Dimensionless Production and Pressure Equations

Dimensionless Pressure for Time Intervals (partially compressible):

$$ p_{wD} = \frac{k h \left[ P_w (t_e) – P_w (t_e + \Delta t) \right] }{141.2 q B \mu} $$

$$ \Delta t_D = \frac{0.0002637 k \Delta t}{\phi \mu c_t L_f^2} $$

Dimensionless Pressure for Compressible Flow:

$$ p_D = \frac{k h}{1422 q_g (T + 460) } \Delta \Psi$$

Pseudotime

$$ t_a = \mu _{gi}c_{ti} \int_0^t \frac{1}{\mu_g (\bar{p}) c_t (\bar{p}) } \text{d} t $$

Material Balance Pseudotime

$$ t_{mba} = \frac{\mu _{gi}c_{ti}}{q(t)} \int_0^t \frac{q(t)}{\mu_g (\bar{p}) c_t (\bar{p}) } \text{d} t $$

Material Balance Pseudotime for PSS

$$ t_{mba} = \frac{\mu _{gi}c_{ti}}{q(t)} \int_0^t \frac{q(t)}{\mu_g (\bar{p}) c_t (\bar{p}) } \text{d} t = \frac{\mu _{gi} c_{ti} z_i}{q(t)} \frac{G (\Psi_i – \bar{\Psi} ) }{2 p_i} $$

Boundary Dominated Flow:

starts at

$$ t_{dAbf} \approx 0.1 $$

where:

$$ t_{dAbf} = \frac{k t}{\phi \mu c_t A} $$

as \( c \approx 1/p \), \( c\) goes goes up as \( p\) goes down, thust \( t_{dAbf} \) goes down as \( p \) goes down. It will take longer to reach boundary dominated flow at low pressures

Pseudo-Steady State Flow

$$ \log\left( t_D \right) = \log \left( \frac{0.00634 k}{\phi \mu c_t r_w^2} \right) + \log \left( t \right) $$

$$ \log\left( p_D \right) = \log \left( \frac{kh}{141.2 q \mu B} \right) + \log \left( \Delta p \right) $$

$$ \log\left( q_D \right) = \log \left( \frac{141.2 \mu B}{kh \Delta P} \right) + \log \left( q \right) $$

Elliptical Flow:

$$ p_D = \ln \left( \frac{A + B}{x_f} \right) $$

$$ t_{De} = \frac{0.0002637 k t}{\phi \mu c_t B^2} $$

where:

$$ A = \sqrt{B^2 + x_f^2} $$

$$ \frac{B^2}{x_f^2} = \frac{t_{Dx_f}}{t_{De}} = \pi t_{Dx_f}$$

$$ t_{Dx_f} = \frac{0.0002637 k t}{\phi \mu c_t x_f^2} $$

It can be seen then that, for an infinite-conductivity fracture flowing at a constant rate in an infinite reservoir

$$ p_D = \ln \left( \sqrt{\pi t_{Dx_f}} + \sqrt{1 + \pi t_{Dx_f}} \right) $$

Fracture Conductivity

$$ C_f = k_f w $$

Dimensionless Fracture Conductivity

$$ F_{CD} = \frac{k_f w}{k_m x_F} $$

$$ x_\text{eff} = \frac{x_\text{flowing}}{1 + \left( \frac{\pi}{2 F_{CD}} \right)} $$

Elliptical Freacture Conductivity

$$ F_E = \frac{2 k_f \epsilon_0}{k_m} $$

where \( \epsilon_0 \) is the fracture width in elliptical coordinates

Radial Flow (classic models)

$$ p_D = \frac{kh \left( p_i – p_{wf} \right)}{141.2 q \mu B} $$

If you don’t have multi-phase fluid compressibility models but do have oil above the bubble point use \( \Delta P\). For oil below the bubble point use \( \Delta P^2 \).

$$ r_D = r_w e^{-s} $$

$$ r_w^\prime = \frac{x_f}{2} $$

$$ t_D = \frac{0.000284 kht}{\phi \mu c_t r_w^2 h} $$

Radial Flow Model from Russell Paper (ref ?):

$$ \Psi_{Dw} = \frac{2 t_D}{\pi} + \frac{1}{6} + \frac{1}{\pi} \sum \frac{1}{n^2} e^{\left( -4n^2 t_D\right)} $$

$$ \frac{p_i – p_{wf}}{q_g} = \frac{141.2 B_g \mu}{k_g h} \left[ \ln \left( \frac{r_e}{r_w} \right) -0.75 + s_a \right] $$

where

$$ s_a = s + D q_g $$

$$ D \approx \frac{9.106 \times 10^{-5} \gamma_g}{k_g^{1/3} \mu_{g,w} r_w} $$

Productivity Index

$$ \text{PI} = J = \frac{q}{\bar{p} – p_{wf}} \propto q_D = \frac{1}{p_D} $$