Solutions for the Diffusivity Equation under Different Flow Regimes

Introduction

The following section describes a segmented approach to flow regime indentification.

Equations for Linear, Bi-linear, Spherical & Pseudo-Steady State:

For gas we use material balance pseudo time.
For oil we use Material Balance Time

	Oil	Gas
Normalized Pressure
Bourdet (Well Testing) derivative
log Bourdet derivative

From the above equation you can see that the log bourdet derivative will have a linear relationship with log time. The Bourdet plot is popular as it is easy to visually identify these linear trends on the plot, and consequently identify flow regimes.

	Oil	Gas
log10 Bourdet derivative

Pseudo-steady state flow regime:

Parameters

	Oil	Gas
m	with field units with consistent units	with field units and assumed standard conditions with consistent units
b	with field units with consistent units

• Note: in Bourdet’s textbook (section 2.3.3) constant in m is 0.234, in Ahmed’s textbook (equation 1.2.117) it is 0.23396. In Blasingame’s course notes it is 0.23395.

Rectangular reservoir. from Wattenbarger et al. (1998)

Constant rate solution pss:

	Oil	Gas
complete form
m	following substitution allows us to arrive at same eq.s as above	following substitution allows us to arrive at same eq.s as above
b	with consistent units

Linear flow regime:

Parameters

from Clarkson (2001) eq. 1.29:

	Oil	Gas
constant rate solution
m	field units: consistent units:	field units: consistent units:
if using channel width instead of xf, use the following substitution
b

Radial flow regime:

	Oil	Gas
ln form	complete form:	complete form:
Bourdet derivative
ln Bourdet derivative
log10 form

parameters

	Oil	Gas
m	field units: consistent units:	consistent units:
b	Field units: consistent units:
S	Field units: consistent units:	Field units: consistent units:

Versions you may find in textbooks:

	Oil	Gas
log10 version
ln

Nomenclature (field units):

• q: rate (STB/D)

• p: pressure (psi)

• p_i: inital reservoir pressure (psi)

• ∆t: time from start of test (hrs)

• k: permeability (mD)

• h: net pay (ft)

• B: formation volume factor: (RB/STB)

• A: reservoir area (ft²)

• C_A: shape factor

• T : temperature (Rankine)

• rw: wellbore radius (ft)

• γ: euler’s constant

• ϕ: porosity

• µ: viscosity (cP)

Reference:

• Ahmed, Tarek, and Paul McKinney. Advanced reservoir engineering. Elsevier, 2011.

• Blasingame course notes: https://blasingame.engr.tamu.edu/z_zCourse_Archive/P648_17A/P648_17A_Lectures/20170214_P648_17A_Lec_05_(Work)_PDD_RadFlw_[PDF].pdf

• Bourdet, Dominique. Well test analysis: the use of advanced interpretation models. Elsevier, 2002.

• Clarkson, C. R. (2021). Unconventional reservoir rate-transient analysis.

• Harmony Enterprise Help Manual

• Wattenbarger, Robert A., et al. "Production analysis of linear flow into fractured tight gas wells." SPE Rocky Mountain Petroleum Technology Conference/Low-Permeability Reservoirs Symposium. SPE, 1998.