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作者简介:

王东英(1992-),女,博士研究生,研究方向为油气渗流理论与油藏数值模拟。E-mail:dongying.upc@hotmail.com。

通讯作者:

姚军(1964-),男,教授,博士,博士生导师,泰山学者攀登计划专家,研究方向为油气田开发工程与油气渗流理论及应用。E-mail:RCOGFR-UPC@126.com。

中图分类号:TE348

文献标识码:A

文章编号:1673-5005(2020)02-0100-08

DOI:10.3969/j.issn.1673-5005.2020.02.013

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目录contents

    摘要

    油水两相相对渗透率和毛管力曲线是表征裂缝中流体流动特性的重要指标。 基于侵入逾渗模型,建立一种考虑变应力影响的粗糙裂缝内油水两相流动数值模拟方法。 通过蒙特卡洛法构建单条粗糙裂缝模型;利用侵入逾渗模型模拟单条粗糙裂缝中的稳态油-水驱替过程,数值求解毛管力及相对渗透率曲线。 通过与 Mualem 解析模型比较验证模型的准确性。 借助裂缝所受有效应力与开度间的解析关系,研究变应力影响下油水两相流动能力的变化规律。 结果表明:油-水驱替过程中,随着裂缝有效应力增加,毛管力曲线呈整体抬升趋势,其中油相相对渗透率明显减小,而水相相对渗透率基本不变,同时发现油相开始流动对应的含水饱和度和水相基本失去流动能力对应的含水饱和度均随有效应力的增加而减小。

    Abstract

    The relative permeability and capillary pressure curves of oil and water flow are commonly used as important indi-cators to describe the flow characteristics in fractures. Considering the effect of dynamic stress, an invasion percolation model was applied to simulate oil and water two-phase flow in fractures with rough surface. In this work, a single fracture model was firstly constructed using a method of Monte-Carlo stochastic simulation. Then, a steady-state displacement process was simu- lated to numerically solve the relative permeability and capillary pressure curves. The accuracy of the new model was verified by comparing with the Mualem model. The impact of dynamic stress upon oil and water flow was characterized using an ana- lytical relationship between the effective stress and fracture aperture. The results indicate that, as the displacement process continues, the capillary pressure curve is moved upward with the increase of effective stress, where the oil relative permeabil- ity decreases but the water relative permeability almost remains unchanged. In addition, during the displacement process, the water saturation corresponding to the initiation of oil phase flow decreases, and the water saturation corresponding to the ces- sation of water phase flow decreases.

  • 低渗油藏、页岩油藏等非常规油藏普遍发育裂缝[1-5] , 且一般需要进行水力压裂实现有效开采[6-12] 。 裂缝的存在会改变储层中流体的流动特征[13-16] ,开展裂缝中油水两相相对渗透率及毛管力曲线的研究对指导裂缝性油藏开发具有重要意义。 Huo 等[17]和廉培庆等[18] 分别测量了不同有效应力下,真实裂缝的油水相对渗透率曲线; Watanabe 等[19]通过测量裂缝中油水的相对渗透率曲线,发现真实裂缝中两相流动存在较强的相互干扰性,提出了“V”型相对渗透率曲线。 不少学者也已建立了描述单条裂缝中两相流体流动能力的解析模型。 如 Romm [20]提出的“X”型相渗模型;Brooks 等[21] 基于多孔介质推导的考虑两相间相互干扰的相渗模型; Fourar 等[22-23]根据理想管流模型提出的考虑黏性力的相对渗透率解析模型。 以上模型所依据的物理模型忽略了真实裂缝的复杂性,不能直接应用于数值模拟研究。 较成熟的求解相对渗透率的解析模型, 如 Burdine 模型[24] 及 Mualem 模型[25] ,可用于考虑开度分布的粗糙裂缝相对渗透率曲线的求解[26] ,但过分依赖于模型参数取值,直接用于相对渗透率曲线的计算缺乏普适性。 侵入逾渗模型考虑毛管力主导的稳态驱替/ 吸吮过程[27-28] ,已被不少学者应用到裂缝的两相流动模拟中[17,29,30] 。 其中 Ye 等[30] 采用侵入逾渗模型研究了裂缝中的两相流体相渗曲线,但模拟过程为吸吮,且模型没有考虑裂缝所受有效应力对流体流动能力的影响。 笔者基于侵入逾渗模型,模拟单条粗糙裂缝中油水两相流体流动的稳态驱替过程,并讨论不同有效应力对裂缝开度分布及油水两相流动能力的影响,揭示真实裂缝在变化应力下的流动规律。

  • 1 模拟方法

  • 1.1 基本原理

  • 侵入逾渗模型最初用于模拟孔隙网络模型孔隙及喉道中的驱替及吸吮过程[28] ,其中驱替过程取决于其喉道的大小,而吸吮过程由孔隙大小控制[31-32] 。 笔者着眼于低渗透油藏中重要的渗流通道———裂缝, 基于侵入逾渗模型对单条粗糙裂缝内油水两相流动的毛管力和相对渗透率曲线进行研究。

  • 采用蒙特卡洛模拟方法,在服从一定高斯分布规律的前提下,随机分布单条粗糙裂缝中的局部开度,流体流动为稳态驱替过程且遵循达西定律。 初始时刻,裂缝被湿相流体饱和,逐步提高驱替压力注入非湿相流体,使非湿相逐渐进入更多的裂缝空间驱替湿相流体。 本研究选取水相为湿相流体,油相为非湿相流体。

  • 采用侵入逾渗模型来描述稳态驱替过程与规律,为求取油水两相相渗曲线,需先确定不同驱替压力下的油水界面移动过程及饱和度分布。 编制程序搜寻某驱替压力下,与入口端相连的驱替相连通团的位置及编号得到油水两相分布,利用有限差分法分别数值求解水相与油相流体的压力场,最终,通过驱替过程中两相流体的有效渗透率与单相流动时绝对渗透率的比值获得两相流体的相对渗透率。

  • 1.2 模型假设

  • 基于 CT 扫描得到的真实裂缝图像获得裂缝开度数据[33] ,图 1 是由真实裂缝数据重构而成的粗糙裂缝形态及开度分布规律。 经过数据分析发现,裂缝开度服从均值为 580 μm、标准差为 230 μm 的高斯分布。

  • 图 1 真实单条粗糙裂缝形态及开度分布

  • Fig. 1 Real single rough fracture pattern and aperture distribution

  • 为简化模型,考虑粗糙度的单条裂缝如图 2 所示。 将裂缝区域进行离散,裂缝局部采用蒙特卡洛模拟方法随机赋予一定开度值,并使其整体服从高斯分布,同时认为离散后每个网格内部开度一致。 出、入口端定压力,上下两端封闭,驱替过程流体流动方向从左至右,

  • f(b)=12πδexp(-12δ2(b-μ)2
    (1)
  • 式中,f(b)为单条粗糙裂缝开度服从的高斯分布函数;μ为裂缝开度分布均值,m; δ为裂缝开度分布标准差,m;b 为裂缝局部开度,m。

  • 图 2 单条粗糙裂缝模型示意图

  • Fig. 2 Schematic diagram of single rough fracture model

  • 同时,模型假设:考虑二维平面流动,忽略垂向流动及重力的作用;油水两相不相溶,流动遵循达西定律;流体不可压缩,裂缝受有效应力作用可压缩; 考虑裂缝内部的非均质性。

  • 1.3 数学模型

  • 首先,采用立方定律描述裂缝内局部渗透率,即

  • k(Joold=b212
    (2)
  • 式中,kf_local(f = o,w)为裂缝内部局部渗透率,m 2

  • 油相的连续性方程为

  • xρokoJbealpoμo+yρakn-loralpoμo=0
    (3)
  • 水相的连续性方程为

  • x(ρwkw-Jocalpwμw)+y(ρwkw-Jlogwpwμw)=0
    (4)
  • 式中,ρo 和 ρ 分别为油相和水相密度,kg / cm 3o 和 μ 分别为油相和水相的黏度,Pa·s;po 和 p 分别为油相和水相压力,Pa。

  • 辅助方程为

  • pcow=pw-po=2σcosθb
    (5)
  • So+S = 1 (6)

  • 式中,pcow为油水两相之间的毛管力,Pa;σ 为油水界面张力,N/ m; θ为水相接触角;So 为油相饱和度; S 为水相饱和度。

  • 初始条件为

  • p0(x,y)= pi (7)

  • 边界条件为

  • p(0_,y_)=pentr
    (8)
  • p(L_x,y)=pexit
    (9)
  • (py)y=0=0
    (10)
  • (py)y=Ly=0
    (11)
  • 式中,p0(x,y)为初始时刻裂缝各处对应的压力,Pa;pi 为裂缝初始压力,Pa;p(0,y)为裂缝模型左端(入口) 压力,Pa;pentry为入口压力,Pa;p(Lx,y)为裂缝模型右端(出口)压力,Pa;pexit为出口压力值,Pa;Lx 为单条裂缝模型长度,m;Ly 为单条裂缝模型宽度,m。

  • 基于侵入逾渗模型,判断不同驱替压力下的油水两相分布情况,此时含水饱和度与驱替压力关系即为毛管力曲线。 采用有限差分方法数值求解上述方程得到油水两相流体的压力场分布,分别计算两相流体流量,得到相对渗透率,

  • kaln-f=QμμfLxA(pexi-peatry)
    (12)
  • keff-f=qμμfLxA(pexit-pemtry)
    (13)
  • KtL=kefff-fkabl--
    (14)
  • 式中,下标 f 为 o 或 w;kabs_f为绝对渗透率,m 2 ;Qf 为油或水单相流动时的流量,m 3 / s;μf 为油或水的黏度, Pa·s;A 为单条粗糙裂缝的横截面积,m 2 ;keff_f为油或水的有效渗透率,m 2 ;qf 为驱替过程中,油水两相流动时油或水的流量,m 3 / s;Kr_f为油或水的相对渗透率。

  • 2 模型可靠性检验

  • 2.1 模型构建

  • 根据真实粗糙裂缝开度的高斯分布结果构建概念模型。单条粗糙裂缝物理尺寸为 0. 1 m×0.1 m, 横截面积为 5. 86 ×10 -5 m 2 ,模型中网格数为 100 × 100,如图 3 所示。 假设裂缝表面亲水,水相接触角为 60°,界面张力取值 0.048 N/ m,油水两相黏度分别为 15 和 1 mPa·s,模型整体处于大气压力环境, 入口与出口压力差为 10 Pa。

  • 2.2 计算结果

  • 结合侵入逾渗模型及上述基本参数对粗糙裂缝中油驱水过程的两相流体流动进行模拟。

  • 图 3 粗糙裂缝模型开度分布示意图

  • Fig. 3 Local aperture distribution of single rough fracture model

  • 图 4 为驱替过程中的油水两相分布,其中红色代表油相,蓝色代表水相。驱替相即油相首先会进入连续的具有较大局部开度的位置驱替湿相流体, 随着驱替压力逐渐提高,才侵入裂缝中更多开度较小的位置。

  • 图 5 为数值求解得到的毛管力曲线和油水两相的相对渗透率曲线。 由求得的相对渗透率曲线可以看出,对于考虑粗糙度的真实裂缝,若采用传统“X” 型光滑裂缝相渗曲线来描述油水两相流动特征会造成较大误差。 通过研究发现,当含水饱和度较高即驱替初始时,由于水相占据较大孔隙并连通,使之渗透率较高,而油相很大程度上为不连续相,因此渗透率极低。 随着驱替过程的进行,油相渗透率升高, 水相渗透率不断减小。 当含水饱和度减小到一定值后,水相渗透率极低,这是由于裂缝表面亲水, 水相被圈闭在不连续的极小孔隙中,不再具有流动能力。

  • 图 4 驱替过程两相分布

  • Fig. 4 Two-phase distribution during displacement

  • 图 5 毛管力及油水相对渗透率曲线

  • Fig. 5 Capillary pressure and oil-

  • 2.3 模型验证

  • 对比目前描述光滑裂缝相对渗透率的“X”型解析模型,Ye 等[30] 通过研究发现,Mualem 模型可以考虑粗糙裂缝中的开度分布,并将水相与油相的有效饱和度与裂缝开度联系起来,从而对相对渗透率进行描述,因此采用 Mualem 模型与本文中的数值计算结果进行对比。 根据 Mualem 模型,水相与油相的相对渗透率可表达为饱和度的幂律分布形式,

  • Sx==Sw-SmSws-Sm
    (15)
  • S=So-SorSos-Sor
    (16)
  • Km=Sw=ηn
    (17)
  • Km=Sw=η0
    (18)
  • 式中,S 为水相有效饱和度;S 为最小水相饱和度; S 为最大水相饱和度;Soe为油相有效饱和度;Sro为最小油相饱和度;Sos为最大油相饱和度;Krw为湿相相对渗透率;Kro为非湿相相对渗透率;η 为求解湿相流体相对渗透率时的经验系数,η = 2.2;ηo为求解非湿相流体相对渗透率时的经验系数,ηo = 2.1。

  • 结合实例计算结果,将数值求解结果与上述 Mualem 模型进行对比验证,结果见图 6。 由图 6 可以看出,二者的计算结果表现出较好的一致性,验证了本文中数值模拟方法的正确性。

  • 图 6 模型验证

  • Fig. 6 Model validation

  • 3 有效应力对裂缝内油水两相流体流动的影响

  • 3.1 有效应力对裂缝开度分布的影响

  • 裂缝性油藏在压裂及开采的过程中所受应力不断变化,相应的裂缝内部流体的流动也会受其影响。 因此本文中采用 Wu 等[34] 提出的描述单条裂缝开度与裂缝所受有效应力之间的解析关系,

  • b=bi+Δb=bi+(Δσ/κ)
    (19)
  • 式中,κ为裂缝的刚度, κ = 1×10 11 Pa / m;bi 为裂缝初始局部开度,m;Δb 为裂缝在不同应力条件下局部开度的变化,m;Δσ 为裂缝所受的有效应力,Pa。

  • 当有效应力为 15 和 30 MPa 时,单条裂缝对应的开度变化分别为 150 和 300 μm。 研究不考虑应力条件及有效应力分别为 15 和 30 MPa 时裂缝开度分布情况(其他参数取值均不变),由于裂缝开度服从高斯分布,有效应力变化后,裂缝开度整体减小并出现部分闭合的现象,图 7、8 分别为裂缝开度变化趋势及不同有效应力条件下对应的裂缝开度分布情况。

  • 图7 不同有效应力条件下单条粗糙裂缝开度概率密度分布

  • Fig. 7 Aperture distribution in single rough fracture under varying effective stress

  • 图 8 不同有效应力条件下单条粗糙裂缝开度分布

  • Fig. 8 Local aperture distribution in single rough fracture under varying effective stress

  • 3.2 有效应力对裂缝中毛管力及相对渗透率曲线的影响

  • 图 9 为不同有效应力条件下的毛管力曲线和相对渗透率曲线。 随有效应力的增加,毛管力曲线呈现整体抬升的趋势,其中平缓段由 185 Pa 增加到 256 和 425 Pa,说明由于应力的增加,裂缝开度减小,驱替过程变得更加困难。 实际生产过程中,油藏孔隙压力不断减小,导致有效应力增加。 从图 9 可以看出,裂缝作为致密及页岩油藏的主要渗流通道, 随有效应力不断增加,其中水相的相对渗透率基本不变,而油相的相对渗透率会明显减弱。

  • 整体来看,有效应力由初始条件增加到 15 和 30 MPa,裂缝的开度分别减小25. 9% 及 51. 7% ,而油相的相对渗透率分别平均减小了 10.7% 和 24. 3% ,裂缝所受有效应力越大,油相渗透率减小幅度越明显。 这是因为裂缝表面的亲水性,导致水相主要占据相对较小的孔隙流动,受应力影响表现不明显,而油相主要在裂缝中较大开度的孔隙流动,随着有效应力增加,油相的流动通道明显被压缩,对应的毛管力显著增加,油相流动阻力增大,因此油相相对流动能力被削弱。 另外,油驱水过程中,随有效应力的增加,油相开始流动时对应的含水饱和度和水相基本丧失流动能力所对应的含水饱和度均略有减小。 这是因为油相普遍在较大孔隙中流动,应力的作用使流动通道明显变窄,驱替初始时,油相开始连续并流动所对应的含水饱和度越小,即随着生产的进行,裂缝所受到有效应力增大,油相越难以形成连续流动通道;同时,有效应力增加导致裂缝部分闭合,油相作为驱替相侵入连续的位置更加困难,当水相已基本无流动能力时,油相饱和度已达到较高水平,表现为相应的含水饱和度减小。 相比于无法考虑应力变化对裂缝油水两相流动影响的光滑裂缝 “X”型相渗模型[35-37] ,本文中提出的方法更能准确反映变应力条件下真实裂缝内部油水两相流动规律。

  • 图 9 不同有效应力条件下的毛管力及油水相对渗透率曲线

  • Fig. 9 Capillary pressure and relative permeability curves of oil and water under varying effective stress

  • 4 结论

  • (1)对于单条粗糙裂缝中油水两相流动,在初始含水饱和度降低阶段,油相流动能力极低,随着驱替过程的进行,油相流动占主导,水相基本失去流动能力。

  • (2)随着有效应力的增加,单条粗糙裂缝被压缩并出现局部闭合现象;裂缝毛管力曲线整体抬升; 油相相对渗透率减小,且有效应力越大,油相相对渗透率减小越明显,而水相相对渗透率基本保持不变; 同时,油相开始流动对应的含水饱和度和水相基本失去流动能力时对应的含水饱和度均随有效应力的增加而减小。

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