﻿ 压力衰竭下疏松砂岩出砂临界生产压差预测方法
 西南石油大学学报(自然科学版)  2020, Vol. 42 Issue (3): 115-122

Critical Drawdown Pressure of Sanding Onset for Unconsolidated Sandstone Reservoirs when Reservoir Pressure Depleted
SHI Xianya, HUANG Xia, SHI Jingyan, ZHAO Guangyuan, XU Liyuan
China Oilfield Services Limited, Binhai New Area, Tianjin 300459, China
Abstract: Offshore reservoirs are generally exploited without energy supplement, and reservoir pressure depletion will cause rock stress changes which results in an increasing likelihood of serious sand production in the unconsolidated sandstone reservoir. This paper aims at offshore unconsolidated sandstone gas reservoirs, analyzes rock stresses of the borehole wall with impact of reservoir pressure depletion on in-situ stresses. Combining with fully-polyaxial rock failure criterion Mogi-Coulomb, a new sanding Critical Drawdown Pressure (CDP) calculating model. Furthermore, an applicable sanding prediction method for pressure depleted gas reservoir in well whole life cycle is proposed. The model was applied in F gas reservoir, the chart of sanding CDP for Well F-1 in the whole production life cycle is plotted. The research results indicate:initially, the sanding CDP is relatively large; with formation pressure depletion, CDP is reduced step by step until depleted to the critical reservoir pressure, and sand will be produced no matter how small the drawdown is. The research outcomes are consistent with actual oilfield production situation, which can provide some theoretical supports for sand production gas wells.
Keywords: unconsolidated sandstone    reservoir pressure depletion    failure criterion of rock    sand production    critical drawdown pressure
引言

(1) 现场观测法是通过观察岩芯疏松程度或DST测试来判断是否出砂，其分析过程简单，适用于现场对出砂预测精度要求不高的情况[4]

(2) 经验公式法则是基于测井数据和岩石力学测试数据，计算声波时差(95 < ∆tc < 105，轻微出砂；∆tc ≥105，严重出砂)，组合模量(15 <Ec≤20，轻微出砂；Ec≤ 15，严重出砂)，出砂指数(出砂指数≤ 20 GPa，出砂)，斯伦贝谢比(斯伦贝谢比≤ 59 MPa2，出砂)来预测地层是否出砂，这种方法简单实用，在国内外被广泛应用[5-6]

(3) 室内实验测试是以未胶结或弱胶结砂岩为对象，模拟井眼及其生产条件，进行流动实验来判断是否出砂[7]。目前常用的实验模型是厚壁空芯圆柱模型，可评价砂岩初始破坏，而无法说明孔眼的扩大及破坏后的稳定性，未考虑射孔产生的破碎带、损伤带等情况。

(4) 数值模拟方法是建立岩石力学与流体力学的动态耦合数值模型，研究出砂油层流固耦合渗流情况下，骨架砂发生剥离的临界生产压差或临界流速来预测地层出砂[8]，模型所需的储层物性参数、地质力学参数及岩石力学参数过多，参数获取困难，且分析过程相当繁琐复杂，所耗时间长。

1 岩石骨架破坏出砂临界压差计算

 图1 井壁岩石骨架破坏出砂示意图 Fig. 1 Rock failure and sanding production diagram
1.1 井壁应力状态分析

 图2 井眼岩体应力微元及坐标轴变换 Fig. 2 Schematic of an inclined wellbore geometry and earth stress conversion
 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _r} = \frac{{r_{\rm{w}}^2}}{{{r^2}}}{p_{\rm{w}}} + \frac{1}{2}\left( {{\sigma _x} + {\sigma _y}} \right)\left( {1 - \frac{{r_{\rm{w}}^2}}{{{r^2}}}} \right) + \frac{1}{2}\left( {{\sigma _x} - {\sigma _y}} \right)\left( {1 + \frac{{3r_{\rm{w}}^4}}{{{r^4}}} - \frac{{4r_{\rm{w}}^2}}{{{r^2}}}} \right)\cos 2\theta \\ + {\sigma _{xy}}\left( {1 + \frac{{3r_{\rm{w}}^4}}{{{r^4}}} - \frac{{4r_{\rm{w}}^2}}{{{r^2}}}} \right)\sin 2\theta }\\ {{\sigma _\theta } = - \frac{{r_{\rm{w}}^2}}{{{r^2}}}{p_{\rm{w}}} + \frac{1}{2}\left( {{\sigma _x} + {\sigma _y}} \right)\left( {1 + \frac{{r_{\rm{w}}^2}}{{{r^2}}}} \right) - \frac{1}{2}\left( {{\sigma _x} - {\sigma _y}} \right)\left( {1 + \frac{{3r_{\rm{w}}^4}}{{{r^4}}}} \right)\cos 2\theta \\ - {\sigma _{xy}}\left( {1 + \frac{{3r_{\rm{w}}^4}}{{{r^4}}}} \right)\sin 2\theta }\\ {{\sigma _z} = {\sigma _z} - \mu \left[ {2({\sigma _x} - {\sigma _y})\left( {\frac{{r_{\rm{w}}^2}}{{{r^2}}}} \right)\cos 2\theta + 4{\sigma _{xy}}\left( {\frac{{r_{\rm{w}}^2}}{{{r^2}}}} \right)\sin 2\theta } \right]}\\ {{\tau _{r\theta }} = \left[ {\frac{1}{2}\left( {{\sigma _x} - {\sigma _y}} \right)\sin 2\theta + \frac{1}{2}{\sigma _{xy}}\cos 2\theta } \right]\left( {1 - \frac{{3r_{\rm{w}}^4}}{{{r^4}}} + \frac{{2r_{\rm{w}}^2}}{{{r^2}}}} \right)}\\ {{\tau _{rz}} = \left[ {{\sigma _{xz}}\cos \theta + {\sigma _{yz}}\sin \theta } \right]\left( {1 - \frac{{r_{\rm{w}}^2}}{{{r^2}}}} \right)}\\ {{\tau _{\theta z}} = \left[ {{\sigma _{yz}}\cos \theta - {\sigma _{xz}}\sin \theta } \right]\left( {1 + \frac{{r_{\rm{w}}^2}}{{{r^2}}}} \right)} \end{array}} \right.$ (1)
 $\left( {\begin{array}{*{20}{c}} {{\sigma _x}}&{{\sigma _{xy}}}&{{\sigma _{xz}}}\\ {{\sigma _{xy}}}&{{\sigma _y}}&{{\sigma _{yz}}}\\ {{\sigma _{yx}}}&{{\sigma _y}}&{{\sigma _{yz}}} \end{array}} \right) = H\left( {\begin{array}{*{20}{c}} {{S_{\rm{H}}}}&0&0\\ 0&{{S_{\rm{h}}}}&0\\ 0&0&{{S_{\rm{v}}}} \end{array}} \right){H^{\rm{T}}}$ (2)
 $H = \left( {\begin{array}{*{20}{c}} {\cos \psi \cos \mathit{\Omega }}&{\cos \psi \sin \mathit{\Omega }{\rm{ }}}&{ - \sin \psi }\\ { - \sin \mathit{\Omega }{\rm{ }}}&{\cos \mathit{\Omega }{\rm{ }}}&0\\ {\sin \psi \cos \mathit{\Omega }{\rm{ }}}&{\sin \psi \sin \mathit{\Omega }}&{\cos \psi } \end{array}} \right)$ (3)

$\mathit{\boldsymbol{\sigma }} = \left( {\begin{array}{*{20}{c}} {{\sigma _r}}&{{\tau _{r\theta }}}&{{\tau _{rz}}}\\ {{\tau _{r\theta }}}&{{\sigma _\theta }}&{{\tau _{z\theta }}}\\ {{\tau _{rz}}}&{{\tau _{\theta z}}}&{{\tau _z}} \end{array}} \right)$描述井壁围岩一点的应力状态，设I为单位矩阵。当r=rw时，有

 $\left| {\mathit{\boldsymbol{\sigma }} - \lambda \mathit{\boldsymbol{I}}} \right| = \left| {\begin{array}{*{20}{c}} {{\sigma _r} - \lambda }&{{\tau _{r\theta }}}&0\\ {{\tau _{r\theta }}}&{{\sigma _\theta } - \lambda }&0\\ 0&0&{{\sigma _z} - \lambda } \end{array}} \right| = 0$ (4)

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _r} = {p_{\rm{w}}} - \alpha {p_{\rm{p}}}}\\ {{\sigma _{{\rm{1m}}}} = 0.5({\sigma _\theta } + {\sigma _z}) + }\\ {\;\;\;\;\;\;\;\;\;\;\;\;0.5\sqrt {{{({\sigma _\theta } - {\sigma _z})}^2} + 4\tau _{\theta z}^2} - \alpha {p_{\rm{p}}}}\\ {{\sigma _{{\rm{2m}}}} = 0.5({\sigma _\theta } + {\sigma _z}) - }\\ {\;\;\;\;\;\;\;\;\;\;\;\;0.5\sqrt {{{({\sigma _\theta } - {\sigma _z})}^2} + 4\tau _{\theta z}^2} - \alpha {p_{\rm{p}}}} \end{array}} \right.$ (5)

 $\begin{array}{l} {\sigma _1} = \max \left\{ {{\sigma _r}, {\sigma _{{\rm{1m}}}}, {\sigma _{{\rm{2m}}}}} \right\}\\ {\sigma _2} = {\rm{median}}\left\{ {{\sigma _r}, {\sigma _{{\rm{1m}}}}, {\sigma _{{\rm{2m}}}}} \right\}\\ {\sigma _3} = \min \left\{ {{\sigma _r}, {\sigma _{{\rm{1m}}}}, {\sigma _{{\rm{2m}}}}} \right\} \end{array}$
1.2 井壁岩石骨架破坏出砂临界压差

 $F = a + b{\sigma _{\rm{m}}}_{, 2} - {\tau _{{\rm{otc}}}}$ (6)
 ${\tau _{{\rm{otc}}}} = \frac{1}{3}\sqrt {{{\left( {{\sigma _1} - {\sigma _2}} \right)}^2} + {{\left( {{\sigma _2} - {\sigma _3}} \right)}^2} + {{\left( {{\sigma _3} - {\sigma _1}} \right)}^2}}$ (7)
 ${\sigma _{{\rm{m}}, {\rm{2}}}} = \frac{{{\sigma _1} + {\sigma _3}}}{2}$ (8)
 $a = \frac{{2\sqrt 2 }}{3}c\cos \phi$ (9)
 $b = \frac{{2\sqrt 2 }}{3}\sin \phi$ (10)

 ${p_{{\rm{CD}}}} = {p_{\rm{p}}} - {p_{\rm{w}}}$ (11)

2 孔隙压力衰竭对地应力影响分析

 $\Delta {\varepsilon _{\rm{H}}} = \Delta {\varepsilon _{\rm{h}}} = 0$ (12)

 $\left\{ {\begin{array}{*{20}{l}} {{\varepsilon _{\rm{H}}} = \frac{1}{E}\left[ {{S_{\rm{H}}}{\rm{ - }}\alpha {p_{\rm{p}}}{\rm{ - }}\mu ({S_{\rm{v}}}{\rm{ - }}\alpha {p_{\rm{p}}} + {S_{\rm{h}}}{\rm{ - }}\alpha {p_{\rm{p}}})} \right]}\\ {{\varepsilon _{\rm{h}}} = \frac{1}{E}\left[ {{S_{\rm{h}}}{\rm{ - }}\alpha {p_{\rm{p}}}{\rm{ - }}\mu ({S_{\rm{v}}}{\rm{ - }}\alpha {p_{\rm{p}}} + {S_{\rm{H}}}{\rm{ - }}\alpha {p_{\rm{p}}})} \right]} \end{array}} \right.$ (13)

 $\left\{ {\begin{array}{*{20}{l}} {{\varepsilon _{{\rm{H1}}}} = \frac{1}{E}\left[ {{S_{{\rm{H1}}}} - \alpha {p_{{\rm{p1}}}} - \mu ({S_{\rm{v}}} - \alpha {p_{{\rm{p1}}}} + {S_{{\rm{h1}}}} - \alpha {p_{{\rm{p1}}}})} \right]}\\ {{\varepsilon _{{\rm{h1}}}} = \frac{1}{E}\left[ {{S_{{\rm{h1}}}} - \alpha {p_{{\rm{p1}}}} - \mu ({S_{\rm{v}}} - \alpha {p_{{\rm{p1}}}} + {S_{{\rm{H1}}}} - \alpha {p_{{\rm{p1}}}})} \right]} \end{array}} \right.$ (14)

 $\left\{ {\begin{array}{*{20}{l}} {{\varepsilon _{{\rm{H1}}}} = {\varepsilon _{\rm{H}}}}\\ {{\varepsilon _{{\rm{h1}}}} = {\varepsilon _{\rm{h}}}} \end{array}} \right.$ (15)

 $\left\{ {\begin{array}{*{20}{l}} {{S_{{\rm{H1}}}} = {S_{\rm{H}}} - \Delta {S_{\rm{H}}} = {S_{\rm{H}}} - \frac{{1 - 2\mu }}{{1 - \mu }}\alpha \Delta {p_{\rm{p}}}}\\ {{S_{{\rm{h1}}}} = {S_{\rm{h}}} - \Delta {S_{\rm{h}}} = {S_{\rm{h}}} - \frac{{1 - 2\mu }}{{1 - \mu }}\alpha \Delta {p_{\rm{p}}}} \end{array}} \right.$ (16)
3 实例应用分析 3.1 岩石强度测试

3.2 储层物性及相关参数

3.3 压力衰竭后全生产周期出砂临界生产压差

(1) 孔隙压力衰减后地应力变化趋势

 图3 地应力和有效水平地应力随孔隙压力变化趋势 Fig. 3 The variation trend of in-situ stress and effective horizontal stress with pore pressure

(2) 孔隙压力衰减后井壁失稳指数变化趋势

 图4 不同孔隙压力下井壁失稳指数 Fig. 4 Borehole instability index under different pore pressure

(3) 压力衰竭后临界生产压差

 图5 储层压力衰竭下全周期出砂临界压差图版 Fig. 5 A life cycle of the critical drawdown pressure of a well when considering the reservoir pressure depletion

4 结论

(1) 基于井壁处岩石应力分析，引进更符合描述井壁处骨架岩石破坏的Mogi-Coulomb准则，建立出砂临界生产压差力学计算模型。

(2) 考虑储层压力衰竭影响，给出了压力衰竭后的地应力计算方式及出砂压力衰竭下临界出砂计算方法，建立生产井全生产周期的临界生产压差模型}。

(3) 应用该模型及相应计算方法，针对南中国海海上衰竭式开采气藏F-1井进行全生产周期出砂预测，计算并绘制出了压力衰竭条件下临界生产压差图版，预测图版与现场实测数据吻合较好，研究结果能为现场生产井配产提供指导意见。

符号说明

tc——声波时差，µs/ft；

Ec——组合模量，GPa；

SH——最大水平地应力，MPa；

Sv——垂向地应力，MPa；

pp——孔隙压力，MPa；

pw——井筒内压力，MPa；

Sh——最小水平地应力，MPa；

ψ——井斜角，（°）；

——井眼轴线与最大水平地应力夹角，（°）；

σrσθ，——rθ——向的正应力，MPa；

σxσyσz——xyz向的正应力，MPa；

σxy, σyzσxz xy面，yz面，xz面上的应力，MPa；

ττrzτθz 面，rz面，θz面上的剪应力，MPa；

r——近井地带一点到井眼中心距离，m；

rw——井眼半径，m；

θ ——井周角，（°）；

H——变换矩阵；

σ——应力矩阵；

α——Boit系数；

σ1mσ2m井周岩石微元除r向外的另外两个有效主应力，MPa；

σ1σ2σ3最大、中间、最小主应力，MPa；

max，median，min——最大、中间、最小值函数；

F——井壁骨架岩石失稳指数，MPa；

ab——系数；

σm, 2最大和最小主主应力平均值，MPa；

τotc——中间变量，MPa；

c ——内聚力，MPa；

ϕ——内摩擦角，（°）；

εH——最大水平主应变之差，无因次；

εh——最小水平主应变之差，无因次；

εHεh——初始最大和最小水平主应变，无因次；

E——杨氏模量，MPa；

µ——泊松比，无因次；

εH1, εh1 ——孔隙压力衰减后最大和最小水平主应变，无因次；

SH1Sh1——孔隙压力衰减后最大和最小水平地应力，MPa；

pp1——衰减后的孔隙压力，MPa；

ΔSH, ΔSh——最大和最小水平主应力变化值，MPa；

Δpp——孔隙压力改变值，MPa。

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