Well Productivity Index

Introduction

The productivity index ($J$) quantifies a well's ability to produce fluid per unit pressure drawdown:

$$J = \frac{q}{\Delta P}$$

Where:

• $q$ = production rate, STB/d
• $\Delta P$ = pressure drawdown, psi
• $J$ = productivity index, STB/(d·psi)

The productivity index depends on:

• Reservoir properties: permeability, thickness, fluid properties
• Well geometry: vertical, horizontal, completion type
• Flow regime: steady-state, pseudosteady-state, or transient

---

Flow Regime Fundamentals

Steady-State Flow

In steady-state flow, pressure at all points in the reservoir remains constant with time. This occurs when the outer boundary maintains constant pressure (e.g., strong aquifer support or gas cap expansion).

Reference pressure: Pressure at the constant-pressure boundary ($P_e$)

Pseudosteady-State Flow

In pseudosteady-state (PSS) flow, pressure declines at the same rate throughout the reservoir after the pressure transient reaches all boundaries. This is the dominant regime in bounded reservoirs during depletion.

Reference pressure: Average reservoir pressure ($\bar{P}$)

Transient Flow

In transient flow, the pressure disturbance has not yet reached the reservoir boundaries. The productivity index varies with time as the drainage area expands.

Reference pressure: Initial reservoir pressure ($P_i$)

---

Vertical Well Productivity Index

Steady-State

For a vertical well in a circular drainage area with constant-pressure outer boundary:

$$J_{SS} = \frac{k \cdot h}{141.2 \cdot B \cdot \mu \cdot \left( \ln\frac{r_e}{r_w} + S \right)}$$

Excel Function: ProdIndexSS

=ProdIndexSS(K, h, Bl, Ul, Re, Rw, S)

Flow Rate from Productivity Index:

$$q = J_{SS} \cdot (P_e - P_{wf})$$

---

Pseudosteady-State

For a vertical well in a bounded (no-flow boundary) circular drainage area:

$$J_{PSS} = \frac{k \cdot h}{141.2 \cdot B \cdot \mu \cdot \left( \ln\frac{r_e}{r_w} - 0.75 + S \right)}$$

The -0.75 term accounts for the difference between boundary pressure and average reservoir pressure in a circular drainage area.

Excel Function: ProdIndexPSS

=ProdIndexPSS(K, h, Bl, Ul, Re, Rw, S)

Flow Rate from Productivity Index:

$$q = J_{PSS} \cdot (\bar{P} - P_{wf})$$

Time to Pseudosteady State

The transition from transient to pseudosteady-state flow occurs when:

$$t_{PSS} = \frac{\phi \cdot \mu \cdot c_t \cdot \pi \cdot r_e^2 \cdot 0.1}{0.000264 \cdot k}$$

Excel Function: TimeToPSS

=TimeToPSS(Re, K, Ul, porosity, Ct)

Returns time in hours.

---

Transient Flow

During transient flow, the productivity index varies with time:

$$J_{TF}(t) = \frac{k \cdot h}{162.6 \cdot B \cdot \mu \cdot \left( \log t + \log\frac{k}{\phi \mu c_t r_w^2} - 3.23 + 0.87S \right)}$$

Excel Function: ProdIndexTF

=ProdIndexTF(time, K, h, Bl, Ul, porosity, Ct, Rw, S)

Flow Rate from Productivity Index:

$$q = J_{TF}(t) \cdot (P_i - P_{wf})$$

---

Horizontal Well Productivity Index

Horizontal wells offer higher productivity than vertical wells by:

• Increasing contact area with the reservoir
• Reducing drawdown per unit production
• Accessing thin reservoirs more effectively

Multiple correlations exist, differing in their treatment of:

• Geometry: Drainage shape (circular, elliptical, rectangular)
• Anisotropy: Vertical vs. horizontal permeability ratio
• Well position: Centered vs. off-center placement

Anisotropy Ratio

For anisotropic reservoirs:

$$\beta = \sqrt{\frac{k_h}{k_v}}$$

Where:

• $k_h$ = horizontal permeability (or $k_{xy}$)
• $k_v$ = vertical permeability (or $k_z$)

Typical values: $\beta$ = 1-10 for most reservoirs

---

Borisov Method (1984)

For isotropic reservoirs with steady-state flow:

$$J = \frac{0.00708 \cdot k \cdot h}{\mu \cdot B \cdot \left[ \ln\frac{4r_{eh}}{L} + \frac{h}{L} \ln\frac{h}{2\pi r_w} \right]}$$

Use when: Reservoir is approximately isotropic ($k_v \approx k_h$).

Excel Function: ProdIndexHorWellBorisov

=ProdIndexHorWellBorisov(L, Rw, Re, h, K, Bl, Ul)

---

Giger-Reiss-Jourdan Method (1984)

For anisotropic reservoirs:

$$J = \frac{0.00708 \cdot k_h \cdot L}{\mu \cdot B \cdot \left[ \frac{L}{h} \ln X + \beta^2 \ln\frac{h}{2r_w} \right]}$$

Where:

$$X = \frac{1 + \sqrt{1 + \left(\frac{L}{2r_{eh}}\right)^2}}{\frac{L}{2r_{eh}}}$$

Excel Function: ProdIndexHorWellGRJ

=ProdIndexHorWellGRJ(L, Rw, Re, h, Kz, Kxy, Bl, Ul)

---

Joshi Method (1988)

The most widely used correlation for horizontal well productivity:

$$J = \frac{0.00708 \cdot k_h \cdot h}{\mu \cdot B \cdot \left[ \ln R + \frac{\beta^2 h}{L} \ln\frac{h}{2r_w} \right]}$$

Where:

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

$$R = \frac{a + \sqrt{a^2 - (L/2)^2}}{L/2}$$

The parameter $a$ represents the semi-major axis of the elliptical drainage area.

Excel Function: ProdIndexHorWellJoshi

=ProdIndexHorWellJoshi(L, Rw, Re, h, Kz, Kxy, Bl, Ul)

---

Renard-Dupuy Method (1991)

An alternative formulation using the inverse hyperbolic cosine:

$$J = \frac{0.00708 \cdot k_h \cdot h}{\mu \cdot B \cdot \left[ \cosh^{-1}\frac{2a}{L} + \frac{\beta h}{L} \ln\frac{h}{2\pi r'_w} \right]}$$

Where:

$$r'_w = \frac{(1 + \beta) r_w}{2\beta}$$

The effective wellbore radius $r'_w$ accounts for anisotropy effects near the wellbore.

Excel Function: ProdIndexHorWellRD

=ProdIndexHorWellRD(L, Rw, Re, h, Kz, Kxy, Bl, Ul)

---

Babu-Odeh Method (1989)

For rectangular (box-shaped) drainage areas with pseudosteady-state flow:

$$J = \frac{\sqrt{k_y k_z} \cdot b}{141.2 \cdot B \cdot \mu \cdot \left[ \ln\frac{\sqrt{ah}}{r_w} + \ln C_H - 0.75 + S_R + S \right]}$$

Where:

• $a$ = drainage area width (Y-direction, perpendicular to well)
• $b$ = drainage area length (X-direction, parallel to well)
• $h$ = reservoir height (Z-direction)
• $C_H$ = shape factor depending on well position
• $S_R$ = partial penetration skin

This method is the most rigorous for box-shaped reservoirs and accounts for:

• Non-central well placement
• Partial penetration (well shorter than reservoir)
• Full 3D permeability anisotropy ($k_x$, $k_y$, $k_z$)

Excel Function (General): ProdIndexHorWellBO

=ProdIndexHorWellBO(sizeX, sizeY, sizeZ, Kx, Ky, Kz, L, Rw, Bl, Ul, S, x1, y0, z0)

Excel Function (Centered Well): ProdIndexHorWellBO2

=ProdIndexHorWellBO2(sizeX, sizeY, sizeZ, Kx, Ky, Kz, L, Rw, Bl, Ul, S)

Assumes well is centered in all three dimensions:

• $x_1 = (b - L)/2$
• $y_0 = a/2$
• $z_0 = h/2$

---

Method Selection Guide

Vertical Wells

Horizontal Wells

Correlation Comparison

For a typical horizontal well with $L$ = 2000 ft, $h$ = 50 ft, $r_w$ = 0.3 ft, $k_h$ = 100 mD, $k_v$ = 10 mD ($\beta$ = 3.16):

---

Common Applications

Skin Factor Interpretation

From the productivity index equation:

$$S = \frac{k \cdot h}{141.2 \cdot B \cdot \mu \cdot J} - \ln\frac{r_e}{r_w} + 0.75$$

Skin factor components:

• Mechanical damage ($S_d$): Drilling/completion damage
• Partial penetration ($S_p$): Incomplete reservoir contact
• Perforation ($S_{perf}$): Perforation geometry effects
• Deviation ($S_\theta$): Wellbore inclination

Effective Wellbore Radius

The skin factor can be expressed as an effective wellbore radius:

$$r'_w = r_w \cdot e^{-S}$$

Excel Function: EffectiveWellboreRadius

Positive skin reduces effective radius (damage); negative skin increases it (stimulation).

Drainage Area Calculations

Drainage Radius from Area:

$$r_e = \sqrt{\frac{A \cdot 43560}{\pi}}$$

Excel Function: DrainageRadius

Horizontal Well Drainage Area (Method 1 - Stadium Shape):

$$A = \frac{\pi b^2 + 2bL}{43560}$$

Excel Function: DrainageAreaHorWell1

Horizontal Well Drainage Area (Method 2 - Ellipse):

$$A = \frac{\pi \cdot b \cdot (L/2 + b)}{43560}$$

Excel Function: DrainageAreaHorWell2

---

Input Validation

---

Related Documentation

• WellFlow Overview — Flow rate calculations and model selection
• PTA Dimensionless Variables — Dimensionless formulations
• PTA Infinite Reservoir — Transient pressure solutions
• Utilities Interpolation — Spline methods for IPR curves

---

References

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

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

3. Economides, M.J., Hill, A.D., Ehlig-Economides, C., and Zhu, D. (2013). Petroleum Production Systems, 2nd Edition. Prentice Hall.

4. Giger, F.M., Reiss, L.H., and Jourdan, A.P. (1984). "The Reservoir Engineering Aspects of Horizontal Drilling." SPE-13024-MS.

5. Renard, G. and Dupuy, J.M. (1991). "Formation Damage Effects on Horizontal-Well Flow Efficiency." Journal of Petroleum Technology, July 1991, pp. 786-789. SPE-19414-PA.

6. Borisov, J.P. (1984). Oil Production Using Horizontal and Multiple Deviation Wells. Nedra, Moscow (translated by J.S. Strauss).