...
Specific case of general multi-phase pressure diffusion assuming the equivalent with 3-phase Oil + Gas + Water fluid model which assumes that pressure diffusion is equivalent to single-phase diffusion with constant specifically averaged dynamic parameters, thus resulting in linear partial differential equation with constant coefficients:
| LaTeX Math Block |
|---|
|
\phi \, c_t \, \partial_t p - \nabla \big( M \cdot ( \nabla p - \rho \cdot \mathbf{g} ) \big) = \sum_k \, q_k(t) \cdot \delta(\mathbf{r}) \delta(-\mathbf{r}_k) |
where
mathblockanchorqtalignment| left | q_t(\mathbf{r}) = q_w + q_o + q_g = B_w \, q_W + (B_o - R_v \, B_g) \, q_O + (B_g - R_s \, B_| body | --uriencoded--%7B\rm r%7D = (x,y,z) |
|---|
|
| reservoir location |
| LaTeX Math Inline |
|---|
| body | --uriencoded--\mathbf%7Br%7D_k |
|---|
|
| |
| LaTeX Math Block |
|---|
| p = \frac{1}{3} \cdot \left( p_w + p_o + p_g \right) |
|
3-phase average reservoir pressure |
| LaTeX Math Block |
|---|
| q_t(\mathbf{r}) = q_w + q_o + q_g = B_w \, q_W + (B_o - R_s \, B_g) \, q_O + (B_g - R_v \, B_o) \, q_G |
| |
| LaTeX Math Block |
|---|
| B_w, \ B_o, \ B_g |
| |
| LaTeX Math Block |
|---|
| \phi(\mathbf{r}) |
| |
| LaTeX Math Block |
|---|
| s(\mathbf{r}) = \{ s_w(\mathbf{r}), \ s_o(\mathbf{r}), \ s_g(\mathbf{r}) \} |
|
reservoir saturation as a function of location
|
| LaTeX Math Block |
|---|
| c_t = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ] |
| |
| |
| LaTeX Math Block |
|---|
| с_w, \ с_o, \ с_g |
| |
| LaTeX Math Block |
|---|
| M = M_w + M_o \big( 1 + R_{sn} \big) + M_g \big( 1 + R_{vn} \big) |
| |
| LaTeX Math Block |
|---|
| M_w = k_a \cdot M_{rw} |
| |
| LaTeX Math Block |
|---|
| M_o = k_a \cdot M_{ro} |
| |
| LaTeX Math Block |
|---|
| M_g = k_a \cdot M_{rg} |
| |
| LaTeX Math Block |
|---|
| M_{rw} = \frac{k_{rw}(s)}{\mu_w} |
| |
| LaTeX Math Block |
|---|
| M_{ro} = \frac{k_{ro}(s)}{\mu_o} |
| |
| LaTeX Math Block |
|---|
| M_{rg} = \frac{k_{rg}(s)}{\mu_g} |
| |
| LaTeX Math Block |
|---|
| k_a(\mathbf{r}) |
|
absolute permeability as a function of location
at reference pressure
|
| LaTeX Math Block |
|---|
| \mu_w, \ \mu_o, \ \mu_g |
| |
| LaTeX Math Block |
|---|
| R_{sn} = \frac{R_s B_g}{B_o} \ , \quad R_{vn} = \frac{R_v B_o}{B_g} |
| |
| LaTeX Math Block |
|---|
| R_{sp} = \frac{\dot R_s B_g}{B_o} \ , \quad R_{vp} = \frac{\dot R_v B_o}{B_g} |
| |
| LaTeX Math Block |
|---|
| \rho = \frac{ M_{rw} \rho_w + M_{ro} (1 + R_{sn}) \rho_o + M_{rg} (1+R_{vn}) \rho_g }{ M_{rw} + M_{ro} (1 + R_{sn}) + M_{rg} (1+R_{vn}) } |
|
1
All the above dynamic properties are calculated at reference pressure
and temperature thus making
| LaTeX Math Block Reference |
|---|
|
a
linear partial differential equation.
...
| LaTeX Math Block |
|---|
|
p_{\rm ref} = p_e |
The above equations Вышеприведенные формулы
| LaTeX Math Block Reference |
|---|
|
–
| LaTeX Math Block Reference |
|---|
|
представляют собой обобщение оригинальной модели is generalization of original model (
[1],
[2] )
на случай летучей нефти.Для случае черной нефти (
| LaTeX Math Inline |
|---|
| body | R_v = R_{vn} = R_{vp} = 0 |
|---|
|
) некторые из вышеприведенных формул упрощаются:to the case of Volatile Oil fluid model.
In case of Black Oil fluid model (
| LaTeX Math Inline |
|---|
| body | R_v = R_{vn} = R_{vp} = 0 |
|---|
|
) some equations are simplified:
| LaTeX Math Block |
|---|
| q_t = B_w \, q_W + B_o \, q_O + B_g \, ( q_G - R_s \, q_O) |
|
| LaTeX Math Block |
|---|
| c_t(s) = c_r |
|
| LaTeX Math Block |
|---|
|
q_t = B_w \, q_W + B_o \, q_O + B_g \, ( q_G - R_s \, q_O) |
суммарный отбор воды, нефти и газа в пластовых условиях | | LaTeX Math Block |
|---|
|
c_t(s) = c_r (1 + R_{sn} s_o ) + c_w s_w + c_o s_o (1+R_{sn}) + c_g s_g + R_{sp} s_o |
эффективная сжимаемость пласта с мультифазным насыщением как функция насыщенности при опорном давлении | | LaTeX Math Block |
|---|
|
\alpha(s) = \Big \langle \frac{k} {\mu} \Big \rangle = \alpha_w(s) + \alpha_o(s) \big( 1 + R_{sn} \big) + \alpha_g(s) |
эффективная проводимость пород как функция насыщенности при опорном давлении | | LaTeX Math Block |
|---|
|
\rho_{\alpha} = \frac{ \alpha_{rw} \rho_w + \alpha_{ro} \rho_o \alpha_{rg} \rho_g }{ \alpha_{rw} + \alpha_{ro} (1 + c_w s_w + c_o s_o (1+R_{sn}) |
|
\alpha_{rg} }
гравитационная компонента потока при опорном давлении | ...
1q_tM(s) = \Big \langle \frac{k} {\mu} \Big \rangle = |
|
B\, q_WB_o q_Oсуммарный отбор воды, нефти и газа в пластовых условияхR_{sn} \big) + \alpha_g(s) |
|
|
0OM3Sc_t(s)c_r + c_w sc_o s_oэффективная сжимаемость пласта с мультифазным насыщением как функция насыщенности при опорном давлении | | LaTeX Math Block |
|---|
|
\alpha(s) = \Big \langle \frac{k} {\mu} \Big \rangle = \alpha_w(s) + \alpha_o(s) |
M_{ro}(1 + R_{sn}) \rho_o + M_{rg} \rho_g }{ M_{rw} + M_{ro} (1 + R_{sn}) + M_{rg} }
|
|
For the 2-phase Oil + Water fluid model (where Linear Perrine multi-phase diffusion mode is the most accurate ) the above equations are getting even simpler:
| LaTeX Math Block |
|---|
| q_t = B_w \, q_W + B_o q_O |
|
| LaTeX Math Block |
|---|
| c_t(s) = c_r + c_w s_w + c_o s_o |
|
|
эффективная проводимость пород как функция насыщенности при опорном давлении | | LaTeX Math Block |
|---|
|
\rho_{\alpha} = \frac{ \alpha_{rw} \rho_w + \alpha_{ro} \rho_o }{ \alpha_{rw} + \alpha_{ro}}
|
гравитационная компонента потока при опорном давлении | | LaTeX Math Block |
|---|
| anchor | Perrine2phase_alpha |
|---|
| alignment | left |
|---|
|
\sigma = \Big \langle \frac{k} {\mu} \Big \rangle |
|
\, hk , h\, \Bigg[ \k}{mu}\frac{k\mu_o} \Bigg] гидропроводность пласта при опорном давлении | LaTeX Math Inline |
|---|
body | P_{\bf ref}
Despite the fact that in many practical cases the Linear Perrine multi-phase diffusion model leads to inaccurate pressure predictions it still:
...
[ Pressure diffusion models ] [ Linear Perrine multi-phase diffusion @model derivation ]
Reference
...
...
...
- Perrine, R.L. 1956. Analysis of Pressure Buildup Curves. Drill. and Prod. Prac., 482. Dallas, Texas: API.
...
- SPE-1235-G,
...
- Martin, J.C.
...
- , Simplified Equations of Flow in Gas Drive Reservoirs and the Theoretical Foundation of Multiphase Pressure Buildup Analyses
...
- , 1959
...
