原文：
SPE 93632
Artificial Neural Network Model for Predicting Bottomhole Flowing Pressure in Vertical Multiphase Flow
Osman, E.A.，SPE，KFUPM，Saudi Arabia，Ayoub，M.A.，SPE，and Aggour，M.A.，SPE，Abu-Dhabi Petroleum Instit.
Abstract
Accurate prediction of pressure drop in vertical multiphase flow is needed for effective design of tubing and optimum production strategies. Several correlations and mechanistic models have been developed since 1950. In addition to the limitations on the applicability of all existing correlations,they all fails to provide the desired accuracy of pressure drop predictions. The recently developed mechanistic models provided little improvements in pressure drop prediction over the empirical correlations. However, there is still a need to further improve the accuracy of prediction for a more effective and economical design of wells and better optimization of production operations.
This paper presents an Artificial Neural Network (ANN) model for prediction of the bottom-hole flowing pressure and consequently the pressure drop in vertical multiphase flow. The model was developed and tested using field data covering a wide range of variables. A total of 206 field data sets collected from Middle East fields; were used to develop the ANN model. These data sets were divided into training, cross validation and testing sets in the ratio of 3:1:1. The testing subset of data, which were not seen by the ANN model during the training phase, was used to test the prediction accuracy of the model and compare its performance against existing correlations and mechanistic models. The results showed that the present model significantly outperforms all existing methods and provides predictions with higher accuracy. This was verified in terms of highest correlation coefficient, lowest average absolute percent error, lowest standard deviation, lowest maximum error, and lowest root mean square error. A trend analysis was also conducted and showed that the present model provides the expected effects of the various physical parameters on pressure drop.
Introduction
A reliable and accurate way of predicting pressure drop in vertical multiphase flow is essential for the proper design of well completions and artificial-lift systems and for optimization and accurate forecast of production performance. Because of the complexity of multiphase flow, mostly empirical or semi-empirical correlations have been developed for prediction of pressure drop.
Numerous correlations have been developed since the early 1940s. Most of these correlations were developed under laboratory conditions and are, consequently, inaccurate when scaled-up to oil field conditions1. The most commonly used correlations are those of (Hagedorn and Brown2; Duns and Ros3; Orkiszewski4; Beggs and Brill5; Aziz and Govier6; Mukherjee and Brill correlation7). Numerous studies were done to evaluate and study the applicability of those correlations under different ranges of data8-15. Most researchers agreed upon the fact that no single correlation was found to be applicable over all ranges of variables with suitable accuracy1. It was found that correlations are basically statistically derived, global expressions with limited physical considerations, and thus do not render them to a true physical optimization.
Mechanistic models are semi-empirical models used to predict multiphase flow characteristics such as liquid hold up, mixture density, and flow patterns. Based on sound theoretical approach, most of these mechanistic models were generated to outperform the existing empirical correlations. The most widely used mechanistic models are those of Hasan and Kabir16; Ansari et al17.; Chokshi et al.18; Gomez et al.19. Other studies were conducted to evaluate the validity of such mechanistic models20-22. Generally, each of these mechanistic models has an outstanding performance in specific flow pattern prediction and that is made the adoption for certain model of specific flow pattern by investigators to compare and yield different, advanced and capable mechanistic models.
However, a statistical study indicated that there is no pronounced advantage for mechanistic models over the current empirical correlations in pressure prediction ability when fallacious values are excluded1.
The recent development and success of applying artificial neural networks (ANN) to solve various difficult engineering problems has drawn the attention to its potential applications in the petroleum industry. The use of artificial intelligence in petroleum industry can be tracked back just almost twenty years23. The use Artificial Neural Network (ANN) in solving many petroleum industry problems was reported in the literature by several authors. Recently, ANN has been applied in the multiphase flow area and achieved promising results compared to the conventional methods (correlations and mechanistic models). With regard to this field, a few researchers applied ANN technique to resolve some problems associated with multiphase problems including pressure drop24-25, flow patterns identification26-27, liquid hold up30, and gas and liquid superficial velocities28.
Experience showed that empirical correlations and mechanistic models failed to provide a satisfactorily and a reliable tool for estimating pressure in multiphase flow wells. High errors are usually associated with these models and correlations. Artificial neural networks gained wide popularity in solving difficult and complex problems, especially in petroleum engineering.
The artificial intelligence (AI) or soft computing shows better performance over the conventional solutions. AI’s aim can be stated as “the development of paradigms or algorithms that require machines to perform tasks that apparently require cognition when performed by humans29. Artificial intelligence techniques are classified into ANN, genetic algorithms, expert systems, and fuzzy logic. ANN is a machine that is designed to model the way in which the brain performs a particular task or function of interest. The system of ANN has received different definitions30. However, a widely accepted term is that adopted by Alexander and Morton31: “A neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use”.
This paper presents an Artificial Neural Network (ANN) model for prediction of the bottom-hole flowing pressure and consequently the pressure drop in vertical multiphase flow. The model was developed and tested using field data covering a wide range of variables. A total of 206 field data sets collected from Middle East fields; were used to develop the ANN model. These data sets were divided into training, cross validation and testing sets in the ratio of 3:1:1. The testing subset of data, which were not seen by the ANN model during the training phase, was used to test the prediction accuracy of the model and compare its performance against existing correlations and mechanistic models.
Model Development
The developed ANN model utilizes multiple-layer feed forward networks, which were selected due to their capabilities of representing non-linear functional mappings between inputs and outputs. The developed model consists of one input layer (containing nine input neurons or nodes), which represent the input parameters (oil rate, water rate, gas rate, diameter of the pipe, length of pipe, wellhead pressure, oil gravity "API", surface temperature, and bottomhole temperature), three hidden layers (the first one contains six nodes, the second and third hidden layer each contains three nodes) and one output layer (contains one node) which is bottomhole pressure. This topology is achieved after a series of optimization processes by monitoring the performance of the network until the best network structure was accomplished (Fig. 1).
Data Acquisition and Pre-processing
A total of 386 data sets were collected from different Middle East fields. The data used for developing the model covers an oil rate from 280 to 19618 BPD, water cut up to 44.8%, and gas oil ratios up to 675.5 SCF/STB. To check the validity of the collected data and remove the suspected outliers, empirical correlations and mechanistic models were used to predict the bottomhole flowing pressures and compare it with the measured value. The mechanistic models of Hasan and Kabir16, Ansari et al.17, Chokshi et al.18, Gomez et al.19, and the correlations of Hagedorn and Brown2, Duns and Ros3, Orkiszewski4, Beggs and Brill5, and Mukherjee and Brill7 were used. Data sets which consistently resulted in poor predictions by all correlations and mechanistic models were considered to be invalid and, therefore, removed. A cut-off-error percentage (relative error) of 15% was implemented for the whole data. After such a screening, a total 206 data sets were used to develop the artificial neural network model. These were randomly divided into three different groups: training, validation, and testing. The training set is used to develop and adjust the weights in a network; the validation set is used to ensure the generalization of the developed network during the training phase, and the testing set is used to examine the final performance of the network and compare the model performance with other correlations and mechanistic models. Different partitioning ratios were tested (2:1:1, 3:1:1, and 4:1:1). The ratio of 4:1:1 (suggested by Haykin30) yielded better training and testing results. Table 1 shows the statistical analysis of the used data.
Results and Discussion
To evaluate a newly developed model, two tests must be performed. First, the model must be tested to prove that it is stable and simulates the physical process; this is done through "trend analysis". Second, the predictive performance of the new model must be compared against existing correlations and models. This is done through cross plots and a group error analysis, using the average absolute percent error as an indicator..
Trend Analysis
A trend analysis was carried out to check whether the developed model is physically correct or not. For this purpose, synthetic sets were prepared where in each set only one input parameter was changed while other parameters were kept constant. To test the developed model, the effects of gas rate, oil rate, water rate, tubing diameter, and pipe length on flowing bottomhole pressure were determined. Figures 2 and 3 show the effect of gas rate and tubing diameter on bottomhole pressure, respectively. The developed model showed the correct trend where the flowing bottomhole pressure decreases as the gas rate and tubing diameter increase.
Some correlations and Gomez model showed a decrease in bottomhole pressure followed by an increase when gas rate increase. The reason is that when the gas liquid ratio becomes very high, additional increase in gas rate results in an increase in frictional and acceleration pressure drop which is more than the decrease in the hydrostatic head. Figures 4 through 6 show the effect of water rate, oil rate, and depth, respectively. The figures show that the present model successfully produced the expected trends; i.e. the bottomhole pressure is increasing with increase in water rate, oil rate, and depth.
Comparison of the ANN Model against Other Models
As mentioned earlier, 41 data sets were used to evaluate the predictive capability of the present artificial neural network model and compare its performance against existing correlations and mechanistic models. The prediction performances of five correlations that have been used by the industry (Hagedorn and Brown2; Duns and Ros3; Orkiszewski4; Beggs and Brill5; Mukherjee and Brill7), and four mechanistic models (Hasan and Kabir16; Ansari et al17.; Chokshi et al.18; Gomez et al.19) were compared against the present model. Table 2 lists the important statistical parameters (defined in Appendix A) for comparative evaluation of the correlations, mechanistic models and the present ANN model.
To demonstrate the robustness of the developed model, the group error analysis was conducted. Average absolute percent (Ea) relative error is used as a good indicator of the accuracy. This effective comparison of all investigated correlations and mechanistic models provides a good means of evaluating models performance. AAPE is utilized in this analysis by grouping input parameter and hence plotting the corresponding values of average absolute relative error for each set. Figures 7 through 11 present the statistical accuracy of flowing bottomhole pressure correlations and models for different groups of the studied parameters. These include oil rate, gas rate, water rate, tubing diameter and depth, respectively. The figures showed that the present model consistently outperformed all correlations and mechanistic models and resulted in the lowest average absolute relative error in all data ranges of the studied parameters.
Cross plots were used to compare the performance of the developed mode and other correlations and mechanistic models. A 45° straight line between the estimated versus actual data points is drawn on the cross plot, which denotes a perfect correlation line. The scattered cloud of data points indicates bad correlation. Figures 12 through 21 present cross plots of predicted versus measured bottomhole pressure actual for the developed model, other empirical correlations and mechanistic models. Investigation of these figures clearly shows that the developed ANN model outperforms all correlations and mechanistic models.
Several observations and conclusions can be made by investigation of Figures 12 to 21 and Table 2. Hasan and Kabir model produced the largest error in predicting the bottomhole flowing pressure (Ea of 9.23% and correlation coefficient of 0.7502). Accuracy of prediction was improved for Ansari et al. model (Ea of 6.75% and correlation coefficient of 0.8178). The other two mechanistic models of Chokshi et al. and Gomez et al. resulted in a similar performance. Surprisingly, the empirical correlations, except for Duns and Ros, performed much better than the mechanistic models. Finally, Mukherjee and Brill correlations outperformed other correlations and mechanistic models (Ea of 4.903% and correlation coefficient of 0.8792). The predicted pressure drop by the present ANN model is compared against the measured values in Figure 21. Investigation of the figure clearly demonstrates the outstanding performance of the present model. The model predicted the 41 values of bottomhole flowing pressure with Ea of 2.165% compared to 9.23% for Hasan and Kabir. The correlation coefficient for the model is 0.9735 compared to 0.9015 for Orkiszewski, and 0.8836 for Chokshi model.
Conclusions
1. Artificial Neural Network model based back-propagation learning algorithm has been used was developed to predict the bottomhole flowing pressure in vertical wells.
2. The new model provided exceptionally accurate predictions over the best available empirical correlations and mechanistic models.
3. The developed model achieved best correlation coefficient (0.9735), the lowest maximum absolute relative error (7.1401%), the lowest root mean squared error (2.8013), the lowest standard error deviation (66.2448), and the lowest average absolute percent error (2.1654%).
4. Trend analysis of the model showed that the model correctly predicted the expected effects of the independent variables on bottomhole flowing pressure. This indicated that the model simulates the actual physical process.
5. The present study clearly demonstrates the power of artificial neural network model in solving complicated engineering problems. The developed model could perform even better if more data were used for training.
6. The new developed model can be used only within the range of used data. Caution should be taken beyond the range of used input variables.
译文：
版权2005年，石油工程师学会公司
Osman, E.A.，SPE， KFUPM，Saudi Arabia， Ayoub， M.A.， SPE， and Aggour， M.A.，SPE，Abu-Dhabi Petroleum Instit.
摘 要
准确预测垂直多相流的压力降是有效的设计油管和最佳生产战略的需要。几个相关和机理模型自1950年以来就已经制定了。除了限制适用于所有现有的相关式以外，它们都未能提供理想的压力降的预测准确性。最近开发的机理模型在压力降预测的实证的相关性方面提供了极少的改善。不过，仍然需要进一步改善预报的准确性，，以便更有效和经济的设计水井和更好地优化生产经营。
本文介绍了一个人工神经网络（ ANN ）模型预测井底流压和垂直多相流的压力降。该模型是用涵盖范围广泛的变数的野外数据来开发和测试的。为了发展人工神经网络模型，总共收集了从中东领域的206场的数据集。这些数据集被分为培训，交叉验证和测试集的比例是３：１：１。其中没有被人工神经网络模型看到的在训练期测试的子集数据，，是用来测试预测模型的准确性，，并与现有的相关式和机理模型比较其性能。结果表明，本模型显着优于所有现有的方法和提供的预测，具有较高的准确性。这是在验证的条款包括，相关系数最高，最低平均绝对误差，最低的标准偏差，最低最大误差，和最低的均方根误差。同时也进行了趋势分析，表明本模型提供了关于压力降的预期的效果的各种物理参数。
导 言
一个可靠和准确的预测垂直多相流压力降的方法，是完井必不可少的适当设计，以及落成、人工电梯系统和进行优化和准确预测的生产性能。由于复杂的多相流，所以为了预测压力降，大部分的实证或半经验的相关式已经研究开发出来了。
很多相关式已自20世纪40年代开发出来。大部分这些相关式的发展都是在实验室条件下，因此当相关式扩大到油田条件时就不准确了。最常用的相关式，是哈格多恩和布朗；顿斯和罗斯；奥克适韦斯基；贝格斯和比利；阿齐兹和古韦；慕克吉和比利相关式。无数的研究根据不同的数据范围对这些相关式做了评估和研究其适用性。大多数研究人员认可的事实，即没有一个单一的相关式被发现是适用于所有范围的变数和获得合适的准确性。结果发现，这些相关式，基本上是统计得出的，用有限的物理因素表达全局，因此不给予他们一个真正的物理优化。
机理模型是半经验模型，可用来预测多相流的特点，比如持液虑，混合密度，流动模式。基于健全的理论方法，大部分这些机理模型产生了超出现有的实证的相关式。使用最广泛的机理模型，是那些哈桑和卡比尔；安萨里等人；弨可士等人；戈麦斯等人。其他的研究则是为了评估机理模型的有效性。一般来说，这些机理模型中的每一个在具体的流型预测有出色的表现，这是取得通过为某型号的具体流态调查比较和产量的不同，先进的和有能力的机械模型。
然而，一项统计研究表明，当谬误的价值观被排除时，机理模型的预测压力能力和目前的实证相关式相比，不存在突出的优势。
最近人工神经网络（ANN）的发展和成功应用表示，以解决工程困难等问题和其潜在的应用都在石油工业引起了注意力。在石油工业中使用人工智能技术还可以追踪回到几乎20年前。在文献中由若干作者报道，使用人工神经网络（ANN）可以解决许多石油工业的问题。最近，人工神经网络已应用在多相流领域，与传统的方法（相关性和机理模型）相比取得了可喜的成果。关于这方面，有几个研究者应用人工神经网络技术来解决一些相关的问题与多相问题，包括压力降，流动形态的判定，持液虑，气体和液体表面的速度。
经验表明，实证的相关性和机理模型未能在多相流井估计压力方面提供令人满意的和可靠的工具。高错误通常是与这些模型和相关式相关的。人工神经网络在解决困难和复杂的问题，特别是在石油工程得到了广泛普及。
该人工智能（AI）或软计算表明，其较传统的解决办法相比有更好的表现。人工智能的目的，可以表述为“发展需要机器来执行任务的范式或算法，由人运行时显然需要认知。人工智能技术可分为人工神经网络，遗传算法，专家系统，模糊逻辑。人工神经网络是一台是设计以脑执行特定任务或功能的兴趣的方式的模型的机器。该人工神经网络系统已经有不同的定义。然而，被广泛接受的任期是通过亚历山大和莫顿的 ：“神经网络是一个大规模并行分布式处理器，有一个自然的倾向储存的知识和经验，并使其可使用”。
本文介绍了一个人工神经网络（ ANN ）预测井底流压垂直多相流的压力降的模型，该模型是开发和测试使用的涵盖范围广泛的变数的野外数据，收集了从中东领域共206组的数据集；被用来发展人工神经网络模型。这些数据集，分为培训，交叉验证和测试集，它们的比例是3：1：1。其中没有被看到由人工神经网络模型在训练期测试的子集数据，，是用来测试预测模型的准确性，，并与现有的相关性和机理模型比较其性能。
模型开发
发达的人工神经网络模型，利用由于他们代表非线性泛函映射投入和产出之间的能力被选定的多层前馈网络。发达的模式，由一个输入层（含9输入神经元或节点） ，其中所代表的输入参数（油率，含水率，燃气率，直径管道，管道的长度，井口压力，原油比重，“空气污染指数” ，表面温度，井底温度） ，三隐层（第一个包含6节点，第二次和第三次隐层的每个所包含的三个节点） ，以及一个是井底压力的输出层（包含一个节点）组成。当监测网络的表现取得了一系列的优化过程，直到最好的网络结构完成（图1 ），拓扑就完成了。
数据采集和前处理
来自中东的不同领域的共386数据集被收集起来。所有数据均用来发展模型，涵盖了石油比率由280至19618（沸点深度曲线），含水率高达44.8% ，石油和天然气的比例高达675.5标准立方英尺/标准桶。检查所收集的数据的有效性，并消除怀疑离群，实证的相关性和机理模型被用来预测井底流动压力，并比较它与实测值。机械模型如哈桑和卡比尔；安萨里等人；弨可士等人；戈麦斯等人，以及相关的哈格多恩和布朗，顿斯和罗斯，奥克适韦斯基，贝格斯和比利，慕克吉和比利等均被使用。其中一贯导致在所有的相关性和机理模型无力的预测数据集，被视为无效，因此被删除。 一15 ％截断误差百分比（相对误差）被实施为整个数据。经过这样的筛选，共有206个数据集被用来发展人工神经网络模型。这些数据被随机分为三个不同的群体：培训，验证和测试。训练集是在一个网络中用来发展和调整的权数；验证一套是在训练阶段用来确保泛化发达的网络；测试集是用来检查网络最后的表现，并与其他的相关性和机理模型比较模型的表现。不同的分割比例（ 2：1：1， 3：1：1，）被进行了测试。4：1：1（哈金所建议的）的比例取得了更好的培训和测试结果。表1显示的统计分析所用的数据。
结果与讨论
评价一种新开发的模型，两项测试都必须执行。首先，模型必须被测试，以证明它是稳定的和可模拟物理过程；这项工作是通过“趋势分析”完成的。第二，新模型预测的表现，必须与现有的相关式和模型相比。这项工作是通过交会图和一组误差分析，使用平均绝对误差作为指标。
趋势分析
为了检查发展模式是正确的物理过程或不是，研究者对趋势进行了分析。为此，在每套只有一个输入参数的改变而其他参数保持恒定的地方准备了综合数据集。测试该发达的模型，气体率，油率，含水率，油管直径和管道长度对流动井底压力的影响被进行了测定。图2和图3分别显示了气体率和油管直径对井底压力的影响效果。该发达的模型表明了正确的趋势，如由于含气率和油管直径增加流动的井底压力下降。
一些相关式和戈麦斯模型表明，当含气率增加时，流动的井底压力下降。原因是，当气液比例变得非常高，额外增加的气体率的结果有所增加，摩擦和加速度压力比在静压头下降的多许多。图4到6分别显示了含水率，油率和深度的影响。这些图显示，本模型成功地产生预期的趋势；即当含水率，油率和深度增加时井底的压力是越来越增加
人工神经网络模型与其他模型的比较
如前所述， 41个数据集被用来评估目前的人工神经网络模型预测的能力，并与现有的相关式和机理模型比较其性能。5个已被工业所用的相关式（哈格多恩和布朗；顿斯和罗斯；奥克适韦斯基，贝格斯和比利，慕克吉和比利）和4个机理模型（哈桑和卡比尔；安萨里等人；弨可士等人；戈麦斯等人）与目前的模型相比其预测性能。表2列出了进行比较评价的相关式，机理模型和目前的人工神经网络模型的重要的统计参数（定义于附录A）。
为了表现发达模型的稳健，该模型的错误分析被指出。平均绝对百分（Ea）相对误差是用来作为一个很好的准确性的指标。这个关于所有的相关式的调查和机理模型有效的比较提供了一个评价模型的表现的良好的手段。美国石油地质学家协会是利用在这方面的分析，分组输入参数，并为每套策划了在平均绝对相对误差方面的相应的价值。图7至图11为不同的群体的研究参数展示了目前流动井底压力的相关式和模型统计的准确性。这些参数分别包括油率，燃气率，含水率，油管直径和深度。这些图片显示，目前的人工神经网络模型，始终优于所有的相关式和机理模型，并在所有研究参数数据的范围产生最低的平均绝对相对误差。
交会图被用来比较该发达的模型和其他的相关式和机理模型的表现。 一条45°的直线是由估计的数据与实际的数据点的交叉点来制定的，这是一个完美的相关线。散落在外的数据点显示为不正确的相关式。图12至21显示为该发达的模型，其他实证的相关式和机理模型预测测量井底压力。调查这些图线清楚地表明，发达的人工神经网络模型优于所有的相关式和机理模型。
由调查图12至21和表2可以作出若干意见和结论。哈桑和卡比尔模型在预测井底流压时产生的最大的错误（ 是9.23 ％和相关系数为0.7502 ）。安萨里等人的模型提高了预测的准确性（ 是6.75厘和相关系数为0.8178 ）。其他两个机理模型弨可士等人和戈麦斯等人的模型产生了一个类似的现象。令人惊讶的是，实证的相关式，除了顿斯和罗斯，表现明显优于机理模型。最后，慕克吉和比利的相关式优于其他的相关式和机理模型（ 是4.903 ％和相关系数为0.8792 ）。由目前的人工神经网络模型预测的压力降与其他的相关式和机理模型相比较，其测量值在图21。调查的图线清楚地表明了目前的模型的出色表现。该模型预测的41组井底流压其为2.165 ％，与其相比，哈桑和卡比尔的 是9.23％。本模型的相关系数是0.9735，与其相比，奥克适韦斯基为0.9015，和弨可士模型为0.8836。
结 论
1. 以反向传播学习算法基础的人工神经网络模型已经在垂直井被用于开发预测井底流压。
2. 新模型超过现有的最佳实证的相关式和机理模型提供了极为准确的预测。
3. 该发达的模型，取得了最好的相关系数（0.9735），最低的最大绝对相对误差（7.1401％），最低的均方根误差（2.8013），最低的标准误差偏差（66.2448）和最低的平均绝对百分率误差（2.1654 ％）。
4. 模型的趋势分析表明，在独立变量对井底流压有预期的影响时，该模型给出了正确预测。这表明，该模型模拟了实际的物理过程。
5. 本研究清楚地表明了人工神经网络模型在解决复杂的工程问题的能力。如果有更多的数据被用于训练，该发达的模型甚至可以更好地执行。
6. 新开发的模型只可用于有范围使用的数据。读者应注意，超越范围使用的输入变量。
