Well Flow Overview

Introduction

Well flow analysis predicts production rates based on reservoir and well properties. This analysis connects:

• Reservoir deliverability — what the formation can supply
• Well inflow — how fluid enters the wellbore
• Outflow constraints — tubing, choke, and surface equipment limits

The intersection of inflow and outflow performance determines the operating point for a well.

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Fundamental Concepts

Inflow Performance Relationship (IPR)

The IPR describes the relationship between bottom-hole flowing pressure ($P_{wf}$) and flow rate ($q$):

                    ▲ Rate (q)
                    │
            q_max   ●━━━━━━━━━━━━━━━━━━━━━┓
                    │                      ┃
                    │                      ┃
                    │      IPR Curve       ┃
                    │                      ┃
                    │                  ┃
                    │              ┃
                    │          ┃
                    │      ┃
                    │  ┃
                    ●━━━━━━━━━━━━━━━━━━━━━━━━━▶ Pressure (Pwf)
                    0                        P_res

Productivity Index (J)

For single-phase liquid above the bubble point:

$$q = J \cdot (P_{res} - P_{wf})$$

The IPR is a straight line with slope $J$.

For two-phase flow below the bubble point, the IPR becomes curved (see Vogel IPR below).

Reference Pressure Conventions

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Flow Regime Selection

Time Domain

Decision Framework

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Available Functions by Category

Productivity Index Functions

📖 Full Documentation: Productivity Index

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Horizontal Well Functions

📖 Full Documentation: Productivity Index

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Flow Rate Functions

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Gas Well Functions

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Utility Functions

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Vogel IPR for Two-Phase Flow

The Challenge

When flowing pressure drops below bubble point:

• Gas evolves from solution
• Two-phase flow occurs near wellbore
• Relative permeability reduces oil mobility
• IPR becomes non-linear

Vogel's Correlation (1968)

For pressures below the bubble point:

$$\frac{q_o}{q_{o,max}} = 1 - 0.2\left(\frac{P_{wf}}{P_{res}}\right) - 0.8\left(\frac{P_{wf}}{P_{res}}\right)^2$$

Where $q_{o,max}$ is the maximum oil rate at $P_{wf} = 0$ (AOF):

$$q_{o,max} = \frac{J \cdot P_{res}}{1.8}$$

Generalized Vogel (Standing Extension)

For reservoirs with $P_{res} > P_b$ (undersaturated at average pressure):

Above bubble point ($P_{wf} > P_b$): $$q_o = J \cdot (P_{res} - P_{wf})$$

Below bubble point ($P_{wf} \le P_b$): $$q_o = J(P_{res} - P_b) + \frac{J \cdot P_b}{1.8}\left[1 - 0.2\frac{P_{wf}}{P_b} - 0.8\left(\frac{P_{wf}}{P_b}\right)^2\right]$$

The FlowRateXXVogel functions implement this generalized approach.

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Gas Well Deliverability

Darcy Flow (Low Rate)

For laminar flow in gas wells:

$$q_g = \frac{k \cdot h \cdot (P^2_{avg} - P^2_{wf})}{1422 \cdot T \cdot \bar{\mu}_g \cdot \bar{Z} \cdot \left[\ln\frac{r_e}{r_w} - 0.75 + S\right]}$$

Excel Function: GasFlowRatePSS

Non-Darcy Flow (High Rate)

At high velocities near the wellbore, turbulence creates additional pressure drop:

$$q_g = \frac{-a + \sqrt{a^2 + 4b(P^2_{avg} - P^2_{wf})}}{2b}$$

Where:

• $a$ = Darcy coefficient (laminar term)
• $b$ = Non-Darcy coefficient (turbulent term)

Excel Function: GasFlowRatePSSNonDarcy

Non-Darcy Coefficient (D-Factor)

The D-factor can be estimated from:

$$D = \frac{2.222 \times 10^{-15} \cdot \gamma_g \cdot k}{r_w \cdot h \cdot h_{perf} \cdot \mu_g}$$

Excel Function: NonDarcyCoefficient

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Workflow: Complete Well Performance Analysis

Step 1: Gather Input Data

Step 2: Determine Flow Regime

Time on production: t = ?
Time to PSS: t_pss = TimeToPSS(Re, K, Ul, φ, Ct)

If t < t_pss → Use transient equations
If t ≥ t_pss → Use PSS equations (or SS if pressure support)

Step 3: Calculate Productivity Index

Vertical well:

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

Horizontal well:

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

Step 4: Generate IPR Curve

Above bubble point:

For each Pwf from P_res to 0:
  q = J × (P_res - Pwf)

Below bubble point (Vogel):

For each Pwf from P_res to 0:
  q = FlowRatePSSVogel(J, P_avg, Pwf, Pb)

Step 5: Determine Operating Point

Intersect IPR with:

• VLP (Tubing Performance) — outflow from wellbore
• Surface constraints — choke, facilities limits

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Method Selection Summary

Vertical vs. Horizontal Wells

Horizontal Well Method Selection

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Related Documentation

Productivity Index Details

• Productivity Index — Complete J calculations for vertical and horizontal wells

Supporting Analysis

• PTA Overview — Well test interpretation
• PTA Dimensionless Variables — Dimensionless formulations
• PVT Overview — Fluid property correlations

Data Processing

• Utilities Interpolation — Spline methods for IPR curves

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References

1. Vogel, J.V. (1968). "Inflow Performance Relationships for Solution-Gas Drive Wells." Journal of Petroleum Technology, January 1968, pp. 83-92. SPE-1476-PA.

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

3. Joshi, S.D. (1991). Horizontal Well Technology. PennWell Books.

4. Brown, K.E. (1984). The Technology of Artificial Lift Methods, Vol. 4. PennWell Books.

5. Golan, M. and Whitson, C.H. (1991). Well Performance, 2nd Edition. Prentice Hall.