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AbstractEstimating effective permeability at the reservoir scale has been a long-standing challenge in carbonate fields. The carbonate depositional and diagenetic history can be very complex, which can lead to permeability that is difficult to characterize. Vuggy or fractured intervals can have perme-ability significantly higher than the matrix permeability measured in core plugs. This “excess permeability” may control reservoir flow paths but is difficult or impossible to predict from static well-log transforms that are calibrated to matrix permeability.

In the Tengiz field, a giant carbonate reservoir in west-ern Kazakhstan, a method has been developed to calculate apparent permeability on the basis of flow rate from produc-tion-logging tools (PLTs). Incorporating this flow-calibrated permeability into the Earth model offers an elegant solution to the long-standing problem of the best method to incor-porate dynamic PLT data into a reservoir model. Recently, a reservoir model built with apparent permeability resulted in more-realistic heterogeneity and a step-change improve-ment over previous methods in which only static log-based permeability transforms were used to populate the Earth model. Fewer changes were required to calibrate the full-field reservoir model. History matching to pulse tests showed that this model has a much-improved prediction of the interwell connectivity. Models containing apparent-permeability data have provided higher-confidence estimates of the future movement of gas injection from the Tengiz platform.

The apparent permeability is calculated by solving Darcy’s law on an interval basis, using our knowledge of flowing and static pressures as input along with well, reservoir, and fluid properties, and then calibrating with the pressure-buildup transient permeability-thickness product, kh. Accuracy of the method is enhanced by the derivation of coarse-scale zonal layer pressures with multirate PLTs. Selecting an appropriate skin factor to use in the calculation is important. The acid

effect on pulsed-neutron-capture (PNC) logs works well in Tengiz and has been helpful in addressing this problem.

IntroductionThe fundamental building blocks of a reservoir model are porosity, permeability, and fluid saturation. Porosity and fluid saturation can be computed from wireline logs with reasonable confidence, but often, the same confidence is not possible for permeability. Permeability can be measured directly from core material or determined with well tests by use of pressure-transient tests (PTTs), pulse tests between wells, and wireline formation tests. Scale differences in these sources of permeability data complicate the permeability esti-mation in a reservoir model (Al-Henshiri et al. 2005; Ahmed et al. 1991). In most reservoirs, the amount of core available for direct permeability measurement is limited, and perme-ability estimates usually are made by proxy by use of wire-line-log responses. The science of permeability prediction has evolved into a high degree of sophistication, with each additional level resulting in an incremental gain in predictive accuracy (Skalinski and Sullivan 2001). Despite the available sophisticated tools, permeability prediction in carbonates remains problematic. The widely varying pore structures (Lucia 1995) and their potential lack of connectivity result in a highly variable porosity/permeability relationship. Vuggy porosity and fractures, if they are connected, can result in extremely high layer flow rates or very low rates, if not con-nected, for the same value of porosity.

Several authors have described improving the accuracy of the reservoir characterization by incorporating dynamic data, such as from PTTs, periodic pressure measurements, and production profiles, into the reservoir-characterization process (Bahar et al. 2005; Sibley et al. 1997; Mezghani et al. 2000). However, the industry has lacked a robust work pro-cess to flow calibrate log-derived permeability and distribute it between wells in a multilayer reservoir model.

This paper summarizes a new method to overcome this obstacle by deriving apparent permeability from PLT flow profiles. The method provides a direct integration of static core and wireline-log data with dynamic pressure-transient and PLT data. It calibrates the wireline-scale permeability to groundtruth pressure-transient results and retains the char-acter of the log data. This method improves the accuracy

DISTINGUISHED AUTHOR SERIES

Copyright 2007 Society of Petroleum EngineersThis is paper SPE 103486. Distinguished Author Series articles are general, descriptive representations that summarize the state of the art in an area of technology by describing recent developments for readers who are not specialists in the topics discussed. Written by individuals recognized as experts in the area, these articles provide key references to more definitive work and present specific details only to illustrate the technology. Purpose: to inform the general readership of recent advances in various areas of petroleum engineering.

Michael J. Sullivan, SPE, is a petroleum engineer coor-dinating reservoir surveillance for the Tengiz field in Kazakhstan. Previously, he worked as a petrophysicist at ChevronTexaco E&P Technology Company in Houston. Sullivan had previous assignments as an openhole petro-physicist and as a cased-hole petrophysicist for Chevron in Angola. He has held a variety of positions in petroleum and production engineering since 1979. Sullivan holds a BS degree in petroleum engineering from Montana College of Mineral Science and Technology.

Permeability From Production Logs—Method and Application Michael J. Sullivan, SPE, Chevron

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of the permeability data used in the model and inherently integrates dynamic data into the static model. The method overcomes difficulties created by differences in scale of the various data types mentioned. The reservoir-characterization process is improved because calibration to dynamic flow data is built in at the beginning.

Use of the method in Tengiz was enhanced by two other surveillance techniques that may be of additional interest. Multirate PLTs are analyzed with a selective-inflow-perfor-mance (SIP) technique to determine the layer pressures of individual reservoir flow units. Additionally, the acid effect on PNC logs is used to assess stimulation effectiveness and refine assumptions of wellbore skin. Readers desiring a detailed description of the method are invited to read paper SPE 102894 (Sullivan et al. 2006).

The method will work equally well in sandstone reservoirs but is viewed as having greater benefit to carbonate reser-voirs because the need for improvement is greater. Because the technique presumes knowledge of fluid viscosity, it will work best in wells with single-phase flow, or where oil and water have similar viscosities. The accuracy of the method may be limited in applications with multiphase flow of fluids having greatly different viscosities. The authors have not studied the potential effect of non-Darcy-flow effects as found in many gas wells, nor have they studied the effect of highly deviated or horizontal wellbores.

Tengiz ReservoirTengiz is a large carbonate reservoir in western Kazakhstan. Tengiz is more than 110 sq km in area at its top and 400 sq km at its base. The oil column is approximately 1600 m thick. Tengiz has much development potential remaining; conse-

quently, a large focus is made on reservoir characterization. Surveillance activities in Tengiz include production-logging surveys, interwell pulse testing, and pressure-buildup tests.

Most of the wells in the Tengiz reservoir are vertical or near-vertical, and all wells flow single-phase oil with a flow-ing bottomhole pressure above the bubblepoint. From this perspective, the conditions for PLT profiling are ideal.

Apparent-Permeability MethodA typical production log in Tengiz includes profiles acquired under both stabilized shut-in and then stabilized flowing conditions. After the flowing survey is completed, the well is shut in for a pressure buildup test until the infinite-acting radial-flow period is reached (Chambers et al. 2001). The profiles then are analyzed with detailed inflow points defin-ing the regions over which permeability will be calculated. The derivative with respect to depth of cumulative-flow profile is calculated, yielding the volumetric inflow on a meter-by-meter basis. This derivative is equivalent to the traditional interval contribution commonly seen on PLT analyses divided by the length of the zonal contribution.

Given the flow contribution per meter from the derivative of cumulative flow, for each meter (h=1), the permeability is calculated by use of the Darcy radial-flow equation:

C·qi·µo·Bo rdkplt = [ln( )+S′] ........................................(1) (pe−pwf) rw

where the constant of proportionality, C, is 141.2 if depth is expressed in feet, or 43.07 if depth is expressed in meters (Tengiz convention); qi is flow rate in STB/D; µo is viscosity in cp; p is pressure in psi; and drainage radius, rd, and wellbore radius, rw, are in consistent units.

PLT permeability (kplt) represents average permeability over that specific inflow zone. The kplt data are accumu-lated over the entire interval to compute khplt, which then is normalized to kh from pressure buildup. This normalized PLT permeability is used as a baseline reference to which the wireline-transform permeability from openhole logs is adjusted by use of a multiplicative factor. This process is repeated, inflow by inflow, from the bottom of the zone upward in a stepwise fashion. This adjusted transform-per-meability curve is called apparent permeability.

Handling Differential Depletion. The Tengiz reservoir comprises three main flow units that behave as different pressure compartments (Fig. 1). The uppermost zone, Zone 1, generally is the most prolific; therefore, it generally is more depleted than the lower two layers (Zones 2 and 3).

The differential depletion in Fig. 1 results in crossflow into Zone 1 when the well is shut in at surface. These differ-ent individual-layer pressures must be accounted for in the calculation of permeability by use of the apparent-perme-ability method (pe in Eq. 1). Individual-layer pressures are measured routinely by use of the SIP technique (Stewart et al. 1981). This technique involves acquiring production profiles at multiple rates and flowing bottomhole pressures.

81

Fig. 1—Pressure-gradient graph showing differential depletion.

Zone 3

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Pressure is plotted against rate for each reservoir-flow unit (Fig. 2). For single-phase liquid flow, when the bottomhole pressures are above the bubblepoint pressure, the points will plot on a straight line. That pressure for which the layer flow rate would be zero defines the layer pressure. By use of this technique, no zone ever has to be static (i.e., a rate of zero). Tengiz wells typically are shut in for several days before logging such that a stabilized shut-in profile is available in most wells. The leftmost point (negative flow rate) on the Zone 1 curve in Fig. 2 is the stabilized rate of crossflow into that zone.

Simplified Approach. If no differential depletion exists in the reservoir and the assumption of uniform skin distribution can be made, a simplified approach may be taken. The accumulat-ed permeability may simply be normalized to well-test kh and distributed vertically according to interval contribution. This simplified approach has proved successful in other areas.*

Significance of Errors Examining the certainty of the input variables in Eq. 1 can assess the accuracy of the apparent-permeability calculation. For our purposes, a value within a factor of two of the “true value” of permeability is adequate for reservoir-characterization purposes and is a significant improvement over a permeability field that is based only on log-to-core transforms. The most important feature of the apparent-permeability method is that it achieves the correct vertical distribution of permeability, help-ing to characterize the flow units in the reservoir correctly.

Uncertainty regarding many of the input variables results in an effect on permeability that is insignificant for this tech-nique. In the calculation of apparent permeability, several simplifying assumptions are made, but the resulting errors are second order in nature and do not have a significant effect on the result.

To illustrate, Fig. 3 is a crossplot of permeability vs. poros-ity from six cored wells. There are approximately 1.5 log cycles of scatter in the raw data (50×). A difference of a factor of two is essentially “within the noise.”

Uncertainties. Potential uncertainties in the variables can affect the apparent-permeability calculations.

Fluid Properties. The product of µo and Bo changes little over the typical range of pressures in Tengiz and usually is known to within 5%. The effect on calculated permeability is insignificant.

Pressures. An error in drawdown will cause a proportional error in calculated permeability. In most cases, it was possible to assess stabilized drawdown within 10% uncertainty.

A potential source of inaccuracy in the SIP technique is lack of stability. Normally, assessing the layer pressure by use of the SIP technique is possible to within approximately 2% of the true value. The result is an equivalent uncertainty in calculated permeability. As shown in Fig. 4, the layer pres-sure can be well constrained if the well is shut in for a suf-ficiently long time before the PLT test. The leftmost points in Fig. 2 represent the operating rates and pressures while the well is shut in at surface with stable downhole crossflow.

DISTINGUISHED AUTHOR SERIES

Fig. 4—Uncertainty in layer pressures usually is low by use of the SIP technique, even when the drawdown has not stabilized, as long as the shut-in passes are at stable pressures.

Pressure

Uncertainty inLayer Pressure

Uncertainty inDrawdown

− Rate +

Fig. 2—SIP graph of zonal rates and pressures.

Zone 3

Zone 1

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8200

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Flow Rate, B/D

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ssu

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* Harrison, C., Personal correspondence: Technique for production-log analysis by use of simplified assumptions.

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The stabilization time required for the flowing tests some-times is an issue. The well is opened to flow with PLTs above all perforations that then monitor rates and pressures for stability with a real-time log-log pressure-derivative graph before commencing flowing tests. When analyzing historical data for which this procedure was not done, pressure sta-bility may be assessed by comparing pass-to-pass pressure repeatability. Even if the flowing bottomhole pressure is not stable, the effect is minor because the resulting kplt is normal-ized later to pressure-buildup kh.

Drainage Radius and Wellbore Radius. A 20% differ-ence in either radius makes only a 1.5% difference in cal-culated permeability.

Flow Rate. When possible, an attempt is made to flow the well through a test separator during a PLT test, but this is not always possible because of operational constraints. The PLT tests that are conducted with stable flow measured with accurate surface meters typically match the surface rates with a difference of less than 10%. Because rate is proportional to calculated permeability, this again has a negligible effect on apparent permeability.

Skin Factor. The assumptions made about skin factor typi-cally receive the greatest amount of attention because it can have a several-fold effect on permeability within the typical ranges of skin in Tengiz (Fig 5).

Understanding the stimulation and workover history is critical. Approximately half of our completions are not com-plicated by borehole problems and have been acid jetted with several reciprocations of coiled tubing. Typically, observed skins are between −4 and −5 on these wells, and it is rea-sonable to assume that this skin is distributed evenly when acid penetration is confirmed by the PNC log and there was no crossflow during the stimulation. Other completions are complicated by borehole crossflow during stimulation, heavy mud losses (possibly with lost-circulation material), post-stimulation workovers with well kills, or scale deposition and may have skins from slightly negative to highly positive. Because there is no “textbook” way to handle these compli-cations, these wells are reviewed on a case-by-case basis by the work team. Uncertainty is communicated to the reservoir engineer by use of a confidence factor. If a significant upward adjustment to the transform permeability is required to

match the PLT-test permeability, there is high confidence the permeability is real and not the product of overstimulation. The lower limit of −5 skin does not permit a 10-fold increase above the wireline-transform permeability (Fig. 5); therefore, the apparently high permeability is real.

Low Permeability vs. Damage. If the productivity of a zone is much less than predicted, then it must be determined whether the permeability is low or whether this layer is damaged or unstimulated. To make this assessment, factors such as drilling history (whether fractures were encountered and the amount of mud losses into those fractures indicate high permeability), workover history, prior flow tests (espe-cially preworkover tests), and stimulation history (type and volumes) must be taken into account. Some wells may have poor openhole-log quality, limited logging suites, or logs made with tools that were poorly characterized by contem-porary standards. In these cases, the confidence in the result-ing predicted permeability is much lower, and downward permeability adjustments would be more acceptable.

Acid Effect With PNC Sigma. A key factor in assessing whether low productivity is caused by low permeability or high skin factor is acid-stimulation effectiveness. Most wells in Tengiz receive acid stimulations, either reciprocating coiled-tubing acid washes or bullhead diverted treatments. Several PNC logs have been run in the central area of the Tengiz field as baseline data in preparation for a future hydro-carbon-miscible flood. Although not the primary reason for running the PNC logs, it was found that the sigma (capture cross section—a primary measurement of the PNC tool) curve is useful in assessing placement of acid stimulations.

The PNC log records the thermal-neutron-capture cross section of the formation by measuring the thermal-neutron rate of absorbtion. Chlorine is one of the strongest neutron absorbers of the common Earth elements; therefore, the chlorine present in the formation water (usually as NaCl) determines the response of the PNC log. The sigma reading also has a pronounced effect caused by chlorine in the acid-reaction products in the form of calcium chloride (Al-Saif et al. 1975). A significant increase in sigma has been observed in most wells in Tengiz after HCl acid stimulations were conducted. The irreducible connate water is approximately 10% pore volume and has a low salinity (approximately 60,000 ppm total dissolved solids). The elevated chlorine concentration of the acid-reaction products imbibes into the irreducible water, increasing its chlorine concentration in the near-wellbore region. The spent-acid water flows back during cleanup, but the elevated chlorine levels in the irreducible water remain and may be detected with a PNC sigma log. It is believed that this effect is permanent, or nearly so, because elevated sigma readings caused by acid remain for several years after the stimulation. With only clean-oil production, there is no mechanism to dilute the chlorine concentration with time in the near-wellbore region.

The wettability of the reservoir rock sometimes is ques-tionable in carbonates. For the acid to have an effect, the rock must be water-wet, although it is possible that strong-water-

Fig. 5—Effect of skin factor, expressed as a proportion of the productivity the well would have with a skin of zero.

-5 0 5 10 15 20

Pro

du

ctiv

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Rat

io

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10X

84 JPT • JULY 2007

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wetting surfactants used in acid mixtures may leave the near-wellbore region water-wet even in an oil-wet reservoir.

A direct correlation has not been found between the amount of acid effect and permeability in Tengiz. Further study is required to understand fully the relationship between the amount of acid effect and acid penetration or stimulation ratio. Perhaps at some point the acid effect may be used to generate a quantitative skin-factor log, but currently, its use is limited sim-ply to assessing whether a zone has been stimulated or not.

Prestimulation sigma logs were not acquired on most Tengiz wells. This problem is overcome by comparing the measured sigma with a synthetic baseline sigma generated with a predictive transform that is accurate within 0.5 to 1 capture units in wells with good-quality openhole logs. The increase in sigma caused by acid effect typically is 3 to 5 capture units. The difference between the measured and synthetic baseline sigma is computed and used as an indica-tor of acid stimulation.

The post-stimulation sigma logs are a great help in validat-ing assumptions about the distribution of acid penetration. Sometimes, these sigma comparisons indicate that the reser-voir has been stimulated unevenly, requiring use of a differ-ent skin factor by zone in computing apparent permeability.

Fig. 6 shows the use of the acid effect in assessing acid distribution. Track 4, on the right, shows the results of a PLT test conducted in 2001. A pressure-transient test at the time yielded a skin factor of +1 (blue curve). After a scale problem was identified, a bullhead acid treatment with no diverter was conducted the following year to restore production. In 2006, the PLT test in Track 3 was conducted. A noticeable reduction in flow contribution was observed below 4287 m, and a vari-able skin of up to +20 (blue curve) had to be used to reconcile the 2006 results with the flow observed in the 2001 PLT test.

A PNC log then was run to assess the effectiveness of the pre-vious acid treatment. The acid effect is posted on porosity in Track 1 and shaded in magenta. It shows that little or no acid went below 4287 m, as might be expected from an undiverted bullhead acid treatment. This information confirmed the pre-vious interpretation of damage in this zone, which probably was caused by scale deposition in the wellbore.

This and other similar examples provided an increased confidence in this diagnostic technique. It helps in comput-ing apparent permeability and with the frequent decision as to whether the lower-than-anticipated flow rate is caused by a lack of stimulation or lower-than-predicted permeability.

Constraining Apparent Permeability With Multilayer-Transient Analysis Another technique used to determine individual-layer per-meability and skin is multilayer-transient-pressure testing (Ehlig-Economides and Joseph 1987). These procedures are operationally more complex, but they further constrain the apparent-permeability solution. Tengiz can have either two or three layers that form separate pressure compartments. Multilayer transient tests are integrated into a multirate PLT procedure (Jackson and Banerjee 2000). During a multilayer transient test, the PLT is positioned above successive stations in the wellbore. At each station, the well is opened to flow at successively higher rates. Pressure-transient data are collected during each rate change. After the transient data are collected, a flow profile is obtained. Sequential analysis of the pressure transients is used to determine individual-layer kh and skin. Layer pressures required for the multilayer analysis are deter-mined by use of the SIP analysis described previously.

Fig. 7 is an example of a well for which the results of a multilayer transient test match well with the apparent-

Fig. 6—Example of PNC acid effect showing poor stimu-lation effectiveness, confirming a previous interpreta-tion of high skin with analysis of time-lapse PLT tests.

Fig. 7—Example of a good match between apparent permeability, acid effect, and multilayer-pressure-transient interpretation.

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permeability interpretation. The porosity in this well was from a Soviet Union era uncalibrated neutron log, and the reading was too low, resulting in the permeability being underpredicted. Track 2 shows transform permeability in red, normalized PLT-test permeability in black, and apparent permeability in blue. Track 3 is the flowing-PLT-test results. Also posted in Track 3 is a log of the calculated skin factor from the multilayer transient test, which was a value of −3 in the upper zone and +1 in the lower zone.

These results are consistent with the acid effect observed in the upper zone, but not observed in the lower zone. The well was crossflowing while shut in during the coiled-tubing acid stimulation, and the acid was swept into the upper perfora-tions, leaving the lower perforations unstimulated.

Restoration of Natural Heterogeneity. Carbonate reservoirs are naturally heterogeneous, with a permeability variance often exceeding 1.5 orders of magnitude for a given porosity value (Fig. 8A). The nature of regression-based transforms reduces this heterogeneity to a single value for a given facies/porosity pair, leading to models that are inherently too homogeneous. The result is predicted flood responses that are too “piston-like,” and actual breakthrough times are ear-lier than predicted. This increased homogeneity is shown in Fig. 8B. The illustrated transform permeability was derived by use of a state-of-the-art facies that is based on a multivari-ate transform; however, the heterogeneity still is notably less than that of the original core data.

Fig. 8C shows the restoration of heterogeneity with appar-ent permeability. There is slightly less scatter because the

apparent-permeability measurement does not capture the “low-side” heterogeneity (e.g., between 0.1 and 0.01 md)—the minor fluid contributions of intervals with this low per-meability cannot be resolved with the PLT. The “high- side” heterogeneity containing the high-permeability intervals contributing most of the flow is captured well.

The introduction of apparent-permeability data also has helped to identify permeability trends in different stratigraph-ic intervals of the reservoir. Specific stratigraphic intervals are observed to have more statistical difference from the conven-tional matrix-permeability transform. A cross section through the central platform area of the Tengiz field (Fig. 9) shows the tendency of Zone 1 to have a positive permeability correction (blue) relative to the conventional matrix transform.

Validation of Apparent-Permeability-Model Improvement With Pulse Tests. The Tengiz injection wells have perma-nent downhole pressure gauges and have been shut in while preparing for gas injection (Fig. 10). Surrounding producers are pulsed, and the pressure response is measured at the injector. The results have shown both higher levels of con-nectivity than previously modeled and different levels of con-nectivity between various well pairs (colored arrows).

Fig. 8—A: Core Permeability. B: Transform permeabil-ity from the same cored well. C: Apparent permeability from the same cored well. D: Apparent permeability vs. porosity in Earth model.

Fig. 9—Difference between apparent-permeability and conventional-permeability transform for one area of the Tengiz field.

Fig. 10—Sour-gas-injection pilot area with results of pulse testing showing relative connectivity between wells. Injectors are in the center of the patterns (red arrows).

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The pressure response to the pulse-test flow sequences was simulated by use of the flow-calibrated apparent-perme-ability-based reservoir model and compared to the simulated results from the previous model, in which the permeability was based solely on transforms from wireline-log data.

A significant improvement in the predictive accuracy was observed with the new model after the introduction of appar-ent permeability. This history-match calibration confirms that the simulation model now has approximately the correct amount of connected pore space and permeability. Fig. 11 shows a comparison of actual pulse-test results (black lines) and simulated pressure response with both the apparent-permeability-based Earth model (red lines) and the previous transform-permeability-based model (blue). The bottom panel shows the flow sequence, the top panel is the pressure response, and the middle panel is the derivative of pressure. This analysis shows that after introducing apparent perme-ability, the model has a much better match to the pulse tests than did the conventional reservoir model.

Conclusions1. Permeability may be calculated with wellbore data by

use of interval contributions from PLT data as input to Darcy’s law.

2. The apparent-permeability technique is especially useful in characterizing permeability in carbonate reservoirs, for which variable rock fabrics make accurate permeability prediction from static wireline logs difficult, particularly for zones with “excess permeability” resulting from vuggy porosity and fractures.

3. The SIP technique is effective for measuring coarse-scale individual-layer pressures. This technique enables accounting for differential depletion correctly when solv-ing for permeability.

4. The effect of uncertainty in the input variables for Darcy’s law was examined. The errors in permeability caused by typical ranges of uncertainty in µo, Bo, rd, and rw are insignificant. The potential errors caused by uncertainty in

layer pressure and skin can be significant, so these uncer-tainties warrant careful consideration.

5. The PNC-tool acid effect can be used to identify acid penetration to justify the skin factors used when solving for permeability.

6. The improvement to geological-model construction by use of the apparent-permeability technique was validated by the improved match between predicted and observed pressure responses during interwell pulse tests.

Nomenclature Bo= oil formation volume factor C= constant of proportionality h= thickness k= permeability kplt= permeability from PLT analysis kh= permeability thickness khplt= permeability thickness from PLT analysis pe= layer pressure pwf= wellbore flowing pressure qi= interval flow rate from PLT rd= drainage radius rw= wellbore radius S′= skin factor φ= porosity µo= oil viscosity

AcknowledgmentsI wish to thank my many colleagues within TengizChevroil reservoir management who supported the development of this process and provided feedback and support. Contributing to the evolution and success of this method were Mark Skalinski, Steve Jenkins, Kaveh Dehghani, Steve Johnson, Peter Dunn, Edward Neubauer, and John Clarke, all with Chevron, as well as Peter Hillock of ExxonMobil. I am especially grateful for the insightful contributions and continued collaboration of David Belanger, and for his review of this manuscript.

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Fig. 11—Simulated vs. actual pulse-test responses with apparent-permeability-based model and previ-ous transform-permeability-based model.

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