Bounded Reservoir Solutions for Pressure Transient Analysis

Overview

Real reservoirs have boundaries that eventually affect well test behavior. Understanding boundary effects is crucial for:

• Reservoir size estimation - Distance to boundaries
• Boundary type identification - Sealing vs. constant pressure
• Production forecasting - Long-term pressure behavior
• Well placement optimization - Understanding drainage patterns

The method of images is the primary analytical technique for modeling boundary effects. An image well is placed on the opposite side of the boundary to mathematically reproduce the boundary condition.

Boundary Types

Boundary Configurations

Petroleum Office supports these boundary geometries:

• Linear - Single boundary (fault or aquifer edge)
• Perpendicular - Two boundaries at 90° (corner)
• Parallel - Two boundaries (channel geometry)
• Intersecting - Two boundaries at 60° angle

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Theory

Method of Images

The superposition principle allows us to represent boundary effects using virtual image wells. The dimensionless wellbore pressure with boundaries is:

$$p_{wD} = p_{wDi} + p_{wDb}$$

where:

• $p_{wDi}$ = Dimensionless pressure from infinite homogeneous reservoir
• $p_{wDb}$ = Additional pressure effect from boundaries

The boundary effect $p_{wDb}$ is calculated using the line source solution at the image well locations.

Line Source Solution for Image Wells

The dimensionless pressure contribution from an image well at distance $L$ from the active well is:

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

where $L_D = L/r_w$ is the dimensionless distance to the image well.

For a sealing boundary at distance $L$ from the well, the image well is at distance $2L$, so $L_D = 2L/r_w$.

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Equations

Linear Sealing Fault

A single sealing (no-flow) boundary at distance $L$ from the well:

$$p_{wDb} = -\frac{1}{2}\text{Ei}\left(-\frac{L_D^2}{t_D}\right)$$

where $L_D = L/r_w$.

Physical interpretation: The image well produces at the same rate, adding to the pressure drop. Late-time derivative doubles.

Linear Constant Pressure Boundary

A single constant pressure boundary (aquifer, gas cap) at distance $L$:

$$p_{wDb} = +\frac{1}{2}\text{Ei}\left(-\frac{L_D^2}{t_D}\right)$$

Physical interpretation: The image well injects at the same rate, reducing the pressure drop. Late-time derivative drops to zero as steady-state is approached.

Two Perpendicular Sealing Faults

Two sealing boundaries at 90°, with distances $L_1$ and $L_2$:

$$p_{wDb} = -\frac{1}{2}\left\{\text{Ei}\left[-\frac{L_{1D}^2}{t_D}\right] + \text{Ei}\left[-\frac{L_{2D}^2}{t_D}\right] + \text{Ei}\left[-\frac{L_{1D}^2 + L_{2D}^2}{t_D}\right]\right\}$$

Physical interpretation: Three image wells are required - one for each boundary plus one in the corner. Late-time derivative quadruples.

Two Perpendicular Constant Pressure Boundaries

Two constant pressure boundaries at 90°:

$$p_{wDb} = +\frac{1}{2}\left\{\text{Ei}\left[-\frac{L_{1D}^2}{t_D}\right] + \text{Ei}\left[-\frac{L_{2D}^2}{t_D}\right] + \text{Ei}\left[-\frac{L_{1D}^2 + L_{2D}^2}{t_D}\right]\right\}$$

Perpendicular Mixed Boundaries

One sealing fault and one constant pressure boundary at 90°:

$$p_{wDb} = -\frac{1}{2}\left\{\text{Ei}\left[-\frac{L_{1D}^2}{t_D}\right] - \text{Ei}\left[-\frac{L_{2D}^2}{t_D}\right] - \text{Ei}\left[-\frac{L_{1D}^2 + L_{2D}^2}{t_D}\right]\right\}$$

where $L_1$ is the distance to the sealing fault and $L_2$ is the distance to the constant pressure boundary.

Two Parallel Sealing Faults (Channel)

Two parallel sealing faults at distances $L_1$ and $L_2$:

$$p_{wDb} = -\frac{1}{2}\sum_{j=1}^{\infty}\left\{\text{Ei}\left[-\frac{\left(\lfloor(j+1)/2\rfloor L_{1D} + \lfloor j/2\rfloor L_{2D}\right)^2}{t_D}\right] + \text{Ei}\left[-\frac{\left(\lfloor(j+1)/2\rfloor L_{2D} + \lfloor j/2\rfloor L_{1D}\right)^2}{t_D}\right]\right\}$$

Physical interpretation: Infinite series of image wells. Late-time behavior shows linear flow (half-slope on log-log derivative).

Two Parallel Constant Pressure Boundaries

Two parallel constant pressure boundaries:

$$p_{wDb} = -\frac{1}{2}\sum_{j=1}^{\infty}(-1)^j\left\{\text{Ei}\left[-\frac{\left(\lfloor(j+1)/2\rfloor L_{1D} + \lfloor j/2\rfloor L_{2D}\right)^2}{t_D}\right] + \text{Ei}\left[-\frac{\left(\lfloor(j+1)/2\rfloor L_{2D} + \lfloor j/2\rfloor L_{1D}\right)^2}{t_D}\right]\right\}$$

Two Parallel Mixed Boundaries

One sealing fault and one constant pressure boundary in parallel:

$$p_{wDb} = -\frac{1}{2}\sum_{j=1}^{\infty}\left\{(-1)^{j/2}\text{Ei}\left[-\frac{\left(\lfloor(j+1)/2\rfloor L_{1D} + \lfloor j/2\rfloor L_{2D}\right)^2}{t_D}\right] + (-1)^{(j+1)/2}\text{Ei}\left[-\frac{\left(\lfloor(j+1)/2\rfloor L_{2D} + \lfloor j/2\rfloor L_{1D}\right)^2}{t_D}\right]\right\}$$

Intersecting Boundaries (60° Angle)

Two boundaries intersecting at 60°:

$$p_{wDb} = -\frac{1}{2}\sum_{j=1}^{5}\left\{\text{Ei}\left[-\frac{a_j^2}{2t_D}\right]\right\}$$

where: $$a_j = \frac{1-\cos(j\theta)}{\sin^2(\theta/2)} L_D^2, \quad j = 1, 2, 3, 4, 5, \quad \theta = 60°$$

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Applicability & Limitations

Boundary Effect Onset

The boundary effect becomes detectable when:

$$t_D \approx 0.1 L_D^2$$

In dimensional terms: $$t_{boundary} \approx \frac{380 \phi \mu c_t L^2}{k}$$ (hours)

Diagnostic Signatures

IARF = Infinite Acting Radial Flow (derivative = 0.5)

Distance to Boundary

The distance to a boundary can be estimated using the intersection method:

$$L = \sqrt{\frac{k \cdot t_x}{948 \phi \mu c_t}}$$

where $t_x$ is the intersection time of the derivative extrapolations from infinite acting and boundary-affected periods.

Physical Constraints

1. $L_D > 10$: Boundary must be far enough for radial flow to develop first
2. $t_D > 100$: Sufficient time for line source approximation to be valid
3. Boundary distances: $L_1, L_2 > 0$

Limitations

1. Homogeneous Reservoir: No permeability variation between well and boundary
2. Idealized Boundaries: Perfect sealing or perfect constant pressure
3. Vertical Well: Fully penetrating, radial flow geometry
4. Single-Phase Flow: No multiphase effects near boundaries
5. Linear Boundaries: Straight-line boundaries only (no curved faults)
6. Fixed Geometry: Boundaries must remain at fixed angles (90° or 60°)

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Related Documentation

Prerequisite Concepts

• Dimensionless Variables - Definitions of $p_D$, $t_D$, $L_D$
• Infinite Reservoir Solutions - Baseline infinite acting behavior

Advanced Topics

• Channel reservoirs (parallel fault systems)
• Corner effects (perpendicular boundaries)
• Mixed boundary systems

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References

1. Tiab, D. and Crichlow, H.B. (1979). "Pressure Analysis of Multiple-Sealing-Fault Systems and Bounded Reservoirs by the Pressure-Derivative Method." SPE Journal, 19(6): 378-392.

2. Tiab, D. and Kumar, A. (1980). "Detection and Location of Two Parallel Sealing Faults Around a Well." Journal of Petroleum Technology, 32(10): 1701-1708.

3. Proano, E.A. and Lilley, I.J. (1986). "Derivative of Pressure: Application to Bounded Reservoir Interpretation." SPE Formation Evaluation, 1(5): 481-486.

4. Ehlig-Economides, C. (1988). "Use of the Pressure Derivative for Diagnosing Pressure-Transient Behavior." Journal of Petroleum Technology, 40(10): 1280-1282.

5. Gringarten, A.C. (1986). "Computer-Aided Well Test Analysis." SPE Formation Evaluation, 1(4): 373-392.

6. Buhidma, I.M. and Chu, W.C. (1992). "The Use of Computer in Pressure Transient Analysis." SPE Formation Evaluation, 7(4): 723-734.

7. 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.

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