Specific case of general multi-phase pressure diffusion with 3-phase Oil + Gas + Water fluid model which assumes that pressure diffusion is equivalent to single-phase diffusion with specifically averaged dynamic parameters, thus resulting in linear partial differential equation with constant coefficients:

(1)	$\begin{array}{l}\displaystyle \phi \, c_t \, \partial_t p - \nabla \big( M \cdot ( \nabla p - \rho \cdot \mathbf{g} ) \big) = \sum_k \, q_k(t) \cdot \delta(\mathbf{r} -\mathbf{r}_k)\end{array}$

where

$\begin{array}{l}t\end{array}$

time

$\begin{array}{l}{\rm r} = (x,y,z)\end{array}$

reservoir location

$\begin{array}{l}\mathbf{r}_k\end{array}$

Well–reservoir contact for $\begin{array}{l}k\end{array}$ -th well

(2)	$\begin{array}{l}\displaystyle p = \frac{1}{3} \cdot \left( p_w + p_o + p_g \right)\end{array}$

3-phase average reservoir pressure

(3)	$\begin{array}{l}\displaystyle q_t(\mathbf{r}) = q_w + q_o + q_g = B_w \, q_W + (B_o - R_s \, B_g) \, q_O + (B_g - R_v \, B_o) \, q_G\end{array}$

total sandface flowrate at reservoir location $\begin{array}{l}\mathbf{r}\end{array}$

(4)	$\begin{array}{l}\displaystyle B_w, \ B_o, \ B_g\end{array}$

formation volume factors at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(5)	$\begin{array}{l}\displaystyle \phi(\mathbf{r})\end{array}$

effective porosity in reservoir location $\begin{array}{l}\bf r\end{array}$ at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$

(6)	$\begin{array}{l}\displaystyle s(\mathbf{r}) = \{ s_w(\mathbf{r}), \ s_o(\mathbf{r}), \ s_g(\mathbf{r}) \}\end{array}$

reservoir saturation as a function of location $\begin{array}{l}\bf r\end{array}$

(7)	$\begin{array}{l}\displaystyle c_t = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ]\end{array}$

total multiphase compressibility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(8)	$\begin{array}{l}\displaystyle с_r\end{array}$

reservoir pore compressibility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$

(9)	$\begin{array}{l}\displaystyle с_w, \ с_o, \ с_g\end{array}$

fluid compressibilities at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(10)	$\begin{array}{l}\displaystyle M = M_w + M_o \big( 1 + R_{sn} \big) + M_g \big( 1 + R_{vn} \big)\end{array}$

total fluid mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(11)	$\begin{array}{l}\displaystyle M_w = k_a \cdot M_{rw}\end{array}$

water mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(12)	$\begin{array}{l}\displaystyle M_o = k_a \cdot M_{ro}\end{array}$

oil mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(13)	$\begin{array}{l}\displaystyle M_g = k_a \cdot M_{rg}\end{array}$

gas mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(14)	$\begin{array}{l}\displaystyle M_{rw} = \frac{k_{rw}(s)}{\mu_w}\end{array}$

relative water mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(15)	$\begin{array}{l}\displaystyle M_{ro} = \frac{k_{ro}(s)}{\mu_o}\end{array}$

relative oil mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(16)	$\begin{array}{l}\displaystyle M_{rg} = \frac{k_{rg}(s)}{\mu_g}\end{array}$

relative gas mobility at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(17)	$\begin{array}{l}\displaystyle k_a(\mathbf{r})\end{array}$

absolute permeability as a function of location $\begin{array}{l}\bf r\end{array}$ at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$

(18)	$\begin{array}{l}\displaystyle \mu_w, \ \mu_o, \ \mu_g\end{array}$

water, oil, gas dynamic viscosity at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(19)	$\begin{array}{l}\displaystyle R_{sn} = \frac{R_s B_g}{B_o} \ , \quad R_{vn} = \frac{R_v B_o}{B_g}\end{array}$

normalized cross-phase exchange ratios at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(20)	$\begin{array}{l}\displaystyle R_{sp} = \frac{\dot R_s B_g}{B_o} \ , \quad R_{vp} = \frac{\dot R_v B_o}{B_g}\end{array}$

normalized cross-phase exchange derivatives at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(21)	$\begin{array}{l}\displaystyle \rho = \frac{ M_{rw} \rho_w + M_{ro} (1 + R_{sn}) \rho_o + M_{rg} (1+R_{vn}) \rho_g }{ M_{rw} + M_{ro} (1 + R_{sn}) + M_{rg} (1+R_{vn}) }\end{array}$

mobility-weighted fluid density at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$

(22)	$\begin{array}{l}\displaystyle g = 9.81 \ \textrm{m} / \textrm{s}^2\end{array}$

standard gravity

(23)	$\begin{array}{l}\displaystyle \big ( \big)^{\LARGE \cdot} = \frac{d}{dp}\end{array}$

differentiation with respect to the pressure

All the above dynamic properties are calculated at reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ and temperature $\begin{array}{l}T_{\rm ref}\end{array}$ thus making (1) a linear partial differential equation.

The reference temperature $\begin{array}{l}T_{\rm ref}\end{array}$ is more or less in practical cases.

The choice of the reference pressure $\begin{array}{l}p_{\rm ref}\end{array}$ depends on the task.

For accurate modelling of early time pressure response (ETR) it is recommended to use initial bottom hole pressure at the moment of the test:

(24)	$\begin{array}{l}\displaystyle p_{\rm ref} = p_{wf}(t=0)\end{array}$

For accurate modelling of late time pressure response (LTR) it is recommended to use current formation pressure:

(25)	$\begin{array}{l}\displaystyle p_{\rm ref} = p_e\end{array}$

The above equations (1) – (23) is generalization of original model ( ^[1], ^[2]) to the case of Volatile Oil fluid model.

In case of Black Oil fluid model ( $\begin{array}{l}R_v = R_{vn} = R_{vp} = 0\end{array}$ ) some equations are simplified:

(26)	$\begin{array}{l}\displaystyle q_t = B_w \, q_W + B_o \, q_O + B_g \, ( q_G - R_s \, q_O)\end{array}$

(27)	$\begin{array}{l}\displaystyle c_t(s) = c_r (1 + R_{sn} s_o ) + c_w s_w + c_o s_o (1+R_{sn}) + c_g s_g + R_{sp} s_o\end{array}$

(28)	$\begin{array}{l}\displaystyle M(s) = \Big \langle \frac{k} {\mu} \Big \rangle = M_w(s) + M_o(s) \big( 1 + R_{sn} \big) + \alpha_g(s)\end{array}$

(29)	$\begin{array}{l}\displaystyle \rho = \frac{ M_{rw} \rho_w + M_{ro}(1 + R_{sn}) \rho_o + M_{rg} \rho_g }{ M_{rw} + M_{ro} (1 + R_{sn}) + M_{rg} }\end{array}$

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:

(30)	$\begin{array}{l}\displaystyle q_t = B_w \, q_W + B_o q_O\end{array}$

(31)	$\begin{array}{l}\displaystyle c_t(s) = c_r + c_w s_w + c_o s_o\end{array}$

(32)	$\begin{array}{l}\displaystyle M(s) = \Big \langle \frac{k} {\mu} \Big \rangle = M_w(s) + M_o(s)\end{array}$

(33)	$\begin{array}{l}\displaystyle \rho = \frac{ M_{rw} \rho_w + M_{ro} \rho_o }{ M_{rw} + M_{ro}}\end{array}$

Despite the fact that in many practical cases the Linear Perrine multi-phase diffusion model leads to inaccurate pressure predictions it still:

provides fair understanding on how pressure responds to multi-phase intakes to or offtakes from formation
introduce a concept of multiphase transmissibility, diffusivity, compressibility, mobility and total sandface flowrate which are very helpful in multiphase dynamic analysis
provides fair understanding on how the above properties depend on reservoir porosity, permeability, saturation, compressibility and PVT/SCAL model

This makes Linear Perrine multi-phase diffusion model a helpful analytical and methodological tool for multiphase dynamic analysis.

One should remember that high content of light oil and gas in reservoir and high drawdowns deteriorate the accuracy of Linear Perrine multi-phase diffusion model.

More accurate pressure estimations are provided by the following pressure diffusion models:

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

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