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:

📖 Full Documentation: Dimensionless Variables

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Model Selection Framework

Decision Tree

Quick Reference Table

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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

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Bounded Reservoir

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

Boundary Types:

Configurations:

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

📖 Full Documentation: Bounded Reservoir Models

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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)

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:

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

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Interpretation Workflow

Step 1: Data Quality Check

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

Step 2: Diagnostic Analysis

1. Plot log-log diagnostic (Δp and derivative vs. Δt)
2. Identify flow regimes from derivative shape
3. 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

1. Semi-log analysis for kh and skin
2. Type curve matching for verification
3. History matching for complex cases

Step 5: Validation

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

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Key Parameters Extracted

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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.

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Related Documentation

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

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References

1. Lee, J., Rollins, J.B., and Spivey, J.P. (2003). Pressure Transient Testing. SPE Textbook Series Vol. 9.

2. Bourdet, D. (2002). Well Test Analysis: The Use of Advanced Interpretation Models. Elsevier.

3. Horne, R.N. (1995). Modern Well Test Analysis: A Computer-Aided Approach, 2nd Edition. Petroway Inc.

4. Earlougher, R.C. Jr. (1977). Advances in Well Test Analysis. SPE Monograph Vol. 5.

5. Matthews, C.S. and Russell, D.G. (1967). Pressure Buildup and Flow Tests in Wells. SPE Monograph Vol. 1.