﻿ 合采井层间干扰现象数学模拟研究
 西南石油大学学报(自然科学版)  2017, Vol. 39 Issue (6): 109-116

1. "海洋石油高效开发"国家重点实验室, 北京 朝阳 100027;
2. 中海石油(中国)有限公司北京研究中心, 北京 朝阳 100027;
3. 中国石油大庆油田第三采油厂, 黑龙江 大庆 163113

Mathematical Simulation Study on Interlayer Interference in Commingled Production
ZHOU Wensheng1,2 , LI Qianru3, GENG Zhanli1,2, WANG Shoulei1,2
1. State Key Laboratory of Offshore Oil Exploitation, Chaoyang, Beijing, 100027, China;
2. Beijing Research Center, CNOOC(China) Co. Ltd., Chaoyang, Beijing, 100027, China;
3. The No.3 Oil Recovery Plant, Daqing Oilfield, Daqing, Heilongjiang 163113, China
Abstract: Offshore oil fields usually adopt a large-interval commingled production strategy. Interlayer interference is relatively serious. To evaluate this phenomenon correctly, based on the theory of seepage mechanics, an interlayer interference evaluation numerical model considering the number of layers, thickness, permeability, porosity, viscosity, formation pressure, supply radius, and so on, was established. The model considered four aspects of the interlayer interference mechanism, including the interlayer heterogeneity, initial pressure gradient, wellbore connectivity, and liquid transfer. Based on the mathematical model, the productivity of each layer and the entire well during commingled production, the productivity of each layer and the entire well during separate production, and the interlayer interference coefficient were calculated. The productivity under the conditions of commingled and separate production was analyzed. The results show that the contributions of the permeability difference ratio, test operation procedures, water saturation stage, hydrostatic pressure, and other parameters to the interlayer interference were 50%, 25%, 15%, and 10%, respectively. Interlayer interference is mainly controlled by interlayer heterogeneity. During the commingling of multiple layers, if the permeability ratio is less than 4, the interlayer interference coefficient can be guaranteed to be less than 0.6. If the permeability ratio is less than 10, the interference coefficient can be guaranteed to be less than 0.6.
Key words: multi-layer reservoir     commingled production     separate production     interlayer interference     mathematical simulation

1 模型的建立 1.1 物理模型

 图1 n层地质模型平面径向流示意图(n=3) Fig. 1 Diagram of n plane radial flow n=3
 图2 n层油藏示意图(n=3) Fig. 2 Diagram of multi-layer n reservoir(n=3)

1.2 数学模型

1.2.1 压力降波及到含油外边界前(单相油流)

 图3 压力降波及到含油外边界之前水驱油示意图 Fig. 3 Diagram of water/oil displacement before productive limit is swept by producing pressure difference
 ${{p}_{\text{wf}i}}\left( {t} \right)={{p}_{\text{e}i}}-\dfrac{{{Q}_{\text{L}i}}\left( {t} \right){{\mu }_{\text{o}i}}}{\text{4 }\!\!{\rm{\pi }}\!\!\text{ }{{K}_{{i}}}{{h}_{{i}}}}\ln\dfrac{\text{2.25}{{\eta }_{{i}}}t}{r_{\text{w}}^{\text{2}}}$ (1)
 ${{f}_{{\text w}i}}\left( t \right)=0$ (2)
 ${{Q}_{\text{o}i}}\left( t \right)={{Q}_{\text{L}i}}\left( t \right)$ (3)

${Q}_{{\text{L}}i}\left( {t} \right)$ $t$ 时刻第 $i$ 层的分层产液量，m3/d；

${\mu }_{{\rm o}i}$ —第 $i$ 层油相的黏度，mPa·s；

$f_{{\rm w}i}$ —第 $i$ 层的含水率，%；

${Q}_{{\text{o}}i}\left( {t} \right)$ $t$ 时刻第 $i$ 层的分层产油量，m3/d。

1.2.2 未见水时期(油水两相流+单相油流)

 图4 未见水时期地层水驱油示意图 Fig. 4 Diagram of water/oil displacement in water free producing period

(1) 渗流阻力

 $\left\{ {\begin{array}{*{20}{l}} {{R_i}\left( t \right){\rm{ = }}\frac{{{\rm{100}}}}{{{\rm{8}}.{\rm{64}}}}\frac{{{\mu _{{\rm{o}}i}}}}{{{\rm{2}}\pi {h_i}{K_i}{K_{{\rm{ro}}i}}\left( {{S_{{\rm{wc}}i}}} \right)}} \cdot }\\ {\quad \left[ {\frac{{{K_{{\rm{ro}}i}}({S_{{\rm{wc}}i}})}}{{\rm{2}}}\int_{{S_{{\rm{wm}}i}}}^{{S_{{\rm{wf}}i}}} {\frac{{\varphi _i^\prime \left( {{S_{\rm{w}}}} \right){\rm{d}}{S_{\rm{w}}}}}{{\xi \left( t \right)}}} {\rm{ + ln}}\frac{{{r_{{\rm{fi}}}}\left( t \right)}}{{{r_{\rm{w}}}}}} \right]}\\ {\xi \left( t \right){\rm{ = }}\left[ {{K_{{\rm{ro}}i}}\left( {{S_{\rm{w}}}} \right){\rm{ + }}{\mu _{{\rm{ow}}i}}{K_{{\rm{rw}}i}}\left( {{S_{\rm{w}}}} \right)} \right] \cdot }\\ {\quad \left[ {\frac{{{\varphi _{\rm{i}}}\left( {{S_{{\rm{wf}}i}}} \right)}}{{{\rm{1 - }}{{\left( {{r_{\rm{w}}}{\rm{/}}{r_{{\rm{ei}}}}} \right)}^{\rm{2}}}}} - {\varphi _i}\left( {{S_{\rm{w}}}} \right)} \right]} \end{array}} \right.$ (4)

${{K}_{\text{ro}i}}\left({S}_{{\rm wc}i} \right)$ —第 $i$ 层束缚水饱和度对应的油相相对渗透率，%；

${S_{{\rm{wc}}i}}$ —第 $i$ 层的束缚水饱和度，%；

${S} _{{\text{wf}i}}$ —第 $i$ 层的前缘含水饱和度，%；

${S} _{{\text{wm}i}}$ —第 $i$ 层边界处含水饱和度，%；

${{\varphi }_{i}}'\left( {{S}_{\text{w}}} \right)$ —第 $i$ 层含水率的二阶导数；

${{S}_{\text{w}}}$ —含水饱和度，%；

${\mu }_{{\rm ow}i}$ —第 $i$ 层油水黏度比，无因次；

${{\varphi }_{i}}\left( {{S}_{\text{wf}i}} \right)$ —第 $i$ 层前缘含水饱和度对应的含水率的一阶导数，无因次；

${{\varphi }_{i}}\left( {{S}_{\text{w}}} \right)$ —第 $i$ 层含水率的一阶导数。

(2) 井底流压

 ${{p}_{\text{wf}i}}\left( t \right)\text{=}{{p}_{\text{e}i}}-{{R}_{{i}}}\left( t \right)\cdot {{Q}_{\text{L}i}}\left( t \right)$ (5)

(3) 含水率

 ${{f}_{\text{w}i}}\left( t \right)=0$ (6)

(4) 产油量

 ${{Q}_{\text{o}i}}\left( t \right)={{Q}_{\text{L}i}}\left( t \right)$ (7)
1.2.3 见水时期(油水两相流)

 图5 见水期地层流动区域示意图 Fig. 5 Diagram of water/oil displacement after water breakthrough

(1) 渗流阻力

 $\left\{ \begin{array}{l} {R_i}\left( t \right) = \dfrac{{100}}{{8.64}}\dfrac{{{\mu _{{\rm{o}}i}}}}{{2\pi {h_i}{K_i}}}\int_{\, {s_{{\rm{wf}}i}}}^{\, {s_{{\rm{we}}i}}} {\dfrac{{{\varphi _i}'\left( {{S_{\rm{w}}}} \right){\rm{d}}{S_{\rm w}}}}{{2\xi \left( t \right)}}} \\[6pt] \xi \left( t \right) = \left[{{K_{{\rm{ro}}i}}\left( {{S_{\rm{w}}}} \right) + {\mu _{{\rm{ow}}i}}{K_{{\rm{rw}}i}}\left( {{S_{\rm{w}}}} \right)} \right] \cdot \\{\kern 40pt}\left[{\dfrac{{{\varphi _i}\left( {{S_{{\rm{wf}}i}}} \right)}}{{1-{{\left( {\dfrac{{{r_{\rm{w}}}}}{{{R_{{\rm{e}}i}}}}} \right)}^2}}}-{\varphi _i}\left( {{S_{\rm{w}}}} \right)} \right] \end{array} \right.$ (8)

${{K}_{\text{rw}i}}\left({S}_{{\rm w}} \right)$ —第 $i$ 层含水饱和度对应的油相相对渗透率，%。

(2) 井底流压

 ${{p}_{\text{wf}i}}\left( t \right)={{p}_{\text{e}i}}-{{R}_{{i}}}\left( t \right)\cdot {{Q}_{\text{L}i}}\left( t \right)$ (9)

(3) 产油量

 ${{Q}_{\text{o}i}}\left( t \right)={{Q}_{\text{L}i}}\left( t \right)\left( 1-{{f}_{\text{w}i}}\left( t \right) \right)$ (10)

(4) 含水率

 ${{W}_{\text{D}}}_{{i}}\left( t \right)=\dfrac{{{w}_{{i}}}\left( t \right)}{\pi \left( {{r}_{\text{e}i}}^{2}-{{r}_{\text{w}}}^{2} \right){{h}_{{i}}}{{\phi }_{{i}}}}$ (11)

${{ w}_{{i}}}\left( t \right)$ —分层累积注水量，m3/d。

 ${{\varphi }_{i}}\left( {{S}_{\text{w}}} \right)=\dfrac{1}{{{W}_{\text{D}}}_{{i}}\left( t \right)}$ (12)

 图6 含水率计算方法图 Fig. 6 Calculation of water cut
1.2.4 产能计算

(1) 合采采油指数

 ${{J}_{}}\left( t \right)=\sum\limits_{i=1}^{n}{\dfrac{{{Q}_{{\rm o}i}}\left( t \right)}{{p}_{{\rm e}i}-{{p}_{\rm wf}}\left( t \right)}}$ (13)

${{p}_{\rm wf}}$ —井底流压，MPa。

(2) 分采采油指数

 ${{J}_{i}}\left( t \right)\text{=}\dfrac{{{Q}_{{\rm o}i}}\left( t \right)}{{{p}_{{\rm e}i}}-{{p}_{{\rm wf}i}}\left( t \right)}$ (14)

1.2.5 层间干扰系数

 $\eta(t)=\dfrac{\sum\limits_{i=1}^{n}{{{J}_{i}}(t)} -{{J}_{}}(t)}{\sum\limits_{i=1}^{n}{{{J}_{i}}(t)}}$ (15)

1.3 模型考虑的主要因素

(1) 层间非均质性

(2) 启动压力梯度

 ${{G}_{i}}=0.6989{{\left( \dfrac{{{K}_{i}}}{{{\mu }_{\text{o}i}}} \right)}^{-1.1147}}$ (16)

(3) 井筒连通

(4) 液量转移

2 无水期层间干扰机理与规律

2.1 层间非均质性

(1) 出液厚度比例

 图7 出液厚度比例变化曲线 Fig. 7 The change of thickness of outputting liquid in water free producing period

 图8 层间干扰系数与层数的关系曲线 Fig. 8 Relationship between interlayer interference coefficient and number of layers

(2) 渗透率级差

 图9 典型井层间干扰系数与渗透率级差关系曲线 Fig. 9 Relationship between interlayer interference coefficient and permeability contrast

2.2 启动压力梯度

 图10 层间干扰系数与启动压力梯度的关系图 Fig. 10 Relationship between interlayer interference coefficient and threshold pressure gradient

2.3 测试工作制度

(1) 生产压差利用所建立的数学模型，在无其他因素影响的情况下，计算不同合采生产压差下的层间干扰系数，绘制了无水期层间干扰系数随生产压差的变化曲线，计算结果见图 11

 图11 层间干扰系数与合采生产压差关系曲线 Fig. 11 Relationship between interlayer interference coefficient and producing pressure difference of commingled production

(2) 产液量

 图12 层间干扰系数与合采产液量关系曲线 Fig. 12 Relationship between interlayer interference coefficient and liquid production of commingled production

3 见水期层间干扰机理与规律

3.1 合采含水率

 图13 层间干扰系数与合采含水率的关系曲线 Fig. 13 Relationship between interlayer interference coefficient and water cut of commingled production

3.2 生产压差

 图14 层间干扰系数与合采生产压差关系曲线 Fig. 14 Relationship between interlayer interference coefficient and producing pressure difference of commingled production

4 结语

(1) 基于渗流力学知识，结合层间干扰主控因素分析结果，建立了层间干扰评价模拟模型，模型考虑了层间非均质性、启动压力梯度、井筒连通以及液量转移。

(2) 层间干扰的主控因素包括储层及流体物性、启动压力梯度、含水阶段以及测试工作指数。无水期，各主控因素通过控制流体渗流阻力，影响油井出液厚度比例，最终形成层间干扰。见水期，各主控因素通过控制流体渗流阻力，影响高含水层产液比例，最终形成层间干扰。

(3) 无水采油期，合采段层数越多，层间干扰越严重；渗透率级差越大，层间非均质性越严重；启动压力梯度越大，低渗透层越难参与流动，层间干扰越严重；随着合采产液量的增加，干扰系数逐渐减小；分采与合采产液量比值越大，层间干扰越严重。见水期，合采含水率越大，层间干扰越严重，高含水层干扰低含水层，由于含水率越高，则低阻力层的液量比例越大，高阻力层(产油量多)液量比例越小，低阻力层对高阻力层形成抑制作用，加剧层间干扰；并且层间干扰主要受高含水层产液比例的影响。增大生产压差，层间干扰加剧的概率将会增加。

(4) 按照测试层段的厚度进行配产，可以减小由于测试工作制度造成的层间干扰；应避免射开厚度小，渗透率大的小层产油量少，以免在见水期形成低效或者无效循环，影响采收率；在无水期，层间干扰程度主要受出液厚度的影响，故可以不划分层系，在合理生产范围内尽可能放大生产压差，提高产液量，增加油层动用厚度，减小层间干扰；见水期，应控制生产压差以减少层间干扰，可以在这一时期采取划分层系、对薄差层实施压裂或者调剖堵水等增产措施。

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