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: 

\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)

where

p = \frac{1}{3} \cdot \left( p_w + p_o + p_g \right)

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

B_w, \ B_o, \ B_g 

\phi(\mathbf{r})

s(\mathbf{r}) = \{ s_w(\mathbf{r}), \ s_o(\mathbf{r}), \ s_g(\mathbf{r})  \}

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) ]

с_r

с_w, \ с_o, \ с_g

M = M_w + M_o \big( 1 + R_{sn} \big) + M_g \big( 1 + R_{vn} \big)

M_w = k_a \cdot M_{rw}

M_o = k_a \cdot M_{ro}

M_g = k_a \cdot M_{rg}

M_{rw} = \frac{k_{rw}(s)}{\mu_w}

M_{ro} = \frac{k_{ro}(s)}{\mu_o}

M_{rg} = \frac{k_{rg}(s)}{\mu_g}

k_a(\mathbf{r})

\mu_w, \ \mu_o, \ \mu_g

R_{sn} = \frac{R_s B_g}{B_o} \ , \quad R_{vn} = \frac{R_v B_o}{B_g}

R_{sp} = \frac{\dot R_s B_g}{B_o} \ , \quad R_{vp} = \frac{\dot R_v B_o}{B_g}

\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}) }

g = 9.81 \ \textrm{m} / \textrm{s}^2

 \big (   \big)^{\LARGE \cdot} = \frac{d}{dp}

All the above dynamic properties are calculated at reference pressure and temperature  thus making  a linear partial differential equation.

The reference temperature   is more or less in practical cases.

The choice of the reference pressure  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: 

p_{\rm ref} = p_{wf}(t=0)

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

p_{\rm ref} = p_e

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

In case of Black Oil fluid model () some equations are simplified:

q_t =  B_w \, q_W + B_o \, q_O + B_g \, ( q_G - R_s \, q_O)

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 

M(s) = \Big \langle \frac{k} {\mu} \Big \rangle = M_w(s) + M_o(s) \big( 1 + R_{sn} \big) + \alpha_g(s) 

\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}  }

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:

q_t = B_w \, q_W + B_o  q_O

c_t(s) = c_r  + c_w s_w + c_o s_o

M(s) = \Big \langle \frac{k} {\mu} \Big \rangle = M_w(s) + M_o(s)

\rho = \frac{ M_{rw} \rho_w + M_{ro} \rho_o   }{ M_{rw}  + M_{ro}}

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:
 

• Pseudo-linear multi-phase diffusion model
   
• Non-linear multi-phase diffusion model
   
• Volatile Oil and Black Oil dynamic flow models

See also

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing

[ Pressure diffusion models ] [ Linear Perrine multi-phase diffusion @model derivation ]

Reference

1.  Perrine, R.L. 1956. Analysis of Pressure Buildup Curves. Drill. and Prod. Prac., 482. Dallas, Texas: API.
   
   
2.  SPE-1235-G, Martin, J.C., Simplified Equations of Flow in Gas Drive Reservoirs and the Theoretical Foundation of Multiphase Pressure Buildup Analyses, 1959