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| \partial_t \bigg [ \phi \ \rho_W \bigg ] + \nabla \bigg ( \rho_w \ \mathbf{u}_w \bigg ) = q_{mW}(\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \rho_O \bigg ] + \nabla \bigg ( {\tilde m}_{Oo} \ \rho_o \ \mathbf{u}_o
+ {\tilde m}_{Og} \ \rho_{g} \ \mathbf{u}_g \bigg ) = q_{mO}(\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \rho_G \bigg ] + \nabla \bigg ( {\tilde m}_{Go} \ \rho_{o} \ \mathbf{u}_o
+ {\tilde m}_{Gg} \ \rho_g \ \mathbf{u}_g \bigg ) = q_{mG}(\mathbf{r}) |
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anchor | DarcyW1 |
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alignment | left |
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| \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \mathbf{g} ) |
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anchor | DarcyO1 |
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alignment | left |
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| \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \mathbf{g} ) |
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anchor | DarcyG1 |
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alignment | left |
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| \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \mathbf{g} ) |
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anchor | CapilarOW1 |
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alignment | left |
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| P_o - P_w = P_{cow}(s_w) |
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anchor | CapilarOG1 |
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alignment | left |
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| P_o - P_g = P_{cog}(s_g) |
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anchor | swsosg1 |
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alignment | left |
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| s_w + s_o + s_g = 1 |
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Подставляя значения плотностей и массовых долей компонент Substituting the values of mass densities and mass shares of fluid components (см. "Модель Летучей Нефти") и разделив каждое уравнение на плотность соотвествующей компоненты в стандартных условиях, получаем наиболее популярную форму записи уравнений движения Летучей НефтиVolatile Oil Model") and dividing each equation by density of corresponding component in standard conditions one gets the most popular form of Volatile Oil flow equations:
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| \partial_t \bigg [ \phi \ \bigg ( \frac{s_w}{B_w} \bigg ) \bigg ] + \nabla \bigg ( \frac{1}{B_w} \ \mathbf{u}_w \bigg ) =
q_W (\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \bigg ( \frac{s_o}{B_o} + \frac{R_v \ s_g}
{B_g} \bigg ) \bigg ] + \nabla \bigg ( \frac{1}{B_o} \ \mathbf{u}_o
+ \frac{R_v}{B_g} \ \mathbf{u}_g \bigg ) = q_O(\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \bigg ( \frac{s_g}{B_g} + \frac{R_s \ s_o}
{B_o} \bigg ) \bigg ] + \nabla \bigg ( \frac{1}{B_g} \ \mathbf{u}_g
+ \frac{R_s}{B_o} \ \mathbf{u}_o \bigg ) = q_G (\mathbf{r}) |
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anchor | DarcyW |
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alignment | left |
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| \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \ \mathbf{g} ) |
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anchor | DarcyO |
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alignment | left |
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| \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \ \mathbf{g} ) |
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anchor | DarcyG |
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alignment | left |
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| \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \ \mathbf{g} ) |
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| LaTeX Math Block |
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anchor | CapilarOW |
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alignment | left |
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| P_o - P_w = P_{cow}(s_w) |
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anchor | CapilarOG |
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alignment | left |
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| P_o - P_g = P_{cog}(s_g) |
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anchor | swsosg |
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alignment | left |
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| s_w + s_o + s_g = 1 |
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Equations В уравнениях
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правые части равны нулю во всем объеме пласта за исключением контакта скважин с пластом, который описывается моделью скважины (см. ниже)....
suggest no sources of flow in the right side except the contacts between wells and reservroir which is specified by well models as boundary conditions (see below).
Initial Conditions
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Начальное условие по температуре задается распределением температурного поля:
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