A Facies Proportional Determination Method Based on the Theory of Confidence Intervals: A Case Study in the M Gas Field in the East China Sea

Abstract

A reservoir is the space and gathering place of underground oil and gas reservoirs. Lithofacies proportion is the key parameter of a reservoir. Due to the heterogeneity of geological bodies, the multi-solution of seismic data and the limited and uneven distribution of logging data, it is unclear how to determine the lithofacies proportion in the study area directly from the available data. The mean is a way to make a quick estimate, but the mean cannot quantify the uncertainty of the lithofacies proportion using the available data. In this paper, a new method for quantifying uncertainty and determining the value of the three levels of the facies proportion is proposed. Taking sandstone proportion as an example, a normality test was carried out to verify the applicability of the sandstone proportion data to this method, and then the mean and variance of samples were calculated. Based on the confidence interval theory, the uncertainty of the mean value and the range of sandstone proportion were determined, that is, the optimistic value and the pessimistic value of the sandstone proportion. The results show that this method provides a new reference for quantifying values for the three levels of the uncertainty, and it uses the petrographic proportion as an example to provide the data basis for subsequent modeling and reservoir research.

1. Introduction

The heterogeneity of geological bodies, limited access to data, and cognitive limitations all have an impact on reservoir uncertainty. It is difficult to predict reservoir properties using limited information, and this difficulty persists throughout the entire reservoir description process [1,2]. Due to the heterogeneity of geological bodies and the multi-solution of seismic data, there is uncertainty in the early stages of oil and gas exploration and development. This uncertainty arises from the uncertainty of geological bodies, geophysical and petrophysical properties, and dynamics. The level of accuracy in the reservoir modeling stage, the choice of various modeling algorithms and parameters, the theoretical underpinnings and practical expertise of reservoir crews, as well as the quality of the reservoir modeling, all contribute to the uncertainty. The objective and the cognition categories of system uncertainty can be distinguished [3,4]. Uncertainty primarily results from incomplete geological data, varying solutions, and reservoir practitioners’ inadequate comprehension of reservoir conceptual models in reservoir characterization [5]. The uncertainties in different stages and directions run through every link of reservoir research. In reservoir research, it is necessary to accurately analyze the source of uncertainty and evaluate the influence of uncertainty on reservoir development.

The development of oil and gas reservoirs has faced several risks and challenges as a result of reservoir uncertainty [6,7]. It mainly focuses on reducing and characterizing reservoir uncertainty. In order to reduce the risk of exploration and development and to provide instructions and suggestions for production, many scholars have studied the uncertainty from engineering, geology, and other aspects to reduce or have attempted to quantify the uncertainty of the reservoir [8,9,10,11,12,13]. Some academics used numerical analysis and seismic data comparison to reduce uncertainty [14,15]. Using dynamic data analysis and other techniques, we can better comprehend the reservoir and lessen its uncertainty [16,17]. The three levels of values—pessimistic, most likely, and optimistic—are currently often used to evaluate or characterize the uncertainty of reservoir parameter values. For instance, Tang et al. extracted the occurrence frequency of the specified sedimentary microfacies in multiple simulated three-dimensional spatial grids and obtained the three-dimensional probability distribution model of the specified sedimentary microfacies in each layer; they then used the Monte Carlo method to generate the cumulative probability distribution map of reserves. The cumulative probability points of 10%, 50%, and 90% are used to find the corresponding three reserve values, and the probability that the real reserves can reach these three reserve values is 90%, 50%, and 10%, respectively. These three reserve values are also known as P90, P50, and P10. They are used to evaluate the uncertainty of model reserves [18]. It should be noted that P10, P50, and P90 are not fixed, and they depend on the method of exceedance probability or non-exceedance probability. Taking P10 as an example, if the exceedance probability is selected, P10 means that the probability of being higher than the cut-off point is 10%, and the cut-off point is a higher value. P10 indicates a 10% probability of falling below this cut-off point, which is a lower value. In order to model the BZ oilfield in Bohai Bay, Chong et al. chose six geological variables, including the lower porosity limit, the sedimentary microfacies change-difference function, the sedimentary microfacies content, the porosity variation function, and the oil–water interface. They adopted maximum, minimum, and average values to represent the uncertainty [19]. Xiao chose several seismic attribute truncation values to represent the uncertainty of the fan-delta border [20]. By calculating the three level values, we can, on the one hand, enhance our control over the real value and, on the other, master the edge value’s range. As a key reservoir parameter, the lithofacies proportion affects the connectivity of sand bodies and the estimation of reserves [21], as well as the net-to-gross (NTG) in reservoir modeling, which indirectly affect the model quality. As for how to evaluate the uncertainty of the petrofacies proportion, the three level values are also generally used for assessment. For example, Huo et al., in analyzing the uncertainty variables affecting geological modeling and reserve calculation, gave the percentage of the microfacies volume of the branch channel as 43.0%, 41.3% and 40.0%, respectively, for the three level values [22]. Xue et al. selected several uncertain variables that had an impact on the reserves of the oil-field geological model for analysis. Using the data analysis function of the modeling software and the planar distribution of sedimentary microfacies, P10, P50, and P90 of different sedimentary microfacies were obtained; among these, the percentage content of sedimentary microfacies of the point bar was 45%, 55%, and 65% respectively [23]. The three corresponding level values are given in the above studies, but the specific calculation steps are not explained in detail. Based on this, this paper describes in detail a method that can provide the lithofacies proportion quickly and reasonably and can estimate the uncertainty of the lithofacies proportion.

A technique for analyzing sample properties based on the confidence interval theory is the N₀ test. To determine the smallest number of samples needed to accurately represent the sample population within a certain margin of error and variation, the variance and mean of the samples are used. In the reservoir modeling, a sparse set of datapoints is employed for the simulation, which is unable to capture the features as a whole. The quantity of calculations will grow if more datapoints are used for the simulation. In addition to maintaining the general geographical characteristics, using the N₀ approach for the estimation lowers the computation expense. The N₀ method can also be used to select the mesh size and calculate the optimal sampling density according to the requirements [24]. Based on the N₀ method, according to the existing data, especially under the condition of fewer wells with stronger uncertainty, the N₀ equation transformation is carried out under the condition of retaining the basic eigenvalues to achieve the following two goals based on the probability distribution: (i) to calculate the value of the three levels of the facies proportion to provide a reasonable range of facies proportion and (ii) to strengthen the control of the edge value. The uncertainty is indicated by the error p% of the mean value.

2. Method

The mean of the facies proportion in a particular reservoir has a specific value. However, the data available is limited, and it is not possible to obtain all the data that can reflect the facts. The proportional mean can only be estimated with limited data. Usually, the calculated mean of the sample is close to the real mean of the population, but the gap between the two cannot be ignored. Quantifying the gap between the “calculated value” and the “real value” is important because the size of the gap will affect the decisions of reservoir operators. This gap can be described by the confidence interval method. The confidence interval provides the degree of confidence in the measured value of the measured parameter as well as the probability that the real value of the parameter will fall within the range of the measurement findings. The range of the confidence interval can be used to describe the uncertainty of the facies proportion since it accurately captures the difference between the “calculated value” and the “real value”. There is an example of this method based on the confidence interval theory, which is called N₀ method. The N₀ method is based on the confidence interval theory and is used to estimate the number of samples representing the overall characteristics of samples under a certain error level. The approach of N₀ is to summarize the interval estimation as an error, and the interval range represents the error range. The error and confidence levels are taken as known quantities or expressed as acceptable values. At this time, the sample is taken as the population, the sample variance is taken as the population variance, and the estimation can represent the minimum sample number N₀ of the current sample characteristics. It is usually used for sample reduction to improve computational efficiency. At this point, in order to determine the appropriate facies proportion, the estimation of the sample number N₀ is converted to the estimation of the sample mean, and the fluctuation of the sample mean is represented by the error. The fluctuation range is the interval estimation of the sample mean under the given confidence level. The error can be calculated by the transformation of the N₀ formula. The following first introduces the N₀ method and then the formula transformation. The range of the facies proportion is determined by the transformed formula.

2.1. N₀ Test

Assuming that there are standard errors in the calculated mean and the selected significance level, the N₀ can estimate the samples required to meet the true mean of these predetermined errors and significance levels. For example, if there are N_x sandstone proportion datapoints, how many N₀ are required to estimate the mean value with the selected error and significance level, where $\overline{k}$ and S are the mean sandstone proportion and standard deviation calculated by N_x samples, respectively. $\overline{k}$/$\frac{S}{\sqrt{N_x}}$ is used for normalization, and $\frac{\overline{k}\sqrt{N_x}}{S}$ is assumed to have a t-distribution; at a certain confidence level (95%), its corresponding quantile is ${\tau }_{N_{0-\mathrm{1,0.975}}}$.

The goal is to find how many observations N₀ are needed to satisfy a predetermined 95% confidence interval for $\overline{k}$ within the error p%. Assuming that the statistic (τ) is based on the observed value N₀, that it satisfies the following relationship, and that $\frac{\overline{k}p\sqrt{N_0}}{100S}$ is half the confidence interval, then the formula is as follows:

$$\frac{\overline{k}p\sqrt{N_0}}{100S}={\tau }_{N_{0-\mathrm{1,0.975}}}$$

(1)

$\mathsf{\Phi }$ is the quantile corresponding to 0.975 in the standard normal distribution. Looking up the table, ${\mathsf{\Phi }}_{0.975}=2$, so assuming that ${\tau }_{N_{0-\mathrm{1,0.975}}}=2$, and

$$\frac{\overline{k}p\sqrt{N_0}}{100S}=2$$

(2)

Manipulating the formula gives:

$$N_0={\left[\frac{200S}{\overline{k}p}\right]}^2$$

(3)

where $\frac{S}{\overline{k}}=C$, and C is the coefficient of variation. Through the above formula, when the error is p% and the confidence is 95%, the required number of N₀ can be calculated.

2.2. Formula Improvement

To estimate the error p% of the mean, the formula in (3) is changed and updated. The known sample number is taken as N₀, and the unbiased estimate of the sample variance is taken as the population variance. The equation then reads as follows:

$$p=\frac{100\tau S}{\sqrt{N_0}\overline{k}}$$

(4)

At this time, the unknown population is estimated with samples, and τ is the corresponding quantile in the t distribution under a given confidence level and degree of freedom (DOF), wherein DOF is the sample number N₀ − 1, the equal-tail confidence interval is used, and $\overline{k}$ is sample mean. The difference between the sample mean and the true mean is represented by p%, and the standard deviation is S. The pessimistic and optimistic estimates of the mean can be calculated to characterize the uncertainty of the facies proportion, according to $\overline{k}(1\pm p\%)$. Under a specific level of confidence, the converted N₀ formula can quickly estimate the range of the mean. The approximate calculation flow of the method is shown in Figure 1.

3. An Application Example

3.1. Geologic Setting

The above workflow is applied to an example to predict the sandstone proportion of the M gas field, which is 439 km southeast of Shanghai and located in the X Depression in the northeast of the East China Sea shelf basin, one of the largest offshore Meso-Cenozoic sedimentary basins in China. The X Depression is mainly composed of Cenozoic sediments, and the large braided river sedimentary system is developed in the central and northern part of the sag, which forms giant thick diara- and riverbed-superimposed sand bodies. The source rocks of the Eocene Pinghu Formation (E2p) and Oligocene Huagang Formation (E3h) are mainly developed, which are the main spaces for oil and gas accumulation, of which the main gas layer H3 sand formation has the best reservoir property. The M gas field is located in the central inversion zone of the X depression and belongs to a low-permeability reservoir (Figure 2). The reservoir was formed under the background of large-scale compressions and contractions during the Oligocene period and belongs to the reversed anticlinal structure. The M gas field as a whole has a large burial depth, and it develops successively from bottom to top in the Eocene Pinghu Formation (E2p), Oligocene Huagang Formation (E3h), Miocene Longjing Formation (N1l), Yuquan Formation (N1y), Liulang Formation (N1ll), and Pliocene Sandan Formation (N2s). Among them, the Huagang Formation is the main effective reservoir with little change in thickness, which can be divided into the upper members of the Huagang Formation (H1–H5) and lower members of the Huagang Formation (H6–H7). The trap area of the target interval ranges from 31.75 to 49.83 km². The M gas field is located in the main channel of the braided-river-delta underwater distributary channel, showing the characteristics of large sand body thickness and multi-stage superposition of channel. The H3b sand formation is the main gas-producing sand formation, and the diara is the most favorable reservoir. The target interval of the study area comprises the H4 and H3 members of the Huagang Formation, and the research data are the lithologic proportion data of five wells and ten beds within the H4 and H3 members. The regional structural geological background of the study area is shown in Figure 2a. The overall regional structural background of this area is not complicated, but there are many structural zones, which lead to complex fault systems and diverse fault combination styles, resulting in an unstable reservoir and migration modes of oil and gas. The paleogeographic overview and sedimentary environment of the study area are shown in Figure 2b. The sedimentary materials are mainly controlled by the NE provenance, and the drilling wells are distributed at the front of the braided river delta. The lithology distribution of the five wells is shown in Figure 2c, which also includes the H4 and H3 intervals. The lithology is mainly medium-fine sandstone and siltstone.

Pebbled coarse sandstone, coarse sandstone, medium sandstone, fine sandstone, siltstone, and argillaceous siltstone are split into sandstone facies, while silty mudstone and mudstone are divided into mudstone facies to make the facies proportion calculation easier. Here, the reservoir is primarily made of sandstone. In order to lay the foundation for the subsequent reservoir characterization, the sandstone facies are used as an example to compute the proportion range of the sandstone facies and to examine the uncertainty of the sandstone proportion.

3.2. Application

It is vital to evaluate or analyze the research data before using the method of calculating facies proportions based on the confidence interval theory since the quality of the data may influence the outcome of the calculations. Declustering and normality tests are among the test methodologies. Since the premise of the approach relies on the statistics of a normal distribution for all the data, the distribution of the data will have an impact on how well the calculations turn out. This method gives values of the three levels directly from the data, so there are certain requirements to ensure data quality. The result might be distorted if certain huge clusters of high or low values exist. A similar occurrence happens frequently when developing reservoirs. For example, in the clastic sedimentary reservoir of continental facies, the facies transition is rapid, and the reservoir heterogeneity is serious. The characteristics of uneven reservoir distribution and severe heterogeneity lead to irregularity of well location design, and a favorable reservoir is the priority target. This results in the irregularity of the distribution of the obtained geological data. In the favorable part of the reservoir, the development wells will be dense, so that a lot of relevant data can be obtained. Due to inaccurate reservoir prediction and other reasons, there are only a few development wells and a few exploration wells in the distribution range of unfavorable reservoirs, so the geological data obtained is very limited. This uneven distribution of geological data may result in statistical results that are not representative of the actual data distribution. More favorable reservoir data skew the results. In less heterogeneous reservoirs, well distribution may primarily explore the edges of the reservoir, but this artificial tendency can also result in uneven well-data distribution. Generally speaking, the more data, the better the statistical results, and the smaller the error. The idea that we want the data distribution to be as even as possible is the underlying assumption. In this example, the number of wells in the studying area is small and the distribution is uniform, so only the normality test is carried out, and no declustering is carried out.

3.2.1. Normality Test

The assumption in the technique’s basic tenet is that the normal distribution’s sampling distribution may provide [25], i.e., the normal distribution’s data samples are relevant to the N₀ method. The distribution of the sample mean and the distribution of the sample variance are merged based on the properties of the normal distribution. The range of the sample mean under a certain level of confidence is computed using the principle of interval estimation. To demonstrate the applicability of this method, it is therefore important to conduct a normality test on the data of the study area prior to its application. H3 and H4 have 10 monolayers; the sandstone proportion of five wells was calculated, and its normality was tested in the working area as an example.

Table 1. Sandstone proportion data of 5 wells.

Wells	Well 1	Well 2	Well 3	Well 4	Well 5
Sandstone proportion (%)	88.928	83.415	87.766	75.550	95.971

displays the information from the five wells.

There are numerous ways to conduct a normality test; the most popular ones include the normal probability paper method, the Shapiro–Wilk test (S–W), the Kolmogorov–Smirnov test (K–S), and the skewness–kurtosis test. In this study, the distribution properties of samples were independently verified using the frequency histogram, Q–Q plot, P–P plot, skewness–kurtosis normality test and S–W normality test.

The frequency histogram is a technique for determining if the sample data follows the normal distribution by comparing its similarity to the normal curve. While the Q–Q plot compares the degree of coincidence between the theoretical quantile and the sample quantile, the P–P plot compares the degree of coincidence between the cumulative frequency of the theoretical normal distribution and the cumulative frequency of the sample data. The datapoints essentially coincide with the theoretical curve if the sample data follows the normal distribution. Skewness and kurtosis use the symmetry and steepness of the data distribution to determine whether the normal distribution is obeyed. The S–W normality test method was employed to evaluate the hypothesis of the sample data since there were fewer than 50 samples [26]. The null hypothesis cannot be rejected when the test result is greater than the significance threshold, meaning that the samples follow the normal distribution. The next five findings of the five normality tests derived from data samples (Figure 3,

Table 2. Skewness and kurtosis normality test table.

Sample Number	Skewness			Kurtosis
	Statistics	Standard Error	Z-Score	Statistics	Standard Error	Z-Score
5	−0.350	−0.913	−0.383	−0.681	2.000	0.341

and

Table 3. Shapiro–Wilke (S–W) normality test table.

Data Category	Statistics	DOF	Significance Level
Well data	0.981	5	0.939

).

The data distribution exhibits a normal pattern with a high frequency of intermediate values, as can be observed from the frequency histogram (Figure 3a). The findings of the P–P plot (Figure 3b) and the Q–Q plot (Figure 3c) demonstrate that the point data and theoretical line fit together nicely, and the data exhibits normal distributional properties. According to the comprehensive graph analysis (Figure 3), the data basically has the trend of a normal distribution, but further analysis is needed to determine the conclusion. The results of the skewness and kurtosis tests were observed, and the statistical values of both were less than 1 (Table 2). In order to facilitate the analysis of the results, the Z-score was introduced here. The Z-score converts a certain original score value into a standard score value, which can make the previously uncomparable values comparable. When the significance level α is 0.05, the absolute values of −0.383 and 0.341 are both smaller than 1.96 (Table 2), so it can be considered that the data basically follow the normal distribution. The data result for the S–W normality test table (Table 3) was 0.939, which was greater than the significance level of α (α = 0.05). It can be considered that the data has the characteristics of a normal distribution. It is concluded that the sandstone proportion data in this study area obey normal distribution by three methods: comprehensive graph judgment, the skewness–kurtosis normality test and the non-parameter test.

3.2.2. Results and Discussion

The computation is based on the drilled well’s petrographic proportion data. According to drilling coring or logging, the distribution of sandstone in each well is categorized, and the average proportion of sandstone is computed using the distribution of sandstone in each well (Table 1). The sample standard deviation in this instance is 7.52, the DOF is 4, and the relative τ value at the 95% confidence level is 2.776. The value of error p% can be calculated using formula 4 and the calculated characteristic values. The mean sandstone proportion’s optimistic and pessimistic estimates can then be determined. These values are listed in

Table 4. Calculation results of the sandstone proportion of the 5 wells.

Mean (%)	S (%)	C	τ	p (%)	Pessimism (%)	Optimism (%)
86.326	7.525	0.087	2.776	10.819	76.987	95.665

.

As a result, three levels of sandstone proportion are established at the 95% confidence level. The optimistic estimate is 95.665%, signifying the greatest sandstone proportion; the pessimistic estimate is 76.987%, signifying the lowest sandstone proportion; the mean value is 86.326%, representing the closest sandstone proportion; and the error p% is around 10.8%. If the proportion of the deterministic facies is determined solely by the mean value, the uncertainty may be disregarded, leading to a single computation result that cannot identify the error. The confidence interval theory-based approach of calculating the sandstone proportion yields an estimate interval. The genuine facies proportion value has a 95% likelihood of being inside the confidence interval, which is made up of optimistic and pessimistic values. The results that are estimated after accounting for error are more valuable and reliable. The intuitive data representation error p% is useful for expressing uncertainty. The three level values provide precise values for the underground sandstone proportion’s boundary range, quantifies some of the uncertainty, provides sandstone proportion values under various level conditions for reserve calculation and reservoir modeling, and provides elastic support for later exploration and development decision deployment. The dataset’s degree of dispersion is indicated by the standard deviation. Therefore, the standard deviation has an effect on the value of the error p%. The ratio of the standard deviation to the mean is known as the coefficient of variation. Indirectly reflecting the size of uncertainty, the size and variation of the coefficient of variation also reflect the fluctuation of error p%.

The closer the relationship between the sample mean and the population mean is in a normal distribution study, the more samples are used. However, this method is based on a t distribution and has low requirements for sample size, so it is also applicable for low samples, and there is no need to worry about the small number of samples without principle support. The underlying presumption, though, is that the raw data is as normally distributed as feasible, which can have a big impact on how satisfied people are with the outcomes. We also need to pay attention to data processing and optimization. In actuality, data distribution is frequently uneven, or even only for a partial data cluster distribution, which will impact the outcome of calculations. The clustering data will raise the weight of the related type because each individual data point has the same weight, which will cause the result to deviate. The distribution of wells is relatively sparse in the early phases of oil exploration and development, but as these phases advance, well density rises, and the impact of well dispersion must be taken into account. If the data distribution is too dense, the dense distribution characteristics of such data may have an impact on the estimation of the mean value. Before using the sample data, it is necessary to carry out the declustering step to eliminate the impact of the dense data on the result. Cell declustering and polygonal declustering are two declustering techniques, and the suitable declustering can be chosen based on various regional characteristics or the type of data. In particular, in the study of reservoir geology, it is crucial to underline that expert knowledge should be maintained in the process of data processing and optimization, where geological knowledge will determine whether or not a trend is retained. A simple fluvial reservoir illustrating the effects of data clustering is constructed using an object-based approach (Figure 4). The proportion of sandstone in the reservoir is set at 40%. The length and width are 15,000 m and the thickness is 25 m. The channel sand body in the reservoir is mainly located at the X-axis at 4000–10,000 m (Figure 4). Due to the influence of oil and gas collection, most drilling is located in the area with a high proportion of sandstone, because sandstone is a favorable place for oil and gas accumulation. By calculating the sandstone proportion of all the wells, the sandstone proportion of the reservoir is estimated to be 53.57%. Obviously, this is higher than the set fraction. The research area is divided into two parts: high proportion and low proportion, and the surface area represents the weight. The highest proportion is the position of the X-axis at 4000–10,000 m, the average is 76.07%, and the weight is about 0.4. The other positions are low-proportion parts, with an average of 19.83% and a weight of about 0.6. At this time, the proportion of sandstone is 42.32%. The latter is closer to the true value of 40%. Declustering is necessary at times.

4. Conclusions

Based on the theory of confidence intervals, the method for determining facies proportions can rapidly calculate the range of facies proportions and provide a definitive confidence interval. Taking the sandstone proportion of five wells in the M gas field as an example, the value of the three levels of the sandstone proportion are calculated as 76.987%, 86.326% and 95.665% for the pessimistic, most likely, and optimistic values, respectively. This provides a method reference for quantifying the uncertainty of data and also provides reasonable data preparation and probability support for subsequent modeling and reserve estimation. The application of this method is outlined below:

The data distribution of the original facies proportion may affect the results of the calculation, such as whether the data follows the normal distribution and whether the data is evenly sampled. However, the geometric form and distribution pattern of reservoirs vary, and the size of this influence needs to be analyzed in combination with the actual situation.

The control of the edge value is strengthened, and the smoothing effect provided by the mean value is weakened, when the optimistic and pessimistic estimations of the sandstone proportion are calculated using the N₀ technique with a certain degree of certainty.

On the basis of the probability distribution, we were able to introduce the error p%, determine the discrepancy between the sample mean and the true value based on the sample of data, and assess the success of replacing the sample mean with the population mean.