西南石油大学学报（自然科学版）  2017, Vol. 39 Issue (1): 124-132

1. 中国石油塔里木油田公司勘探开发研究院, 新疆 库尔勒 841000;
2. "油气藏地质及开发工程"国家重点实验室·西南石油大学, 四川 成都 610500

Pressure Variation Characteristics in Bead-shaped Fractured Vuggy Carbonate Gas Reservoirs
WANG Hai1, LIN Ran2 , ZHANG Chenyang1, HUANG Binguang2, MIU Changsheng1
1. Institute of Petroleum Exploration and Development, Tarim Oilfield Company, PetroChina, Korla, Xinjiang 841000, China;
2. State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China
Abstract: This paper establishes a mathematical model for gas flow through bead-shaped fractured vuggy carbonate gas reservoirs based on material balance and fracture flow equations. This model could calculate the pressure variation in each vug during the gas production stage. A series of numerical calculations were conducted on the mathematical model, and then a double logarithmic curve of the bottom hole pressure(BHP) and a semi-logarithmic curve of pressure derivative were plotted. According to the semi-logarithmic curve, pressure derivative curves of vugs initially increase and then decrease; meanwhile, the derivative curves of the nearest vug from the well increases and decreases most rapidly, and its turning point presents earlier than other vugs. The double logarithmic curve could be divided into four stages-wellbore storage effect reaction stage, fracture reaction stage, vug reaction stage, and boundary reaction stage. Finally, we studied the characteristics of the double logarithmic curve from a series of models with different number/volume/arrangement of vugs and different apertures/arrangements of fractures, and it was found that the characteristics of double logarithmic curves change with the different above-mentioned parameters.
Key words: bead shaped fractured vuggy carbonate     mathematic model     reservoir pressure variation     well test analysis     fracture flow

1 数学模型建立

 图1 串珠状缝洞储层数学模型示意 Fig. 1 Mathematical model of beads shaped fractured vuggy reservoir

(1) 模型内流体为单相非理想可压缩气体，流动为等温过程，满足流体力学原理。

(2) 由于岩石基质相对于缝洞储集体的低渗低储特性，故忽略流其内部的流体流动。

(3) 边部裂缝中有一口生产气井，并以定产量的生产制度进行产气。

(4) 由于溶洞内部完全由气体充填，不符合渗流规律，故将溶洞视为一个等势体[6, 21, 31-32]，即其内部压力处处相等。

(5) 考虑井筒储集效应，忽略缝洞储集体的压缩性、井筒压降和井底表皮效应。

1.1 裂缝内气体流动方程

 图2 裂缝流动示意 Fig. 2 Fracture flow

 ${p_1}A - {p_2}A - \rho {\rm{g}}LA\sin \alpha - {\bar \tau _0}CL = 0$ (1)

A—裂缝截面积，m2

$\rho$—气体密度，kg/m3

g—重力加速度，g=9.8 m/s2

L—裂缝长度，m；

$\alpha$—裂缝倾斜角度，(°)；

C—裂缝过流截面湿周，m；

${\bar \tau _0}$—裂缝壁面对气体的平均切向力，Pa。

 ${\bar \tau _0} = \dfrac{{\int_0^C {\tau {\rm{d}}C} }}{C}$ (2)

 $\dfrac{{{p_1}}}{{\rho {\rm{g}}}} - \dfrac{{{p_2}}}{{\rho {\rm{g}}}} - {z_2} + {z_1} = {\bar \tau _0}\dfrac{{CL}}{{\rho {\rm{g}}A}}$ (3)

 ${h_{\rm{f}}} = \left( {{z_1} + \dfrac{{{p_1}}}{{\rho {\rm g}}}} \right) - \left( {{z_2} + \dfrac{{{p_2}}}{{\rho {\rm g}}}} \right)$ (4)

 ${h_{\rm{f}}} = {\bar \tau _0}\dfrac{{CL}}{{\rho {\rm g}A}}$ (5)

 ${\bar \tau _0} = {C_{\rm{f}}}\rho \dfrac{{{v^2}}}{2}$ (6)

${C_{\rm{f}}}$—气体摩擦系数，无因次。

 ${h_{\rm{f}}} = {C_{\rm{f}}}\dfrac{{CL}}{A}\dfrac{{{v^2}}}{{2{\rm g}}}$ (7)

 $\dfrac{{{\rm{d}}p}}{\rho } + {\rm{gd}}z + v{\rm{d}}v + \dfrac{{\tau C}}{{\rho A}}{\rm{d}}s = 0$ (8)

s—移动距离，m。

 $\dfrac{{\tau C}}{{\rho A}} = {\rm{g}}\dfrac{{{h_{\rm{f}}}}}{L}$ (9)

 $\dfrac{{\tau C}}{{\rho A}} = \dfrac{{{C_{\rm{f}}}}}{{A/C}}\dfrac{{{v^2}}}{2}$ (10)

 $\dfrac{{{\rm{d}}p}}{\rho } + {\rm{gd}}z + v{\rm{d}}v + \dfrac{{{C_{\rm{f}}}}}{{A/C}}\dfrac{{{v^2}}}{2}{\rm{d}}s = 0$ (11)

 $\dot m = \rho Av$ (12)

 $\rho = p/Z{\rm R'}T$ (13)

$\rm R'$—气体常数，Pa·m3/(kg·K)；

T—温度，K。

 $v = \dfrac{{\dot mZ{\rm R'}T}}{{pA}}$ (14)

 $- \left( {\dfrac{{2{A^2}}}{{{{\dot m}^2}Z{\rm{R'}}T}}} \right)p{\rm{d}}p = \dfrac{{{C_{\rm{f}}}}}{{A/C}}{\rm{d}}s + \dfrac{{2{\rm{d}}v}}{v}$ (15)

 $\dot m = A\sqrt {\dfrac{{ {p_1^2 - p_2^2} }}{{Z{\rm{R'}}T\left( {{C_{\rm{f}}}\dfrac{L}{{A/C}} + 2\ln \dfrac{{{p_1}}}{{{p_2}}}} \right)}}}$ (16)

 $\dot m = Hd\sqrt {\dfrac{{\left( {p_1^2 - p_2^2} \right)}}{{Z{\rm{R'}}T\left[{{C_{\rm{f}}}\dfrac{{2L(H + d)}}{{Hd}} + 2\ln \dfrac{{{p_1}}}{{{p_2}}}} \right]}}}$ (17)

d—裂缝开度，m。

 ${q_{{\rm{sc}}}} = \dfrac{{Hd}}{{{\rho _{{\rm{sc}}}}}}\sqrt {\dfrac{{\left( {p_1^2 - p_2^2} \right)}}{{Z{\rm{R'}}T\left[{{C_{\rm{f}}}\dfrac{{2L(H + d)}}{{Hd}} + 2\ln \dfrac{{{p_1}}}{{{p_2}}}} \right]}}}$ (18)

${\rho_{{\rm{sc}}}}$—标况下气体密度，kg/m3

 ${C_{\rm{f}}} = f/4$ (19)

 ${\mathop{ Re}\nolimits} = \dfrac{{\rho vd}}{\mu }$ (20)

$Re$—雷诺数，无因次。

 $v = \dfrac{{202000ZT{q_{{\rm{sc}}}}}}{{273\left( {{p_1} + {p_2}} \right)A}}$ (21)

 $f = \dfrac{{64}}{{{\mathop{ Re}\nolimits} }}$ (22)

 $\dfrac{1}{{\sqrt f }} = 1.14 - 2\lg \left (\dfrac{E}{d} + \dfrac{{21.25}}{{{{{\mathop{Re}\nolimits} }^{0.9}}}}\right )$ (23)

1.2 溶洞内气体物质守恒方程

 $pV = nZ{\rm R}T$ (24)

n—气体物质的量，mol；

R—普适气体常数，R = 8.314 J/(k·mol)。

1.3 串珠状缝洞模型气体流动方程

 $q_{1\_0} = {常数}$ (25)
 \begin{align} & {{q}_{{{i}\_{(}}i-1)}}=\frac{{{H}_{{{i}\_{(}}i-1)}}{{d}_{{{i}\_{(}}i-1)}}}{{{\rho }_{\text{sc}}}}\sqrt{\frac{p_{i}^{2}-p_{i-1}^{2}}{{{{\bar{Z}}}_{{{i}\_{(}}i-1)}}{R}'T\left[ {{f}_{{{i}\_{(}}i-1)}}\frac{{{L}_{{{i}\_{(}}i-1)}}}{2{{d}_{{{i}\_{(}}i-1)}}}+2\ln \frac{{{p}_{i}}}{{{p}_{i-1}}} \right]}} \\ & \left( i=2,3,\cdots ,N \right) \\ \end{align} (26)

${q_{{{i\_(i}} - {{1)}}}}$—${{{i\_(i}} - {{1)}}}$号裂缝内的气体流量(标况)，m3/s；

${H_{{{i\_(i}} - {{1)}}}}$—${{{i\_(i}} - {{1)}}}$号裂缝的高度，m； $p_{i}$—i号溶洞内压力，Pa；

${{\overline Z }_{{{i\_(i}} - {{1)}}}}$—${{{i\_(i}} - {{1)}}}$号裂缝内气体的偏差因子，无因次；

${f_{{{i\_(i}} - {{1)}}}}$—${{{i\_(i}} - {{1)}}}$号裂缝内气体的摩阻系数，无因次；

${L_{{{i\_(i}} - {{1)}}}}$—${{{i\_(i}} - {{1)}}}$号裂缝的长度，m；

${d_{{{i\_(i}} - {{1)}}}}$—${{{i\_(i}} - {{1)}}}$号裂缝的开度，m；

N—溶洞数。

 ${p_{i}}{V_{i}} = {n_{i}}{Z_{i}}{\rm R}T{，}{\kern 10pt} \left (i = 2,3,\cdots,N \right)$ (27)

$n_{i}$—i号溶洞内气体物质的量数，mol；

$Z_{i}$—i号溶洞内气体偏差因子，无因次。

 ${\rm{d}}{n_i} = \dfrac{{ - {q_{i\_(i - 1)}}{\rm{d}}t}}{{22.4/1000}}{，}{\kern 10pt} \left (i = 2,3,\cdots,N \right)$ (28)

2 数值计算

3 结果分析

 图3 各溶洞压力导数半对数曲线 Fig. 3 Semi-logarithmic curve of pressure derivative of different vugs

 图4 井底压力双对数曲线 Fig. 4 Semi-logarithmic curve of pressure and pressure derivative

3.1 溶洞数量的影响

 图5 不同溶洞数量时井底压力双对数曲线 Fig. 5 Double logarithmic curve of BHP with different number of vugs

3.2 溶洞体积与排列顺序的影响

 图6 不同体积溶洞排列时井底压力双对数曲线 Fig. 6 Double logarithmic curve of BHP with different arrangement of vugs
 图7 不同体积溶洞时井底压力双对数曲线 Fig. 7 Double logarithmic curve of BHP with different volume of vugs

3.3 裂缝开度与排列顺序的影响

 图8 不同裂缝开度时井底压力双对数曲线 Fig. 8 Double logarithmic curve of BHP with different aperture of fractures
 图9 不同裂缝排列时井底压力双对数曲线 Fig. 9 Double logarithmic plot of BHP with different arrangement of fractures

4 结论

(1) 建立了串珠状缝洞型储层中气体流动的数学模型，并通过数值方法求解出了定产量生产时储层内压力变化规律。

(2) 模型中各溶洞压力导数半对数曲线呈现出先上升后下降的趋势；初期阶段，由于靠近井筒的溶洞对产气量贡献更大，所以其压力导数的上升和下降时间都更早。末期阶段，各溶洞流量趋于平衡，压力导数趋于一致。

(3) 压力导数双对数曲线可以分为4段：井筒储集反应段、裂缝反应阶段、溶洞反应阶段、边界反应阶段。

(4) 模型中缝洞参数的变化对井底压力双对数曲线的对应阶段特征都有比较明显的影响。所以，理论上可以通过对比井底压力双曲线变化特征，判断模型的参数情况。

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