西南石油大学学报（自然版）  2015, Vol. 37 Issue (1): 141-145

1. 长江大学地球科学学院, 湖北 武汉 430100;
2. 陕西延长石油(集团)有限责任公司研究院, 陕西 西安 710075

Tong's Water Drive Curve Improvement and its Application
Xue Ying1, Shi Lihua2 , Xi Tiande2
1. School of Geosciences, Yangtze University, Wuhan, Hubei 430100, China;
2. Research Institute of Shaanxi Yanchang Petroleum(Group) Co. Ltd., Xi'an, Shaanxi 710075, China
Abstract: A new generalized water drive curve proposed by Tong Xianzhang is of simple shape, and reflects different laws of water cut and it is easy to obtain its parameters. This water driving curve has played an important role in predicting oil production and recoverable reserves. On this basis, further studies were made and improved beta-water drive curves were derived according to the two-phase seepage theory. The improved water drive curves show that seepage characteristic determined the water drive curves. By using the improved water drive characteristic formula more accurate results can be obtained in geological reserves calculation. The improve water drive curve provides a new method for more accurate water-flooding reserves calculation in high water cut stage and has important practical significance to mine production pratice.
Key words: generalized water drive curve     geological reserves     seepage characteristics     relative permeability curve     recovery ratio

1 童氏公式的不足

 $\lg \left( {\frac{{{f_{\rm{w}}}}}{{1 - {f_{\rm{w}}}}}} \right) = \frac{{{N_{\rm{p}}}}}{B} + A$ (1)

 $\lg \left( {\frac{{{f_{\rm{w}}}}}{{1 - {f_{\rm{w}}}}}} \right) = \frac{{7.5{N_{\rm{p}}}}}{{7.5B}} + A = 7.5R + A$ (2)

 $A = \lg \left( {\frac{{{f_{\rm{w}}}}}{{1 - {f_{\rm{w}}}}}} \right) - 7.5R$ (3)

fw=98%时

 $A = 1.69 - 7.5{R_{\rm{m}}}$ (4)

 $\lg \left( {\frac{{{f_{\rm{w}}}}}{{1 - {f_{\rm{w}}}}}} \right) = 7.5\left( {R - {R_{\rm{m}}}} \right) + 1.69$ (5)

 $R = - \frac{A}{{7.5}}$

fw=98%时

 $R = \frac{1}{{7.5}}\lg \left( {\frac{{0.98}}{{1 - 0.98}}} \right) - \frac{A}{{7.5}} = 0.225 - \frac{A}{{7.5}}$

fw=98%时与fw=50%时二者的R值相差0.225。其物理意义表明：水驱特征曲线出现直线段后，含水率由50%上升到98%时，采出程度均为22.5%，这在一定程度上不符合油田的实际情况。童宪章通过大量的注水开发油田实际资料的统计，得到水驱油田动用地质储量与水驱曲线直线斜率的倒数成正比，相关系数约为7.5，这个数值不适用于所有油藏。

2 改进公式

 $F = \frac{{{\mu _{\rm{o}}}}}{{{\mu _{\rm{w}}}}} \cdot \frac{{{K_{{\rm{rw}}}}\left( {{{\bar S}_{{\rm{wf}}}}} \right)}}{{{K_{{\rm{ro}}}}\left( {{S_{{\rm{wi}}}}} \right)}}$ (6)
 $\frac{{{K_{{\rm{rw}}}}\left( {{{\bar S}_{{\rm{wf}}}}} \right)}}{{{K_{{\rm{ro}}}}\left( {{S_{{\rm{wi}}}}} \right)}} = C{10^{ - D{s_{\rm{w}}}}}$ (7)

 $F = \frac{{{\mu _{\rm{o}}}}}{{{\mu _{\rm{w}}}}} \cdot C{10^{ - D{S_{\rm{w}}}}}$ (8)

 $f({S_{\rm{w}}}) = \frac{{{q_{\rm{w}}}}}{{{q_{\rm{w}}} + {q_{\rm{o}}}}} = {\left( {1 + \frac{{{\mu _{\rm{o}}}}}{{{\mu _{\rm{w}}}}} \cdot C{{10}^{ - D{s_{\rm{w}}}}}} \right)^{ - 1}}$ (9)

 $f'({S_{\rm{w}}}) = 2.303Df\left( {{S_{\rm{w}}}} \right)\left[{1-f\left( {{S_{\rm{w}}}} \right)} \right]$ (10)

 ${{\bar S}_{\rm{w}}} = {S_{\rm{w}}} + \frac{{1 - f\left( {{S_{\rm{w}}}} \right)}}{{f'\left( {{S_{\rm{w}}}} \right)}}$ (11)

 $R = \frac{{\left( {{{\bar S}_{\rm{w}}} - {S_{\rm{w}}}_{\rm{i}}} \right)}}{{{S_{{\rm{oi}}}}}}$ (12)

 ${S_{\rm{w}}} = R{S_{{\rm{oi}}}} + {S_{{\rm{wi}}}} - \frac{1}{{2.303D \cdot f\left( {{S_{\rm{w}}}} \right)}}$ (13)

 $\frac{1}{F} = \frac{{{\mu _{\rm{w}}}}}{{{\mu _{\rm{o}}}}}C{10^{ - D\left[{R{S_{{\rm{oi}}}} + {S_{{\rm{wi}}}}-\frac{1}{{2.303Df\left( {{S_{\rm{w}}}} \right)}}} \right]}}$ (14)

 ${F_{\rm{g}}} = \frac{{{q_{\rm{w}}}}}{{\frac{{{\rho _{\rm{o}}}}}{{{B_{\rm{o}}}}} \cdot {q_{\rm{o}}}}} = \frac{{{B_{\rm{o}}}}}{{{\rho _{\rm{o}}}}}F$ (15)

 $\lg {F_{\rm{g}}} = \frac{{{N_{\rm{P}}}}}{B} + A - \frac{1}{{2.303\frac{{{\rho _{\rm{o}}}}}{{{B_{\rm{o}}}}}{F_{\rm{g}}}}}$ (16)

 $R = \frac{{{N_{\rm{p}}}}}{{{N_{\rm{o}}}}} = \frac{1}{{D{S_{{\rm{oi}}}}}}\lg {F_{\rm{g}}} - \frac{A}{{D{S_{{\rm{oi}}}}}}$ (17)

 $R{|_{{F_{\rm{g}}} = 1}} = - \frac{A}{{D{S_{{\rm{oi}}}}}}$ (18)

Fg=49(fw=98%)时

 $R{|_{{F_{\rm{g}}} = 49}} = \frac{{1.69}}{{D{S_{{\rm{oi}}}}}} - \frac{A}{{D{S_{{\rm{oi}}}}}}$ (19)

 $R{|_{{F_{\rm{g}}} = 49}} - R{|_{{F_{\rm{g}}} = 1}} = \frac{{1.69}}{{D{S_{{\rm{oi}}}}}}$ (20)

3 实际应用

 图1 不同方式计算的地质储量对比 Fig. 1 Comparison of geological reserves with different formulas

4 结语

(1)标准童氏图版是利用多个油藏统计出来的经验公式，其主要是计算中高渗透油田注水开发中后期含水率与采出程度的关系统计规律的图版，不能较准确地确定每一个油藏的水驱地质储量，具有很大的局限性，如直接应用于采收率低的低渗透注水开发油藏，会加大水驱地质储量计算的不准确性。因此，在使用经验性的公式时，应当判断该公式是否适应，以免造成开发决策失误。

(2)在童式公式的基础上，从油、水两相渗流特征出发，导出了水驱特征曲线理论公式，计算的水驱地质储量，偏差更小，精度更高，适用于任一注水开发油田，为油田高含水阶段准确估算水驱储量提出了新的思路，不但具有理论意义，而且对矿场生产实践具有重要的实际意义。

fw-含水率, %;

Np-累积采油量, ×104 t;

B-直线段斜率的倒数的系数, ×104 t;

R-采出程度, %;

Rm-最终采出程度, %;

A-系数, 无因次;

F-水油比, 无因次;

f-分流方程, 无因次;

qw-水产量, mL;

qo-油产量, mL;

Kro-油相相对渗透率, %;

Krw-水相相对渗透率, %;

µo-油相黏度, mPa·s;

µw-水相黏度, mPa·s;

Swf-平均前沿含水饱和度, %;

Swi-原始含水饱和度, %;

Sw-含水饱和度, %;

Fg-质量水油比, 无因次;

C, D-系数, 无因次;

Sw-平均含水饱和度, %;

Soi-原始含油饱和度, %;

ρo-原油密度, g/cm3;

Bo-原油体积系数, m3/m3;

No-原油地质储量, ×104 t;

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