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# Estimating permeability from well log data

Many approaches to estimating permeability exist. Recognizing the importance of rock type, various petrophysical (grain size, surface area, and pore size) models have been developed. (See links to these in Single phase permeability). This page explores techniques for applying well logs and other data to the problem of predicting permeability [k or log(k)] in uncored wells.

If the rock formation of interest has a fairly uniform grain composition and a common diagenetic history, then log(k)-Φ patterns are simple, straightforward statistical prediction techniques can be used, and reservoir zonation is not required. However, if a field encompasses several lithologies, perhaps with varying diagenetic imprints resulting from varying mineral composition and fluid flow histories, then the log(k)-Φ patterns are scattered, and reservoir zonation is required before predictive techniques can be applied.

## Multiple linear regression

A widely used statistical approach is multiple linear regression. Linear regression techniques are popular for establishing predictors of geological variables because the methods are effective at predicting mean values, are fast computationally, are available in statistical software packages, and provide a means of assessing errors.

### Predictors with one or two input variables

When a straight-line relationship between log(k) and Φ exists, as it does in Figs. 1 and 2, the computation of a predictor for log(k) by Eq. 1 is straightforward and merits little discussion. Curvature in the log(k)-Φ relationship is treated by adopting a polynomial in Φ. Increased accuracy is also afforded by dividing the field by area or vertically and computing regression coefficients for each area. In one area, curvature in the statistical predictor may be rather pronounced; in another, curvature may be absent. ....................(1)

### Predictors with several input variables

The quality of the predictor can often be enhanced by adding a variable such as gamma ray response or depth normalized to top of formation. As variables are added to Eq. 1, families of curves are required to present graphically the effect of combinations of variables. When one or two parameters are varied, the curves sweep out a large area on the log(k)-Φ plot. Predictive power can be increased by adding other parameters. Predictive accuracy does not increase indefinitely as parameters are added but instead usually reaches a limit after several (anywhere from two to six) parameters are included in the regression (see Fig. 17 of Wendt et al. for an example).

### Predictors using computed parameters

Computed logs such as shale volume and differences between porosities from different logs can be included as independent variables. In this way, petrological information can also be incorporated into the predictive relationships. A petrological parameter (cement or gravel) is first "predicted" from well logs using core observations as "ground truth." The predicted petrological parameters can then be included in a relationship to estimate permeability.

As the complexity of the log(k)-Φ plot increases (i.e., as the data deviate from a linear trend), more variables must be incorporated into the predictive model to maintain predictive accuracy, although instability can result from having too many variables. The better the understanding is of petrological controls on permeability, the more effective the predictor and its application will be. Other complications with regression methods are mentioned by various authors.. These complications include:

• Underestimation of high-permeability zones
• Overestimation of low-permeability zones

At some point, it becomes necessary to adopt a method of zoning the reservoir.

## Database approach

A database approach equivalent to an n -dimensional lookup table can also be used for predicting permeability within a field or common geology. In this approach, the user must first select the logs or log-derived variables that offer sufficient discriminating power for permeability. One must also choose a suitable bin size for each variable on the basis of its resolution. Then, a database is constructed from the core permeability values and associated log values. Each n-dimensional bin or volume is bounded by incremental log values and contains mean and standard deviation values of permeability plus the number of samples. In application, permeability estimates are extracted from a bin addressed by the log values. An interpolation scheme is used to extract an estimate from an empty bin. Like the regression method, the database approach can be used only when adequate core data are available to build the model, and results generally cannot be transferred to other areas.

## Fuzzy clustering techniques

Fuzzy clustering techniques provide a means of determining the number of clusters (bins in the preceding paragraph) and their domains. The term "fuzzy" indicates that a given input/output pair can belong (partially) to more than one cluster. Finol and Jing applied the technique to a shaly sandstone reservoir in which permeability ranged from 0.05 to 2,500 md. Six clusters were defined. In each cluster, permeability is determined by ....................(2)

where:

• Φ is porosity
• Qv is the cation exchange capacity per unit pore volume

The final determination of log(k) is a weighted sum of the six log(ki), with weights determined by the degree of membership of Φ and log(Qv) in their respective clusters. An average correlation coefficient of 0.95 was obtained on test sets. Implementation of Eq. 2 in uncored wells requires that Qv be determined from a porosity log and requires an estimate of grain density and shale fraction (Vsh).

## Artificial neural networks

Artificial neural networks are a third method of establishing a predictor specific to an area of interest. A back-propagation neural network is optimized on a training set in which the desired output (permeability at a given depth) is furnished to the network, along with a set of inputs chosen by the user. Rogers et al. established a predictor for a Jurassic carbonate field using only porosity and geographic coordinates as inputs. For each value of permeability to be predicted, porosity values spanning the depth of the desired permeability value were provided as inputs, rather than a single porosity value at a single depth. Permeability values predicted by the neural network in test wells were generally closer to the core measurements than were the values predicted by linear regression.