Pressure Transient Analysis Overview

Introduction

Pressure Transient Analysis (PTA) uses pressure response measurements during flow rate changes to characterize reservoir and well properties. By analyzing how pressure propagates through the formation, engineers can determine:

Permeability — formation flow capacity
Skin factor — near-wellbore damage or stimulation
Reservoir boundaries — faults, aquifers, drainage limits
Drainage area — connected pore volume
Formation pressure — initial and average reservoir pressure

Petroleum Office provides analytical solutions for well testing interpretation based on the diffusivity equation.

Fundamental Concepts

The Diffusivity Equation

All PTA solutions derive from the diffusivity equation for slightly compressible fluid flow:

\frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} = \frac{\phi \mu c_t}{k} \frac{\partial p}{\partial t}

This equation describes how pressure disturbances propagate radially from a wellbore through a porous medium.

Dimensionless Variables

Working in dimensionless form simplifies analysis and enables type curve matching:

Variable	Definition	Physical Meaning
$p_D$	$\frac{kh(p_i - p)}{141.2 q B \mu}$	Normalized pressure drop
$t_D$	$\frac{0.0002637 k t}{\phi \mu c_t r_w^2}$	Normalized time
$r_D$	$r / r_w$	Normalized radial distance
$C_D$	$\frac{0.8936 C}{\phi c_t h r_w^2}$	Normalized wellbore storage

📖 Full Documentation: Dimensionless Variables

Model Selection Framework

Decision Tree

                         Start Here
                             │
                             ▼
                 ┌───────────────────────┐
                 │  Boundary Effects?    │
                 └───────────────────────┘
                      │            │
                     No           Yes
                      │            │
                      ▼            ▼
              ┌────────────┐  ┌────────────────┐
              │  Infinite  │  │ Boundary Type? │
              │ Reservoir  │  └────────────────┘
              └────────────┘       │      │      │
                    │          Sealing  Const.  Mixed
                    │           Fault    Pres.
                    ▼              │       │      │
              ┌──────────┐         ▼       ▼      ▼
              │Wellbore  │    ┌────────────────────┐
              │Storage & │    │  Bounded Reservoir │
              │  Skin?   │    │      Models        │
              └──────────┘    └────────────────────┘
                 │   │
               Yes   No
                 │    │
                 ▼    ▼
            With WS  Line
            & Skin   Source

Quick Reference Table

Scenario	Early Time	Middle Time	Late Time	Model
Ideal infinite	-	Radial flow	Continues	Line Source
With storage/skin	Unit slope	Radial flow	Continues	VW with S, C
Single fault	-	Radial → doubled slope	Doubled slope	Linear sealing
Channel	-	Radial → linear	Linear flow	Parallel boundaries
Closed	-	Radial → PSS	PSS decline	Bounded rectangle

Available PTA Models

Infinite-Acting Reservoir

The fundamental solution for a well in an infinite homogeneous reservoir.

Line Source Solution:

p_D(r_D, t_D) = -\frac{1}{2} \text{Ei}\left(-\frac{r_D^2}{4t_D}\right)

At the Wellbore (with skin and storage): Solutions use Laplace transform and Stehfest numerical inversion.

Diagnostic Features:

Semi-log straight line after wellbore effects
Slope gives permeability-thickness (kh)
Intercept gives skin factor

📖 Full Documentation: Infinite Reservoir Solution

Bounded Reservoir

Real reservoirs have boundaries that affect pressure behavior. The method of images handles various boundary configurations.

Boundary Types:

Type	Physical Example	Pressure Effect
Sealing fault	Impermeable barrier	Pressure drops faster
Constant pressure	Aquifer, gas cap	Pressure stabilizes
Mixed	Fault + aquifer	Combined effects

Configurations:

Linear fault — Single sealing boundary
Perpendicular faults — Corner (90°)
Parallel faults — Channel geometry
Constant pressure boundaries — Pressure support

📖 Full Documentation: Bounded Reservoir Models

Flow Regime Identification

Diagnostic Plot Features

The log-log diagnostic plot (pressure change and derivative vs. time) reveals flow regimes:

log(Δp), log(Δp')
      │
      │    ●●●●●●●●●●●●●●●          Derivative plateau = radial flow
      │   ●
      │  ●                          
      │ ●    Transition from        
      │●     wellbore storage       
      │●●●●●                        Unit slope = wellbore storage
      │
      └─────────────────────────→ log(Δt)

Derivative Signature	Flow Regime	Indicates
Unit slope (45°)	Wellbore storage	Early-time storage effects
Horizontal plateau	Radial flow	Infinite-acting period
Half slope (1/2)	Linear flow	Fracture or channel
Doubled plateau	Boundary effect	Sealing fault nearby
Falling derivative	Constant pressure	Aquifer support

Semi-Log Analysis

The Horner plot (or MDH plot for drawdown) identifies radial flow and extracts parameters:

p_{wf} = p_i - m \log(t) + \text{const}

Where slope $m$ gives:

kh = \frac{162.6 qB\mu}{m}

Interpretation Workflow

Step 1: Data Quality Check

Verify rate history accuracy
Check for gauge resolution and drift
Identify operational events (shut-ins, rate changes)

Step 2: Diagnostic Analysis

Plot log-log diagnostic (Δp and derivative vs. Δt)
Identify flow regimes from derivative shape
Determine appropriate model

Step 3: Model Selection

Based on diagnostic features:

Infinite reservoir if derivative stabilizes
Bounded if derivative deviates late-time
Wellbore effects if early unit slope

Step 4: Parameter Estimation

Semi-log analysis for kh and skin
Type curve matching for verification
History matching for complex cases

Step 5: Validation

Check material balance (average pressure)
Compare with geology/seismic
Validate against production data

Key Parameters Extracted

Parameter	Symbol	From	Significance
Permeability-thickness	$kh$	Semi-log slope	Flow capacity
Skin factor	$S$	Semi-log intercept	Wellbore condition
Wellbore storage	$C$	Early unit slope	Wellbore volume effects
Initial pressure	$p_i$	Extrapolation	Original reservoir pressure
Average pressure	$\bar{p}$	Horner extrapolation	Current reservoir pressure
Distance to boundary	$L$	Deviation time	Boundary location

Function Categories

Dimensionless Pressure Functions

Calculate $p_D$ for various reservoir configurations.

Real Pressure Functions

Convert dimensionless solutions to field units.

Derivative Functions

Calculate pressure derivative for diagnostic plots.

Boundary Models

Handle sealing faults, constant pressure, mixed boundaries.

Detailed Model Theory

Dimensionless Variables — pD, tD, CD definitions and conversions
Infinite Reservoir Solution — Line source, Ei function, wellbore effects
Bounded Reservoir Models — Method of images, boundary effects

Supporting Functions

Exponential Integral (Ei) — Special function for line source
Oil Viscosity — Required for mobility calculations
Oil Compressibility — Required for total compressibility

References

Lee, J., Rollins, J.B., and Spivey, J.P. (2003). Pressure Transient Testing. SPE Textbook Series Vol. 9.
Bourdet, D. (2002). Well Test Analysis: The Use of Advanced Interpretation Models. Elsevier.
Horne, R.N. (1995). Modern Well Test Analysis: A Computer-Aided Approach, 2nd Edition. Petroway Inc.
Earlougher, R.C. Jr. (1977). Advances in Well Test Analysis. SPE Monograph Vol. 5.
Matthews, C.S. and Russell, D.G. (1967). Pressure Buildup and Flow Tests in Wells. SPE Monograph Vol. 1.