Horizontal Well Productivity

Overview

Horizontal wells offer significant advantages over vertical wells:

• Larger contact area with reservoir
• Lower drawdown for same production rate
• Delayed water/gas breakthrough
• Better drainage of thin, layered reservoirs
• Access to naturally fractured reservoirs

Horizontal well productivity is governed by:

1. Horizontal permeability (kh) — Flow perpendicular to wellbore
2. Vertical permeability (kv) — Flow from above/below wellbore
3. Anisotropy ratio (kv/kh) — Critical parameter
4. Well length (L) — Longer = higher productivity
5. Reservoir thickness (h) — Thin reservoirs benefit most
6. Wellbore position (zw) — Vertical placement in pay zone

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Productivity Index Correlations

Five major correlations are implemented, each with specific advantages:

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Joshi Method (1988)

Productivity Index

$$J_h = \frac{k_h h}{141.2 B \mu \left[\ln\left(\frac{a + \sqrt{a^2 - (L/2)^2}}{L/2}\right) + \frac{h}{L} \ln\left(\frac{h}{2\pi r_w \sqrt{k_h/k_v}}\right)\left(\frac{\sqrt{k_h/k_v}}{1 + \sqrt{k_h/k_v}}\right) + s\right]}$$

Where:

$$a = \frac{L}{2} \sqrt{1 + \left(\frac{2r_{eh}}{L}\right)^4} \approx \frac{L}{2}\left[0.5 + \sqrt{0.25 + \left(\frac{r_{eh}}{L}\right)^4}\right]$$

Parameters:

• $L$ = horizontal well length, ft
• $h$ = reservoir thickness, ft
• $r_{eh}$ = drainage radius in horizontal plane, ft
• $r_w$ = wellbore radius, ft
• $k_h$ = horizontal permeability, md
• $k_v$ = vertical permeability, md
• $s$ = skin factor

Physical interpretation:

• First term: horizontal drainage (major contribution)
• Second term: vertical drainage correction
• Accounts for anisotropy (kv ≠ kh)

Drainage Area

Joshi provides two methods:

Method 1 (Ellipse):

$$A = \pi r_{eh} r_{ev}$$

Where:

$$r_{ev} = r_{eh} \sqrt{k_v / k_h}$$

Method 2 (Rectangle with rounded ends):

$$A = L \times W + \frac{\pi W^2}{4}$$

Where W = drainage width.

Typical values:

• L = 2000 ft, reh = 1000 ft → A ≈ 10-15 acres

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Borisov Method (1984)

Productivity Index (Isotropic: kv = kh)

$$J_h = \frac{k h}{141.2 B \mu \left[\ln\left(\frac{4r_e}{L}\right) + s\right]}$$

Where:

• Assumes isotropic reservoir (kv = kh = k)
• Simplest correlation
• re = equivalent circular drainage radius

When to use: Sandstone with uniform permeability in all directions.

Limitation: Cannot handle anisotropy (most reservoirs are anisotropic!).

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Giger-Reiss-Jourdan (GRJ) Method (1985)

Productivity Index

$$J_h = \frac{2\pi k_h h}{141.2 B \mu \left[\ln\left(\frac{L}{r_w}\right) + \frac{2\pi k_h}{\beta L}\left(\frac{L}{2a}\right) + s\right]}$$

Where:

$$\beta = 2\pi \sqrt{k_h k_v}$$$$a = \text{major axis of elliptical drainage}$$

Features:

• Elliptical drainage assumption
• Accounts for anisotropy via $\beta$
• Good for long horizontal wells (L > 1000 ft)

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Renard-Dupuy (RD) Method (1991)

Productivity Index

$$J_h = \frac{k_h h}{141.2 B \mu \left[\ln\left(\frac{(x_e - x_w)}{r_w'}\right) + s\right]}$$

Where:

$$r_w' = r_w \sqrt{\frac{k_h}{k_v}}$$

Features:

• Rectangular drainage (box-shaped)
• xe = horizontal extent of drainage
• xw = well position from edge
• Good for wells in elongated rectangular reservoirs

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Babu-Odeh (BO) Method (1989)

Full General Form (BO)

Most complex but most accurate for any well position in reservoir:

$$J_h = \frac{2\pi k_h h}{141.2 B \mu \left[\ln\left(\frac{I_y}{I_x}\right) + \frac{h}{L}\ln\left(\frac{h\beta}{2\pi}\right) + F + s\right]}$$

Where $I_x$, $I_y$ are shape factors depending on:

• Well location (xw, yw, zw) relative to drainage boundaries
• Anisotropy ratio (kv/kh)
• Drainage shape (xe, ye, h)

Function: ProdIndexHorWellBO — Full positioning

Simplified Form (BO2 — Centered Well)

For well centered in reservoir (xw = xe/2, yw = ye/2):

$$J_h = \frac{k_h h}{141.2 B \mu \left[C_1 + C_2 \frac{h}{L} + C_3 \ln(L) + s\right]}$$

Where C₁, C₂, C₃ are correlation constants.

Function: ProdIndexHorWellBO2 — Simplified for centered wells

When to use:

• BO (full) — Well near edges or boundaries
• BO2 — Well centered, simpler calculation

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Comparison of Methods

Prediction Accuracy (Relative to Numerical Simulation)

Recommendation:

• General use: Joshi (widely validated, simple)
• Best accuracy: Babu-Odeh (if positioning data available)
• Quick estimate: Borisov (isotropic only)

Effect of Anisotropy

For typical carbonate (kv/kh = 0.1):

Conclusion: Must account for anisotropy in carbonates, shales, and layered reservoirs.

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Drainage Area Estimation

DrainageAreaHorWell1 (Joshi Ellipse)

$$A = \pi a b$$

Where:

• $a$ = major axis (calculated from Joshi's equations)
• $b = a \sqrt{k_v/k_h}$ = minor axis

DrainageAreaHorWell2 (Joshi Rectangle)

$$A = L \times W + \frac{\pi W^2}{4}$$

Where W is estimated drainage width.

Use case: Well spacing design, interference analysis.

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Related Documentation

• WellFlow Overview — Well performance concepts
• Productivity Index — Vertical well PI (detailed)
• Vogel IPR — Two-phase IPR (horizontal adaptation exists)
• Vertical Flow Correlations — Tubing hydraulics

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References

1. Joshi, S.D. (1988). "Augmentation of Well Productivity with Slant and Horizontal Wells." Journal of Petroleum Technology, 40(6), pp. 729-739. SPE-15375-PA.

2. Borisov, J.P. (1984). Oil Production Using Horizontal and Multiple Deviation Wells. Moscow: Nedra Publishing (translated by J.Strauss, R&D Library Translation).

3. Giger, F.M., Reiss, L.H., and Jourdan, A.P. (1984). "The Reservoir Engineering Aspects of Horizontal Drilling." SPE-13024-MS, presented at SPE Annual Technical Conference, Houston, TX, September 16-19.

4. Renard, G. and Dupuy, J.M. (1991). "Formation Damage Effects on Horizontal-Well Flow Efficiency." Journal of Petroleum Technology, 43(7), pp. 786-869. SPE-19414-PA.

5. Babu, D.K. and Odeh, A.S. (1989). "Productivity of a Horizontal Well." SPE Reservoir Engineering, 4(4), pp. 417-421. SPE-18298-PA.

6. Economides, M.J., Brand, C.W., and Frick, T.P. (1996). "Well Configurations in Anisotropic Reservoirs." SPE Formation Evaluation, 11(4), pp. 257-262. SPE-27980-PA.

7. Joshi, S.D. (1991). Horizontal Well Technology. Tulsa, OK: PennWell Publishing Company.