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一种考虑物理过程信息的油气渗流深度学习新模型

黄朝琴¹
，年凯¹
，王斌¹
，巩亮¹
，胡慧芳²
，张世明²
， CHUNG Eric ³

1. 中国石油大学(华东)油气渗流研究中心,山东青岛 266580； 2. 中国石化胜利油田勘探开发研究院,山东东营 257000； 3. 香港中文大学数学系,香港 999077；

A novel deep learning model with physical process information for prediction of flow behaviors in oil and gas reservoirs

HUANG Zhaoqin¹
， NIAN Kai¹
， WANG Bin¹
， GONG Liang¹
， HU Huifang²
， ZHANG Shiming²
， CHUNG Eric³

1. Research Center of Multiphase Flow in Porous Media, China University of Petroleum (East China), Qingdao 266580 , China； 2. Shengli Oilfield Exploration and Development Research Institute, SINOPEC, Dongying 257000 , China； 3. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong 999077 , China；

作者简介:

黄朝琴(1981-),男,副教授,博士,研究方向为复杂介质渗流理论及其数值模拟｡E-mail:huangzhqin@upc.edu.cn｡

中图分类号:TE319;TE312

文献标识码:A

DOI:10.3969/j.issn.1673-5005.2020.04.006

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目录contents

摘要Abstract
关键词Keywords
1 考虑物理过程信息的油气渗流深度学习模型
2 单相渗流场的深度学习
2.1 定注采关系下的渗流场学习与预测
2.1.1 一维单相渗流问题
2.1.2 二维单相渗流问题
2.2 变注采关系下的渗流场学习及预测
3 两相渗流场深度学习
3.1 Buckley-Leverett强非线性方程
3.2 二维两相渗流问题
4 结论
参考文献

摘要

目前机器学习仍属于纯数据驱动下的单一数据学习,其结果和学习过程的物理可解释性有待提高｡在油气田开发中,井网密度较小,井点数据属于稀疏训练样本数据,即使采用深度学习,其预测效果仍欠佳｡油气流动物理过程一般较为明确,即满足渗流方程,将渗流方程作为约束条件加入深度学习损失函数项,建立一种考虑物理过程信息的油气渗流深度学习新模型｡通过单相和两相流算例验证模型的正确性和高效性｡结果表明:在数据样本充足情况下,无论是传统模型还是新建模型均能获得良好的学习和预测效果;随着数据样本的减少,传统模型的学习和预测结果误差也随之增大,但新建模型仍能保持较高精度,即使在强非均质和注采关系反转条件下也能保证预测精度｡

Abstract

At present, machine learning still belongs to pure data-driven learning. The results and the physical interpretability of the learning process need to be improved. In the development of oil and gas fields, the density of the well pattern is small, and the well production data is of sparse training sample data. Therefore, if deep learning could be applied, the prediction re- sults would be still not as expected. However, in the process of oil and gas flow in oil reservoirs, the physical model is clear and can be uniquely described by the Darcy’s law, so the machine learning methods can be applied to study flow behaviors in porous media. In this regard and in this study, the percolation physics equation was added as a constraint condition to the deep learning loss function term, and a new type of deep neural network machine learning model was proposed. The correctness and efficiency of the model were verified through single-phase and two-phase flow case studies. The results show that, with suffi- cient training data samples, both the traditional model and the new model can obtain good learning and prediction results. Hov- ever, with the decrease of training data samples, the errors from the traditional model increase, while the new model can still provide high prediction accuracy, even under strong heterogeneity and in the process of injection-production well inversion

关键词

深度神经网络；物理过程信息；油气渗流；渗透率场

Keywords

deep neural network ； physical process information ； oil and gas flow in porous media ； permeability field.

Hinton等^[1]于2006 年创建了深度信任网络训练模型,并成功应用至数据降维和图像识别中。随后,深度神经网络模型迅速发展并成为研究热点^[2-12],相比于传统人工神经网络具有更强的数据关系挖掘能力,此外深度神经网络亦可以多函数求解,同步预测,大大提高了计算效率^[13-18]。但现有的深度神经网络模型对小数据样本的学习预测结果不佳,且预测结果缺乏物理可解释性。针对油气勘探开发问题,李燕生等^[19]提出了一种神经网络模式识别方法,结合地震反射资料来预测井间储层参数; 张雁等^[20]根据渗流原理,利用人工神经网络算法建立了砂岩储层渗透率预测模型;宋志军等^[21] 集成多种神经网络模型及算法,基于油水井动静态数据对储层含油饱和度进行了预测研究。上述模型考虑了流量、压力等生产参数,但仅将其视为渗透率及饱和度的影响因素,未考虑渗流物理过程信息,其本质仍类似于传统的数理统计方法。最近,Takbiri-Boruje-ni等^[22]开发了流动数值模拟代理模型,实现了基于固体颗粒的几何形状和孔径分布对流场和渗透率预测;但模型结果仅对观测井局部区域适用,对于井网稀疏的小数据样本预测能力不佳。 Raissi等^[23]提出了一种基于物理信息的神经网络模型,并应用于Burgers等方程参数的识别和预测研究,但该模型对于强非线性问题的学习效果欠佳。对此,笔者将渗流物理过程信息有机融合到损失函数项中建立一种油气渗流深度学习新模型。
1 考虑物理过程信息的油气渗流深度学习模型
考虑单相渗流通用控制方程为

μ C_{t} \frac{\partial p (x, t)}{\partial t} - \nabla \cdot [k (x) \nabla p (x, t)] = 0

(1)

式中, $p$ 为储层压力,Pa;t为时间变量,s;x为空间变量,m;k为储层渗透率, $μ$ m ²; $μ$ 为流体黏度,mPa· s; $C_{t}$ 为综合压缩系数,Pa ^-1。
由式(1)左端定义函数 $f (x, t)$ 为

f (x, t) = μ C_{t} \frac{\partial p (x, t)}{\partial t} - \nabla \cdot [k (x) \nabla p (x, t)]

(2)

在此定义深度神经网络输出 $\tilde{p}$ (x,t)和 $\tilde{k}$ (x)用以逼近压力p( x,t)和渗透率k( x),定义考虑渗流物理信息的神经网络输出 $\tilde{f}$ ( x,t),该网络可以由 $\tilde{p}$ (x,t)和 $\tilde{k}$ ( x)应用自动微分^[24] 获得。深度神经网络模型简图如图1 所示。
图1 深度神经网络模型简图
Fig.1 Schematic of a minimal deep neural network
为简单起见,图1 中设置了2 层深度神经网络, 每层仅包含1 个神经元,输入为X,则输出h可以表示为

h = W^{(3)} φ (W^{(2)} (φ (W^{(1)} X + b^{(1)})) + b^{(2)}) + b^{(3)}

(3)

式中, $φ$ 为双曲正切激活函数; $W^{(1)}$ 和 $b^{(1)}$ 分别为第i层的权重矩阵和基向量。
在本研究中,设置了9 层深度神经网络,每个隐藏层网络包含20 个神经元和1 个双曲正切激活函数,如图2 所示。
神经网络 $\tilde{p}$ (x,t)、 $\tilde{k}$ (x)和 $\tilde{f}$ (x,t)之间的共享参
图2 深度神经网络结构
Fig.2 Structure diagram of deep neural network
数可以通过最小化均方差损失函数来学习:

L = L_{p} + L_{k} + L_{f}

(4)

其中L表示损失函数,本文中采用均方差,下标p、k、 f分别表示压力、渗透率和残差函数f(x,t), $L_{p} = \frac{1}{N} \sum_{i = 1}^{N} {| p (x_{p}^{i}, t_{p}^{i}) - p^{i} |}^{2}$ 和 $L_{k} = \frac{1}{N} \sum_{i = 1}^{N} {| k (x_{k}^{i}) - k^{i} |}^{2}$ , 为神经网络与目标函数的均方差,该部分对与样本存在偏差的预测值施加惩罚,是传统纯数据深度学习损失函数项, $L_{f} = \frac{1}{N} \sum_{i = 1}^{N} {| f (x_{f}^{i}, t_{f}^{i}) |}^{2}$ ,为基于渗流物理信息的神经网络的均方差,该部分强制预测结果满足式(1),即预测结果满足渗流物理过程,是本文中模型的创新之处,选用L-BFGS算法^[25] 优化损失函数L以得到目标函数p(x,t) 和k(x)。
2 单相渗流场的深度学习
2.1 定注采关系下的渗流场学习与预测
2.1.1 一维单相渗流问题
对式(1)设置初边值条件,为计算简便,取值为单位变量,初始时刻压力为1 Pa,左端以1 Pa定压注入,右端以0 Pa定压开采,储层长度1 m,模拟开发时间1 s。由于注采井数和观测井数的限制,通常只能获得个别空间位置随时间变化的流场信息,因此本研究选取固定点压力和渗透率作为训练样本, 即 ${x_{p}^{i}, t_{p}^{i}, p^{i}}_{i = 1}^{N}$ 为p(x,t)在固定点的整个时间域内训练数据样本, ${x_{f}^{i}, t_{f}^{i}}_{i = 1}^{N}$ 为f( x,t)与之对应的数据点, ${x_{k}^{i}, k^{i}}_{i = 1}^{N}$ 为k(x)对应位置的数据点。对式(1) 进行数值求解,其中空间步数为200,时间步数为100,渗透率取值为经过光滑处理的随机函数,最大值为1 $u$ m ²,极差为50,形成200×100 个压力数据点和200 个渗透率数据点。本研究中,从200 个空间步数里等距选取8 个观测井点(包含2 个注采边界点),将观测井点对应的8 个渗透率值和8×100 个压力值作为训练样本,实现对全域内的压力和渗透率预测。将损失函数式(4)中的Lf项移除,仅对训练样本进行纯数据深度学习,图3、4、5 分别为本文新模型和纯数据深度学习的压力、压力切片(t=0.5 s)和渗透率预测结果。结果对比表明,考虑物理过程信息的油藏渗流场深度学习模型具有良好的准确性和高效性。
图3 一维单相渗流压力预测结果对比
Fig.3 Comparison of 1D single-phase flow pressure prediction
图4 一维单相渗流压力切片对比(t=0.5 s)
Fig.4 Comparison of 1D single-phase flow pressure slice(t=0.5 s)
为进一步说明考虑物理过程信息的油气渗流深度学习模型的预测效果,对不同渗透率极差下,不同观测井点数的压力和渗透率预测展开研究,其中观测井点均匀分布。表1 和表2 分别为不同渗透率极
图5 一维单相渗流渗透率预测结果对比
Fig.5 Comparison of 1D single-phase flow permeability prediction
差、不同观测井点数下,应用本文模型和纯数据深度学习得到的压力和渗透率预测误差对比,其中k_r 为渗透率极差。
表1 不同渗透率极差下一维单相渗流压力预测误差对比
Table 1 Error of 1D single-phase flow pressure prediction under different permeability ratios
表2 不同渗透率极差下一维单相渗流渗透率预测误差对比
Table 2 Error of 1D single-phase flow permeability prediction under different permeability ratios
压力预测误差和渗透率预测误差定义为

E_{Φ} = \sqrt{\frac{{(Φ_{prediction} - Φ_{real})}^{2}}{Φ_{real}^{2}}}

(5)

式中, $Φ$ 为压力或渗透率。
表1、2 结果表明,当观测井点数足够多,即训练样本足够多时,两种学习方法预测误差均很小。对于同一渗透率极差,随着观测井点数减少,整体的预测误差增大,但本文模型预测结果优于纯数据深度学习预测结果,这体现了本文方法的准确性和高效性;对于同一观测井点数,随着渗透率极差的增大, 整体的预测误差增大,但本文模型预测结果优于纯数据深度学习。
此外,研究观测井点分布对于预测结果的影响。分别从200 个空间步数里均匀选取和随机选取8 个观测井点,采用两种观测井点选取方式进行预测,图6、7、8 分别为不同观测井分布方式下的压力、压力切片和渗透率预测结果。对比结果表明,尽管观测井点数相同,但观测井点均匀分布时预测结果较好, 因此为了实现高效精准的预测目标,观测井点应尽可能均匀分布,避免局部区域观测井点位置过于稀疏或密集造成样本学习数据的缺失或重复。
图6 不同观测井点分布方式下压力预测结果对比
Fig.6 Comparison of pressure prediction under different observation well point distributions
图7 不同观测井点分布方式下压力切片对比(t=0.5 s)
Fig.7 Comparison of pressure slice prediction under different observation well point distributions
图8 不同观测井点分布方式下渗透率预测结果对比
Fig.8 Comparison of permeability prediction under different observation well point distributions
2.1.2 二维单相渗流问题
对式(1)设置初边值条件,研究区域为1 m×1 m,初始时刻压力为1 Pa,左下端以0 Pa定压开采, 右上端以1 Pa定压注入。对式(1)进行数值求解, 空间划分为20×20 个网格,时间步数为100,渗透率取经过光滑处理的随机函数,最大值为1 $u$ m ²,极差为2,形成20×20×100 个压力数据点和20×20 个渗透率数据点。本研究中,从20×20 个网格中均匀取25 个观测井点(包含2 个注采边界点),将观测井点对应的25 个渗透率值和25×100 个压力值作为训练样本,实现对全域内的压力和渗透率预测,本文模型压力预测误差为2.6%,渗透率预测误差为7.2%。将损失函数式(4)中的L_f 项移除,仅对训练样本进行纯数据深度学习,压力学习误差为12%,渗透率学习误差为11.3%。图9、10 分别为二维单相压力和渗透率预测结果。 “·”表示观测井点位置。结果表明,对于二维单相渗流问题,本文模型依然具有良好的准确性和高效性。
同样地,对不同渗透率极差下,不同观测井点数的压力和渗透率预测展开研究,其中观测井点位置在平面内均匀分布。表3、4 给出了不同渗透率极差、不同观测井点数下应用本文模型和纯数据深度学习得到的压力和渗透率预测误差对比。从对比结果可以得到和一维单相渗流一致的结论。
图9 二维单相渗流压力预测结果对比
Fig.9 Comparison of 2D single-phase flow pressure prediction
图10 二维单相渗透率预测结果对比
Fig.10 Comparison of 2D single-phase flow permeability prediction
表3 不同渗透率极差下二维单相渗流压力预测误差对比
Table 3 Error of 2D single-phase flow pressure prediction under different permeability ratios
表4 不同渗透率极差下二维单相渗流渗透率预测误差对比
Table 4 Error of 2D single-phase flow permeability prediction under different permeability ratios
2.2 变注采关系下的渗流场学习及预测
实际的油气开采,由于油气分布和生产要求的改变,需要对生产措施进行实时调整,注采呈动态变化,因此进一步探究该模型对于注采压力动态变化时的适用性。对式(1)设置初边值条件,初始时刻压力为1 Pa,左端以1 Pa定压注入,右端以0 Pa定压开采,t=0.5 s时注采反转,储层长度为1 m,模拟开发总时长1 s。从200 个空间步数里等距选取8 个观测井点进行预测。图11、12、13 分别为变注采关系下压力、压力切片( t=0.5 s)和渗透率预测。结果表明对于改变注采动态,本文模型具有良好的准确性和高效性。
图11 变注采关系下压力预测结果对比
Fig.11 Comparison of pressure prediction under variable injection-production relationship
图12 变注采关系下压力切片对比(t=0.5 s)
Fig.12 Comparison of pressure slice prediction under variable injection-production relationship(t=0.5 s)
图13 变注采关系下渗透率预测结果对比
Fig.13 Comparison of permeability prediction under variable injection-production relationship
3 两相渗流场深度学习
3.1 Buckley-Leverett强非线性方程
一维油水两相渗流饱和度控制方程及初边值条件为

{\begin{matrix} \frac{\partial S_{w} (x, t)}{\partial t} + \frac{q}{φ A} \frac{d f_{w} (S_{w}) \partial S_{w} (x, t)}{d S_{w}} = 0, t \in (0,1) \\ x \in (0, x_{f}) \\ \frac{d f_{w} (S_{w})}{d S_{w}} = \frac{\frac{K_{r d} μ_{w}}{K_{m} μ_{o}} (\frac{n_{w}}{S_{w} - S_{w c}} + \frac{n_{o}}{1 - S_{w} - S_{w c}})}{{(1 + \frac{K_{r o} μ_{w}}{K_{r w} μ_{o}})}^{2}} \end{matrix}

(6)

{\begin{matrix} S_{w} (x, 0) = S_{w c}, x \in [0,1]; \\ S_{w} (0, t) = 1 - S_{o r}, t \in (0,1] \\ S_{w} (t, x_{f}) = S_{w c}, t \in (0,1] \end{matrix}

(7)

式中, $S_{w}$ 为含水饱和度; $q$ 为注采流量,取0.1 m/s; $φ$ 为孔隙度,取0.25;A为储层截面面积,取1 m ²; $K_{r o}$ 和 $K_{r w}$ 分别为油相和水相相对渗透率,最大值取0.8; $u$ _o 和 $u$ 分别为油相和水相黏度,取1 mPa·s; $n_{o}$ 和 $n_{w}$ 分别为油相和水相渗透率指数,取2; $S_{w c}$ 为束缚水饱和度,取0.1; $x_{f}$ 为t时刻对应的油水前缘位置; $S_{o r}$ 为残余油饱和度,取0.1。
饱和度控制方程定义在(0, $x_{f}$ )上,因此饱和度分布的学习需要先确定生产时间为t时油水前缘位置 $x_{f}$ ,再学习确定(0, $x_{f}$ )上的含水饱和度分布,并补充( $x_{f}$ ,1)含水饱和度值(该部分含水饱和度即为束缚水饱和度)。
对式(6)、(7)进行数值求解,其中空间步数为400,时间步数为200,随机选取60 个初始和边界数据点进行预测,图14 为本文模型和纯数据深度学习的饱和度预测结果对比,图15 为t=0.5 s时本文模型和纯数据深度学习饱和度预测结果对比。结果表明本文中模型应用于Buckley-Leverett强非线性方程学习及预测仍具有良好的准确性和高效性。
图14 一维两相渗流饱和度预测结果对比
Fig.14 Comparison of 1D two-phase saturation prediction
图15 一维两相渗流饱和度切片对比
Fig.15 Comparison of 1D two-phase flow saturation slice prediction
3.2 二维两相渗流问题
二维油水两相渗流控制方程为

\frac{\partial}{\partial x} (ρ_{o} \frac{k_{x} K_{r o} \partial p_{o}}{μ_{o} \partial x}) + \frac{\partial}{\partial y} (ρ_{o} \frac{k_{y} K_{r o} \partial p_{o}}{μ_{o} \partial y}) + q_{o} = \frac{\partial}{\partial t} (φ ρ_{o} S_{o})

(8)

\frac{\partial}{\partial x} (ρ_{w} \frac{k_{x} K_{r w} \partial p_{w}}{μ_{w}}) + \frac{\partial}{\partial y} (ρ_{w} \frac{k_{y} K_{r w} \partial p_{w}}{μ_{w}} \partial y) + q_{w} = \frac{\partial}{\partial t} (φ ρ_{w} S_{w})

(9)

p_{c} = p_{o} - p_{w}, S_{w} + S_{o} = 1

(10)

式中, $S_{o}$ 为油饱和度;x和y为空间变量,m; $q_{o}$ 和 $q_{w}$ 分别为油、水注采流量;φ 取0.2;k_x 和k_y 分别为x和y方向上的渗透率,设置储层各向同性,k_x=k_y=1 $u$ m²; $ρ_{o}$ 为油密度,取初始值为850 kg/m 3; $ρ_{w}$ 为水密度,取初始值为1000 kg/m 3; $p_{o}$ 和 $p_{w}$ 分别为油、水压力; $p_{c}$ 为毛管压力,可忽略。
对式(8)、(9) 设置初边值条件,研究区域为200 m×200 m,初始时刻压力为20 MPa,左下端以定流量100 m²/d注入水,右上端以定流量100 m ²/d开采。对式(8)、(9)进行数值求解,空间划分为20 ×20 个网格,时间步数为100,形成20×20×100 个压力和含水饱和度数据点。本研究中,从20×20 个网格中均匀取25 个观测井点( 包含2 个注采边界点),将观测井点位对应25×100 个压力值和含水饱和度值作为训练样本,实现对全域内的压力和含水饱和度预测,本文模型压力预测误差为2.2%,渗透率预测误差为5.9%。仅对训练样本进行纯数据深度学习,压力学习误差为10.1%,渗透率学习误差为13.4%。图16、17 分别为使用本文模型及纯数据深度学习的压力和饱和度预测结果,“·”表示观测井点位置。结果表明,对于二维两相渗流问题,本文模型具有良好的准确性和高效性。
图16 二维两相渗流压力预测结果对比Fig.16 Comparison of 2D two-phase flow pressure prediction
图17 二维两相渗流饱和度预测结果对比
Fig.17 Comparison of 2D two-phase flow saturation prediction
4 结论
(1)对于同一渗透率极差,随着观测井点数的减少,整体的预测误差增大,但本文模型预测结果优于纯数据深度学习预测结果。对于改变注采关系问题,纯数据深度学习预测误差明显,基于物理信息的深度学习预测渗透率方法依然具有良好的准确性和高效性。
(2)对于同一观测井点位数,随着渗透率极差的增大,整体的预测误差增大,但本文模型预测结果优于纯数据深度学习预测结果。当观测井点位数足够多,即训练样本足够多时,两种学习方法预测误差均很小。
(3)为了实现高效精准的预测目标,观测井点应尽可能均匀分布,避免局部区域观测井点位置过于稀疏或密集造成样本学习数据的缺失或重复。
参考文献
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基本信息

中图分类号: TE319;TE312
文献标识码: A
DOI: 10.3969/j.issn.1673-5005.2020.04.006

基金信息

国家重大科技专项(2017ZX05009-001, 2016ZX05060-010)；
中央高校基本科研业务费专项(18CX05029A, 17CX06007)；
中石化大数据前瞻项目(20191115200418146)；

引用信息

黄朝琴,年凯,王斌,等. 一种考虑物理过程信息的油气渗流深度学习新模型 [J]. 中国石油大学学报(自然科学版),2020,44(4):47-56.

HUANG Zhaoqin, NIAN Kai, WANG Bin, et al. A novel deep learning model with physical process information for predic- tion of flow behaviors in oil and gas reservoirs [ J]. Journal of China University of Petroleum(Edition of Natural Science), 2020,44(4):47-56.

稿件历史

收稿日期: 2019-12-31

参考文献

[1] HINTON G E,OSINDERO S,TEH Y W.A fast learning algorithm for deep belief nets[J].Neural Computation,2006,18(7):1527-1554.
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[3] RANZATO M,BOUREAU Y L,LECUN Y.Sparse fea-ture learning for deep belief networks [J].Advances in Neural Information Processing Systems,2008,20:1185-1192.
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[5] WULSIN D F,GUPTA J R,MANI R,et al.Modeling electroence phalography waveforms with semi-supervised deep belief nets:fast classification and anomaly measure-ment[J].Journal of Neural Engineering,2011,8(3):036015.
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一种考虑物理过程信息的油气渗流深度学习新模型

A novel deep learning model with physical process information for prediction of flow behaviors in oil and gas reservoirs

摘要

Abstract

关键词

Keywords

1 考虑物理过程信息的油气渗流深度学习模型

(1)

(2)

(3)

(4)

2 单相渗流场的深度学习

2.1 定注采关系下的渗流场学习与预测

2.1.1 一维单相渗流问题

(5)

2.1.2 二维单相渗流问题

2.2 变注采关系下的渗流场学习及预测

3 两相渗流场深度学习

3.1 Buckley-Leverett强非线性方程

(6)

(7)

3.2 二维两相渗流问题

(8)

(9)

(10)

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献