Dimensionless Variables for Pressure Transient Analysis

Overview

Dimensionless variables are fundamental to pressure transient analysis (PTA). By converting physical quantities (pressure, time, distance) into dimensionless form, we achieve:

1. Universal Type Curves: Solutions become independent of specific reservoir properties
2. Generalized Solutions: One solution covers infinite combinations of parameters
3. Simplified Interpretation: Pattern matching on log-log plots becomes possible

The key dimensionless groups in PTA are:

• $p_D$ - Dimensionless pressure
• $t_D$ - Dimensionless time
• $r_D$ - Dimensionless radius
• $L_D$ - Dimensionless distance (to boundaries)
• $C_D$ - Dimensionless wellbore storage

Historical Context

The use of dimensionless variables in petroleum engineering traces back to Van Everdingen and Hurst (1949), who applied Laplace transforms to solve the radial diffusivity equation. The approach was later refined by Agarwal, Bourdet, and others to develop modern well test analysis methods.

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Theory

The Diffusivity Equation

All well test analysis is based on the radial diffusivity equation for single-phase flow through porous media:

$$\frac{\partial^2 p_D}{\partial r_D^2} + \frac{1}{r_D}\frac{\partial p_D}{\partial r_D} = \frac{\partial p_D}{\partial t_D}$$

This dimensionless form applies to all reservoirs regardless of their specific properties. The boundary and initial conditions determine the particular solution.

Initial Condition

$$p_D(r_D, 0) = 0$$

The reservoir is initially at uniform pressure.

Inner Boundary Conditions

Skin Effect: $$p_{wD} = \left[p_D - S\left(\frac{\partial p_D}{\partial r_D}\right)\right]_{r_D=1}$$

Wellbore Storage: $$C_D\frac{dp_{wD}}{dt_D} - \left[r_D\frac{\partial p_D}{\partial r_D}\right]_{r_D=1} = 1$$

Outer Boundary Condition (Infinite Reservoir)

$$\lim_{r_D \to \infty} [p_D(r_D, t_D)] = 0$$

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Equations

Dimensionless Pressure

For constant-rate production, the dimensionless pressure is defined as:

$$p_D = \frac{kh(p_i - p)}{141.2 \, q \, B \, \mu}$$

where:

• $k$ = Permeability (mD)
• $h$ = Net pay thickness (ft)
• $p_i$ = Initial reservoir pressure (psi)
• $p$ = Pressure at time t (psi)
• $q$ = Production rate (STB/d)
• $B$ = Formation volume factor (bbl/STB)
• $\mu$ = Viscosity (cp)

The constant 141.2 provides unit consistency for oilfield units.

Dimensionless Time

$$t_D = \frac{0.0002637 \, k \, t}{\phi \, \mu \, c_t \, r_w^2}$$

where:

• $t$ = Elapsed time (hours)
• $\phi$ = Porosity (fraction)
• $c_t$ = Total compressibility (1/psi)
• $r_w$ = Wellbore radius (ft)

The constant 0.0002637 provides unit consistency when time is in hours.

Dimensionless Radius

$$r_D = \frac{r}{r_w}$$

where $r$ is the radial distance from the wellbore center (ft).

Dimensionless Distance

$$L_D = \frac{L}{r_w}$$

where $L$ is the distance from wellbore to a boundary or point of interest (ft).

Dimensionless Wellbore Storage

$$C_D = \frac{0.8936 \, C}{\phi \, c_t \, h \, r_w^2}$$

where:

• $C$ = Wellbore storage coefficient (bbl/psi)

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Applicability & Limitations

Typical Value Ranges

Flow Regime Identification

Physical Constraints

• $p_D \geq 0$ (pressure drop is positive for production)
• $t_D > 0$ (time must be positive)
• $C_D \geq 0$ (wellbore storage cannot be negative)
• $0 < \phi < 1$ (porosity is a fraction)

Limitations

1. Single-Phase Flow: Equations assume single-phase oil flow
2. Homogeneous Reservoir: No permeability variation assumed
3. Constant Properties: μ, B, ct treated as constants
4. Radial Geometry: Wellbore is vertical and fully penetrating

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Related Functions

Infinite Homogeneous Reservoir

Boundary Effects

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References

1. Van Everdingen, A.F. and Hurst, W. (1949). "The Application of the Laplace Transformation to Flow Problems in Reservoirs." Petroleum Transactions, AIME, 186: 305-324.

2. Bourdet, D., Whittle, T.M., Douglas, A.A., and Pirard, Y.M. (1983). "A New Set of Type Curves Simplifies Well Test Analysis." World Oil, May 1983, pp. 95-106.

3. Bourdet, D., Ayoub, J.A., and Pirard, Y.M. (1989). "Use of Pressure Derivative in Well-Test Interpretation." SPE Formation Evaluation, 4(2): 293-302. SPE-12777-PA.

4. Abass, E. and Song, C.L. (2012). "Computer Application on Well Test Mathematical Model Computation of Homogeneous and Multiple-Bounded Reservoirs." IJRRAS, 11(1): 41-52.

5. Stehfest, H. (1970). "Algorithm 368: Numerical Inversion of Laplace Transforms." Communications of the ACM, 13(1): 47-49.

6. Lee, J. (1982). Well Testing. SPE Textbook Series, Vol. 1. Society of Petroleum Engineers.