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Abstract A systematic method is proposed to better understand the relation between permeability, porosity, saturation, and clay content, to get representative permeability distributions as part of reservoir characterization.

This method utilizes the combination of a well-recognized permeability-porosity relationship and the Archie equation, as a predictive link between cores and well logs. This approach mathematically bonds the hydraulic (dynamic) and geometric (static) properties of the rocks. Permeability and rock types from well logs are obtained by careful theoretical and practical selection of parameters. The methodology converts a multidimensional correlation problem into sets of smaller correlations.

The procedure leads to a system of simultaneous equations, where the slope of one of the linear equations constitutes an exact calibration parameter between permeability and saturation. The equations allow a sequential transformation process from permeability-saturation, to permeability-porosity, and rock type-depth coordinates. The equations are solved for rock types, using reference grids constructed in all three coordinate systems. A calibration/diagnostic crossplot (slope-rock type) enables to find permeability, assess the integrity of the model, identify outliers, and perform consistency checks at different depths.

The practical implementation consists of transform tables, and correlations. Results can be extended to shaly sections, transition zones, and swept intervals. Some examples are presented using data from diverse reservoir classes.

Advantages: 1. Calculates permeability honoring and integrating

conventional core analysis, capillary pressure, and logs

2. Requires log curves generally available in every well 3. Quantitative in uncored sections

4. Follows a deterministic method, so results can be easily extrapolated to uncored wells. No black box approach

5. Based on well-documented engineering and geological theory (flow zones, rock fabric, pore geometry).

6. Formulation permits a connection to the geology through rock fabrics, for interwell extrapolation.

Previous models lack a good nexus between cores and well logs. Neural networks, fuzzy logic, multiple regression, or a combination of statistical methods establish this connection, usually with difficulty and not very effectively. The creation of an effective direct link and accuracy are some of the strengths of the proposed procedure. Introduction Permeability (k) is one petrophysical parameter, which has proved hard to obtain over the years. An actual “permeability log” does not exist. Numerous authors have tried to get permeability from well logs with limited success. Some of the models are theoretically sound, but they would be more useful if all wells were cored, which would be expensive.

Modern techniques use rock typing to find permeability, and need permeability (the unknown) to determine rock types, creating a circular calculation problem in uncored sections. Until a true permeability tool is invented, the “real permeability problem” is that a reliable mathematical link between cores and well log data (with respect to rock types and permeability) is still missing. The proposed method in this paper attempts to solve this problem. Current Permeability Models. Nelson published a paper1 in 1994, with a comprehensive discussion of the methods available at that time. He showed that the most successful models can be characterized by a linear relationship in the log-log permeability-porosity coordinate system, with the following generic form:

+)]= blog(Ø [aLog(k) ...….…………………..….(1)

where k and Ø are the permeability and porosity of the porous media, and a and b are calibration parameters.

The list includes Windland, Katz and Thompson, Pittman, Timur, Sen et al., Berg, Van Baren, Lucia, Timur, and Coates models, among others. Carman-Kozeny relationship, with its sophisticated mathematical form, also conforms to this rule for practical purposes, because it plots very close to a straight line on double logarithmic coordinates,2 for the porosity range found in hydrocarbon reservoirs.

SPE 89516

The Perfect Permeability Transform Using Logs and Cores Carlos F. Haro, SPE, Occidental Oil & Gas

2 SPE 89516

Models developed to date differ in the way a and b are used and interpreted (Table 1). The a slope is constant in some models (Carman-Kozeny, Windland), while a is variable in others (Lucia, Civan) as illustrated in Figs. 1, 2 and 3. Data from around the world presented by Nelson,1 exhibit variable a, sometimes with very steep slopes. The a values can be large according to Berg and Timur, or small like in the Winland model. Bourbié et al. indicated that a value of a=3 appeared appropriate for very clean materials as Fontainebleau sandstone, whereas a=4 or a=5 might be adequate for more general materials.3,4 They observed values of a=7 or a=8 for low porosity Fontainebleau sandstone. They reported diverse a slopes for granular limestones, shaly sandstones, micritic sandstones, and silts.4

Amaefule5 et al. interpreted the b intercept as the flow zone indicator (FZI). Civan6 showed that b corresponds to an interconnectivity parameter. The b intercept is variable and defines different rock types as noted by several authors. Lucia permeability-porosity lines join in a common point at an approximate porosity of 3.5%. The different a lines represent various carbonate rock fabrics/particle sizes.2 Nelson postulated some explanations for the diversity in the a values.1

Differences in the models are parameterization options between porosity and permeability, and it does not mean that any model in particular is wrong, according to Jennings and Lucia.2 Most authors probably used simplifying assumptions, and expressed permeability as a function of the dominant rock characteristic of the area under study. This was very likely dictated by the necessity/objective, prevailing geology, depended on the dataset, and on the available technology. The dominant rock characteristic was grain geometry (size, sorting, shape, orientation, packing, pore surface area) sometimes, and pore space geometry (pore body and/or pore throat shape/dimensions) other times. None of these rock characteristics are directly measured by well logs. This explains the difficulty in obtaining a reliable permeability transform. Some of the models symbolize an oversimplification of a difficult problem. Capillary Pressure and Permeability. Most authors1,7,8 have recognized the importance of capillary pressure curves as a good mechanism to predict permeability. Capillary pressure curves are direct indicators of pore geometry of the rocks, which controls permeability. Some authors use the “knees” or the flat portion of the capillary pressure curves (Windland, Swanson, Katz and Thompson) to establish a mathematical connection between rock dimensions/parameters and permeability. Others prefer to use the oblique/vertical section (Timur, Granberry and Keelan, Purcell, Lucia), as depicted in Fig. 4.

Permeability is generally obtained from capillary pressure curves as a function of either one or two parameters. One parameter is deemed enough when using the segment above the transition zone, because water saturation tends to a fixed asymptotic value at irreducible conditions (Swi). Two parameters are needed to specify one point in the “knees” or in the flat part of the capillary pressure curve, because the point falls in the transition zone, where saturation and capillary pressure keep changing. This point in the transition zone is set by a water saturation value (Sw), and a pore throat radius (r).

A typical example is r35, which is a pore throat radius at 35% non-wetting fluid saturation.

Various capillary pressure curves can pass through a point. Only a curve (several points) uniquely defines a pore throat size distribution. This distribution can be bimodal. A family of capillary pressure curves that group similar rock types is averaged when using the J function. Thus, capillary pressure curves can be modeled in different ways. The preferred way of parameterization depends on the specific application given to the capillary pressure data. Capillary pressure curves –in the way they are generally used- reflect the equilibrium between gravity (fluid density differences) and capillary forces (saturation gradient). This corresponds to a static condition.

The interest of professionals in capillary pressure curves differs, according to Pittman. Reservoir engineers and petrophysicists utilize this data to estimate permeability.7 This requires a quantitative approach. Geoscientists employ pore dimensions from mercury injection in defining the sealing capacity of the cap rocks, and to find displacement pressures required for hydrocarbon migration.7 These objectives are obtained in a more qualitative manner.

The apex, entry, displacement and threshold pressures, points in the middle and initial part of the mercury injection curves, are terms of great importance for geologists. They need to define the minimum continuous interconnected pore system, the “hydrocarbon filament” through the rock that would permit fluid migration over geologic time.7 As such, it is their utmost interest to establish correlations, between the “knees” and the flat part of the capillary pressure curve, with permeability. Only the major pore system with least resistance that dominates migration flow (and delineates the trapping mechanism) is considered in this portion of the curve. It is in this context that geologists sustain that the flat sector of the curves corresponds to the flow capacity of each pore type, while the vertical segment defines hydrocarbon storage.

Pittman tested the flat section thoroughly, generating correlations between permeability, porosity and pore throat radius at different mercury saturation values (10-75%).7 His intention was to obtain a complete pore throat distribution in that part of the capillary pressure curves. His results can be interpreted as an experimental demonstration of the variability and dependence of the porosity exponent (a slope) on saturation.

One important reference value in engineering is effective oil permeability at irreducible water saturation conditions (Koswi). Thus, engineers are mainly interested in the behavior of the oblique/vertical portion of the capillary pressure curves at irreducible conditions. This behavior necessarily involves the major and much smaller pore systems contribution to flow (flat and vertical section, respectively). Higher pressures are required to enter or drain the smaller pores, to overcome the natural resistance of the flow paths.

Buoyancy and hydrodynamic forces (driving forces), and capillary pressure (resistance force) control migration of hydrocarbons. Much larger forces are usually employed in hydrocarbon production and exploitation. Larger magnitude of pressure drawdown applied to the rocks would cause fluids withdraw and flow through smaller pore bodies and pore throats. The limit of this behavior is set by irreducible conditions, when no additional small pore system contribution

SPE 89516 3

to flow can occur. This signifies a true permeability definition, which is independent of pressure changes. This also indicates that both the flat and vertical segments of the capillary curves (the whole distribution) symbolize the flow capacity of each pore type.

Due to the diverse necessities, geologists and engineers have different views of fluid flow. Geologists are describing a slow drainage process: hydrocarbons entering the pore system by displacing the original water. Engineers are referring to hydrocarbons rapidly (relatively speaking) exiting the pore system through surface films/lenses of irreducible water. Permeability parameterization, in terms of r35 (Windland) or the apex (Swanson),1 would give more benefit to geoscientists for exploration purposes. Relationships as a function of Swi would better help engineers in hydrocarbon production calculations and reservoir characterization. Role of Mathematics, Statistics and Geology in Engineering Flow Problems Engineering relies on mathematical expressions to understand and solve problems. The more these expressions relate to fluid flow theory, to the physics, and to the geology of the rocks, the better their ability to describe real flow phenomena and predict permeability. Prediction can be achieved using a mathematical equation (correlation), or using statistics. In either case, there is generally a gap between correlations/statistics and prediction/probability, since datasets are not exhaustive.9

While models should fully describe the problem, they should be simple and practical enough to remain mathematically solvable. Some unsolvable problems in the past can now be numerically modeled and solved due to technology advancement. Solutions can be obtained with simple arithmetic approaches, or with very sophisticated methods like porous media networks.6

Mathematical simplification might be required in difficult cases. Examples of simplification in fluid flow problems are linearization of equations, discretization of differential equations, clustering, and main component analysis, among others. In extreme cases, only the main drivers of the problem are taken into account by making several assumptions, and secondary parameters are not considered. In our quest for results, we sometimes overlook the assumptions we have made, and ignore the limitations of the solutions.

Coordinate transformations and parametric equations are other good mathematical alternatives to simplify equations handling. Complicated situations can be described with just a simple parameter obtained at specific conditions. A good example is capillary pressure data. This curve is represented either by an equation, by a whole pore throat size distribution, by the irreducible water saturation (Swi) value, or by a pore throat radius given at a fixed water saturation in the transition zone.

Accuracy and predictability might appear as opposite concepts in calculating permeability. A line is mathematically accurate for two data points. The line -while fully defined- might be poor in terms of prediction, because of lack of enough data points to be statistically representative. Predictability will be more robust for a cloud of data points, but accuracy will diminish for every single data point, because

only a few points will be exactly modeled by the corresponding correlation. Rock type correlations that transgress the natural data trends, might gain in accuracy but lose statistically (ability of prediction). In a given dataset, the smaller the bin size, the better the results in terms of accuracy, but the worst in terms of prediction. Accuracy and predictability need to be balanced or combined to obtain meaningful/useful results.

In geological terms, permeability depends on textural parameters like rock fabric (orientation/packing), size, sorting, roundness, and sphericity of the grains. Orientation might have a dramatic impact on permeability.10 Textural parameters are more associated with rock grain characteristics. Grain properties are more related to sedimentary facies. Effects of deposition on the rock tend to be more organized and predictable. Post depositional events, like compaction, are in general more linked to pore space changes. Diagenetic changes on the rock tend to be less organized and less predictable. Permeability is directly connected to pore space which permits flow, and indirectly to rock grains which oppose flow. Maybe, this is why sometimes permeability looks so unpredictable.

The ingredients of a successful permeability model: The rock and fluid perspective Darcy law governs the movement of fluids in porous media. This law implies that fluid flow results from a balance of viscous, gravity, capillary, and other external forces. Flow, by definition, denotes dynamic conditions. Permeability is defined as a dynamic property by Darcy law, relating fluid characteristics, and fluid-rock interaction. Darcy law, in the way it is used in permeability transform calculations, is an isotropic formulation5 that assumes average spatially-uniform rock properties. Fluid flow throughout the reservoir is controlled/modified by pore geometry, which is a static property that better reflects the anisotropy of the media. A combination of static and dynamic relationships is necessary to fully describe permeability in reservoirs.

Permeability should be parameterized using all the main variables associated with fluid flow for a geologic area, whenever practical and possible. Calculations should include rock pore space geometry, fluid-rock interaction, lithology/mineralogical composition, deposition and diagenesis effects to account for the different evolutionary k/Ø paths of the rock.1 Some simplifications are necessary like assuming laminar flow, no friction, no slip, incompressible single-phase fluid flow, no chemical reaction between the fluids and the rock, no thermodynamic changes, and that the minimum homogenization volume criteria has been met (no problems of scale).4

A hydraulic average radius (rh), pore body (R)/pore throat (r) radii, or another parameter (like grain size or surface area) that correlates with one of the above, generally characterizes pore geometry. Irreducible water saturation (Swi), as a point from the capillary pressure curves, can be such parameter. Reservoir zones, with larger pore bodies and pore throats, have lower irreducible water saturation. Some of these parameters might be more effective than others, depending on the prevailing geology of the area. Pore throat radius is

4 SPE 89516

usually accepted as the most effective, while grain size might be the least effective.1

Geometry is restricted to spherical shapes with most of current models, when using pore body/pore throat radii, grain size or surface area. Rock internal geometry is seldom that shape.4 Grains shape is approximated as spheres, and the pore space is approximated, either as spheres or as a series of capillary tubes in a current simplification scheme. Meniscus/crevices within the grains are not considered. Pore space of a grain boundary pore (GBP) is star-like shape for a square arrangement (packing angle of 90 degrees) of spherical grains.11 Some models define a shape factor to compensate for this sort of limitation.5

Pores are naturally laid out in complicated patterns within the rock framework. Pores intricate geometry requires tortuosity and branchiness12 to fully model permeability. Tortuosity defines the winding characteristic of the capillary tubes. Branchiness is the degree of bifurcation/connection amongst capillary tubes (Fig. 5). It is related to coordination numbers, which represent interconnectivity of pores or the number of pore throats emanating from a pore body.6

Permeability transform values from well logs are generally calibrated to air permeability (after Klinkenberg and overburden corrections), and usually assumed “equivalent” to liquid permeability.13 Air is preferred in permeability measurements for its convenience and availability. Air is a relative inert fluid toward the core material.13 Contact angle is equal to zero for air permeability, assuming a strongly water wet system and a clean smooth wetting surface.14 Wettability is considered unimportant for routine porosity/permeability core analysis. Liquid permeability is only appropriate to model aquifers. Gravity and capillary forces –as utilized in the Darcy equation- are zero in aquifers, because there is no fluid density difference, and there is no saturation gradient.

Base permeability needed in engineering studies is an end point permeability, which denotes flow of one single fluid at Swi conditions (KoSwi). As such, effective oil permeability actually should be used in calibrations. Permeability should be calculated with well log data acquired above the transition zones, where irreducible conditions prevail. Despite the differences, liquid (absolute permeability), effective oil, and air permeability are used somewhat interchangeably with correlations. Both liquid and effective oil permeability values generally show a strong correlation to air permeability, as reported in the literature.15 The free exchange of these values introduces inconsistencies in permeability calculations.

Effective oil permeability (as a point from the relative permeability curves) depends on wettability, which constitutes fluid/rock interaction. Wettability is not considered in an explicit manner in any permeability model, although authors utilize the Poiseuille equation and capillary pressure curves as ingredients to calculate permeability. Both capillary pressure curves and Poiseuille equation are related to wettability. In fact, this equation characterizes laminar flow in a circular pipe/tube with smooth surfaces, where the velocity profile of the fluid is a paraboloid.3 This displacement profile results from the adhesion forces at the fluid/rock interface, viscosity of the fluid, and by applying a constant pressure gradient. Adhesion -in presence of two fluids- corresponds to wettability. These forces oppose fluid flow.

The wetting fluid completely fills the smallest pores when the pore system is in equilibrium. It will be in contact with the majority of the rock surface, if the saturation of the wetting fluid is sufficiently high.16 The wetting fluid is occupying a space, which otherwise would be available for non-wetting fluid flow. Thus, areosity6 (Ap) -areal porosity open to flow- is equal to Ø(1-Swi), both in the pore bodies and in the pore throats. Wettability is not zero under these conditions.

Wettability dictates the relative position of fluids within the rock, thus it controls their relative ability to flow. Wettability should be another ingredient in permeability calculations, since rock adhesion forces change -from one depth to another- due to either lithology (clastics or carbonate), cementation (e.g.: quartz overgrowth), or angularity and roughness variations of the rock. Changes in salinity or pH of the brine and oil composition between zones also alter rock adhesion forces.

Pore geometry, tortuosity, branchiness and wettability should be essential parts of permeability models, which reflect more closely the static rock framework. These ingredients should be combined with dynamic relationships for a full permeability characterization. Non-Archie rocks. Additional permeability variation is observed due to the presence of different clay types, and clay distribution1 in the rocks. Kaolinite, chlorite, smectite, or illite have diverse effects on permeability. Clay can be present in a rock, either as a detrital matrix or as authigenic cement. Clay distribution might be structural, laminated, or dispersed. Structural clay does not affect permeability much, as long as it remains below a reasonable percentage. Laminar clay mainly influences the anisotropy of the rock, the vertical to horizontal permeability ratio (kv/kh). Dispersed clay causes a decrease in permeability, even if clay is present in small amounts. Distribution mode of dispersed clays might be pore filling (kaolinite), pore lining (chlorite) and pore bridging (illite). The latter is the most detrimental to permeability, causing various orders of magnitude drop.17

Clay effects and cementation are important in permeability modeling. Permeability is influenced by the percentage of the matrix/cement material. Pore morphology in carbonates can be completely modified due to diagenesis. Effects like secondary porosity, leaching, dolomitization, and karsting add other complications to permeability calculations. Shaly sands and carbonates with moldic secondary porosity are in the realm of non-Archie rocks.3

Permeability calculation: Well logs perspective A good correlation generally exists between irreducible water saturation (Swi) and permeability, as indicated by several authors.13,18 Swi has an excellent correlation with the flow zone indicator (FZI), as reported by Amaefule.5

Resistivity (Rt) correlates with permeability too, mainly explained by Ohm law and Darcy equation similarities.13 Cluster analysis -conducted for carbonate and clastic rocks- confirms that resistivity is good at rock type/electrofacies discrimination, as presented in Fig. 6. Effective porosity (Ø) is one of the most common parameters found to correlate with permeability.1,2,4,5,19

SPE 89516 5

These three well log parameters can be historically catalogued as the best parameters to correlate with permeability. Swi and Ø are the two constituents in the proposed definition of areosity (areal porosity open to flow), taking into account wettability effects. It makes sense to use Swi, Rt, and Ø to calculate permeability from the well logs standpoint. Authors like Saner et al.20 use resistivity (Rt) and irreducible water saturation (Swi) to derive permeability.

Exceptions to these good correlations have also been found. Problems arise with resistivity correlations, whenever the rock electric characteristics are modified due to changes in resistivity/salinity (Rw) of the formation water, or due to the presence of clay minerals. It is more appropriate to use the Rt/Rw ratio to handle Rw variations, and minimize calculation problems.

The SwiØ product, bulk volume irreducible water (BVI), has been observed to remain constant in front of zones of nil water production for a given rock type.8,18 In line with this idea, constant BVI values (Buckles number) can be used as criteria to define rock types. Buckles number has been used by Tixier, Timur, and Coates-Dumanoir in the published Schlumberger permeability charts.21 It is also an integral part of the game board recommended by Borbas8 to predict reservoir performance. Non-Archie rocks. Clay can exhibit different resistivity and cation exchange capacity (CEC) values, which further complicate resistivity correlations. Total porosity and effective porosity coincide in clean zones, but they are different in shaly sections. Effective porosity does not include non-connected vuggy porosity. Effective porosity should be used in permeability estimation because it constitutes interconnected/permeable pore space.2,5,19 Permeability calculations should account for the amount of clay material. Clay volume (VCL) is another parameter recommended in the literature for permeability calculations.22 Comparison between models: Evolution perspective Nelson has already performed an extensive comparison of numerous permeability models.1 The intention in this paper is to compare very few models, emphasizing the way a simple geometric model (constant a, one equation) has evolved into other more sophisticated models (variable a, two equations) that admit -in a more explicit manner- the complexity of porous media. Technology has facilitated the evolution.

1. Windland model. This model (1980) is considered excellent to assess reservoir performance.8,23 The Windland model aims to capture the geometry of the rocks. As such, it reflects more closely the static conditions of porous media. This model appears to be more popular in the geoscientist community.

It was derived using capillary pressure curves to obtain real rock geometry, represented by pore throat radius (r35) at 35% non-wetting fluid saturation. While pore throat radius can be an important parameter to find permeability, it is not the only one. Furthermore, r35 (a point that belongs to the capillary pressure curves) mathematically is the necessary condition to identify a given pore throat size distribution, but it is not sufficient to define a unique distribution. It could correspond to various distributions, passing through that point,

with different permeability values. The same limitation applies to Swi values, or to any other single point on the capillary pressure curve. This implies that either a full capillary pressure curve (several points) or additional parameters are needed to uniquely determine permeability.

The Windland model assumes a series of straight, smooth, circular, non-communicating capillary tubes/pore throats. No explicit terms in the r35 equation directly relate to tortuosity, wettability, or branchiness of the pore system. A low range of air permeability values slippage corrected, and some others uncorrected were used to generate this model.7 The employed dataset might explain the small a slope. Pore throat radius is obtained in the transition zone, without a clear physical justification for the 35% non-wetting fluid saturation value.7 This model is not consistent with irreducible conditions.

Porous media aspect ratio is not taken into account, because only a representative pore throat radius (not pore body and pore throat radii) is considered. Aspect ration have small ranges in intergranular and intercrystaline pore systems. Non-Archie rocks have large aspect ration.24 There is no shape factor to correct for non-circular capillary tubes/pore throats. No direct link between cores and well logs within the model exists, thus core data is needed to apply it correctly. There is only one governing equation, which makes clear the requirement of a fixed value of a, one of the unknowns in Eq. 1 (a≈1.5 as depicted in Fig. 2). The pore throat radius a lines generally transgress the natural data trends.2

This model was created to find pore throat radius in defining rock types in wells without capillary pressure data, not to predict permeability.2,7 This is the way it is most of the time used.2, 23 Rocks classification by pore throat size as megaport (>10 µm), macroport (10-2 µm), mesoports (2-0.5 µm), and microport (<0.5 µm) is more a qualitative than a quantitative approach. For its simplicity, this model is useful to anticipate rock productivity, especially when used in the context of the game board and with limited datasets.8,23

Results of this model might look very appealing, because it is not difficult to show the relation of pore geometry and permeability through core photographs.

2. Carman-Kozeny model. This model (1937) has been the basis of numerous permeability models. It captures more closely the hydraulic behavior of the rocks, reflecting the dynamic conditions of the porous media. It is more accepted in the engineering community. It is considered theoretically sound by most authors, because it was derived using Poiseuille and Darcy laws, assuming a bundle of capillary tubes.5 These are tortuous, parallel, smooth, circular capillary tubes that do not communicate with each other (Fig. 5). An average hydraulic radius (rh) characterizes the radii of the tubes. A factor corrects for non-circular shapes.

One of the primary assumptions is that porosity must be effective.5 Only one equation controls this model, which justifies the necessity of assuming a fixed value of a, as shown in Fig. 1. The literature specifies that the Carman-Kozeny model with a≈3 is applicable to clean rocks.3

Civan6 and Larson12 et al. have reported that this model has inherent limitations. In addition, no direct link between core data and well logs exists for this model. Carman-Kozeny model, as used by Amaefule (1993), heavily relies on the ability of identifying flow zones with well logs in uncored

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sections. According to Amaefule et al., only a qualitative prediction of hydraulic units for the uncored sections can be expected.5 Errors in assigning the correct flow zones often occur, and the model becomes inaccurate.25 There are no means in the model to correct for this inaccuracy. The FZI lines generally transgress the natural data trends.

Larson12 et al. indicate that the “bundle of capillary tubes” is inadequate in at least three ways, as a description of capillary invasion of porous media: (1) the model does not describe the wettability behavior of real porous media; (2) the model ignores the irregular geometry of real porous matrices; (3) the model ignores branchiness of real porous media. Civan6 states that real porous materials may significantly deviate from the bundle of capillary tubes model, also because Carman-Kozeny relationship does not consider cementation effects.

3. Civan model. Civan utilizes fractal theory to generate a permeability model (1996), based on the premise that porous media is the end product of a complicated/irregular pore distribution, difficult to assess with classic (Euclidean) geometry. Civan adds that porous networks would be more appropriate for describing porous media, but at the same time, networks are computational intensive and disadvantageous for simulation studies.6

In line with fractal theory, Civan proposes a power law flow unit equation with a cementation exclusion factor (α). This approach overcomes the Carman-Kozeny oversimplification, while maintaining a practical implementation of permeability calculations, according to Civan. The equation honors branchiness, and it stands for a bundle of tortuous leaky hydraulic tubes that communicate with each other. Branchiness is mainly handled by the variation of the a slope from Eq. 1. Civan indicates that a varies from 0 to ∞.26 Civan equation is similar in form to the Carman-Kozeny relationship when the cementation exclusion factor is equal to 1. Parameters in the Civan equation are experimentally determined.

4. Lucia model. This model (1999) was specifically designed for carbonates, using core data from several fields worldwide. Carbonates are complex rocks, which exhibit small-scale variability in both porosity and permeability.2 The a slope is variable in the Lucia model, to handle the complexity of carbonate rocks, and follow the natural data trends.

The model uses particle size/rock fabric to categorize rock types (modified Dunham classification) as depicted in Fig. 3. Particle size is catalogued as an indirect geometric method, not very effective to calculate permeability, according to Nelson.1 Permeability is a petrophysical parameter more associated to the void/pore space of the rocks open to flow, than to the grain/solid volume opposing fluid flow. In this sense, it would be desirable the differentiation of rock types (to enhance permeability) using pore space characteristics. This might prove cumbersome in carbonates, caused by the diversity of porosity types in these rocks: interparticle, intraparticle, intercrystal, moldic, fenestral, fractures, vuggy, etc.

In theory, a and b can be found in this model because there are two governing equations. A direct link between well logs and cores is obtained by means of rock fabric, which has a

dual petrophysical and geological significance, according to Lucia. This model was derived through correlations, and lacks a more theoretical background, as compared to the Carman-Kozeny model.

Lucia indicates that the model, with some global-default parameters, can be applied as a first pass to calculate reasonable permeability values, if porosity values are higher than 5% and much lower than 30%. Proper calibration of the model requires petrological data obtained from thin sections. Lucia emphasizes the requirement of interparticle (effective) porosity in calculations.

Most of the permeability data required in simulation models is between wells. The Lucia model attempts to comply with this necessity, by using geological principles to guide the 3-D permeability interpolation. Lucia indicates that, in carbonates, rock fabric changes tend to be organized in a predictable manner within a sequence stratigraphic framework.2 In this sense, the Lucia model is an ambitious model. The merit and the limitation of this model are the mixing of two different scales: the macro/geological scale, and the small/porosity-permeability scale.

5. Results of the comparison. It should be clear at this point that the a slope must be variable to honor the complexity of porous media.1,2,3,4,6 Eq. 1 has two unknowns a and b, thus two equations are needed to get the solutions. Due to the lack of a second equation, the a slope has been almost always assumed a fixed constant value. A second relationship is required to account for a variability.

Among all the equation options,1 Carman-Kozeny is the most thorough and has good theoretical basis. For the rest of the discussion, Eq. 1 will be Carman-Kozeny-Amaefule relationship, with variable a slope as recommended by Civan. Proposed method and mathematical implications A new methodology is proposed in this paper, which incorporates most of the ideas already explained. This methodology involves a series of coordinate transformations to get usable correlations. It directly relates cores and well logs. It actually transforms a difficult multidimensional correlation problem into a set of smaller correlations. It puts rock characteristics in terms of commonly available well logs, which have proven to have good correlation with permeability.

The above is possible considering that, in the same fashion as Eq. 1, the Archie equation (1941) can also be plotted as a line in double logarithmic coordinates as shown in Eq. 2:

)log(1)log()log(t

wwi R

Rm

Smn −Φ=− ...….……….(2)

where m and n are the cementation factor and saturation exponent, respectively. For simplification, notice that the value of a17 from the Archie equation is equal to 1, and it is not in any way related to the a slope used in this paper.

Dividing by a, Eq. 1 can be written as follows:

ab

akLog +Φ= )log()(

……………………………...(3)

The corresponding coefficients in Eqs. 2 and 3 can be equated, in order to calibrate permeability and well log

SPE 89516 7

responses. This is the only theoretical-mathematical way in which the two equations can be exactly matched, taking into account the form of the equations, and the analysis of Eqs. 4 and 5. This is equivalent to the method of undetermined coefficients used for partial fraction decomposition of rational functions.27 This gives:

)log()/1(w

t

RR

mab = ……..…………………………..(4)

)log()log(

wiSk

nma −= ...……….………………….……..(5)

Eqs. 4 and 5 are not optional. They are a mathematical requirement to calibrate logs, if we admit that the formations being analyzed must comply with Eqs. 2 and 3. The a slope is not and cannot be a fixed value. It is a mathematically exact calibration parameter between core permeability and irreducible water saturation in cored sections, according to Eq. 5. The a slope and b intercept can be easily obtained using Eqs. 5 and 3 in cored sections; a, b, and k are the unknowns in uncored sections.

Eq. 2 reflects more closely the geometry (static property) of the porous media, whereas Eq. 3 is more related to its hydraulic (dynamic) characteristics. As such, a connection between two important physical properties of porous media is established by combining the two equations. A direct link between well logs and core permeability is created too. Analysis of the proposed equations This analysis is only appropriate for Archie rocks, since the Archie equation is being used. Although this condition appears very restrictive, the literature reports that the Archie equation is applicable to a remarkably wide range of rocks.3 Archie rocks are clean rocks exhibiting intergranular non-vuggy porosity. Non-Archie rocks are shaly sands, rocks with moldic secondary porosity, rocks with isolated micro porous grains,3 and fractured rocks among others.

Throughout the analysis, all Archie equation parameters are assumed known. Swi, implying irreducible conditions, has replaced Sw. The Archie equation includes the three parameters Swi, Ø, Rt with the best-reported correlations with permeability.

The right hand side of Eq. 4 –for brevity referred as the resistivity ratio- entirely depends on well logs, and not on permeability. This value can be used to generate electrofacies (rock). This electrofacies definition, also equal to the b/a ratio, can be depicted in a modified Pickett plot (resistivity ratio versus Ø), which resembles some of the approaches proposed by Aguilera.28 Use of a modified Pickett plot (as shown in Fig. 7) is necessary in some fields to properly handle extreme variations in formation water resistivity (Rw).

In Eq. 4, a stands for branchiness; b correlates with the flow zone indicator (FZI) as defined by Amaefule,5 and/or with the interconnectivity parameter from the Civan model.6 The right hand side of Eq. 4 is equal to the inverse of the product [(Swi)(Ø)] whenever n=m. This product is the Buckles number, or constant BVI. Thus, the proposed electrofacies definition has a rock type meaning, which has been used in

several publications,21 and that is consistent with irreducible conditions.

Eq. 5 has all the “rock framework” ingredients of a successful permeability model: tortuosity and wettability of the rocks strongly correlate to cementation factor m29 and saturation exponent n30 respectively, as reported by several authors. Again, the a slope characterizes branchiness. Swi –as a point that belongs to the capillary pressure curve- represents pore geometry.

The Archie saturation exponent becomes almost independent of wettability, when the brine saturation is sufficiently high that the brine is continuous.31 Wettability effects are very important when the brine saturation is lowered, and saturation exponent might exhibit some non-Archie rock behavior.11,31 Experimentally, saturation exponent values for oil-wet rocks have been found substantially higher (n≈2.5 to 9.5) than for water-wet media.3, 4

The proposed model resembles more closely the parameterization of permeability by Lucia,2 the Carman-Kozeny modification proposed by Civan,26 the equations by Pittman1 and Bourbié,4 since a is not constant. In theory, a can change from 0 to ∞ (Eq. 5), which coincides with the range found by Civan. In practice, the a slope upper limit is lower than ∞, because irreducible water saturation cannot reach 100%.

Two governing equations are used in this model, thus a and b can be determined. Most of the current models (derived from Eq. 1) can be interpreted as a subset of the proposed model. It symbolizes the general case, taking into account its a and b variability. It does not include any cementation exclusion factor like the Civan model.26 Model theoretical implementation Archie rocks. Eq. 5 shows that permeability (k) can be calculated, just by knowing the appropriate value of the a slope. Eq. 4 shows that rock only depends on well logs. Thus, there is a double benefit when relating a and rock. A sequence of coordinate transformations from the log-log permeability-saturation space to the a-rock coordinates can be performed, the advantage being that in the latter coordinates, permeability does not need to be explicitly known. Data behavior in the log-log permeability-saturation coordinates has also to be understood in the a-rock coordinates to implement the transformations.

The characteristics outlined above can be accomplished by creating grids, as described below. The grids shall signify the “perfect transform,” because the grids will be built with perfect/full compliance of the equations and assumptions. The grids shall serve to compare/calibrate the “real world” well logs. The primary objectives of the grids are to enable the calculation of electrofacies (rock), in a manner that is consistent with the theory explained in this paper, and to ensure a linear a-rock relationship within the grids.

1. Grids construction. It is common in Engineering to use unit systems or reference coordinates to compare different quantities. In line with this idea, a good/practical step is to construct grids in the log-log permeability-saturation (Fig. 8), log-log permeability-porosity (Fig. 9), and (b/a) ratio-depth (Fig. 10) coordinate systems. The grids and data must

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conform to all simultaneous equations 2 through 5 at every depth. Archie rocks data points are already matching Eq. 2.

Eq. 5 represents a family of lines in the permeability-saturation coordinates. All grids obeying Eq. 5 must pass through the point (1, 1). This introduces significant mathematical simplification. Only one additional point is necessary to define a line, or find a in this coordinate system. The b value is the intercept in the permeability-porosity coordinates according to Eq. 3; a is the slope in both coordinate systems, according to Eqs. 3 and 5.

A fixed reference value of permeability must be assumed to start creating calibrated grids. It can be mathematically proved (combining Eqs. 1 and 5) that the selected permeability value does not matter, as long as it remains fixed during the grids construction in the three coordinate systems. The assumed value cancels out when the b/a ratio is calculated. This floating permeability scale is a desirable situation, because permeability is not always known. This characteristic becomes important when comparing the actual log data with the b/a grids. Rock calculations (b/a ratio) will not depend on permeability as in other models,5,7 avoiding this way a circular calculation problem.

Both minimum and maximum water saturation values must also be assumed. The corresponding amin and amax can be calculated using Eq. 5. The assumed value of k (vertical axis), Swi minimum, and Swi maximum (horizontal axis) establish the effect of scale.

The grids can be equally spaced as follows:

binsaaa minmax −=∆ …………………………………….(6)

where bins is the desired/needed number of rock types. Rock values are parametrically defined in the permeability-

saturation grids. Rock values are bracketed within the corresponding a values, as shown in Fig. 8. Notice that a and rock definitions establish a linear relationship between both parameters in the grids, because rock are numbered in the same sequential order as the a values. This completes the setup of the permeability-saturation coordinates, where amin, amax, ∆a, and rock are already properly calibrated.

It is possible now to start making calculations at every depth. Using a=amin, b can be calculated with Eqs. 3. The corresponding b/a ratio can be easily obtained. While a and k remain constant, b and b/a ratio change at every depth because porosity is changing. The second b and b/a values can be calculated using (amin+∆a), the third set of values using (amin+ 2∆a), and so forth. The expression [amin+(rock)∆a] is used in the general case. Equivalently, the interval between the first and second b/a values corresponds to rock=1, the interval between the second and third b/a values corresponds to rock=2, etc. in the b/a ratio-depth plot. The rock value is bracketed within its corresponding pair of b/a values as shown in Fig. 10. This completes the steps to create calibrated grids in all coordinate systems.

2. Data behavior in the a-rock coordinates. The grids are (by construction) the locus of the points that comply simultaneously with Eqs. 2, 3 and 5. Therefore, a series of data points that plot as a line within two consecutive grids (same rock) in the log-log permeability-saturation coordinates,

will comply with Eqs. 2, 3 and 5, and with the following interpolation rule:

multiplierarockaarocka

a][])1[( minmin ∆++∆−+

= …...(7)

In fact, the multiplier, amin, and ∆a will be constants for a linear dataset. For reference, the multiplier should be equal to 2, if it equidistant in a terms from two a grid lines. These constants can be combined in coefficients like a1 and b1:

a=[a1(rock)]+b1……..…………..…..………………….(8)

A series of data points that plot as a line across the grids (different rocks) in the log-log permeability-saturation coordinates, will comply with Eqs. 2, 3 and 5, and with the following extrapolation rule:

multiplierarockaacrocka

a][])[( minmin ∆++∆−+

= …..(9)

where c can be any value >1. Again, Eq. 9 will plot as a line –equivalent to Eq. 8- when the multiplier, amin, ∆a and c are constants for a given dataset. In summary, data points that plot as a line, in both the log-log permeability-saturation and log-log permeability-porosity coordinates, will plot as a line in the a-rock coordinates, as Eq. 8 demonstrates.

3. Rock determination using the grids. Direct comparison, between the resistivity ratio (right hand side of Eq. 4) and the b/a ratio from the grids (left hand side of Eq. 4), establishes the data point rock number. This operation (mathematically equivalent to Eq. 4) can be performed at every depth, no matter if it is cored or uncored section. Results can be visualized versus depth, observing the resistivity ratio (obtained from logs) bracketed within two consecutive loci of b/a values (Fig. 10). Other permeability methods5,7 would require rock to be estimated by neural networks, fuzzy logic, statistical inferences, or multiple regression.

4. The a slope calibration with core data. Rock has been univocally defined by means of the grids and well logs comparison (Fig. 10). The a values from actual data points are still uncalibrated. Calibration can be obtained taking into account that a can be calculated in cored sections using Eq. 5, using the corresponding core permeability. Up to this point, a and rock have been determined independently of each other. It is now necessary to relate these parameters.

5. The a and rock correlation. The calibrated a values can be plotted versus rock in cored sections as shown in Fig. 11, to find the a1 and b1 coefficients of Eq. 8 by the least-squares method. The same linear equation, with the same coefficients, can be applied in uncored zones. This is valid because the rock definition is the same in cored and uncored sections, and it only depends on well logs.

The better the core data and well logs simultaneously comply with Eqs. 2 through 5, the closer a versus rock will plot as a line. Relate this result to decline curve analysis, where a continuous comparison with a reference line helps to determine (by trial and error) the appropriate decline constant.32 A refinement of cementation factor m and

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saturation exponent n can be obtained with such technique, in combination with Sanyal and Ellithorpe procedure to find (m - n).28,33 This can be beneficial to mitigate the problem that m and n are reported as the biggest sources of uncertainty in permeability determination.34 This is also a powerful tool to find outliers, which are difficult to recognize with other methods, especially if a black box approach (like neural networks) is used.

6. Quality control. Some outliers will always be found, because it appears an illusion to find an “universal” relation in permeability modeling.34 These outliers correspond to points that do not conform to the validity rules. Either a different model is necessary for these points, or additional data preparation and screening about the quality of the data is required. The number of outliers was small in our datasets. They were found in places where rock characteristics were changing rapidly with depth, and the lack of vertical resolution of the logging tools was evident. Core values need upscaling to compensate for this difference. Non-Archie rocks. The same correlation tactic applied in the clean sections was tried in the shaly sections, using small increments of clay volume (VCL) at a time. For example, all points complying with VCL > 0 and VCL ≤ 5 percent were plotted as shown in Fig. 12. The method worked quite well for clay volumes up to 40 percent. Samples were insufficient to confirm validity of the approach above that value; a1 and b1 were different for every clay volume (VCL) level. This result most probably can be explained since a linear response is also obtained in a Pickett plot for equal-VCL values, as described by Aguilera.33 This was demonstrated, considering that any of the shale correction algorithms currently available can be assimilated in a common shale group relationship.33

Comparison of Figs. 11 and 12 illustrates the variation of the a1 and b1 parameters. This approach can be improved, accounting for the different clay types and their distribution in the pore space, using some of the methods proposed by Aguilera in regard to m, n and Swi.33 An alternative is to use either Waxman-Smits or the dual-water saturation equations, to extend the applicability to higher VCL values.1 However, some other problems arise in such implementation.

Lucia35 has already shown how to solve vuggy carbonates, by means of special relationships to calculate interparticle porosity and cementation factor. Once such methodology has been applied, then it is possible to proceed with the techniques explained in this paper. Transition and swept zones. Capillary pressure curves are needed to find permeability in the transition zones. Some of these curves are depicted in Fig. 13. Rock types in the transition zones can be determined using the elevation (h) above the free water level, and water saturation (Sw), bracketed within two consecutive capillary pressure curves. These parameters directly relate to depth and Sw from well logs. Then, Swi can be calculated for every rock type using capillary pressure curve extrapolation. Permeability can be obtained in the same way already described, once Swi is determined.

In swept zones, Swi can be calculated either using an electrical balance equation36 wherever a mixture of two water salinities exists, or extrapolating capillary pressure curves.37

Model practical implementation The practical implementation of the proposed approach is simple. Transform tables or databases38 can be constructed in cored sections using Ø, rock and a values obtained with Eqs. 3, 4 and 5. Since a values can be exactly calculated in cored sections, permeability values in cored sections can be exactly reproduced (Fig. 14) as well. This characteristic resembles krigging algorithms, where estimates honor the data values at their location (exactitude property). The values of Ø, rock and a included in the lookup tables, correspond to all parameters needed in Eq. 3 to calculate permeability. Thus, querying of the tables can be used to predict permeability in uncored sections. These tables implement the accuracy of the model.

Core data sets are not exhaustive,9 and interpolation in cored and uncored sections can be achieved with correlations obtained in cored sections using Eq. 8. Results can be enhanced using porosity as a third variable/discriminator, to honor Eq. 3. This combination of equations enables the predictability power of the model.

A similar procedure is discussed in the literature as table interpolation, which is especially suited when mathematical algorithms to fit the data are difficult to obtain.39 The entire process assumes that a given set of log values will predict the same permeability value, wherever the same set of log values is found.38 This is a basic and fundamental assumption that is employed in every permeability or facies prediction using well logs. This leads to the concept of separability: the ability to discriminate. The electrofacies definition (rock) should be sufficient to differentiate, using well logs, the geological imprints on permeability over a wide dynamic range.38

The actual application of the proposed model is simple and very fast, once a program to create the grids is available. A permeability transform can be created in a matter of hours for big fields with lots of data, as compared to weeks with other methods that involve lengthy hours of trial and error. This model can be used with any size datasets. However, smaller datasets will not be statistically as robust. Results and Data Validation Example 1. This example corresponds to a prolific sandstone reservoir, with dramatically changing formation water salinity (200 ppm to 50 Kppm NaCl). The connate water is different from the aquifer water. Determination of Rw values from one formation to the other is very challenging. More than 20 fold changes have been observed in an 800 ft depth span. Misjudgment would result in bypassing of good oil zones, or unnecessary testing of water zones. Sandstones have very good porosity, good permeability, and are unconsolidated. Kaolinite chocking the pore throats, carbonate tight stringers, clay laminations, and bioturbation are some of the problems found in this reservoir.

Permeability was obtained using core data from 9 wells with more than 1200 core data points. Fig. 15 shows the significant scatter of the data. The good separability of the proposed rock definition permitted an exact core-well logs fit of permeability. Fig. 14 shows the good match obtained in the 9 wells. Most of the points plot in a 45 degrees line, with very few outliers.

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Well 1, with the best quality core, is a good well for data validation. This well is located at the crest of the structure. Water saturations were at irreducible conditions at the time of discovery of the field. Well 1 was drilled before production started in the area, so water saturations were at original conditions at the time of coring and logging. Well 1 was cored with oil-based mud, thus connate water saturations were preserved.

Several water extractions were performed in cores from well 1, using the centrifuge technique. The goal was to obtain representative samples of connate water. Rw values -from water extractions- coincided with well log calculations and with values obtained from samples collected in nearby wells.

Cementation factor and saturation exponent average values, estimated from logs, were compared to cores. Results indicated validity. Fig. 16 shows that log porosities (continuous line) fit well core porosities (dots). Core water saturations compare well with log calculated saturations. Core permeability values match log derived permeability, over six decades of variation (Fig. 14).

The significance of well 1 data from example 1 is: 1. All the different parameters of Archie equation were

obtained both from core data and from well log calculations. Complete agreement was obtained

2. Figs. 11 and 12 demonstrate compliance of the data with Eqs. 2 through 5, because a linear response is observed

3. Fig. 14 validates the method proposed in this paper, and shows that Eq. 1 is appropriate for permeability calculations. This is the same equation several authors have recommended, and that is in current use in the industry

The good match in another well is illustrated in Fig. 17. This well includes sections of higher clay content, demonstrating that the methodology can be extended to non-Archie intervals. Example 2. This example is from another good quality sandstone reservoir, located in an area of varying formation water salinity (4 Kppm to more than 100 Kppm NaCl). Rocks were deposited in fluvial and shallow marine environments. Dispersed kaolinite chocking the pore throats is found in places, which can be mistaken as water bearing. This happens because those zones exhibit good porosity and low resistivity values. The presence of tight carbonate intervals, intrusives, swelling clays, and glauconite complicates reservoir evaluation. Hydrocarbon sections change from intervals with good quality oil to tar mats, evidencing possible episodes of biodegradation.

Permeability was calculated using core data from 2 wells with about 150 core data points. Fig. 18 illustrates the good match obtained in the 2 wells, with very few outliers, covering six decades of permeability variation. The importance in this case is that a good fit is also obtained in a transition zone and in front of intervals with increasing shaliness, as depicted in Fig. 19. Swi values were determined by means of the capillary pressure curves in the transition zone. Rock types were defined first, and extrapolation was then employed, as shown in Fig. 13. This verifies that the method is also appropriate for smaller datasets.

Example 3. This example is extracted from a carbonate reservoir with varying degrees of reservoir quality. Very saline water (more than 240 Kppm NaCl) is found throughout the reservoir. Capillary pressure curves show that irreducible water saturation (Swi) does not change much, whereas entry pressures confirm big variation. In this sense, this reservoir is the opposite as compared to reservoirs from examples 1 and 2. This formation is characterized by good porosity, low irreducible water saturation, and low permeability values mostly. This is in line with observations reported in the literature,24 because these are rocks with intercrystalline, micro and minor separate-vug porosity. A tar mat is present in the lowermost high permeability streaks. Part of the challenge in this reservoir is to locate the high permeability intervals with good oil.

Permeability was obtained using core data from 12 wells (Fig. 20) with more than 1400 core data points. Fig. 21 shows the good match obtained in the 12 wells, with very few outliers, covering various decades of permeability variation. The importance in this case is that a good permeability match is also obtained in zones with varying lithologies, as depicted in Fig. 22. Sequences of limestone, dolomite and anhydrite are typically found. Swi values were determined by means of the Archie equation. Water saturation calculations compare fairly well with capillary pressure data. Lithology changes were anticipated using the density-neutron separation, and the photoelectric effect curve (PEF). Summary and Discussion Eq. 1 has been interpreted in the past, either as a hydraulic (dynamic, e.g.: FZI) or as a geometric (static, e.g.: r35) representation of porous media, closely related to engineering and geologic principles, respectively. In both cases, the a slope traditionally has been a constant/fixed value, due to the absence of additional relationships. Only one (b) of the two unknowns (a, b) of Eq. 1 could be independently determined.

The hydraulic definition is more applicable for engineering purposes, but it lacks the effect of the bundle of capillary tubes communicating with each other. This effect, denoted as branchiness,12 has been added by the Civan model using fractal theory, accepting a slope variability from 0 to ∞.6

A new technique to calculate permeability is presented in this paper, which permits to solve for the two unknowns of Eq. 1. This is performed establishing the mathematical connection between Eq. 1 and the Archie equation, which is facilitated based on the following theoretical considerations:

1. Both equations represent a line in log-log coordinates as a function of porosity. As such, it is mathematically legitimate to equate the corresponding coefficients, to calibrate logs to permeability

2. The a slope is variable,6 which enables the mathematical bond of both equations

3. Swi, as a point from the capillary pressure curve, is an indicator of pore geometry, which controls permeability

4. Eq. 5 can be explained by means of the current understanding of a successful permeability model:5,6,12 Swi, m, n, a stand for pore geometry, tortuosity, wettability, and branchiness, respectively

SPE 89516 11

5. Eq. 4 corresponds to a categorization of rock types (b/a ratio), consistent with irreducible conditions, and closely related to the Buckles number (ØSwi product)

6. Effective oil permeability, defined at irreducible conditions, is the base permeability needed in engineering studies, not air or liquid permeability. Thus, irreducible conditions are a necessity in permeability modeling

7. Effective oil permeability, as a point in the relative permeability curve, depends on wettability of the rock

8. Swi, Ø, Rt, main parameters of the Archie equation, have historically demonstrated good correlation with permeability. This is in line with areosity (areal porosity open to flow), redefined as Ø(1-Swi). Irreducible water is occupying an space, which otherwise would be available for non-wetting fluid flow

The association of Eq. 1 and Archie equation symbolizes a connection between hydraulic and geometric properties of porous media. It also establishes a link between cores and well logs. This permits a robust calibration of well logs, using core permeability data.

Permeability and water saturation are considered two of the most uncertain and difficult parameters to get in reservoir characterization. The combination of Eqs. 1 and 2 creates a strong relationship between these two parameters, and diminishes the degree of freedom and uncertainty in their calculation.

The model implementation relies on the generation of reference grids on the permeability-saturation, permeability-porosity, and rock-depth coordinate systems. The grids are referred as the perfect transform since the grids are constructed with perfect/full compliance of the equations and assumptions.

Rock type and permeability calibration (of well logs using cores) is performed by comparison of the log responses against the grids. The calibration goodness is assessed, judging the degree of linearity of the a-rock correlation. This linearity check is a powerful tool to recognize outliers. This also permits an improvement of m and n calculations. The procedure enables consistency checks at different depths, which converts an underdetermined into an over determined system, as explained by Hawkins.40

The level and distribution of permeability are the most difficult reservoir characteristics to find.41 Petrophysical parameters should be preferably defined, based on primary principles and theory. Models based merely on correlations are prone to inconsistencies, have local application, and they are not as robust. One of the aims of this paper has been to get a theoretical justification for the different ingredients utilized in the new procedure, and use consistent parameters as much as possible.

The a slope and b intercept are calculated from actual data, they are not obtained from pre-established correlations. There is no clear indication that either the a slope or the rock definition should strictly follow the natural data trends (Fig. 23), because this also depends on how exhaustive is the dataset. Core data (Swi and k) supports an a slope variation

from 3 to 12 for the excellent rock quality reservoir of well 1-example 1, as depicted in Fig. 23.

The literature indicates that permeability is more effectively determined using the pore space, rather than using textural parameters.1 Permeability is a petrophysical property that is more associated to the void/pore space of the rocks open to flow, than to the solid/grain volume opposing fluid flow. Both Eq. 1 (in the way proposed in this paper) and Eq. 5 fulfill the void space principle.

Current permeability models, based on irreducible water saturation or resistivity, are only applicable above the transition zone.1,2 Results in this paper have extended permeability calculations to the transition zones, using capillary pressure curves. Height above the free water level (h) and water saturation (Sw) are used to reconstruct Swi values in the transition zones, because h and Sw are directly extracted from well logs.

Permeability can be obtained in partially depleted zones, using an electrical balance equation36 or capillary pressure curves extrapolation,37 to get an estimate of Swi.

Well logs and core data agreement suggest that the process can also be applied to shaly sections, despite the fact that permeability calculations are based on the Archie equation (clean zones). Techniques proposed by Lucia2 to calculate Ø and m would permit using the proposed method in non-Archie carbonate rocks.

How many rock types are too many? Detailed binning of the data (several rock types) might be mathematically necessary to gain accuracy. Once a good permeability prediction is obtained, data can be regrouped into sets that make more geological and engineering sense, and are more manageable.

Lucia has already shown the connection between rock fabrics (calculated as a continuum),2 sequence stratigraphy and Eq. 1. This opens the possibility of mathematically linking the hydraulic and geometry properties of porous media to the geology of the area, to populate permeability data between wells. The corresponding conversion numbers between rock and rock fabrics can be readily calculated.

A percolation porosity can be incorporated both in Eqs. 1 and 2 according to the literature,3 if deemed necessary. The proposed methodology is still applicable under these conditions. Conclusions It has been mathematically verified that the Archie equation can become a natural correlation relationship, between well logs and core analysis, with respect to permeability. This gives a new perspective to the Archie equation.

Permeability calculations indicate that a line trend in a log-log plot of permeability and porosity (Eq. 1) is not a bad assumption, as many authors have found in different fields worldwide. Some outliers will exist in every dataset, which might need a different model/additional data preparation/data quality screening.

The bond of Archie equation and Eq. 1 is the basis for the proposed method. This permeability approach establishes a direct calibration scheme between well logs and core values. It permits a perfect match in cored sections, due to the good

12 SPE 89516

separability of the electrofacies/rock definition in combination with the lookup tables/database method.38,39

Electrofacies/rock types (necessary to enhance permeability calculations) are defined in the permeability-saturation coordinates, not in the permeability-porosity crossplots.1,2,5,7 These electrofacies have true rock properties meaning, and are directly associated with BVI values. This eliminates one of the limitations of a database approach as reported by Nelson.1

Results found in this paper favor a variable a slope, which can be easily found and is fully defined in the log-log permeability-saturation coordinate system (Eq. 1). The b intercept is also variable, and can be obtained in the log-log permeability-porosity coordinate system.

The proposed method enables a quantitative approach in uncored sections, as opposed to “qualitative results only” with other techniques.5 The main difference is that previous models can be inaccurate because of errors in assigning rock types in uncored sections,25 while there is not ambiguity with the proposed method.

The interpolation/prediction over sections of non-continuous core permeability data is facilitated using the a slope versus rock correlations. These correlations are of easier implementation/control, as compared to statistical procedures used by other authors.5,38 Prediction (in wells without permeability measurements) is obtained using a combination of querying the tables/database, and using the a versus rock correlations.38,39

Rank correlation/principal component analysis25 are not needed to select the permeability estimators when using the proposed method. The estimators are well log curves (Ø, VCL, Rt, Swi), which should be available in every field as a minimum. The user defined grids are used in lieu of cluster analysis25 and/or binning.

The proposed method is a deterministic method, not a black box approach. It does not rely on any pre-established empirical relationship: the a and b coefficients are not fixed,1 they are data driven. It permits generalization and integration of most of the methods to calculate permeability developed to date. Acknowledgments The author thanks Occidental Oil & Gas for access to their data, and permission to publish this paper. My gratitude to those colleagues who sacrificed their leisure time to help correcting this manuscript. REFERENCES

1. Nelson, P.: “Permeability-Porosity Relationships in Sedimentary Rocks,” The Log Analyst (May-June 1994) 35, No. 3, 39.

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3. Mavko, G., Mukerji, T., Dvorkin, J.: The Rock Physics Handbook, Cambridge University Press (1998) 260.

4. Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media, Editions Technip (1987) 13.

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16. Anderson, W.G.: “Wettability Literature Survey-Part 1: Rock/Oil/Brine Interactions and the Effects of Core Handling on Wettability,” Journal of Petroleum Technology (1986).

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20. Saner, S., Kissami, M., Al Nufaili, S.: “Estimation of Permeability from Well Logs Using Resistivity and Saturation Data,” paper SPE 26277 available from SPE, Richardson, Texas (1993).

21. Log Interpretation Principles/Applications Schlumberger Publication (1989) 10-3.

22. Topham, B.D., Liu, K., Eadington, P.J.: “Relationships Between V-Shale, Petrographic Character and Petrophysical Data from the Jurassic Reservoir Sandstones in the Southern Vulcan Sub-Basin,” Petrophysics (January-February 2003) 44, No. 1, 36.

23. Martin, A.J., Solomon, S.T., Hartmann, D.J.: “Characterization of Petrophysical Flow Units in Carbonate Reservoir,” AAPG Bulletin (1997) 81, No. 5, 734.

24. Holtz, M.H., Major, R.P.:”Integrated Geological and Petrophysical Characterization of Permian Shallow-Water Dolostone,” SPE Reservoir Evaluation & Engineering (February 2004) 7, No. 1, 47.

25. Mathisen, T., Lee, S. H., Datta-Gupta, A.: “Improved Permeability Predictions in Carbonate Reservoirs,” paper

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SPE 70034 presented at the 2001 SPE Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 15-16 May.

26. Civan, F.: “Fractal Formulation of the Porosity and Permeability Relationship Resulting in A Power-Law Flow Units Equation –A Leaky- Tube Model,” paper SPE 73785 presented at the 2002 SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, Louisiana, February 20-21.

27. Jeffrey, A.: Handbook of Mathematical Formulas and Integrals, Second Edition, Academic Press, San Diego, California (2000) 63.

28. Aguilera, R., Aguilera, M.: “The Integration of Capillary Pressures and Pickett Plots For Determination of Flow Units And Reservoir Containers,” paper SPE 71725 presented at the 2001 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 30-October 3.

29. Salem, H. S.: “Derivation of The Cementation Factor (Archie’s Exponent) and The Kozeny-Carman Constant From Well Log Data, and Their Dependence on Lithology and Other Physical Parameters,” paper SPE 26309 available from SPE, Richardson, Texas (1993).

30. Sharma, M.M., Garrouch, A., Dunlap, H.F.: “Effects of Wettability, Pore Geometry, and Stress on Electrical Conduction in Fluid-Saturated Rocks,” The Log Analyst (September-October 1991) 32, No. 5, 511.

31. Anderson, W.G.: “Wettability Literature Survey-Part 3: The Effects of Wettability on the Electrical Properties of Porous Media,” Journal of Petroleum Technology (1986).

32. Slider, H. C.: Worldwide Practical Petroleum Engineering Methods, Revised Printing, Pennwell Books, Tulsa, Oklahoma (1983) 536.

33. Aguilera, R.: “Extensions of Pickett Plots for the Analysis of Shaly Formations by Well Logs,” The Log Analyst (September-October 1990) 31, No. 5, 304.

34. Balan, B., Mohaghegh, S., Ameri, S., West Virginia University: “State of the Art in Permeability Determination from Well Log Data: Part 1- A Comparative Study, Model Development,” paper SPE 30978 presented at the 1995 SPE Eastern Regional Conference and Exhibition, Morgantown, West Virginia, September 17-21.

35. Lucia, F. J.: Carbonate Reservoir Characterization, Springer (1999) 68.

36. Yi-ren Fan, Shao-gui Den, Can-Can Zhou, Bing-kai Liu: “Experimental Study and Theoretical Analyses of Formation Resistivity Under Fresh Water Flooding,” SPWLA 42th Logging Symposium (2001).

37. Harrison, B., Jing, X.D.: “Saturation Height Methods and Their Impact on Volumetric Hydrocarbon in Place Estimates,” paper SPE 71326 presented at the 2001 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 30-October 3.

38. Nicolaysen, R., Svendsen, T.: “Estimating the Permeability for the Troll Field Using Statistical Methods Querying a Field-wide Database,” paper presented at the 1991 SPWLA 32nd Annual Logging Symposium, June 16-19.

39. Ertekin, T., Abbou-Kassem J., King, G.: Basic Applied Reservoir Simulation, SPE Textbook Series Vol. 7, Richardson, Texas (2001) 381.

40. Hawkins, J.M.: “Integrated Formation Evaluation with Regresion Analysis,” paper SPE 28244 presented at the 1994 SPE Annual Technical Conference and Exhibition, Dallas, Texas, July 31-August 3.

41. Mattax, C.C., Dalton, R.I.: Reservoir Simulation, SPE Monograph Volume 13, Richardson, Texas (1990) 31.

Nomenclature a = Porosity exponent/branchiness a1 = Tuning parameter Ap = Areal porosity open to flow, fraction b = FZI or interconnectivity parameter b1 = Tuning parameter bins = Desired number of rock types BVI = Bulk volume irreducible water c = Natural number > 1 CEC = Cation exchange capacity FZI = Flow zone indicator GBP = Grain boundary pore h = Height above the free water level, ft k = Permeability, md KoSwi = Effective oil permeability @ Swi, md kh = Horizontal permeability, md kv = Vertical permeability, md multiplier = Correlation factor m = Cementation factor n = Saturation exponent PEF = Photoelectric effect factor R = Pore body radius r = Pore throat radius r35 = Pore throat radius at 35% non-wetting saturation rh = Average hydraulic radius rock = Type of rock, electrofacies Rt = Formation resistivity, ohm-m Rw = Formation water resistivity, ohm-m Sw = Water saturation, fraction Swi = Irreducible water saturation, fraction VCL =Clay volume, fraction ∆a = Evenly grid spacing Ø = Porosity, fraction Subscripts CL = clay i = irreducible min = minimum max = maximum o = oil w = water Table 1 – Values of the a slope according to different authors/models Model a slope Windland ~ 1.5 Pore dimension models ~ 2 Amaefule ~ 3.0 Bourbié et al. ~ 3 – 8 Surface area models ~ 4 Timur 4.4 Grain-based models ~ 5 Berg 5.1 Lucia ~ 4 – 9 Civan 0 - ∞ This model 0 - ∞

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Figure 1 – Carman-Kozeny (Amaefule) model.5 It plots very close to linear response on double logarithmic coordinates.2

Figure 2 – Windland model (pore geometry).7,8,23

Figure 3 – Lucia model (for carbonates).2

Figure 4 – Capillary pressure curve permeability parameterization according to several authors.1

Figure 5 – Cartoon to illustrate branchiness, tortuosity, pore geometry of porous media.11 Comparison with the bundle of tortuous tubes that do not communicate.

Figure 6 – Cluster analysis demonstrating the discrimination power of resistivity values in carbonates. Similar discrimination response is also observed in clastics.

Porosity

Resistivity

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Figure 7 – Modified Pickett Plot to handle significant water salinity variations.

Figure 8 – Permeability-saturation log-log coordinates and reference grid system. Assumed k=100 md, Swimin=0.01, Swimax≈0.80. The amin, amax, rock determination is illustrated.

Figure 9 – Permeability-porosity log-log coordinates and reference grid system. An example when a=2, Ф=0.18, k=100 md. is illustrated. The reference k value is kept fixed between coordinates (Fig. 8 and Fig. 9).

Figure 10 – Data point rock number determination: resistivity ratio (logs calculation) versus (b/a) ratio obtained from the grid reference systems. Values are plotted versus depth.

Figure 11 – a slope versus rock crossplot. Actual data points in front of clean sections.

Figure 12 – a slope versus rock crossplot. Actual data points in shaly sections.

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Figure 13 – Capillary pressure curves with good spread in terms of Swi values. No variation in entry pressures. Example 1.

Figure 14 – Core versus well logs permeability comparison. Very few outliers. Sandstone reservoir. Example 1.

Figure 15 – Dataset with significant scatter of the data. Permeability-porosity crossplot. Sandstone reservoir. Example 1 with ~1200 data points.

Figure 16 – Core versus well logs porosity, water saturation and permeability comparison. Very few differences observed on the permeability track. Sandstone reservoir. Example 1.

Figure 17 – Core versus well logs porosity and permeability comparison. Sandstone reservoir. Very few differences observed on the permeability track. Example 1.

Figure 18 – Core versus well logs permeability comparison. Very few outliers. Sandstone reservoir. Example 2.

~ 1200 data points

AIR BRINE Vs WETTING PHASEM2A-M2B

0

10

20

30

40

50

60

0 20 40 60 80 100 120

WETTING PHASE

Ht a

bove

FW

L Ft

CL02-8200'

CL02-8194'

CL02-8188'

CL02-8185'

LY06-8245

ly06-8284'

Data point

Swi determination

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Figure 19 – Core versus well logs porosity and permeability comparison. A transition zone is included, where capillary pressure extrapolation is required. Very few differences observed on the permeability track. Example 2.

Figure 20 – Dataset with significant scatter of the data. Permeability-porosity crossplot. Carbonate reservoir. Example 3 with ~ 1450 data points.

Figure 21 – Core versus well logs permeability comparison. Very few outliers. Carbonate reservoir. Example 3.

Figure 22 – Core versus well logs porosity, saturation and permeability comparison. Carbonate zones with changing lithology are included. Very few differences observed on the permeability track. Example 3.

Figure 23 – Core irreducible water saturation and core permeability comparison. Sandstone reservoir. Example 1.