A reservoir simulator provides a means for analyzing horizontal-well tests that are a challenge because of their anisotropy and multiflow regimes.Mabruk MethnaniPRC Inc.

Doha, Qatar

To demonstrate the capabilities of a reservoir simulator for this type of analysis, a simplified well test was simulated using Oasis, PRC Inc.'s in-house desktop reservoir simulator.

Important flow events and their timings are highlighted and cross-checked with analytical predictions. Effects of layering and vertical barriers are also simulated and some test design guidelines are noted.

### Base case

A clean-cut well test was simulated using Oasis. The main aim was to reproduce some of the main flow regimes characterizing reservoir response under a horizontal well test and check on event timing. In the base case, a homogeneous model is assumed. Table 1 [8,775 bytes] lists the relevant data.The well was assumed flowing for 7 days at 2,000 st-tk b/d before being shut in.

To keep matters simple, the model assumes a storage-free well bore and an infinite-acting reservoir. Fig. 1a [150,989 bytes] is a "ln-ln" plot of the pressure buildup and its derivative.

### Flow regimes

Within the first hour of buildup, a brief vertical radial flow stabilization can be observed on the pressure derivative curve. In this case, this regime is very brief because of the relative closeness of the top and bottom boundaries.As soon as these boundaries are reached, a linear flow regime is established for a few hours before leveling off towards a pseudoradial flow stabilization.

Expressed in field units, the build-up slope for pressure vs. logarithm of shut-in time is estimated for the vertical and pseudoradial flow regimes, respectively, as shown in Equations 1 and 2 (see equation box).

The respective slope values obtained by the simulation for the two regimes are 3.7 and 10, compared to 3.13 and 9.9 obtained with Equations 1 and 2.

The 20% discrepancy in the vertical slope is due partly to incomplete stabilization of the early radial flow in view of the short interval between the top and bottom boundaries, and partly to errors in the early numerical pressure derivative values.

These errors are characteristic of the early build-up period and could be reduced by decreasing the time steps. The late radial buildup slope, however, is closely matched by the simulation.

### Event timing

As an interpretation guide, one can check the timing of the different flow events. For the start of early linear flow, use Equation 3. And for the start of late radial flow, use Equation 4.For the vertical radial flow, the value obtained by Equation 3 is 0.5 hr, which closely matches that predicted by the simulation. For the late radial flow, however, Equation 4 gives a value of 40 hr, a factor twice as large as that predicted by the simulation. It is worth noting that the formulas used for event timing are somewhat subjective, as the transition from flow regime to the next is expected to be gradual rather than abrupt.

### Layering effects

To check the effect of layering on the build-up behavior, the simulation was rerun with six layers, as listed in Table 2 [5,154 bytes]. The arithmetically averaged porosity and permeability values are equivalent to the base case.Fig. 1b shows the resulting build-up and derivative curves. The trend for early and late radial flow stabilization is similar but wavier as a result of the layering.

The late radial slope is also 25% lower, reflecting the bias towards higher permeability layers in the lower half of the reservoir.

### Vertical barrier effects

To check the effect of vertical barriers on the build-up behavior, the simulation was rerun, assuming a less communicating barrier between Layer 3 and 4.Fig. 1c shows that the build-up and derivative curves after the vertical permeability for this boundary was set to 0.1 md. One can also note that the late radial plateau has decreased in slope by some 50%, again reflecting the bias towards higher permeabilities below the barrier.

If the well was completed above the barrier, an increase in slope would have been expected. A shortening of the vertical linear flow period is also noticed for this case, compared to the previous two cases, as a result of the tighter vertical communication.

### Test design guidelines

Using the reservoir data previously presented, a reservoir pressure at 2,400 psi at 5,600 ft, and taking a drawdown equivalent to 15% of that pressure (360 psi), one would obtain a pseudosteady-state flow of 1,750 st-tk b/d, for a 50-md permeability reservoir.This translates into a 20 psi/cycle build-up slope on the ln scale.

In case of surface shut-in, well bore storage is expected to influence the early build-up response. Storage-influenced time can be estimated with Equation 5.^{2 3}

For falling liquid in the well bore, the well coefficient is calculated with Equation 6.

Assuming an oil density of 58 lbm/cu ft, a well bore-storage coefficient of 0.178 bbl/psi is obtained.

Entering this value into Equation 5, and assuming again a 50 md, 85-ft thick reservoir, the time of well bore storage influence is calculated to last about 4 hr.

Another important design factor is to estimate the required shut-in time. A recommended criteria is that the shut-in pressure must reach a value equal to 99% of the initial reservoir pressure.

Based on an infinite-acting transient build-up solution, Equation 7 can be used where Parameter x is defined by Equation 8.

Applying Equation 8, one obtains an estimate of 32 hr for a 99% pressure recovery after shut-in. However, in case faulting and barriers exist in the neighborhood, the time period needed may be longer. This can be taken into consideration by reducing the permeability value used in Equation 8.

Finally, it is useful to assess the radius of investigation for the build-up period. This parameter will put a limit on the range of boundaries detected by the test and is estimated by Equation 9.

With a 72 hr shut-in time as an example, a 1,600-ft radius of investigation would be reached.

### References

- Methnani, M., "OASIS, a Desktop Reservoir Simulation Model," SIAM Conference on Mathematical and Computational Issues in the Geosciences, San Antanio, February 1995.
- Du, K.F., and Stewart, G., "Analysis of Transient-Pressure Response of Horizontal Wells in Bounded Reservoirs," SPEFE, March 1994.
- Horne, R.N., Modern Well Test Analysis.

### The Author

Mabruk Methnaniis a principal consultant at PRC Inc. (Petro Reservoir Characterization). His areas of interest include reservoir engineering and well test analysis. He has also developed computational tools for reservoir simulation and gas network analysis.Methnani holds a PhD in nuclear engineering from the University of Michigan.

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