Permeability profiles provide new reservoir management tools

Dan Georgi, Ekrem Kasap, Mehmet Altunbay, Xiaoming Tang
Western Atlas Logging Services
Houston

New tools and interpretation techniques provide continuous well permeability profiles without coring or extensive well testing.

These profiles obtained from wire line logs are ideally suited for reservoir description and management.

With these permeability profiles, reservoir engineers can quantify heterogeneity and tailor completions to maximize recovery rates and efficiencies.

Permeability can be obtained from acoustic data collected with a multiple array acoustilog (MAC) tool, from nuclear magnetic resonance (NMR) T2 relaxation data collected with a magnetic resonance imaging log (MRIL) system, and from pressure-vs.-time data collected with wire line formation test tools, including the reservoir characterization instrument (RCI) service and formation multitester (FMT) tool.

The robust analysis of pressure-vs.-time data provides permeabilities that can be used to calibrate the permeabilities obtained from MRIL and MAC data.

Estimating reserves

Reservoir description, saturation, and some key reservoir fluid data, such as the oil formation volume factor (B _o), define the in-place hydrocarbons. Seismic survey data (3D), coupled with well-log porosities, are key for developing a reservoir-porosity model. However, developing a corresponding permeability model is generally much more difficult.

Often, one estimates average permeabilities from limited core data or from a few well tests. Permeability heterogeneity data previously were available only from core data. Core data, however, are often limited, and thus, since the first wire line logs were run, an effort has been made to derive permeability from logs. Recent advances have made continuous and accurate permeability profiles routinely available to reservoir engineers.

Rosenbaum's theoretical work¹ suggested that the Stoneley wave data carried permeability information. It took another decade before the first convincing examples of permeability from acoustic logs were published.²

Since the introduction of nuclear magnetic resonance (NMR) logging, a relationship of logs to permeability has been sought. Today, in many basins and formations, the magnetic resonance imaging log provides accurate and continuous permeability profiles.

Although wire line formation test tools were introduced to measure formation pressure and collect formation fluid samples, analysts have sought to estimate permeability from the drawdown and buildup data. These early efforts were of limited success because of tool limitations and interpretation shortcomings. Today, new devices, such as the reservoir characterization instrument, and new interpretation methods are making formation permeability measurements routine and reliable.

By combining estimates from the different sources, the reliability and robustness of computed wire line permeability profiles are greatly increased.

Permeability profiles can be integrated over the producing interval to obtain kH, a measure of a well's flow capacity. The higher the kH the higher the production rate.

Permeability profiles, when viewed as a continuous integral and integrated from the bottom of the producing interval, predict production profiles. The continuous permeability profiles directly measure permeability heterogeneity, which is needed for predicting ultimate recovery, well spacing, and optimized perforation placement, and for designing primary and secondary recovery schemes.

Average production rate

Generally, as the well bore pressure is lowered, flow from the reservoir into the well increases. The production rate, in st-tk b/d/psi of drawdown, is a direct measure of a well's performance. This familiar measure is known as the productivity index or PI, given by Equation 1 (see equation box).

P_e and P_wf are the initial reservoir and well bore flowing pressures, respectively.

For single-phase, steady-state flow and radial geometry, Equation 2 expresses the PI.

From this equation, it is apparent that PI, the well productivity, is directly proportional to the average permeability, k, and the reservoir thickness, H.

PI can be established from production tests or estimated from permeability data. Generally, there is insufficient core coverage to obtain meaningful kH estimates. Wire line-based permeability profiles, however, are ideally suited for computing an appropriate kH.

Recovery efficiency

Recovery efficiency and ultimate recovery depend on the phase of and production mechanism. Generally, we speak of three production phases: primary, secondary, and tertiary recovery.

Primary production relies on the energy stored in the reservoir. Recovery during the primary production phase is limited, and generally it is beneficial to provide some pressure support. Thus, water or gas injection is common for maintaining reservoir pressure and increasing hydrocarbon recovery during the secondary recovery.

Since the mid-1980s, there has been less focus on tertiary recovery processes, enhanced oil recovery, and more emphasis on improved oil recovery. Improved recovery can be achieved by early initiation of secondary recovery methods or even by combining primary and secondary recovery. Improved recovery may require larger initial investments, but it yields better recovery at a lower total investment.

To minimize the risk associated with larger initial investments, it is essential to quantify both the production rate and the recovery efficiency. Detailed reservoir description, in particular detailed reservoir heterogeneity and permeability data, are required to estimate recovery efficiency.

During primary recovery, the hydrocarbons are easily pulled from the higher-pressure reservoir toward the low-pressure producing well, essentially passing over and around permeability obstructions. The secondary process essentially attempts to push the oil by injecting water. When the oil encounters a flow barrier, the fluids are diverted. Thus, the primary recovery efficiency does not depend on reservoir permeability or permeability heterogeneity, while the secondary recovery process is controlled by permeability heterogeneity.

The recovery efficiency during primary oil production depends on whether the reservoir pressure is less than or greater than the bubblepoint. If the reservoir pressure is greater than the bubblepoint pressure, the pressure at which gas will evolve from the liquid hydrocarbons, then the recovery efficiency can be computed from the oil compressibility (Equation 3). Here, DP is the difference between the bubblepoint pressure and P*, the initial reservoir pressure.

If we assume that DP is 3,000 psi and C is 13.6 3 ^10-6/psi, a typical recovery efficiency for many oil reservoirs is only 4.1%.

Once the reservoir pressure drops below the bubblepoint pressure (saturated reservoir), then additional pressure support is available as the secondary gas phase evolves.

Typically, recovery efficiency for depletion gas drives ranges from a low of a few percent to a maximum of 20%. Although recovery efficiencies are low for primary production, it must be stressed that they do not depend on the average reservoir permeability and are only somewhat dependent on permeability heterogeneity.

Horizontal permeability

Ultimate recovery from a multilayered reservoir depends on many factors. Some factors are fixed reservoir attributes, some are fluid properties, and some result from often irreversible actions by the operator. Generally, holding all other attributes constant, a more heterogeneous reservoir will have less ultimate recovery.

The dependence of secondary processes on reservoir heterogeneity, especially vertical variations of horizontal permeability, can be illustrated with a simple two-layer reservoir. If two layers are waterflooded, the recovery efficiency depends primarily on the ratio of the layer permeabilities. Ignoring relative permeability and mobility contrast effects and assuming piston-like displacement and Darcy flow, the ratio of the layer permeabilities will then control the recovery efficiency at the time water breaks through in the high-permeability layer.

Darcy's law (Equation 4) states that fluid velocity (u) in a simple rectilinear system is proportional to the absolute permeability. If one injects water in one well with two layers and produces from the surrounding wells from these two layers, then both layers will be subject to the same pressure gradient. Hence, the flow rates and velocities in the high-permeability layer will be greater and the water/oil front will advance faster from the injection well to the producing well in the high-permeability layer.

When injected water in the high-permeability layer arrives at the producing well, only a fraction of the oil will have been swept from the low-permeability layer. When the water breaks through in the high-permeability layer, the front in the low-permeability layer will have traversed a distance X, where X = k₂/k₁ * L, leaving much of low-permeability layer unswept.

Recovered oil is equal to the sum of the oil recovered from the two layers (Equations 5 and 6). The DS_o is the decrease in the oil saturation attributable to the waterflood. Recovery efficiency (E_R) for the waterflood is the ratio of the recovered oil to oil originally in place (Equation 7).

For layers of equal thickness and porosity, Equation 7 reduces to Equation 8. For a 10:1 permeability and a 70% oil saturation change, Equation 8 yields a 39% recovery efficiency. The greater the permeability contrast, the less is the recovery from the low-permeability layer.

This concept can be generalized and was the motivation of Stiles' recovery efficiency estimates based on permeability heterogeneity.³ The quantification of permeability heterogeneity, including the Lorenz coefficient and the Dykstra-Parsons coefficient of permeability variation were originally intended for use with core data, but these heterogeneity measures are even better suited for use with porosities and permeabilities from wire line logs.

It is instructive to review the computation of the Lorenz coefficient from foot-by-foot porosity and permeability data. The first step is the ordering of the permeability-porosity data pairs in decreasing permeability. One then forms the running sum of permeability, Skh, and porosity, Sfh. Cumulative permeability thickness is sometimes referred to as flow capacity, kH, and is the same kH that appears in the productivity index. The cumulative porosity, fH, is the storage capacity. If the running sum of Skh and Sfh are divided by the cumulative flow capacity and cumulative storage capacity, respectively, then one obtains a diagnostic plot of a stratified reservoir's heterogeneity (Fig. 1 [12,885 bytes]). For a homogeneous reservoir, plotting Skh vs. Sfh results in a diagonal line running from the lower left-hand corner to the upper right-hand corner. The Lorenz coefficient is the ratio of the area above the diagonal to the area below the diagonal as shown in Equation 9.

The Lorenz coefficient varies from zero for a homogenous zone (no permeability variation), to unity for a zone with extreme variability. The more uniform the permeability distribution, the closer the capacity curve moves to the diagonal (line with unit slope) and the higher the recovery efficiency (Fig. 2 [10,242 bytes]).

Vertical communication

Clearly, the layer-to-layer variations in permeability impact secondary-recovery efficiency, especially the vertical-sweep efficiency of waterfloods. However, for some other processes, the ratio of vertical-to-horizontal permeability, k _v/k _h, is also very critical. For example, in reservoirs with a bottom water drive or gas cap expansion or in production from horizontal wells, effective vertical communication is crucial for efficient hydrocarbon recovery.

The k_v/k_h ratio also determines whether coning is likely to occur. The k_v/k_h may be estimated from core plugs or from vertical interference tests. Generally, estimates based on core plugs must be reduced significantly to obtain good matches between reservoir simulation and production data.

Most likely, reductions in the core-based estimate of k_v/k_h are due to the undersampling of low-permeability streaks that ultimately dominate the reservoir scale k_v/k_h. On a reservoir scale, vertical permeability is governed by the location of thin tight streaks, clay drapes, and cemented hard grounds. These low-permeability layers are rarely sampled with 1-in. core plugs and have little impact on horizontal fluid flow, but will impact vertical flow and fluid migration.

Estimates of k_v/k_h can be obtained from wire line permeability profiles by computing the arithmetic and harmonic means of the observed permeabilities for the horizontal and vertical permeabilities, respectively. Depending on the lateral extent of the low-permeability layers, a more representative kv/kh ratio may be obtained if the minimum observed permeability is substituted for the harmonic-mean permeability.

The permeability profile and the k_v/k_h ratio are not just critical to production optimization (control of drawdown) and the estimation of recovery efficiency and performance prediction with reservoir simulation, but they are also important to the completion strategy. When there is a large degree of stratification, especially fine-scale lamination, evident on image logs, it is crucial to increase the perforation shot density. It may also be desirable to increase shot density in low-permeability layers to increase drawdown and recovery.

Reservoir engineers recognized early on that heterogeneity within reservoir units, permeability variations on a foot-by-foot basis, can influence ultimate recovery. The effects of reservoir heterogeneity, in particular permeability heterogeneity, often are measured with Lorenz or Dykstra-Parsons coefficients, coefficients that are then related to ultimate recovery. Although difficult to do and not necessarily accurate, such quantification is necessary to guide economic development and reservoir management.

Permeability sources

Permeability is key to reservoir development and management, and sources for this parameter include:

• Well and drillstem tests
• Wire line formation tests
• Conventional
• Whole cores
• Core plugs
• Probe perameters
• Sidewall rotary cores
• Wire line logs
• NMR
• Stoneley waves.

Well tests are a common source of kH needed to estimate reservoir productivity and flow capacity. These tests come closest to replicating the conditions under which the well will be produced, and thus, these PI and flow capacity estimates are generally believed to be the most relevant.

Furthermore, well tests can also provide information on well bore damage, and if run sufficiently long, can also be used to infer reservoir characteristics far from the well bore. However, well test and drill stem tests provide little or no information on reservoir heterogeneity. There is no means for uniquely deconvolving the measured kH to obtain a detailed permeability profile.

Wire line formation test tools are similar to well test and drill stem tests but differ in some significant ways. The difference in size and packer configuration requires different analysis techniques. As with well tests, however, the permeabilities derived from the pressure response are based on Darcy's law.⁴

Compared to a well test or drill stem test, wire line formation tests investigate a much smaller volume and, hence, are not well suited for estimating a well's flow capacity even when these tools are used to collect 100 or more data points.

Because of time and cost limitations, it is not feasible to obtain a detailed permeability profile with these tools.

Core permeability

In Spanish, cores sometimes are referred to as "testigo" because they are the witnesses for reservoir properties. Cores are considered the best permeability data source, and in the 1950s before the advent of reliable porosity tools, it was not unusual to core an entire reservoir interval and collect porosity and permeability data on a foot-by-foot basis.

It is still common to compare permeability data obtained from wire line logs with core data. However, it is important to realize that there are many different types of core permeabilities stemming from the many different ways of measuring permeability (Fig. 3 [10,806 bytes]).

In the laboratory, permeability was, and often still is, measured with air. Air permeability, k_air, is not an absolute measure of rock permeability, because the measured value depends on the mean measurement pressure. At low mean pressures, the apparent permeability is enhanced by gas slippage (Klinkenberg effects, see Reference 5 for a detailed discussion). Because k_air is easy to measure and is generally greater than k_liquid or k_infinity, k_air continues to be measured and reported by a majority of core analysis laboratories.

Definitions of k_air, k_infinity, and k_liquid are useful for comparing different reservoir rocks and assessing reservoir quality. In most reservoirs, however, hydrocarbons occur in conjunction with water, and if the reservoir is undersaturated, it will also contain gas. Thus, the appropriate measure of fluid flow capacity is effective permeability, k_eff (Equation 10), where k_r, the relative permeability of the flowing phase, is a fraction ranging from 0 to 1.

Further, the dependence of k_r on fluid saturation is indicative of wettability.⁶ Of course, both the flowing fluid type (oil, gas, or water) and the saturation state (oil, water, and/or gas saturations) need to be specified when defining k_eff. The situation becomes significantly more complicated if one deals with the presence and/or flow of three phases, in other words three-phase relative permeability.

Most reservoirs exhibit permeability anisotropy. The anisotropy can arise on the microscopic grain-size scale, or it may be imposed by reservoir layering. The tensorial nature of permeability needs to be taken into account when making permeability measurements. When measuring permeability on core plugs, the pressure gradient is established across the sample, and permeability is a measure of hydraulic conductivity in the direction of the applied pressure gradient.

For permeability analysis, a standard laboratory practice is to drill small plugs that are either parallel or perpendicular to bedding. From measurements on these oriented plugs, one obtains the permeability parallel to (k_h) and perpendicular to (k_v) bedding.

In some laboratories, but especially in those that analyze carbonate cores, it is customary to determine permeability of whole cores. These whole core samples are typically 3-4 in. (7.6-10cm) in diameter and up to 10 in. (25cm) in length. Generally, three permeability measurements are made on the whole core sample: one along the length of the sample and two, at right angles, across the sample. These three measurements are generally reported as k_v, k_max, and k_90°(Fig. 4 [9,959]). Of course, they can be transformed to obtain the three components of the permeability: k_v, k_hx, and k_hy.

Wire line profiles

Three approaches are available for determining permeability from wire line logs: NMR-derived permeabilities, Stoneley wave permeabilities, and wire line formation tester pressure transient-based permeabilities. Undoubtedly, the most significant advance in formation evaluation of the 1990s has been the introduction of reliable, laboratory-quality downhole NMR technology.

MRIL NMR-logging is based on the unique gradient field technology introduced by Numar Corp. The gradient field has significant advantages over conventional homogeneous-field tools and permits both rapid and flexible NMR data acquisition. When the MRIL tool is run in combination with conventional porosity and induction tools, heretofore difficult formation evaluation problems are easily solved. Furthermore, MRIL service combined with MAC and RCI services provides cost effective calibrated permeability data.

Several MRIL outputs are relevant for determining producibility. The MRIL service provides mineralogy-independent porosity, and recently introduced hardware and software permit complete pore-space characterization. With these, it is now possible to determine directly the volume of clay-bound water, the effective porosity, and the porosity available to moveable fluids.

This has proven extremely beneficial, not only for evaluating low-resistivity formations, easily overlooked when only conventional log data are evaluated, but also for determining flow capacity.

The MRIL tool does not directly measure permeability, and the data interpretation does not follow from Darcy's law. NMR permeabilities are inferred from the decay of the echo amplitudes recorded at each depth. The echo amplitudes start at a maximum, the NMR porosity, and then monotonically decrease with time.

The echo amplitude decay rate measures pore space surface-to-volume ratio. Small pores have a large surface-to-volume ratio, while large pores have a small surface-to-volume ratio.

The decomposition of the recorded echo train into a number of underlying exponential decaying terms determines the relative proportion of small and large pores. The NMR pore-size distribution data are the key for determining irreducible water saturation, moveable fluid saturation, and permeability.

Generally, rocks with only large pores, small surface-to-volume ratios, are very permeable. Rocks with mixed pore sizes are less permeable, and rocks with extremely small pores and high irreducible water saturations are tight. It is easy to appreciate NMR data for permeability determination.

The key ingredients in permeability computation from thin sections are porosity, pore size, and pore connectivity (Fig. 5 [11,296 bytes]). Conventional porosity logs provide only one of these three ingredients. NMR T2 distributions measure pore size and, thus, NMR permeability estimates are generally superior to permeabilities based on simple porosity-permeability correlations.

If, however, there is no good correlation between pore size and pore connectivity, then it will be difficult to compute reliable permeabilities solely from NMR porosity and T2 distributions. Fortunately, there often is a good correlation between connectivity and pore size and porosity, and hence (especially in clastics) NMR-based permeability estimates are reliable.

Coates' equation has successfully computed permeability from MRIL data for many formations. The constant C, in the Coates equation (Equation 10) is formation specific. In the equation, f is the porosity in % and MBVM and MBVI are the MRIL bulk volume moveable and bulk volume irreducible fluids, respectively.

When core (conventional or rotary sidewall) data are available, it is a relatively straightforward procedure to adjust C and obtain a match of the NMR and core permeabilities. Ideally, the NMR permeabilities are adjusted to effective core permeabilities. However, core permeability often is not available and other means of calibrating the NMR-based permeabilities are required.

Pressure transient permeabilities, derived from formation tests (either RCI or FMT data) provide a means for determining the constant C. This source of permeability calibration data is particularly attractive as it is an effective, in situ permeability, fundamentally linked to the Darcy equation.

Fig. 6 [46,204 bytes] shows the RCI permeabilities and the permeability profiles computed with different values of C for a North Sea well. Also shown is a plot depicting the error minimization procedure used to determine the optimum value for C.

Stoneley wave log analysis is another means for determining permeability and permeability profiles. As with the formation test permeabilities, the derived permeabilities are related to fluid movements and pressure perturbations and can be derived from earlier work.

Early theoretical work^{1 7} established the connection between permeability and Stoneley wave attenuation and velocity. Both laboratory,⁸ and field observations² demonstrated the connection between formation permeability and Stoneley wave properties. Other factors, unfortunately, contribute significantly to the Stoneley wave velocity and attenuation, which must be fully accounted for if permeability is estimated from these logs.

Meaningful inversion of measured Stoneley wave properties required the development of simplified theories^{9 10} and fast computational methods.¹¹ The inversion process accounts rigorously for purely elastic effects and borehole diameter changes. It simultaneously forward models both the velocity and attenuation of the Stoneley wave.

Fig. 7 [45,505 bytes] shows both the Stoneley wave and MRIL permeabilities. There is good correspondence in the permeability profiles. There is, however, an order of magnitude difference in the permeability values.

MRIL permeabilities were computed with the Coates equation, and the constant C (equal to 10) was not adjusted. No effort was made to adjust the Stoneley wave permeabilities (transmissibilities) by varying the assumed formation fluid parameters (density, viscosity, and acoustic velocity).

By combining the RCI, NMR, and Stoneley wave permeabilities, it is possible to obtain calibrated permeabilities. Furthermore, it should also be possible to separate permeability contributions arising from the matrix and fractures.

Profiles application

To evaluate possible completion strategies, the reservoir interval shown in Fig. 6 was subdivided into five horizons. Fig. 8 [34,684 bytes] shows the heterogeneity computations, which are summarized in Table 1 [20,682 bytes]. The cumulative flow capacity kH is tabulated for each interval. It is proportional to the productivity index and the Lorenz coefficient.

From a primary production point, the most productive intervals will be Zones 3 and 5. Zone 2 is the most homogeneous, smallest Lorenz coefficient, and should, when waterflooded, have the highest oil recovery at water breakthrough.

From a secondary waterflood point of view, there appears to be little benefit in completing the lower interval (Zone 5) as two separate intervals (Zones 4 and 5). Of course, one would also need to consider the increased cost of perforating the entire interval when deciding whether to complete Zones 3 and 4 as a single zone or sequentially as two zones.

The permeability and porosity data in Fig. 8 also were used to compute kH and the Lorenz coefficient (Fig. 9 [30,432 bytes]). The logged interval was subdivided into two intervals: x275-x406 ft and x406-x610 ft. The shallower zone is by far the more attractive zone. Its PI is twice that of the lower zone and the Lorenz coefficient indicates that the secondary recovery efficiency will be significantly better than in the lower zone.

The primary objective in perforating a cased well is to maximize productivity. Several key parameters that control productivity in perforated homogeneous formations are shot density, perforation diameter, perforation penetration distance, and shot phasing.12

In laminated and anisotropic formations, it is important to increase shot density to increase the likelihood that productive sands are connected with the well bore. In zones where permeability decreases with depth, it is beneficial to increase shot density with depth to achieve uniform inflow performance. Conversely, where permeability increases with depth, it may be desirable to decrease shot density with depth.

Except for operational limitations, it is possible to modify inflow performance and maximize the recovery by changing the perforation diameter and penetration depth, commensurate with the measured permeability profile.

References

1. Rosenbaum, J.H., "Synthetic Micro-seismograms: Logging in Porous Formations," Geophysics, Vol. 39, 1978, pp. 14-32.
   
2. Williams, D.M., Zemanek, J., Angona, F.A., Denis, C.L., and Caldwell, R.L., "The Long Space Acoustic Logging Tool," Professional Well Log Analysts 25th Annual Log Symposium, 1984.
   
3. Stiles, W.E., "Use of Permeability Distribution in Waterflood Conformance," Transactions AIME, Vol. 186, 1949, pp. 9-13.
   
4. Kasap, E., Altunbay, M., and Georgi, D., "Flow Units from Integrated WFT and NMR data," 4th International Reservoir Characterization Conference, Houston, Mar. 2-4, 1997.
   
5. Jones, S.C., "Two-Point Determinations of Permeability and PV vs. Net Confining Stress," SPE Formation Evaluation, March 1988, pp. 235-41.
   
6. Craig, F.F., The Reservoir Engineering Aspects of Waterflooding, SPE Monograph, Dallas, 1971.
   
7. Biot, M.A., "Mechanics of Deformation and Acoustic Wave Propagation in Porous Media," Journal of Applied Physics, Vol. 33, 1962, pp. 1482-98.
   
8. Winkler, K.W., Liu, H.L., and Johnson, D.L., "Permeability and Borehole Stoneley Waves: Comparison Between Experiment and Theory," Geophysics, Vol. 54, 1989, pp. 66-75.
   
9. Tang, X.M., Cheng, C.H., and Toksoz, M.N., "Dynamic permeability and borehole Stoneley waves: A simplified Biot-Rosenbaum model," J. Acoust. Soc. Am., Vol. 90, 1991, pp. 1632-46.
   
10. Tang, X.M., and Cheng, C.H., "Effects of a Logging Tool an the Stoneley Waves in Elastic and Porous Boreholes, The Log Analyst," September-October, 1993, pp. 46-56.
    
11. Tang, X.M., and Cheng, C.H., "Fast Inversion of Formation Permeability from Stoneley Wave Logs using a simplified Biot-Rosenbaum Model," Geophysics, Vol. 61, No. 3, pp. 639-45.
    
12. Locke, "An Advanced Method for Predicting the Productivity Ratio of a Perforated Well," JPT, December 1981, pp. 2481-88.
    

Bibliography

1. Blair, S.C., Berge, P.A., Berryman, J.G., "Using Two-Point Correlation Functions to Characterize Microgeometry and Estimate Permeability of Sandstones and Porous Glass," Journal of Geophysical Research, Vol. 101, No. 89, pp. 20359-75.
2. Dake, L.P., Fundamentals of Reservoir Engineering, Amsterdam, 1978.
3. Schmalz, J.E., and Price, H.S., "The Variation of Waterflood Performance with Variation in Permeability Profile," Production Monthly, Vol. 15, No. 9, 1950, pp. 9-12.

The Authors