the southeast" pattern consistent with the target interval, aligns with the northeast sediment provenance and the associated northeast-southwest channel trend of the upper Pinghu Formation, and realistically depicts the lateral facies variations and sandstone/mudstone distribution. This validated output provides a reliable basis for subsequent reservoir evaluation and development planning. To quantitatively assess the prediction accuracy, Well Q1 in the southern study area was held back as a blind test. The measured sand-to-shale ratio at Q1 is 45.2%. The predictions from different methods at this location were: Simulation without seismic trend: 21.2% (deviation: 24 percentage points) Direct interpolation of well data: 25.6% (deviation: 20 percentage points) Proposed method (simulation with RMS trend): 51.6% (deviation: 6.4 percentage points) The proposed method reduces the prediction error by approximately two-thirds compared to the traditional approaches, conclusively demonstrating its superior accuracy and reliability for sand-to-shale ratio prediction. 4 Discussion 4.1 Methodological Advantages and Innovations Traditional sand-to-shale ratio prediction is often constrained by inaccurate variograms stemming from sparse well data, which is a longstanding bottleneck limiting reservoir characterization accuracy. Our study introduces a threefold innovative design to overcome this challenge. Enhanced Seismic Attribute Screening : We established a rigorous, three-step screening workflow: spatial matching, quantitative modeling, and correlation ranking. In the Pinghu Formation of the Xihu Sag, for example, RMS amplitude exhibited a strong negative correlation with the sand-to-shale ratio (R² > 0.8), outperforming other candidate attributes. This key attribute serves a dual purpose: as the primary data source for populating virtual wells and as a soft constraint in the subsequent Gaussian simulation, effectively compensating for the limitations of relying solely on hard well data. Virtual Well Design Targeting Data Gaps Virtual wells were strategically designed on grids with varying spacings (500–2000 m) and populated with the optimized attribute values, creating a "virtual well dataset." This approach directly addresses the core issues of large well spacing and sparse distribution, while also supplementing data in structural lows and other undersampled areas. Integrated Hard and Soft Constraints for Geologically Reasonable Predictions By integrating hard constraints (well-based sand-to-shale ratios) with soft constraints (seismic attribute trends) in Gaussian simulation, the resulting model exhibits a geologically plausible "higher in the northwest, lower in the southeast" sand-to-shale ratio pattern. This pattern aligns with the expected fluvial-deltaic depositional setting and demonstrates superior capability in delineating features such as braided channels. 4.2 Comparison with Existing Studies Current sand-to-shale ratio prediction methods primarily exhibit two limitations. First, methods based solely on geostatistics from actual wells are heavily influenced by well distribution (e.g., a maximum spacing of 35 km in this study area). The insufficient number of spatial point pairs for reliable experimental variogram calculation often leads to predictions that lack geological realism. Second, methods relying on simple well-data interpolation yield relatively low prediction accuracy, as they fail to account for the underlying spatial structure of geological variables. Our proposed method effectively circumvents these shortcomings. By supplementing sparse well data with virtual wells, it overcomes the limitation of finite data points, allowing variogram parameters to be estimated from a much denser, spatially representative dataset. Furthermore, the "hard + soft" dual-constraint framework ensures that predictions honor both the geological reality captured by wells and the spatial continuity captured by seismic data, avoiding the bias inherent in single-approach methods. Application results confirm these advantages. The predictions show excellent agreement with the northeast sediment provenance and channel trends, while also providing richer geological detail than traditional methods. Compared to other multi-data fusion studies, our method directly targets the practical problem of "variogram optimization in sparsely drilled areas," offering stronger technical focus and greater practical value for solving real-world prediction challenges. 4.3 Study Limitations Despite the promising performance within the study area, the proposed methodology presents several limitations that warrant further investigation. First, the relationship between the seismic attribute (RMS amplitude) and the sand-to-shale ratio is treated as spatially stationary. The current workflow does not account for potential variations in this relationship across different sedimentary microfacies, such as channels and mouth bars. Incorporating facies-dependent calibration could improve the accuracy of virtual well population and the resulting model. Second, the spherical variogram model was selected based on its suitability for the fluvial-deltaic system in this case. Its applicability to other depositional environments (e.g., deep-water turbidites or aeolian systems) remains untested, thereby limiting the method's generalizability until validated in diverse geological settings. 4.4 Future Research Directions To address the aforementioned limitations, future research will focus on the following aspects. First, a facies-constrained modeling approach will be developed by establishing separate seismic attribute-to-sand-to-shale ratio relationships for key sedimentary microfacies (e.g., channels, mouth bars). Virtual wells will then be populated with attribute values conditioned by these facies-specific models. This strategy is designed to reduce prediction bias introduced by facies heterogeneity and improve the geological reliability of the results. Additionally, the method’s robustness and the suitability of the spherical variogram will be tested in other depositional settings (e.g., turbidite or lacustrine systems) to evaluate and enhance its generalizability. 5 Conclusions This study presents an integrated workflow for predicting sand-to-shale ratio in sparsely drilled areas by optimizing variograms using seismic-attribute-derived virtual wells. The main conclusions are as follows: 1.Root-mean-square (RMS) amplitude is identified as the optimal seismic attribute, exhibiting a strong negative correlation with the sand-to-shale ratio (R² > 0.8). It serves as a reliable soft data trend for both populating virtual wells and constraining the subsequent geostatistical simulation. 2.A multi-scale virtual well network (with spacings of 500 m, 1000 m, 1500 m, and 2000 m) enables robust calibration of the spherical variogram model. The derived parameters including a major range of 3425 m, a minor range of 2537 m, and a principal direction of 25° closely align with the geological architecture and sediment provenance trend of the study area. 3.Sequential Gaussian simulation, conditioned by hard data from wells and the RMS amplitude soft trend, generates a high-resolution sand-to-shale ratio distribution that realistically reflects the fluvial-deltaic depositional system, characterized by a distinct “higher in the northwest, lower in the southeast” pattern. Validation using a blind well (Q1) shows a prediction error of only 6 percentage points, markedly lower than errors from conventional methods (20–24 percentage points). 4.The resulting predictive model provides a reliable basis for key practical applications, including reservoir delineation, quality assessment, and development well planning, thereby offering a robust approach to mitigate risk and optimize investment in exploration and development. Declarations Code/Data availability Data not available due to commercial restrictions. Funding Supported by Fujian Provincial Natural Science Foundation of China (NO. 2023J05053). Competing Interests The authors have no relevant financial or non-financial interests to disclose. Author Contributions Qinghui Xie: Conceptualization, Data curation, Funding acquisition, Investigation, Methodology, Project administration, Software, Supervision, Formal analysis, Writing – original draft, Writing – review and editing.Chuanjin Li: Visualization, Writing – review and editing. References Chen, W. H., Wang, Z. Z., Liu, Y. T., Gao, Y., Li, Z. (2019) Multi-scale horizontal variogram estimation in reservoir stochastic simulation. Oil Geophys Prospecting 54(1):154-163, 174. Feng, G. Q., Mo, H. S., Wu, B. F. (2024) Obtaining variation function parameters in modeling geological attributes based on U-Net and CNN deep learning. Oil Geophys Prospecting 59(4):692-701. Jia, A. L. (2010) Fine reservoir description and geological modeling technology. Petroleum Industry Press. Li, Z., Wu, H. B., Liu, Q. J., Huang, L., Wang, J., Yu, J. (2024) Large-scale fracture prediction of coal seams based on post-stack seismic geometric attributes fusion. Chinese Journal of Engineering Geophysics 21(2):297-304. Li, H. Y., Gong, W., Han, J., Yin, Z., Zhang, Y. (2025) Application of a five-dimensional seismic prediction method based on amplitude attributes in the Shunbei Well X area. Bulletin of Geological Science and Technology 44(3):363-372. Liu, C., Xie, C. L., Li, Y. P., Luo, Q., Ma, S. Z. (2015) The principle and method to enhance geological constraint by inserting virtual wells in facies-controlling stochastic reservoir modeling. Natural Gas Geoscience 26(4):616-624. Liu, J., Cui, F., Wang, R., Wang, Y., Gao, J., Tang, J. (2021) Research on geological modeling of tight sandstone gas reservoirs in Huainan area by geostatistical inversion. China Coal 47(5):7-12. Ou, C. H., Feng, G. Q., Li, B. (2018) Geological modeling for oil and gas reservoir development. Petroleum Industry Press. Sun, H. Q. (1990) Geostatistics and its applications. China University of Mining and Technology Press. Wang, R. D., Hu, G. D. (1988) Linear geostatistics. Geological Publishing House. Wang, Y. Z., Fu, J. G., Qu, Z. Y., Wang, Y., Chen, J. T., Li, Z. (2010) The application of variogram in reservoir geological modeling. Science Technology and Engineering 10(29):7147-7150. Wang, D. C., Liu, Y. K., Fu, Q. C., Xu, Z. P., Ma, X. H. (2020) Data analysis method based on variogram theory in attribute modeling. Journal of Chengdu University of Technology (Science & Technology Edition) 47(6):700-707. Wei, D., Sun, Z. Q. (2021) Calculation method of variation function for predicting sandstone-type uranium ore bodies by geostatistical inversion. Oil Geophys Prospecting 56(6):1381-1390. Yue, D. L., Li, W., Du, Y. S., Zhang, J., Wang, C., Li, X. W. (2022) Review on optimization and fusion of seismic attributes for fluvial reservoir characterization. Earth Science 47(11):3929-3943. Zhang, X. Y., Tan, Y. (2010) The automatic fitting and implementation of the spherical model of variogram. Geophysical and Geochemical Exploration 34(2):253-257. Zhang, L. (2017) Research and implementation of coal quality prediction and three-dimensional visualization of working faces in underground coal mines. Xi’an University of Science and Technology (Master’s thesis). Zhao, L. F., Yu, S. Y., Li, S. H. (2024) An automatic fitting method for a variogram based on deep learning. Geophysical and Geochemical Exploration 48(5):1359-1367. https://doi.org/10.11720/wtyht.2024.1522. Zhou, Y., Cheng, S. Q., Zhang, M. (2010) Setting of variogram parameters in reservoir modeling. Journal of Xi’an Shiyou University (Natural Science Edition) 25(5):25-27, 32. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8736617","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":590755489,"identity":"bb3dcfd8-ff77-4bc4-b461-932fe06e0578","order_by":0,"name":"Qinghui Xie","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzUlEQVRIiWNgGAWjYDACZjCSYGBjYD7AkECiFrYEIrVAdIEAjwFxyvmO8x5+XdhmkdjH3vP5w8Mddgz87d34LZM8zJdmPbNNwpiN5+w2icQzyQwSZ85uwKvF4DCPmTFvm4Qcm0TuNobENmYGA4lc4rTwsMm/efwhsa2eKC3GjyG28DBIJLYdJqxFEmgLM885kF/SzIBajvMQ9Avf+TPGn3nK6hLntx9+/PFnW7Ucf3svfi0MBxjYJJD5PPiVQ7QwfyCsahSMglEwCkY0AAD8Bj7aC0vaFQAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-1039-3739","institution":"Fujian University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Qinghui","middleName":"","lastName":"Xie","suffix":""},{"id":590755490,"identity":"5b787bf0-60b4-406f-b224-f214d9257456","order_by":1,"name":"Chuanjin Li","email":"","orcid":"","institution":"Fujian University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Chuanjin","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2026-01-30 03:38:57","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8736617/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8736617/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102991118,"identity":"24a82bca-6c64-4f5f-8f37-8063fcbb1d63","added_by":"auto","created_at":"2026-02-19 11:30:39","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":116815,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of variogram\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/dba492a48265f7b7cd94443b.png"},{"id":102991117,"identity":"c6b83391-7d6c-4053-8199-b280a39c1221","added_by":"auto","created_at":"2026-02-19 11:30:39","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":279392,"visible":true,"origin":"","legend":"\u003cp\u003eFlow chart of the method\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/d7005f2218576a499d3abde9.png"},{"id":103503964,"identity":"ffc84ec3-dd12-4da4-8baf-af336ee2d43c","added_by":"auto","created_at":"2026-02-26 13:05:49","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":731422,"visible":true,"origin":"","legend":"\u003cp\u003ePlan view of seismic attributes.\u003c/p\u003e\n\u003cp\u003eThe attributes displayed include RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/24fb49d0bf285fa9396ce4ed.png"},{"id":102991119,"identity":"650b5d9b-dc9b-49ae-8e78-e16efbae89a3","added_by":"auto","created_at":"2026-02-19 11:30:39","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":511611,"visible":true,"origin":"","legend":"\u003cp\u003eCross-plots of sand-to-shale ratio versus seismic attributes.\u003c/p\u003e\n\u003cp\u003eThe analyzed attributes include RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/6f7a2aa7852be3d483584914.png"},{"id":103049287,"identity":"f15375f4-ed83-4c28-a5a7-f4d4e5f47ae2","added_by":"auto","created_at":"2026-02-20 07:39:34","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":144609,"visible":true,"origin":"","legend":"\u003cp\u003eExperimental variogram fitted with well data\u003c/p\u003e\n\u003cp\u003e(a: major range; b: minor range)\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/2bd7ffd9466979e4b102734a.png"},{"id":103050000,"identity":"8bbf1d08-e5a4-4314-9126-43c9788a3636","added_by":"auto","created_at":"2026-02-20 07:47:36","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":176138,"visible":true,"origin":"","legend":"\u003cp\u003ePlanar variogram derived from sensitive seismic attributes\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/936969911ce0f26d0d305670.png"},{"id":103050005,"identity":"ab080af7-bfe2-47b4-9e38-a118078f54f4","added_by":"auto","created_at":"2026-02-20 07:47:37","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":301571,"visible":true,"origin":"","legend":"\u003cp\u003eFitting graph of major direction variogram with different virtual well spacings (a, b: fitting graphs with small well spacing; c, d: fitting graphs with large well spacing)\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/948c927f286a777d0f64fefb.png"},{"id":102991121,"identity":"6433a216-c99f-4631-908e-305a0687eb96","added_by":"auto","created_at":"2026-02-19 11:30:39","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":569916,"visible":true,"origin":"","legend":"\u003cp\u003eSand-to-shale ratio planar distribution map\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/cb4ec038810783a8767c78ff.png"},{"id":106961616,"identity":"e0d167df-f132-4bf3-848a-7c37ae809df1","added_by":"auto","created_at":"2026-04-15 09:26:17","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3975398,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8736617/v1/f530e8eb-3aaa-4278-b09d-8077e68a639f.pdf"}],"financialInterests":"","formattedTitle":"A Sand-to-Shale Ratio Prediction Method for Variogram Optimization by Constructing Virtual Wells Using Seismic Attributes","fulltext":[{"header":"0 Introduction","content":"\u003cp\u003eThe sand-to-shale ratio, defined as the ratio of sandstone thickness to total stratigraphic thickness within a given interval, is a critical indicator of sand body development in reservoirs. It reflects the abundance of reservoir sand bodies and the scale of effective pore space. A higher sand-to-shale ratio generally suggests greater sandstone content, better pore development, and more favorable reservoir flow properties; conversely, a lower ratio indicates a higher proportion of non-reservoir rocks, which diminishes exploration and development potential. Predicting the sand-to-shale ratio is significant for hydrocarbon exploration and development. It serves as a key parameter for reservoir evaluation, such as delineating reservoir boundaries and assessing quality, and guides decision-making in exploration and development, thereby helping to reduce risks and optimize investments (Jia, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2010\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eConventional sand-to-shale ratio prediction typically involves defining sandstone-mudstone cut-offs from actual well data, calculating sandstone and total thicknesses for each interval, and subsequently deriving the ratio statistically. This ratio is then interpolated between wells under geological and seismic constraints to generate a spatial distribution map. However, the spatial distribution of wells is often sparse and uneven, primarily due to practical limitations in drilling placement and operations. Wells tend to be clustered in prioritized areas such as structural highs and early exploration success zones (Zhou, et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Feng, et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Wei \u0026amp; Sun, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), leaving structural lows, basin margins, and geologically complex areas with very low drilling density or even creating \"data gaps.\" This sampling bias directly compromises the reliability of variogram parameters calibrated from well data. Sparse or clustered well points provide insufficient spatial point pairs for robust variogram analysis, particularly in data-gap regions or areas with large well spacing. Moreover, the uneven distribution increases the variogram's sensitivity to local anomalies, preventing it from accurately capturing the true regional spatial structure.\u003c/p\u003e \u003cp\u003ePrevious research on variogram optimization has primarily focused on three aspects: (1) enhancing the calculation algorithms to improve accuracy (Wang \u0026amp; Hu, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1988\u003c/span\u003e; Sun, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1990\u003c/span\u003e); (2) refining model-fitting methods to ensure they effectively capture reservoir spatial architecture (Zhang \u0026amp; Tan, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2010\u003c/span\u003e); and (3) exploring the application of variograms in reservoir modeling (Wang, et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Liu, et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In recent years, attention has shifted toward utilizing seismic data to constrain horizontal variograms. For instance, Liu et al. and Chen et al. proposed methods to optimize variograms by constructing virtual wells from seismic impedance data. However, in the process of virtual well construction, existing studies are hard to deeply integrate the advantage of lateral continuity of seismic data with the supplementary spatial sampling capability of virtual wells.\u003c/p\u003e \u003cp\u003eTo bridge this gap, this paper proposes a sand-to-shale ratio prediction method based on variogram optimization via seismic-attribute-derived virtual wells. The method is grounded in geostatistical principles and leverages the complementary strengths of two data types: the extensive lateral continuity of seismic data and the enhanced spatial sampling provided by virtual well modeling. The workflow consists of three key steps: (1) establishing a mapping relationship between seismic attributes and the sand-to-shale ratio through attribute extraction and sensitivity analysis; (2) optimizing the variogram model using the denser sample set from virtual wells; and (3) performing Gaussian random function simulation, constrained by hard data from actual wells and soft data from the optimized seismic attribute, to predict the sand-to-shale ratio across the entire study area.\u003c/p\u003e"},{"header":"1 Theoretical Methodology","content":"\u003cp\u003eThe variogram (also referred to as the semi-variogram or structural function) is defined as half the variance difference between the values of a regionalized variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{Z}\\text{(}\\text{x}\\text{)}\\)\u003c/span\u003e\u003c/span\u003e at two spatial locations \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{x}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{x}\\text{+h}\\)\u003c/span\u003e\u003c/span\u003e separated by a lag distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{h}\\)\u003c/span\u003e\u003c/span\u003e. It serves as a fundamental geostatistical tool for quantifying spatial variability and characterizing the underlying spatial structure of regionally distributed random variables (Sun, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1990\u003c/span\u003e). Its mathematical expression is:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\text{\u0026gamma;}\\text{(h)=}\\frac{\\text{1}}{\\text{2}}{\\text{E}\\text{[}\\text{Z}\\text{(}\\text{x}\\text{)\u0026minus;}\\text{Z}\\text{(}\\text{x}\\text{+h)]}}^{\\text{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe experimental (or sample) variogram is calculated as:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\text{\u0026gamma;}}^{\\text{\u0026lowast;}}\\text{(h)=}\\frac{\\text{1}}{\\text{2}\\text{N}\\text{（h）}}\\sum\\:_{\\text{k}\\text{=1}}^{\\text{N}\\text{(h)}}{\\text{[}\\text{Z}\\text{(}{\\text{x}}_{\\text{k}}\\text{)\u0026minus;}\\text{Z}\\text{(}{\\text{x}}_{\\text{k}}\\text{+h)]}}^{\\text{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{N}\\text{(h)}\\)\u003c/span\u003e\u003c/span\u003e is the number of data pairs separated by lag distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{h}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{\u0026gamma;}}^{\\text{\u0026lowast;}}\\text{(h)}\\)\u003c/span\u003e\u003c/span\u003e is the value of the experimental variogram.\u003c/p\u003e \u003cp\u003eBased on known reservoir parameter values at the well locations, a series of experimental variogram values γ\u0026lowast;(\u003cem\u003ehi\u003c/em\u003e​) can be calculated for different lag distances \u003cem\u003ehi\u003c/em\u003e​ (\u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 2, \u0026hellip;, n) along a given direction. Plotting \u003cem\u003eh\u003c/em\u003e on the abscissa and γ\u0026lowast;(\u003cem\u003ehi​\u003c/em\u003e) on the ordinate yields a set of points (\u003cem\u003eh\u003c/em\u003e, γ\u0026lowast;(\u003cem\u003ehi​\u003c/em\u003e)), known as a variogram plot (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe variogram model is typically characterized by several key parameters: the range \u003cem\u003ea\u003c/em\u003e, the nugget effect \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{c}\\)\u003c/span\u003e\u003c/span\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e​, the sill \u003cem\u003ec\u003c/em\u003e, and the partial sill \u003cem\u003eC\u003c/em\u003e (also referred to as the structural variance).\u003c/p\u003e \u003cp\u003eRange (\u003cem\u003ea\u003c/em\u003e) defines the spatial correlation distance of the regionalized variable. Data points separated by a distance less than aa are spatially correlated, whereas those beyond aa are not.\u003c/p\u003e \u003cp\u003eNugget effect (\u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e​\u003c/sub\u003e) represents microscale variability and measurement error, manifesting as a discontinuity at the origin. A higher nugget indicates poorer short-range continuity and a greater purely random component.\u003c/p\u003e \u003cp\u003eSill (\u003cem\u003ec\u003c/em\u003e) denotes the total variance of the data, representing the maximum variability reached at or beyond the range. A larger sill corresponds to greater overall variability in the parameter.\u003c/p\u003e \u003cp\u003ePartial sill (\u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e) is the sill minus the nugget (\u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e=c\u0026minus;c\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e​\u003c/em\u003e). It quantifies the spatially structured variance observable at the scale of measurement. When the nugget is zero, the sill equals the partial sill.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eCommonly used variogram models include the exponential, spherical, and Gaussian models (Zhang \u0026amp; Tan, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Wang, et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). The selection of an appropriate model depends on the underlying geological continuity:\u003c/p\u003e \u003cp\u003eThe exponential model is often applied to fluvial or channelized deposits, reflecting a relatively high degree of local randomness in spatial variation.\u003c/p\u003e \u003cp\u003eThe spherical model is suitable for representing spatial structures in environments like large-scale channel belts or relatively stable deltaic settings, characterized by moderate randomness.\u003c/p\u003e \u003cp\u003eThe Gaussian model typically describes phenomena with strong, smooth spatial continuity, commonly found in stable, widespread depositional environments such as marine or lacustrine settings.\u003c/p\u003e \u003cp\u003eThe mathematical expressions for these models are:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\text{\u0026gamma;}\\text{(h)=}\\left\\{\\begin{array}{c}\\text{0}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{h}\\text{=}\\text{0}\\\\\\:{\\text{c}}_{\\text{0}}\\text{+}\\text{c}\\text{(}\\frac{\\text{3h}}{\\text{2}\\text{a}}\\text{\u0026minus;}\\frac{\\text{3}{\\text{h}}^{\\text{3}}}{\\text{2}{\\text{a}}^{\\text{3}}}\\text{)}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{0}\\text{\u0026le;}\\text{h}\\text{\u0026le;}\\text{a}\\\\\\:{\\text{c}}_{\\text{0}}\\text{+}\\text{c}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{}\\text{h}\\text{\u0026gt;}\\text{a}\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe primary goal of calculating a variogram is to characterize the spatial correlation structure of geological variables using the principal, secondary, and vertical range parameters. Core calculation settings include the search radius, bandwidth, lag distance (or step size), and angular tolerance (Liu, et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). The search radius should not exceed the maximum distance between data points within the study area to maintain the validity of spatial analysis. Typical default values are often set as follows: the bandwidth is approximately twice the average well spacing; the lag tolerance is half the average well spacing; and the angular tolerance is suggested to be π/8. The nugget effect quantifies short-range variability and heterogeneity. The range reflects the spatial scale of continuity, with the principal range representing the maximum correlation distance along the dominant direction of anisotropy. Determining the principal direction requires a synthesis of geological understanding, such as sedimentary facies trends, and analysis of attribute continuity to ensure consistency with the depositional model.\u003c/p\u003e"},{"header":"2 Application Example","content":"\u003cdiv id=\"Sec4\"\u003e\n \u003ch2\u003e2.1 Regional Geological Setting\u003c/h2\u003e\n \u003cp\u003eThe study area (Area P) is situated within the Xihu Sag, part of the eastern depression zone of the East China Sea Shelf Basin. The stratigraphic sequence, from oldest to youngest, comprises the Paleocene, the Eocene Baoshi and Pinghu formations, the Oligocene Huagang Formation, the Miocene Longjing\u0026ndash;Liulang formations, the Pliocene Santan Formation, and the Quaternary Donghai Group. The target interval, the Eocene Pinghu Formation, was deposited in a restricted marine environment during the late syn-rift stage. Influenced by Pacific Plate compression coupled with fluvial and tidal processes, a tide-dominated delta system developed extensively. In the upper member of the Pinghu Formation, water shallowing reduced tidal influence, leading to a gradual transition to a fluvial-dominated delta system.\u003c/p\u003e\n \u003cp\u003eThe target interval is characterized predominantly by braided-channel deposits, with a principal sediment source from the northeast. Consequently, sand body distribution generally trends from northeast to southwest. A total of 25 wells have been drilled in the area, with an average spacing of about 1 km and a maximum spacing reaching 35 km. These wells are clustered primarily on structural highs in the western part of the study area, resulting in uneven spatial coverage, sparse data in the eastern sector, and localized data gaps.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\"\u003e\n \u003ch2\u003e2.2 Method Workflow\u003c/h2\u003e\n \u003cp\u003eThis study proposes a workflow centered on geostatistics, integrating seismic geophysical interpretation with virtual well modeling, to address the common challenge of low variogram reliability caused by sparse and unevenly distributed well data in conventional sand-to-shale ratio prediction, thereby enabling more accurate reservoir prediction (Fig.\u0026nbsp;2).\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e1) Data Preparation\u003c/h3\u003e\n\u003cp\u003eFor the study area, a combination of well logs was used as the discriminant criterion. Typical log responses include low gamma ray, high density, and low acoustic velocity for sandstone, and high gamma ray, low density, and high acoustic velocity for mudstone. A threshold was determined from a gamma-ray-density cross-plot. Sandstone and mudstone thicknesses were then calculated for each stratigraphic interval, yielding the sand-to-shale ratio at each well location.\u003c/p\u003e\n\u003ch3\u003e2) Seismic Attribute Extraction\u003c/h3\u003e\n\u003cp\u003eA multi-dimensional suite of seismic attributes was extracted from the 3D seismic volume using specialized geophysical software. The extracted attributes encompassed kinematic, dynamic, geometric, and instantaneous types to capture comprehensive subsurface information (Chen, et al., 2019; Liu, et al., 2015; Ou, et al., 2018).\u003c/p\u003e\n\u003ch3\u003e3) Sensitive Attribute Optimization\u003c/h3\u003e\n\u003cp\u003eA three-step methodology including spatial matching, quantitative modeling, and correlation analysis was employed to identify the most sensitive attribute. First, seismic attribute values were spatially matched and paired with the sand-to-shale ratio at corresponding well locations, visualized via cross-plots. A linear regression was then performed to establish a quantitative relationship between the sand-to-shale ratio and each attribute, from which the coefficient of determination (R\u0026sup2;) was calculated. Finally, the attribute with the highest R\u0026sup2; value was selected as the optimal predictor, with this statistical result being validated against the known sedimentary facies distribution.\u003c/p\u003e\n\u003ch3\u003e4) Virtual Well Construction and Sampling\u003c/h3\u003e\n\u003cp\u003eA regular grid of virtual wells with varying spacing was superimposed on the study area. The optimized seismic attribute value was extracted and assigned to each virtual well location. This process generated a combined dataset of actual wells and dense virtual wells, effectively mitigating the original sample sparsity issue.\u003c/p\u003e\n\u003ch3\u003e5) Variogram Optimization\u003c/h3\u003e\n\u003cp\u003eTo enhance the reliability of variogram parameter estimation, key calculation parameters were carefully designed. The search radius was constrained to the maximum inter-well distance within the study area to capture the full range of spatial correlation. Other parameters, including lag distance and bandwidth, were set to ensure a sufficient number of data pairs for stable statistical calculation. Variograms were then computed and plotted for the combined well dataset (actual and virtual) across multiple scales. Finally, the key variogram parameters (nugget, sill, range) were derived from these plots for subsequent simulation.\u003c/p\u003e\n\u003ch3\u003e6) Gaussian Random Function Simulation\u003c/h3\u003e\n\u003cp\u003eThe simulation was performed under a dual-constraint framework. The sand-to-shale ratio values from the actual wells served as hard data, ensuring the simulation honored the measured data points. The optimized seismic attribute, calibrated to the sand-to-shale ratio, was used as a soft data trend to guide the spatial distribution between wells, ensuring geological realism. Using the optimized variogram model parameters, sequential Gaussian simulation was run multiple times to generate a set of equiprobable realizations of the sand-to-shale ratio distribution. The ensemble of realizations was analyzed, and any statistically anomalous outcomes were filtered out to produce a final, robust prediction map.\u003c/p\u003e\n\u003cdiv id=\"Sec12\"\u003e\n \u003ch2\u003e2.3 Sand-to-Shale Ratio Data Preparation\u003c/h2\u003e\n \u003cp\u003eThe lithology discrimination for the study area was established using a combination of well logs. Characteristic log responses were defined as follows: sandstone exhibits low gamma ray, high density, and low acoustic velocity, whereas mudstone exhibits high gamma ray, low density, and high acoustic velocity. The lithologic cut-off was determined from a gamma-ray versus density cross-plot. Sandstone and mudstone thicknesses were then calculated for each stratigraphic interval, resulting in the sand-to-shale ratio for all 25 wells (Table 1). These well-derived ratios provide the hard data constraints for the subsequent geostatistical simulation, ensuring the results are anchored to reliable geological observations.\u003c/p\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 1\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eSand-to-Shale Ratios of Target Interval\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eWell name\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSand-to-Shale Ratio\u003c/p\u003e\n \u003cp\u003e(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eWell name\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSand-to-Shale Ratio\u003c/p\u003e\n \u003cp\u003e(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eWell name\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSand-to-Shale Ratio\u003c/p\u003e\n \u003cp\u003e(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eB6S\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e34.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e66.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eS8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e24.63\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eB3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e66.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e32.74\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eT1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32.85\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eB4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e68.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP113\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e54.96\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eT2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e72.33\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eF1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eS3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e13.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eT3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e76.18\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eF1S\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e31.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eS4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e43.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eT4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e51.31\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e13.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e11.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eT5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e68.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e74.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTT6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e54.78\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32.88\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e60.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e46.94\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eP9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e18.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\"\u003e\n \u003ch2\u003e2.4 Seismic Attribute Extraction and Optimization\u003c/h2\u003e\n \u003cp\u003eFirst, guided by the geological context of the fluvial-deltaic system in the upper Pinghu Formation, a suite of seismic attributes indicative of sandstone reservoir characteristics was extracted. These included attributes such as RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration. The spatial distributions of these attributes are displayed as attribute maps (Fig. 3).\u003c/p\u003e\n \u003cp\u003eThe attributes displayed include RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration.\u003c/p\u003e\n \u003cp\u003eSubsequently, to quantify the relationship between seismic response and reservoir quality, cross-plots were constructed pairing the sand-to-shale ratio at each well with the corresponding seismic attribute value at that location. Linear regression was applied to each attribute to establish a quantitative relationship and calculate the coefficient of determination (R\u0026sup2;) (Fig. 4). The analysis revealed a strong negative correlation between RMS amplitude and the sand-to-shale ratio (R\u0026sup2; \u0026gt; 0.8), identifying it as the optimal predictor. In contrast, maximum duration showed the weakest correlation (R\u0026sup2; \u0026lt; 0.2), rendering it unsuitable for sandstone characterization. The remaining attributes exhibited moderate correlations, with R\u0026sup2; values ranging from 0.2 to 0.6. Based on this quantitative analysis, RMS amplitude was selected as the primary soft data constraint for subsequent modeling.\u003c/p\u003e\n \u003cp\u003eThe analyzed attributes include RMS amplitude, maximum amplitude, average trough amplitude, average energy, average envelope amplitude, arc length, average instantaneous frequency, and maximum duration.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\"\u003e\n \u003ch2\u003e2.5 Variogram Calculation\u003c/h2\u003e\n \u003cp\u003eVariogram parameters were estimated at three distinct scales of data support. The primary scale was derived directly from the sand-to-shale ratio values at the actual well locations. The secondary scale utilized the spatial trend captured by the optimized seismic attribute map. A tertiary scale was introduced through the construction of a dense grid of virtual wells.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003e1) Variogram from Actual Well Data\u003c/h3\u003e\n\u003cp\u003eFor the calculation based solely on the 25 actual wells, key parameters were set as follows (Table 2): a search radius of 35 km (constrained by the maximum inter-well distance in the study area), a lag distance equal to the average well spacing (~\u0026thinsp;1 km), a bandwidth of twice the average spacing, a lag tolerance of half the average spacing, and an angular tolerance of \u0026pi;/8. These settings aimed to balance the capture of spatial structure with the generation of a statistically sufficient number of point pairs. After statistical analysis and iterative adjustment of the principal anisotropy direction, the following best-fit model parameters were obtained for the target interval: a major range of 2700 m, a minor range of 1800 m, a sill of 0.98, a nugget of 0.28, a major direction azimuth of 26\u0026deg;, and a minor direction azimuth of 154\u0026deg; (Table 3, Fig. 5).\u003c/p\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 2\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eVariogram parameter design\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eWell Spacing /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDirection\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSearch Radius/km\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBasic Lag /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDistance Tolerance /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAngular Tolerance\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBandwidth\u003c/p\u003e\n \u003cp\u003e/m\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMajor Direction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026pi;/8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2000\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMinor Direction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e400\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026pi;/8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1600\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 3\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eFitting parameter table of sand-to-shale ratio variogram\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAverage Well Spacing\u003c/p\u003e\n \u003cp\u003e/m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMajor Range\u003c/p\u003e\n \u003cp\u003e/m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMinor Range\u003c/p\u003e\n \u003cp\u003e/m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSill\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eNugget Value\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2700\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003e(a: major range; b: minor range)\u003c/p\u003e\n\u003ch3\u003e2) Variogram from the Seismic Attribute Map\u003c/h3\u003e\n\u003cp\u003eA variogram was also calculated directly from the spatial distribution of the optimized RMS amplitude attribute map. This analysis provided estimates for the structural ranges and orientations: a major range of 5680 m, a minor range of 4250 m, with azimuths of 25\u0026deg; and 125\u0026deg;, respectively (Fig. 6). However, as the attribute map represents a continuous, interpreted trend rather than direct point measurements of the target variable, it does not yield estimates for parameters such as the nugget effect or sill.\u003c/p\u003e\n\u003ch3\u003e3) Variogram from Virtual Well Construction\u003c/h3\u003e\n\u003cp\u003eUsing the optimized RMS amplitude attribute, a virtual well dataset was generated by sampling the attribute grid at four different spacing intervals: 500 m, 1000 m, 1500 m, and 2000 m. Prior to variogram estimation, key parameters were defined (Table 4): a search radius of 35 km, a lag distance equal to the tested spacing, a bandwidth of twice the spacing, a lag tolerance of half the spacing, and an angular tolerance of \u0026pi;/8.\u003c/p\u003e\n\u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 4\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eVariogram parameter design\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eWell Spacing /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDirection\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSearch Radius /km\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBasic Lag /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eDistance Tolerance /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAngular Tolerance\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBandwidth /m\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMajor Direction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e250\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026pi;/8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMajor Direction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026pi;/8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2000\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMajor Direction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e750\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026pi;/8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3000\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMajor Direction\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026pi;/8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4000\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003eFitting variograms to the data from each virtual well set (Fig. 7) yielded the following key model parameters (Table 5): an average major range of 3425 m (ranging from 3250 to 3600 m), an average minor range of 2537 m (ranging from 2400 to 2650 m), and a principal direction azimuth of 25\u0026deg;, which aligns with the known northeast sediment provenance. The average nugget value calculated from the dense virtual well sets was 0.11; however, a value of 0 was applied during the final simulation to enhance the spatial continuity of the model.\u003c/p\u003e\n\u003ctable id=\"Tab5\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 5\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eFitting variogram parameter table of virtual well seismic attribute\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eVirtual Well Spacing /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMajor Range /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMinor Range /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSill\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eNugget Value\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3250\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2400\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3600\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2650\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3350\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2600\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eAnalysis of the results leads to three main conclusions regarding sample spacing and variogram reliability:\u003c/p\u003e\n\u003cp\u003e1. Impact of Well Spacing on Parameter Accuracy: At smaller well spacings (e.g., Fig. 7a and 7b), the experimental variograms are defined by a sufficient number of point pairs, enabling robust parameter estimation. As spacing increases, the number of point pairs decreases significantly. At larger spacings, the scarcity of point pairs at short lag distances (near the origin) critically compromises the accuracy of range estimation, as these short-distance pairs are most sensitive for defining the variogram structure.\u003c/p\u003e\n\u003cp\u003e2. Nugget Effect from Dense Sampling: The dense virtual well grids (500 m and 1000 m spacing) reveal a measurable nugget effect, with an average value of 0.11, indicating inherent short-scale variability.\u003c/p\u003e\n\u003cp\u003e3. Consistency of Anisotropy Structure: Despite variations in well spacing, the derived spatial structure remains consistent: the major range varies between 3250\u0026ndash;3600 m (avg. 3425 m), the minor range between 2400\u0026ndash;2650 m (avg. 2537 m), and the principal direction is stable at 25\u0026deg; (with a perpendicular minor direction at ~\u0026thinsp;155\u0026deg;), corroborating the northeast-southwest depositional trend (Table 6).\u003c/p\u003e\n\u003cdiv\u003e\n \u003ctable id=\"Tab6\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 6\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eFitting variogram parameters from sand-to-shale ratio of different data sources\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eData Source\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMajor Range /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMinor Range /m\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSill\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eNugget Value\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePrincipal Direction\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eActual Wells\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2700\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e26\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eSeismic Attribute Map\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5680\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4250\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e/\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e/\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eVirtual Wells\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3425\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2537\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3\u003e4) Comparative Analysis and Final Parameter Selection\u003c/h3\u003e\n\u003cp\u003eA comparative analysis of the variograms derived from the three data sources (actual wells, seismic attribute map, and virtual wells) reveals key insights:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnisotropy Direction\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe principal anisotropy directions estimated by all three methods are in close agreement and align with the known northeast-southwest sediment provenance direction.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnisotropy Ratio\u003c/strong\u003e: The major range consistently exceeds the minor range across all methods, with an average ratio of approximately 3:2. This confirms a well-defined, elongated spatial continuity of the sand bodies along the principal direction.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eScale Discrepancy\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe major range estimated from the sparse actual well data (~\u0026thinsp;2700 m) is substantially shorter than that derived from the continuous seismic attribute map (~\u0026thinsp;5680 m).\u003c/p\u003e\n\u003cp\u003eGiven the limitations of the actual well data (sparse, uneven distribution) and the overly smoothed, non-point-support nature of the attribute map, the variogram model calibrated from the dense virtual well dataset was selected for the final simulation. This model provides an optimal balance, mitigating the undersampling bias of the wells while preserving more realistic spatial structure than the attribute map.\u003c/p\u003e\n\u003cp\u003eFor the subsequent sequential Gaussian simulation, the final model parameters were set as follows: a major range of 3425 m, a minor range of 2537 m, a principal direction of 25\u0026deg;, and a sill derived from the well data variance. To achieve the dual objectives of (1) strictly honoring the hard well data and (2) generating a spatially continuous model, the nugget value was set to 0. These parameters were used to generate the final sand-to-shale ratio distribution map, providing a critical input for subsequent reservoir evaluation.\u003c/p\u003e"},{"header":"3 Application Results","content":"\u003cp\u003eFor the fluvial-deltaic system of the upper Pinghu Formation, the spherical variogram model was selected due to its suitability for representing spatial structures in large-scale channel and stable deltaic settings. Using the optimized variogram parameters (major range: 3425 m, minor range: 2537 m, azimuth: 25\u0026deg;), a sequential Gaussian simulation was performed under a dual-constraint framework: hard data from the actual well sand-to-shale ratios and a soft data trend from the RMS amplitude attribute (R\u0026sup2; \u0026gt; 0.8). This process generated a high-resolution realization of the sand-to-shale ratio distribution (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe resulting model demonstrates significant improvements over conventional approaches:\u003c/p\u003e \u003cp\u003eCompared to a simulation using only well data (no seismic trend), the new map more accurately reflects the known regional \"higher in the west, lower in the east\" geological trend.\u003c/p\u003e \u003cp\u003eCompared to a simple interpolation of the well data, the new map captures significantly richer geological detail, clearly resolving smaller-scale sedimentary heterogeneities while remaining consistent with the depositional facies model.\u003c/p\u003e \u003cp\u003eThe final sand-to-shale ratio distribution conforms to the expected geological framework: it shows a \"higher in the northwest, lower in the southeast\" pattern consistent with the target interval, aligns with the northeast sediment provenance and the associated northeast-southwest channel trend of the upper Pinghu Formation, and realistically depicts the lateral facies variations and sandstone/mudstone distribution. This validated output provides a reliable basis for subsequent reservoir evaluation and development planning.\u003c/p\u003e \u003cp\u003eTo quantitatively assess the prediction accuracy, Well Q1 in the southern study area was held back as a blind test. The measured sand-to-shale ratio at Q1 is 45.2%. The predictions from different methods at this location were:\u003c/p\u003e \u003cp\u003eSimulation without seismic trend: 21.2% (deviation: 24 percentage points)\u003c/p\u003e \u003cp\u003eDirect interpolation of well data: 25.6% (deviation: 20 percentage points)\u003c/p\u003e \u003cp\u003eProposed method (simulation with RMS trend): 51.6% (deviation: 6.4 percentage points)\u003c/p\u003e \u003cp\u003eThe proposed method reduces the prediction error by approximately two-thirds compared to the traditional approaches, conclusively demonstrating its superior accuracy and reliability for sand-to-shale ratio prediction.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"4 Discussion","content":"\u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Methodological Advantages and Innovations\u003c/h2\u003e \u003cp\u003eTraditional sand-to-shale ratio prediction is often constrained by inaccurate variograms stemming from sparse well data, which is a longstanding bottleneck limiting reservoir characterization accuracy. Our study introduces a threefold innovative design to overcome this challenge.\u003c/p\u003e \u003cp\u003e \u003cb\u003eEnhanced Seismic Attribute Screening\u003c/b\u003e: We established a rigorous, three-step screening workflow: spatial matching, quantitative modeling, and correlation ranking. In the Pinghu Formation of the Xihu Sag, for example, RMS amplitude exhibited a strong negative correlation with the sand-to-shale ratio (R\u0026sup2; \u0026gt; 0.8), outperforming other candidate attributes. This key attribute serves a dual purpose: as the primary data source for populating virtual wells and as a soft constraint in the subsequent Gaussian simulation, effectively compensating for the limitations of relying solely on hard well data.\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eVirtual Well Design Targeting Data Gaps\u003c/strong\u003e \u003cp\u003eVirtual wells were strategically designed on grids with varying spacings (500\u0026ndash;2000 m) and populated with the optimized attribute values, creating a \"virtual well dataset.\" This approach directly addresses the core issues of large well spacing and sparse distribution, while also supplementing data in structural lows and other undersampled areas.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eIntegrated Hard and Soft Constraints for Geologically Reasonable Predictions\u003c/strong\u003e \u003cp\u003eBy integrating hard constraints (well-based sand-to-shale ratios) with soft constraints (seismic attribute trends) in Gaussian simulation, the resulting model exhibits a geologically plausible \"higher in the northwest, lower in the southeast\" sand-to-shale ratio pattern. This pattern aligns with the expected fluvial-deltaic depositional setting and demonstrates superior capability in delineating features such as braided channels.\u003c/p\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Comparison with Existing Studies\u003c/h2\u003e \u003cp\u003eCurrent sand-to-shale ratio prediction methods primarily exhibit two limitations. First, methods based solely on geostatistics from actual wells are heavily influenced by well distribution (e.g., a maximum spacing of 35 km in this study area). The insufficient number of spatial point pairs for reliable experimental variogram calculation often leads to predictions that lack geological realism. Second, methods relying on simple well-data interpolation yield relatively low prediction accuracy, as they fail to account for the underlying spatial structure of geological variables.\u003c/p\u003e \u003cp\u003eOur proposed method effectively circumvents these shortcomings. By supplementing sparse well data with virtual wells, it overcomes the limitation of finite data points, allowing variogram parameters to be estimated from a much denser, spatially representative dataset. Furthermore, the \"hard\u0026thinsp;+\u0026thinsp;soft\" dual-constraint framework ensures that predictions honor both the geological reality captured by wells and the spatial continuity captured by seismic data, avoiding the bias inherent in single-approach methods.\u003c/p\u003e \u003cp\u003eApplication results confirm these advantages. The predictions show excellent agreement with the northeast sediment provenance and channel trends, while also providing richer geological detail than traditional methods. Compared to other multi-data fusion studies, our method directly targets the practical problem of \"variogram optimization in sparsely drilled areas,\" offering stronger technical focus and greater practical value for solving real-world prediction challenges.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec23\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Study Limitations\u003c/h2\u003e \u003cp\u003eDespite the promising performance within the study area, the proposed methodology presents several limitations that warrant further investigation.\u003c/p\u003e \u003cp\u003eFirst, the relationship between the seismic attribute (RMS amplitude) and the sand-to-shale ratio is treated as spatially stationary. The current workflow does not account for potential variations in this relationship across different sedimentary microfacies, such as channels and mouth bars. Incorporating facies-dependent calibration could improve the accuracy of virtual well population and the resulting model.\u003c/p\u003e \u003cp\u003eSecond, the spherical variogram model was selected based on its suitability for the fluvial-deltaic system in this case. Its applicability to other depositional environments (e.g., deep-water turbidites or aeolian systems) remains untested, thereby limiting the method's generalizability until validated in diverse geological settings.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Future Research Directions\u003c/h2\u003e \u003cp\u003eTo address the aforementioned limitations, future research will focus on the following aspects. First, a facies-constrained modeling approach will be developed by establishing separate seismic attribute-to-sand-to-shale ratio relationships for key sedimentary microfacies (e.g., channels, mouth bars). Virtual wells will then be populated with attribute values conditioned by these facies-specific models. This strategy is designed to reduce prediction bias introduced by facies heterogeneity and improve the geological reliability of the results. Additionally, the method\u0026rsquo;s robustness and the suitability of the spherical variogram will be tested in other depositional settings (e.g., turbidite or lacustrine systems) to evaluate and enhance its generalizability.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eThis study presents an integrated workflow for predicting sand-to-shale ratio in sparsely drilled areas by optimizing variograms using seismic-attribute-derived virtual wells. The main conclusions are as follows:\u003c/p\u003e\n\u003cp\u003e1.Root-mean-square (RMS) amplitude is identified as the optimal seismic attribute, exhibiting a strong negative correlation with the sand-to-shale ratio (R² \u0026gt; 0.8). It serves as a reliable soft data trend for both populating virtual wells and constraining the subsequent geostatistical simulation.\u003c/p\u003e\n\u003cp\u003e2.A multi-scale virtual well network (with spacings of 500 m, 1000 m, 1500 m, and 2000 m) enables robust calibration of the spherical variogram model. The derived parameters including a major range of 3425 m, a minor range of 2537 m, and a principal direction of 25° closely align with the geological architecture and sediment provenance trend of the study area.\u003c/p\u003e\n\u003cp\u003e3.Sequential Gaussian simulation, conditioned by hard data from wells and the RMS amplitude soft trend, generates a high-resolution sand-to-shale ratio distribution that realistically reflects the fluvial-deltaic depositional system, characterized by a distinct “higher in the northwest, lower in the southeast” pattern. Validation using a blind well (Q1) shows a prediction error of only 6 percentage points, markedly lower than errors from conventional methods (20–24 percentage points).\u003c/p\u003e\n\u003cp\u003e4.The resulting predictive model provides a reliable basis for key practical applications, including reservoir delineation, quality assessment, and development well planning, thereby offering a robust approach to mitigate risk and optimize investment in exploration and development.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eCode/Data availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData not available due to commercial restrictions.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Supported by Fujian Provincial Natural Science Foundation of China (NO. 2023J05053).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Qinghui Xie: Conceptualization, Data curation, Funding acquisition, Investigation, Methodology, Project administration, Software, Supervision, Formal analysis, Writing – original draft, Writing – review and editing.Chuanjin Li: Visualization, Writing – review and editing.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eChen, W. 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Journal of Xi\u0026rsquo;an Shiyou University (Natural Science Edition) 25(5):25-27, 32.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Sand-to-shale ratio prediction, Virtual well, Variogram, Seismic attribute, Gaussian random function simulation","lastPublishedDoi":"10.21203/rs.3.rs-8736617/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8736617/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAiming at the low accuracy of variograms caused by sparse and uneven drilling data in traditional sand-to-shale ratio prediction, this paper proposes a sand-to-shale ratio prediction method that integrates geostatistics, seismic geophysical technologies, and the concept of virtual well modeling. With geostatistics as the core, this method achieves accurate prediction of reservoir sand-to-shale ratio through a comprehensive workflow, including basic data preparation, seismic attribute extraction and optimization, virtual well construction, variogram optimization, and Gaussian random function simulation. Taking the upper member of the Pinghu Formation in the study area as a case study, the results indicate that the RMS attribute exhibits the strongest correlation with the sand-to-shale ratio. The variogram optimized using virtual well data exhibits a spherical model, where the major range and minor range are in good agreement with the distribution characteristics of geological bodies within the study area.), Gaussian random function simulation was performed with actual drilling data as hard constraints and seismic attributes as soft constraints. The generated planar distribution map of sand-to-ground ratio aligns well with the sedimentary facies pattern of \"higher in the northwest and lower in the southeast\" and demonstrates superior detail accuracy compared to traditional methods.. The detailed accuracy of this map outperforms that of traditional methods, providing reliable support for reservoir evaluation and development scheme design.\u003c/p\u003e","manuscriptTitle":"A Sand-to-Shale Ratio Prediction Method for Variogram Optimization by Constructing Virtual Wells Using Seismic Attributes","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-19 11:30:34","doi":"10.21203/rs.3.rs-8736617/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b2efcb34-9ef7-462c-9906-bf9ef3582c34","owner":[],"postedDate":"February 19th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-14T17:59:45+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-19 11:30:34","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8736617","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8736617","identity":"rs-8736617","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}