Inflow from horizontal and fractured wells

The Tamarack 04-22 horizontal well is nothing like the vertical well of lesson 2. Its lateral runs about 2,080 m through the tight northern flank of the Cardium, where permeability is about 0.5 mD, and it was fractured in 26 stages. 'Vogel was built for vertical wells in a tank,' Marisol says. 'Your horizontal well drains a long, thin slab of rock through more than a hundred fractures, and its productivity is still changing. You need a different way of thinking about its inflow.'

This lesson explains how horizontal wells and hydraulic fractures change the geometry of flow, how their productivity is estimated with Joshi's equation and the effective wellbore radius of a fracture, and why a multistage fractured horizontal well in tight rock has a transient IPR that shrinks for months or years. It ends with how production engineers use the well's own data to build an IPR they can trust.

Why geometry matters

In a vertical well, flow converges radially. The pressure drop is concentrated close to the well, where the area open to flow is smallest, which is why skin has such a large effect and why productivity grows only with the logarithm of the drainage radius. A horizontal well changes this picture. Flow near the lateral is roughly linear rather than radial, the area open to flow is the length of the lateral times the formation thickness, and the pressure drop near the well is much smaller for the same rate. Hydraulic fractures go further, creating planes of very high conductivity that extend tens or hundreds of metres into the rock and turn almost all of the flow into linear flow toward the fracture faces.

These geometries change both the size of the IPR and how it evolves with time. In low-permeability rock, the pressure disturbance from a well spreads slowly. A vertical well in such rock may take years to reach pseudosteady state, and a fractured horizontal well may spend most of its life in transient linear flow, with its productivity falling as the square root of time.

Horizontal wells: Joshi's equation

Several engineers derived steady-state and pseudosteady-state productivity equations for horizontal wells in the 1980s and 1990s. The most widely used is Joshi's, which treats the drainage volume as an ellipse around the lateral and adds a term for the vertical convergence of flow into the wellbore.

Here L is the lateral length, a is the half major axis of the drainage ellipse, which depends on L and the drainage radius, and β is the square root of the ratio of horizontal to vertical permeability. The first logarithm describes flow toward the lateral from the far reservoir; it is small because the lateral is long. The second term describes the convergence of flow from the formation thickness into the small wellbore, and it grows when vertical permeability is low. Laminations and shale streaks, common in many tight formations, cut vertical permeability and hurt unfractured horizontal wells more than any other factor.

Joshi's equation and its relatives assume an undamaged lateral with full contact along its length, steady or pseudosteady flow and a homogeneous reservoir. Real laterals are often damaged by days of exposure to drilling mud, may be partly blocked by cuttings beds or water, and drain unevenly because of friction along the lateral. Their measured productivity is often well below the ideal value.

Hydraulically fractured vertical wells

A single vertical fracture acts like a much larger wellbore. Prats showed that a fracture of infinite conductivity with half-length xf behaves, in pseudosteady state, like a well with an effective wellbore radius of xf/2. Substituting that radius into the radial equation gives the productivity of a fractured vertical well. For a fracture of finite conductivity, the effective radius is smaller and depends on the dimensionless fracture conductivity CfD, as PET-330 showed. In skin terms, a good fracture gives a strongly negative skin, typically between about −3 and −6.

Comparing the geometries

# Horizontal and fractured-well inflow: Joshi for an unfractured lateral, Prats for a fractured vertical well,
# and a transient IPR for the multistage-fractured horizontal well calibrated to its own early test
import math
SI = True   
# rock and fluid data in field units for the equations (ft, md, cp)
if SI:   # Cardium tight flank, 04-22
    k, h, mu, B, rw, L, reh, kvkh = 0.5, 21.3, 0.8, 1.29, 0.25, 6824.0, 1640.0, 0.1
    J30, pr_t, pb, tel = 52.0 / 4350.0, [(1, 14600), (6, 13600), (12, 12900), (24, 11800)], 12800.0, 45
    pu, ru, conv = "kPa", "m3/d", 6.2898 * 6.89476   # 1 m3/d per kPa = 43.4 STB/d per psi
else:    # Bakken, 4-22H
    k, h, mu, B, rw, L, reh, kvkh = 0.01, 45.0, 0.4, 1.50, 0.25, 10000.0, 1320.0, 0.1
    J30, pr_t, pb, tel = 1050.0 / 1000.0, [(1, 5300), (6, 4700), (12, 4200), (24, 3600)], 2600.0, 540
    pu, ru, conv = "psi", "STB/d", 1.0
re_v = reh
Jv = k * h / (141.2 * B * mu * (math.log(re_v / rw) - 0.75))            # vertical well, no skin
a = (L / 2) * math.sqrt(0.5 + math.sqrt(0.25 + (2 * reh / L) ** 4))
beta = math.sqrt(1 / kvkh)
Jh = k * h / (141.2 * B * mu * (math.log((a + math.sqrt(a * a - (L / 2) ** 2)) / (L / 2)) + beta * h / L * math.log(beta * h / (2 * rw))))
xf = 200.0
Jf = k * h / (141.2 * B * mu * (math.log(re_v / (xf / 2)) - 0.75))      # infinite-conductivity fracture, rw' = xf/2
print(f"unfractured vertical well J = {Jv / conv:.4f}  (in {ru} per {pu})")
print(f"unfractured horizontal well J (Joshi, L = {L:,.0f} ft) = {Jh / conv:.4f}, {Jh / Jv:.1f} times the vertical well")
print(f"vertical well with one {xf:.0f} ft fracture (Prats rw' = xf/2) J = {Jf / conv:.4f}, {Jf / Jv:.1f} times")
print(f"multistage fractured horizontal well, measured at day 30: J = {J30:.4f}, {J30 * conv / Jv:.0f} times the vertical well")
print("transient IPR of the multistage well (J falls as 1/sqrt(t) until fractures interfere, then levels off):")
for m, pr in pr_t:
    t = 30.4 * m
    J = J30 * math.sqrt(30.4 / min(t, tel))
    qb = J * max(pr - pb, 0); aof = qb + J * min(pr, pb) / 1.8
    print(f"   month {m:2d}: pr {pr:6,} {pu}  J {J:.4f}  AOF {aof:7.1f} {ru}")

unfractured vertical well J = 0.0002  (in m3/d per kPa)
unfractured horizontal well J (Joshi, L = 6,824 ft) = 0.0062, 29.6 times the vertical well
vertical well with one 200 ft fracture (Prats rw' = xf/2) J = 0.0008, 3.9 times
multistage fractured horizontal well, measured at day 30: J = 0.0120, 57 times the vertical well
transient IPR of the multistage well (J falls as 1/sqrt(t) until fractures interfere, then levels off):
   month  1: pr 14,600 kPa  J 0.0120  AOF   106.5 m3/d
   month  6: pr 13,600 kPa  J 0.0098  AOF    77.7 m3/d
   month 12: pr 12,900 kPa  J 0.0098  AOF    70.9 m3/d
   month 24: pr 11,800 kPa  J 0.0098  AOF    64.4 m3/d

The comparison for the Cardium flank is instructive. An unfractured vertical well in 0.5 mD rock would have a J of only about 0.0002 m³/d per kPa, too low to be economic. A single 200 ft fracture multiplies it by about 4. An ideal unfractured 2,080 m lateral would multiply it by about 30, and the 04-22 as measured at day 30 has about 57 times the vertical well's productivity. In rock of this permeability, much of the gain comes from lateral length; the fractures bypass damage and low vertical permeability and speed up early production, which is why the fractured well beats the ideal lateral.

Transient inflow: an IPR that shrinks

In tight rock, the IPR of a fractured horizontal well is not a fixed curve. Early in its life the well draws on the rock right next to each fracture face. The pressure disturbance moves away from the faces with time, the region feeding each fracture grows more slowly than the time, and the productivity falls roughly in proportion to one over the square root of time. This period of transient linear flow ends when the disturbances from neighbouring fractures meet, after which the region between fractures depletes as a whole and the well enters a boundary-dominated period in which productivity is nearly constant but reservoir pressure falls steadily.

The time to the end of linear flow depends on the permeability and the spacing between fractures. In rock of a few tenths of a millidarcy with fractures 20 m apart, interference comes within weeks; in rock of a few microdarcies, it can take years. Many unconventional wells are still in linear flow when they are first put on artificial lift, so the lift design must allow for inflow that keeps falling.

For the 04-22, the program calibrates J to the day-30 test and assumes that fracture interference in the 0.5 mD flank ends after about 45 days. After that, J is nearly constant at about 0.0098 m³/d per kPa, but the reservoir pressure keeps falling, and the AOF shrinks from about 107 m³/d at month 1 to about 78 at month 6, 71 at month 12 and 64 at month 24. These AOFs are well above what the well actually makes, because the tubing cannot draw the bottomhole pressure down to zero. Lesson 6 combines these curves with the tubing to find the actual rates.

Building an IPR from production data

Because analytical equations for fractured horizontal wells depend on many uncertain inputs, such as effective fracture half-length, the number of fractures that actually contribute and the permeability between them, production engineers lean heavily on each well's own data. The key measurement is the flowing bottomhole pressure, either from a permanent downhole gauge or calculated from wellhead pressure and a tubing model. With rate and pressure recorded daily, the engineer can plot a rate-normalized pressure, the drawdown divided by the rate, against the square root of time. A straight line confirms linear flow, its slope gives the product of fracture area and the square root of permeability, and a bend upward marks the end of linear flow.

These analyses, known as rate transient analysis, are covered in detail in PET-360. For the production engineer, the practical result is a set of J values at different times and a model of how J will evolve, which is all that a lift design needs.

Ethan notes that the 04-22 has a pressure gauge at the heel, installed by Tom's completions team in PET-330. That single decision makes his work much easier: he can watch J fall in real time and see whether the 45-day interference estimate holds.

Flow along the lateral

One more effect matters in long laterals: the pressure drop along the lateral itself. Fluid entering near the toe must travel thousands of metres or feet to reach the heel. At high rates, friction along the lateral reduces the drawdown at the toe, so the toe contributes less than the heel. In toe-up laterals, liquid can collect in low spots and gas can collect in high spots, causing slugging; in toe-down laterals, water can settle near the toe. Production logs and tracers from PET-330 show that in the Tamarack Lake field, heel stages typically contribute more than toe stages. Lift equipment is usually set in the vertical section or the curve above the lateral, so the lateral must carry fluid to the pump intake by itself.