GRAVITY FACTOR CORRECTS HORIZONTAL WELL PI CALCULATION

Ming-Ming Chang, James F. Pautz
National Institute for Petroleum & Energy Research
Bartlesville, Okla.

When a reservoir fluid's viscous force is not significantly greater than the gravity force, a gravity correction factor must be included in the productivity index (PI or J) calculations for horizontal wells located vertically away from the middle of the pay zone.

For both steady state and pseudosteady-state production stages, in simulations, the PI of horizontal wells increases as the horizontal lateral's distance from the top of the pay zone becomes greater. For example, in some instances, the PI near the bottom of a pay zone can exceed twice the PI near the top.

Several methods can quickly estimate a horizontal well's productivity under different production stages including steady state and pseudosteady state.

Among the analytical equations, Equations 1-7 1-6 (see equation box) all underestimated the eccentricity effect on the PI of horizontal wells because the gravitational force was not included in their derivation.

Equation 1 provided values closest to numerical simulation results among the three PI equations, Equations 1-3, that are suitable for steady-state stages.1 PIs from Equation 2 closely matched results from numerical calculations when horizontal well length was less than three quarters of the drainage diameter.2

Based on the cases studied, PI values calculated from Equations 4, 5, and 7 agreed reasonably well with results at the start of pseudosteady state from numerical simulations of horizontal wells with well bores in the middle of pay zones.3-6

Especially for horizontal wells with long laterals, simulation shows that PI values continue to decrease after the pseudosteady state is reached.

The Boast-VHS numerical simulator acted as the standard in evaluation and comparison of the seven analytical equations for calculating the single-phase PI of horizontal wells.

To study the eccentricity effect and calculate PI values of horizontal wells, a horizontal well was assumed to produce from a square reservoir with 2,000 ft sides and 100 ft thickness.

ECCENTRICITY EFFECT

Eccentricity effects on horizontal well PIs are not completely accounted for in Equations 1-7.

Numerical simulations investigated this effect on a 1,200 ft horizontal well in one of seven layers. PIs derived from simulations are shown in Fig. 1a for steady state and in Fig. 1b for pseudosteady-state production stages.

As shown in Fig. 1, PI values increase with well bore distances from the top of the pay zone for both production stages. This increase is due to the gravity force that drives reservoir fluids above the well bore into the production tubing. Therefore, under the same reservoir pressure drop, well bores near the bottom of pay produce more fluid than well bores near the top of pay.

In Fig. 1, the PI value of a well bore in the bottom layer (Layer 7) is more than twice the PI in the top layer (Layer 1) in both steady state or pseudosteady-state production.

The gravity pressure drop between central and bottom layers in this case is 15.3 psi. This is comparable to the viscous pressure drops of 21.6 and 14.8 psi in the horizontal direction for steady state and pseudosteady state, respectively.

To account for the gravity effect, a correction factor, Cg, should be used to multiply J from existing PI equations, as follows:

[SEE FORMULA]

Fig. 2a shows the PIs calculated at steady state for horizontal well bores located at seven different depths within the pay. Equation 3 is the only PI equation that contains a term for the eccentricity effect at steady state.7

For pseudosteady state, Fig. 2b illustrates the PI values at horizontal well bores located at different depths.

In Fig. 3a, Cg is plotted against the ratio of gravity pressure drop (DELTA Pg) to viscous pressure drop (DELTA P) for horizontal well production. Gravity pressure drop in Fig. 3a is defined as the subtraction of the pressure of a center-located well bore from that of an off-center well bore.

Values of Cg vary from less than 0.6 to more than 3. Based on this case study, in Fig. 3b for well bores close to the top or bottom of pay, Cg is replotted vs. flow rates.

When the flow rate is high, the gravity effect on PI values is insignificant. The Cg factor becomes important when the flow rate is low or the viscous pressure drop is low.

Therefore, it is essential to incorporate the gravity correction factor into PI calculations when the reservoir viscous force is not significantly greater than the gravity force.

Such cases include:

Mature field production
Gravity drainage production
Limited production rates due to tubing sizes or production facilities
Light oil production from relatively thick, permeable reservoirs.

STEADY STATE

In simulations of steady-state production, a horizontal producer was assigned in the middle, and two horizontal injectors were assigned at two opposite edges of the reservoir model. The production rate was set at 2, 000 b/d and the injection rate at 1,000 b/d for each of two injection wells. The horizontal well reached steady state within 0.02 pore volumes of production and maintained a constant PI value.

Fig. 4a compares steady state PI values calculated from numerical simulations and analytical equations at various well bore lengths. Reservoir vertical permeability is 100 md or Kv/Kh equals 1.

Equations 1 and 3 agreed with simulation results reasonably well for the entire range of the well bore length studied.

Equation 2's PI values matched simulation values closely until the well bore length increased to 1,400 ft. For horizontal wells longer than 1,400 ft in this example, Equation 2 overestimated PI values compared to simulations.

A decrease in vertical permeability reduces the productivity of horizontal wells. Equation 1 does not consider vertical permeability. Muskat's approach was applied to Equation 1 to account for the 2 permeability anisotropy.2

Fig. 4b compares horizontal well PI values from simulations and analytical equations at reduced vertical permeabilities of 10 and 1 md, respectively. Simulation-derived PI values compared favorably to PIs calculated by both Equations 1 and 2 for both Kv/Kh ratios of 0.1 and 0.01. In contrast, Equation 3 PI values were conservative for all well bore lengths studied.

PSEUDOSTEADY STATE

For calculating horizontal well PI values at pseudosteady state, simulations were made of production depletion from horizontal producers located in the middle of pay. The start of pseudosteady state can be determined from the pressure derivative 5 or the sharp change of PI values as production continues.

At the pseudosteady state, PI values decrease significantly from PIs at the transient state. Because of close boundaries to the top and bottom of the pay, horizontal wells have an extremely short transient production period.

Simulation results show that PI values continue to decrease after the pseudosteady state is reached, especially for horizontal wells with long laterals.

In horizontal wells located in the center of a drainage area, PI values calculated by Equations 4 and 5 match well with PI simulations at the start of pseudosteady state.

For a vertical permeability of 100 md (or Kv/Kh equals 1), simulation results at the start of pseudosteady state provide the same or slightly higher PI values than PIs calculated by Equations 4 and 5 (Fig. 4c).

PIs from Equation 7 agreed with simulation results much better than Equation 6, its original form.

A comparison of horizontal wells located away from the center of a drainage area is depicted in Fig. 4d. PIs are calculated for wells with centers located at 1,000 ft and 500 ft, and 500 ft and 500 ft.

From simulations of wells located at 1,000 ft and 500 ft, PI values at the start of pseudosteady state agree reasonably well with PIs from Equations 4 and 5.

For wells located at 500 ft and 500 ft, simulations predict PI values slightly lower than both Equations 4 and 5. In general, numerical simulations showed a greater effect of well bore locations on PI values than analytical equations.

SIMULATOR VALIDATION

Boast-VHS, the horizontal well productivity simulator developed by the National Institute for Petroleum & Energy Research (Niper) for the U.S. Department of Energy, was used in this study.8 9 For validating Boast-VHS, simulation results were compared to those from the seventh SPE comparative problems.10

Boast-VHS is a black oil reservoir simulator for modeling production from vertical, horizontal, or slant wells.

The comparative simulation project (CSP) modeled the production from a horizontal well in a reservoir with coning tendencies due to the bottom aquifer drive. Black-oil fluid properties, relative permeabilities, the reservoir grid system, and the horizontal well geometry are the same as those specified in the seventh SPE CSP.

An equivalent well bore radius of 3.784 ft11 or productivity index of 59,510 md-ft was used in the well model for the horizontal producer.

Case problems 1, 2, and 3 in the seventh SPE CSP were simulated. Simulation results of Boast-VHS agreed well with those from 14 participant models in comparative problems.

The calculated cumulative oil production agreed with the mean values in the seventh SPE CSP within one standard deviation (Table 1). The automatic time-step control based on saturation change allowed in each time step was used in simulating the three case problems (Table 1).

The good agreement of Boast-VHS simulation results to those in the seventh SPE CSP suggests that Boast-VHS is an adequate reservoir simulator for horizontal well production.

HORIZONTAL WELL MODELS

The horizontal well models assume that a well lies in the middle and parallel to one boundary of a square reservoir. The horizontal permeability is 100 md. PI values were calculated for wells both off-center and in the center of the drainage area in the horizontal direction for pseudosteady-state conditions.

The uniform grid size was used in three-dimensional reservoir simulations for calculating the PI. The reservoir model was dimensioned at 17 x 10 x 7 for horizontal wells located in the middle of the drainage area. A grid system of 18 x 10 x 7 was used for horizontal wells located horizontally off the center of the drainage area.

Single-phase simulations were developed by assigning identical PVT properties for oil and water phases in two-phase simulations. Fluid properties were assumed as follows:

Fluid viscosity: 1 cp
Fluid density: 51.5 lb/cu ft (40.3 API)
Formation volume factor: 1.0.

All simulations used two straight-line relative permeability curves where the relative permeability values to two phases total one at all saturations.

For calculating the effective well-block radius for a horizontal well model, PI simulations used the equation derived in References 11 and 12. A uniform grid system is required for obtaining the effective well-block radius for coupling well block and well bore pressure in numerical simulations.

Some PI values were obtained by using either rate constraint or pressure constraint in production simulations. The constant rate constraint was chosen for this study because of better resolution or less oscillation than with pressure constraint.

PI EQUATIONS

The productivity index, defined as the ratio of the production rate (st-tk b/d) to the pressure drawdown (psi) at the well bore, measures the well's ability to produce. Horizontal wells are capable of providing high productivity due to the long well bores exposed to the pay zone.

Equation 1 was developed in 1964 for horizontal wells producing at steady-state conditions. This PI equation considers the single-phase flow from a centrally located horizontal well in an isotropic, homogeneous, and bounded reservoir.

Because of increasing interest in horizontal well production, Equations 2 and 3 were developed in 1984 and 1986 for horizontal well production at steady state.

Equations 2 and 3 are based on the same assumptions as Equation 1 except that anisotropic permeability in the vertical direction is accounted for.

Equation 3 also takes into account the eccentricity effect for off-centered horizontal wells in the vertical direction.

Production at pseudosteady state describes part of the inflow performance of reservoir depletion production. In 1988, Equations 4, 5, and 6 were derived for horizontal wells producing at pseudosteady state. All three PI equations assume that a horizontal well produces single-phase fluid from an anisotropic, homogeneous, rectangular, and bounded reservoir.

The horizontal well can be located any place within, but has to be parallel to one boundary of the rectangular drainage area. The eccentricity effect is accounted for off-centered horizontal wells in all three equations.

Equation 6 was modified by adding a constant of 1.386 into the denominator to give Equation 7.

All of the seven equations neglect the gravity force. Equations 3-7 take into account off-centered well bores in the vertical direction. In contrast to Equations 5 and 6, Equation 4 provides a simplified form that does not involve infinite series and can be readily done by hand calculations.

ACKNOWLEDGMENT

The authors acknowledge the financial support of this work by the Department of Energy through the Bartlesville project office and thank Bill Linville for help in preparing this article.

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