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:

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

Productivity Index Correlations

Five major correlations are implemented, each with specific advantages:

Correlation	Year	Best For	Key Feature
Joshi	1988	General purpose	Anisotropic, any well position
Borisov	1984	Isotropic reservoirs	Simplest, kv = kh only
Giger-Reiss-Jourdan (GRJ)	1985	Anisotropic, long wells	Elliptical drainage
Renard-Dupuy (RD)	1991	Box-shaped drainage	Rectangular reservoirs
Babu-Odeh (BO)	1989	Any well position	Most general, complex

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

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!).

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)

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

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

Comparison of Methods

Prediction Accuracy (Relative to Numerical Simulation)

Method	Centered Well	Near Boundary	Anisotropic
Joshi	Good (±15%)	Good	Excellent
Borisov	Fair (±25%)	Poor	Cannot handle
GRJ	Good (±10%)	Fair	Good
RD	Good (±15%)	Good	Good
Babu-Odeh	Excellent (±5%)	Excellent	Excellent

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):

Well Length	Joshi J	Borisov J	Error if Ignore Anisotropy
1000 ft	25 STB/d/psi	45 STB/d/psi	+80% overprediction!
2000 ft	45 STB/d/psi	75 STB/d/psi	+67% overprediction!

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

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.

Functions Covered

The following functions implement horizontal well productivity correlations. See each function page for detailed parameter definitions, Excel syntax, and usage examples.

Productivity Index Functions

Function	Method	Description
ProdIndexHorWellJoshi	Joshi	Anisotropic, general purpose
ProdIndexHorWellBorisov	Borisov	Isotropic only (kv = kh)
ProdIndexHorWellGRJ	GRJ	Elliptical drainage, anisotropic
ProdIndexHorWellRD	Renard-Dupuy	Rectangular drainage
ProdIndexHorWellBO	Babu-Odeh	Full positioning, most accurate
ProdIndexHorWellBO2	Babu-Odeh	Simplified, centered well

Drainage Area Functions

Function	Method	Description
DrainageAreaHorWell1	Joshi	Elliptical drainage area
DrainageAreaHorWell2	Joshi	Rectangular + rounded ends

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

References

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.
Borisov, J.P. (1984). Oil Production Using Horizontal and Multiple Deviation Wells. Moscow: Nedra Publishing (translated by J.Strauss, R&D Library Translation).
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.
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.
Babu, D.K. and Odeh, A.S. (1989). "Productivity of a Horizontal Well." SPE Reservoir Engineering, 4(4), pp. 417-421. SPE-18298-PA.
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.
Joshi, S.D. (1991). Horizontal Well Technology. Tulsa, OK: PennWell Publishing Company.

Horizontal Well Productivity

Overview

Productivity Index Correlations

Joshi Method (1988)

Productivity Index

Drainage Area

Borisov Method (1984)

Productivity Index (Isotropic: kv = kh)

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

Productivity Index

Renard-Dupuy (RD) Method (1991)

Productivity Index

Babu-Odeh (BO) Method (1989)

Full General Form (BO)

Simplified Form (BO2 — Centered Well)

Comparison of Methods

Prediction Accuracy (Relative to Numerical Simulation)

Effect of Anisotropy

Drainage Area Estimation

DrainageAreaHorWell1 (Joshi Ellipse)

DrainageAreaHorWell2 (Joshi Rectangle)

Functions Covered

Productivity Index Functions

Drainage Area Functions

Related Documentation

References

Related Excel Functions