Mixture of relevance vector regression experts for reservoir properties prediction☆
Introduction
Reservoir characterization is a process to quantitatively predict and describe the reservoirs by using multidisciplinary information. One of the main tasks is to simulate the spatial distribution of reservoir parameters (including elastic attributes, lithofacies, porosity, permeability, etc.) by using various observed data. Predicting reservoir properties is imperative to reservoir engineers for appraising reservoirs, determining optimal well locations, and promoting production. Therefore, reservoir characterization research is significant in the exploration, development, and evaluation of oil and gas fields (Fournier and Derain, 1995, Alvarez et al., 2003, Feng et al., 2018, Babasafari et al., 2020).
Geophysicists are tasked to estimate unknown reservoir properties in the extensive inter-well area by utilizing the limited known target and other information. Seismic data contain lateral and vertical changes that are the signatures of the response caused by a reservoir. It is essential while difficult to directly obtain reservoir properties from seismic data because the relationship between reservoir properties and seismic data is influenced by many interacting factors (Grana and Rossa, 2010, Baron and Holliger, 2011, Saggaf et al., 2003). Seismic elastic attributes can be used to estimate reservoir properties (Tetyukhina, 2011, Wang, 2012, Yu et al., 2020). Traditional methods can achieve reservoir properties prediction by establishing petrophysical models under some assumptions or building empirical formulas. Then, they are applied to seismic elastic attributes (Babasafari et al., 2021, Bashir et al., 2021). However, the accuracy will be negatively impacted when the assumptions cannot be completely satisfied.
Machine learning can achieve the reservoir properties prediction by training models with the known data and applying the model to unknown data, while it is independent of the traditional pertophysical hypothesis (Liu et al., 2021a). The first step is training a regression model by using the observed data. Then, the learned model performs prediction on incoming data. Scholars have invented a wide range of techniques for geophysical application (Zhang et al., 2018, Liu et al., 2021b). Support vector regression (SVR) and neural networks are two of the most-used machine learning algorithms in geophysics for reasons of computational tractability. Li et al. (2005), and Zhong and Carr (2019) used the support vector machine to detect the reservoirs. Saggaf et al. (2003), and Ahmed et al. (2010) estimated the reservoir properties from seismic data by using neural network approaches. However, the two algorithms are very sensitive to parameters such as penalty parameters (which are used to balance the empirical risk and structural risk), and learning rate (Suykens and Vandewalle, 1999, Wohlberg et al., 2005, Liu et al., 2020). Inappropriate parameters will result in an unsatisfactory prediction result and will enlarge uncertainty. Many publications have investigated how to optimize the related parameters by a global optimization algorithm, such as genetic algorithms (Li et al., 2018), particle swarm algorithms (Zhang and Liu, 2008) and quantum particle swarm algorithms (Liu et al., 2019), and many others. However, it is time-consuming to find the ideal parameters (Liu et al., 2020).
Relevance vector regression (RVR) is an available alternative Bayesian based method that is first described in Tipping (2001). It can, in principle, overcome the drawbacks of SVR. RVR does not rely on human experience to set the penalty parameter. RVR involves dramatical fewer key vectors than SVR with a comparable accuracy, thus the learning model of RVR is more sparse, resulting in a shorter prediction time on test data (Bishop and Tipping, 2000). Through kernel learning, RVR also projects the original data into a high dimensional space, which makes the data be more easily forecasted. Apart from that, RVR does not require the kernel function meets Mercer’s condition, and any function can be exploited as kernel in theory (Burden and Winkler, 2015). RVR has been successfully applied to many fields, whereas applications in geophysics have rarely been reported.
In practice, the distribution of the data in different wells is not completely consistent. In a well, the distribution of data for different lithofacies may also be inconsistent. The issue of data distribution can be addressed by training an individual machine learning model with a large number of training samples. Unfortunately, the training data is limited in practice. That is to say, a single expert usually cannot account for the whole features in the case of limited samples. In this situation, we can train multiple models that are responsible for different parts of the data to enhance accuracy. The mixture of experts (ME) can achieve this goal. Initial ME is a neural network that trains multiple models for local regions of the input data (Jacobs et al., 1991, Meeds and Osindero, 2006, Jain, 2019). It is competitive to regression and classification for non-stationary data. In addition, ME is flexible because most machine learning algorithms can be combined theoretically (Lima et al., 2007, Chao and Neubauer, 2008, Yuksel and Gader, 2010). Each model represents an expert and all experts are integrated by a gating function. ME is a compromise between a single global learning model and multiple local learning models (Meeds and Osindero, 2006, Kim-Anh et al., 2010). ME allows each expert to specialize on different smaller parts of a complex problem (Kim-Anh et al., 2010). The gating function is responsible for making partitions of the input dataset and assigning regions for the individual experts.
We develop a reservoir properties prediction method based on ME to divide and conquer the original dataset. We employ RVR as experts to attain the learning model that is a weighted sum of experts by a gating function. The proposed mixture of relevance vector regression experts method (MR-VRE) decomposes a complex large prediction problem into several small regression problems, whose structure is shown in Fig. 1. The presented method does not assume that the data are stationary. It also does not depend on the rock physics modeling that is used to transform elastic attributes into reservoir properties in many traditional methods.
The key contributions of this manuscript are: (1) The superiority of the union of the mixture of experts and relevance vector regression algorithm is demonstrated in the aspect of likelihood of fit and accuracy; (2) The method for reservoir properties prediction has been successfully applied to well and seismic datasets; (3) The proposed method could potentially be used to incorporate other experts, and would stimulate more investigations into this learning strategy in the geological and geophysical field.
In the following sections, we will introduce the mathematical formulation and the related algorithms in detail, and then show some applications of the novel method on well and seismic data.
Section snippets
The input and output data
We aim to predict reservoir properties in wells and seismic areas where reservoir properties are not observed or interpreted. For each sample, the input is a vector consisted of input features, such as logging measurements and elastic attributes. The training data contain input attributes and target reservoir properties, while the test data only contain input attributes. Assuming a training dataset with samples where denotes the input elastic attributes (density, P-
Test on well data
Initially, we test the method on a well dataset from a work area in China. There are only three available wells (A, B, and C), where well A is used as training data with a total of 4000 samples and well B with a total of 3800 samples is used as a validation well to analyze the performance of MRVRE with varying quantities of experts. Well C contains 5440 samples. The space distance between well A and B is about 3500 m, and well C is far away from Well A with a distance of 2700 m. Then, the
Discussion
The computational time of the method is mainly spent on the training process, which is associated with the number of samples. The method includes three iterate processes. The first iteration (Step.3) is for the whole model and the iteration number in all examples of this paper is about 5 to 15. Too few iterations will produce unsatisfactory predictions whereas too many iterations extend the training time. The inner iteration involves the training of experts (Step.3.2) and the calculation of
Conclusion
We propose a novel quantitative reservoir properties prediction method that employs a mixture of relevance vector regression experts model. The major advantage of the proposed method is that it can use multiple experts to dynamically divide and conquer the data by cooperating with a gating function, which is more targeted than using an individual learning model. Therefore, it allows capturing the incomprehensible relations between reservoir properties and elastic attributes, then improving the
CRediT authorship contribution statement
Xingye Liu: Conceptualization, Methodology, Software, Writing. Guangzhou Shao: Resources, Data curation, Writing – review & editing. Cheng Yuan: Reviewing, Resources. Xiaohong Chen: Investigation, Supervision, Validation. Jingye Li: Supervision, Validation. Yangkang Chen: Visualization, Writing - review & editing.
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This work is financially supported by the Fundamental Research Funds for the Central Universities, CHD (300102261504) and Natural Science Basic Research Program of Shaanxi Province, China (2021JQ-561).