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In many practical cases the pressure response analysis is performed over a short period of time during which reservoir saturation can be considered constant.
If spatial reservoir pressure and capillary pressure gradients are not that high then hydraulic reservoir energy can be quantified by average phase pressure.
Under some basic conditions
(26) –
(29) the basic equations of
Error rendering macro 'mathblock-ref' : Page Volatile/Black Oil reservoir flow model @model could not be found.
–
Error rendering macro 'mathblock-ref' : Page Volatile/Black Oil reservoir flow model @model could not be found.
simplify to one equation on
average phase pressure for some effective single-phase fluid (see
derivation here):
(1) |
\phi \, c_t \, \partial_t P - \nabla \big( M \cdot ( \nabla P - \rho \cdot \mathbf{g} ) \big) = \sum_k \, q_t(\mathbf{r}) \cdot \delta(\mathbf{r} -\mathbf{r}_k) |
where
| time |
| |
| |
(2) |
p = \frac{1}{3} \cdot \left( p_w + p_o + p_g \right) |
|
3-phase average reservoir pressure |
(3) |
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_o) \, q_G |
| |
(4) |
B_w = B_w(p), \ B_o = B_o(p), \ B_g = B_g(p) |
| |
(5) |
\phi(\mathbf{r}, \ p) = \phi_0(\mathbf{r}) \exp \left[ - \int_{p_i}^p c_r(p) \, dp \right] |
| effective porosity as a function of reservoir location
\bf r and reservoir pressure at this location
p({\bf r})
|
(6) |
s(\mathbf{r}) = \{ s_w(\mathbf{r}), \ s_o(\mathbf{r}), \ s_g(\mathbf{r}) \} |
| reservoir saturation as a function of location
\bf r
|
(7) |
c_t(s,p, 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) ] |
| |
| |
(9) |
с_w(p, T), \ с_o(p, T), \ с_g(p, T) |
| |
(10) |
M(s, p, T) = M_w + M_o \big( 1 + R_{sn} \big) + M_g \big( 1 + R_{vn} \big) |
| |
(11) |
M_w(s,p, T) = k_a \cdot M_{rw}(s,p, T) |
| |
(12) |
M_o(s,p, T) = k_a \cdot M_{ro}(s,p, T) |
| |
(13) |
M_g(s,p, T) = k_a \cdot M_{rg}(s,p, T) |
| |
(14) |
M_{rw}(s, p, T) = \frac{k_{rw}(s)}{\mu_w(p, T)} |
| |
(15) |
M_{ro}(s,p, T) = \frac{k_{ro}(s)}{\mu_o(p, T)} |
| |
(16) |
M_{rg}(s,p, T) = \frac{k_{rg}(s)}{\mu_g(p, T)} |
| |
(17) |
k_a(\mathbf{r}, \ p, \ \nabla p) = k_a^{\circ} (\mathbf{r}) \cdot k_p (p, \ \nabla p) |
| |
(18) |
k_a^{\circ} (\mathbf{r}) |
| absolute permeability as a function of location
\bf r at Initial formation pressure, pi and absence of spatial reservoir pressure gradient
|
| absolute permeability correction factor for reservoir pressure and reservoir pressure gradient
|
(20) |
\mu_w(p, T), \ \mu_o(p, T), \ \mu_g(p, T) |
| |
(21) |
R_{sn}(p, T) = \frac{R_s B_g}{B_o} \ , \quad R_{vn}(p, T) = \frac{R_v B_o}{B_g} |
| |
(22) |
R_{sp}(p, T) = \frac{\dot R_s B_g}{B_o} \ , \quad R_{vp}(p, T) = \frac{\dot R_v B_o}{B_g} |
| |
(23) |
\rho(p, T) = \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}) }
|
| |
(24) |
g = 9.81 \ \textrm{m} / \textrm{s}^2 |
| standard gravity |
(25) |
\big ( \big)^{\LARGE \cdot} = \frac{d}{dp} |
| differentiation with respect to the pressure
|
This model accumulates significant error when saturation is changing noticeably during the modelling period and in this case the pressure modelling should be conducted along with saturation modelling by solving the original equations of Volatile/Black Oil dynamic flow model.
Table 1. Pressure Diffusion Model Validity Scope
(26) |
T(t, \mathbf{r}) = T(\mathbf{r}) |
|
Reservoir fluid temperature is not changing over time |
(27) |
s_w(t, \mathbf{r}) = s_w(\mathbf{r}),
\quad s_o(t, \mathbf{r}) = s_o(\mathbf{r}),
\quad s_g(t, \mathbf{r}) = s_g(\mathbf{r}) |
|
Reservoir saturation is not chaging over time |
(28) |
| \nabla B_o | \sim 0, \quad |\nabla B_g | \sim 0 |
|
No high fluid pressure gradients over reservoir volume |
(29) |
|\nabla P_{cow}(s)| \sim 0, \quad | \nabla P_{cog}(s)| \sim 0 |
|
No high capillary pressure gradients over reservoir volume |
See also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing
[ Multi-phase pressure diffusion ] [ Volatile/Black Oil dynamic flow models @model ] [Non-linear multi-phase diffusion derivation @model ] [ Linear Perrine multi-phase diffusion @model ]