Henderson  Petrophysics

Limitations of Petrophysics Log Analysis in Heterogeneous Reservoirs
Lessons from Stochastic Modelling

Introduction

In this article we investigate the limitations of conventional wireline log analyis in hetergeneous reservoirs.

In the following sections we;

• Describe a computer stochastic model of heterogeneous reservoirs,
• Develop simple models of wireline log responses that have very different volumetric response functions, and,
• Investigate the hydrocarbon reserves impact of using conventional wireline log analysis methods to evaluate heterogeneous reservoirs.

Our computer modelling reveals some severe limitations of using conventional wireline log analysis methods in heterogeneous reservoirs. In these reservoirs, our modelling has shown that hydrocarbon volumes (reserves) are always over-estimated. This results mainly from combining wireline log data from tools that investigate far different formation volumes. For example, pad-type porosity tool measurements when combined with deep reading induction tools.

Brief (extremely) Review of Conventional Wireline Log Analysis Methods

Here we present an extremely brief review of conventional wireline log ananalysis methods.

Hydrocarbon saturation is calculated from wireline log data by comparing expected 100 percent water saturated 'wet' resistivity (Ro) with the wireline measured resistivity (Rt). The presence of hydrocarbons is indicated where Rt > Ro. Hydrocarbon saturation (fraction of pore volume) is calculated from the ratio Ro:Rt.

Wet resistivity, Ro is estimated using wireline log data that measures rock properties (shaliness and porosity) in a small volume close to the wellbore. Rt is measured with tools that investigate several feet into the formation, beyond the zone altered by drilling fluids. These Rt measuring wireline logs measure the properties of a much larger rock volume than those used to estimate Ro.

Stochastic Modelling of Heterogeneous Reservoirs

The objective of our modelling is to investigate the impact on reserves estimates by the use of wireline log data with different volumetric response functions.

Our reservoir model is very simple, consisting of only water saturated clean sand. Reservoir heterogeneity is provided by the natural variability of the porosity values. We have simulated both unimodal and bimodal probability distributions (figure 1). The bimodal porosity model is meant to represent reservoirs with zones where porosity has been reduced by diagenetic cement. This model can also be used to represent low porosity (tight sands) with higher porosity 'sweet spots'.

	Example Porosity Distributions; High Porosity: Average: 13 percent St. Dev.: 2 percent Low Porosity: Average: 7 percent St. Dev.: 1 percent
Figure 1: Example porosity population probability distributions

The reservoir model was created using Visual basic for Applications (VBA) macros. The following parameters can be specified in the model;

• Number of cells in the model,
• Mean and standard deviation for the low and high porosity populations,
• Proportions of high and low porosity cells in the model,
• Cementation exponent (m), saturation exponent(n) and brine resistivity (Rw)

The reservoir model consists of a two dimensional array of cells. The columns emulate various distance from a borehole and the rows of cells emulate a vertical reservoir section. Figure 2 shows a very small portion of one implementation of the model.

	Cell annotation is porosity Yellow cells: "High" porosity cells Purple cells: "Low" porosity cells Speckled shading: Cells measured by Rt log Hatched shading: Cells measured by Ro log
Figure 2: Example of reservoir model porosity grid

The following method is used to populate the reservoir model. For each cell in the model;

• Generate a random number (range: 0 - 1) to determine if a reservoir cell will contain high or low porosity rock. For example, if the model specified 10 percent low porosity material, then this population would be sampled if the random number is equal to or less than 0.1,
• Generate another random number to sample the Standard Normal cumulative probability function (figure 3). Use this number to obtain the corresponding 'Z' value from the Standard Normal probability distribution,
• Calculate porosity using: Porosity = Z x Standard Deviation + Population average
  

	Standard Normal Distribution: Mean (average): 0 (defined) Standard deviation: 1 (defined)
Figure 3: Standard Normal Distribution

• Cell conductivity is calculated using the Archie equation;
        Co = Brine conductivity times porosity raised to the power m
• The high resolution log response is the average conductivity of 3 cells closest to the measure point (see figure 4). Apparent resistivity is 1000 divided by the average conductivity. This method is meant to emulate the aparent Ro data that would be generated by using data from a pad-type porosity tool such as the density log.
• The low resolution log response (Rt) is the average conductivity of 42 cells around the measure point (see figure 4). Apparent resistivity is 1000 divided by the average conductivity. This method is meant to emulate a deep-reading conductivity tool in a water saturated reservoir. Since the model does not contain any hydrocarbons wet resistivity Ro = Rt.

	Yellow cells: "High" porosity cells Purple cells: "Low" porosity cells Speckled shading: Cells measured by Rt log Hatched shading: Cells measured by Ro log
Figure 4: Example of reservoir model grid

Modelling Results

Since each cell in the model is generated using random (stochastic) processes every run of the reservoir modelling program results in a unique data set. Figure 4 shows a small portion of one simulation data set. Cells with porosity higher than 8 percent have yellow shading. Lower porosity cells have purple shading.

Figure 5 shows an example frequency distribution plot of porosity data generated by the simulation model. The model consists of 2400 cells with 10 percent of the cells from the low porosity population.

Figure 5: Example porosity frequency distribution from simulation model

Figure 6 shows a expanded portion of one modelling run. The model does not contain any hydrocarbons, so we expect that the Ro (shallow) and Rt (deep) curves will overly. The effect of using measurement data with different volumetric response functions is that Ro and Rt are rarely equal. Hydrocarbon saturation cannot be less than zero, so we normally set Sh equal to zero where Ro > Rt. Elsewhere, the log response indicates significant hydrocarbon saturation. Experienced petrophysicists will normally examine the logs for visual patterns that indicate hydrocarbons. The example model in figure 6 shows some classic log hydrocarbon responses where Rt increases and Ro decreases (see the intervals 1080 - 1090, 1095 - 1100 and 1130 - 1135).

Figure 6: Deep and shallow apparent resistivity - detail :

Apparent hydrocarbon saturation is calculated by applying the Archie equation;

Sh = 1 - [(Ro / Rt) ^ (1 / n)]

Hydrocarbon pore-vlolume thickness is the depth-integrated product of porosity and hydrocarbon saturation. When multiplied by the reservoir area this number gives the in-place hydrocarbon volume.

Modelling Results - Case 1: Unimodal Porosity

Figure 7 shows the results of modelling a reservoir consisting of a unimodal porosity distribution. This is the most simple modelling scenario. Significant hydrocarbons are indicated using conventional log analysis methods.

	Average Porosity: 13 percent Standard Deviation: 2 percent Hydrocarbon Pore Volume Thickness: 0.553
Figure 7: Deep and shallow apparent resistivity: Unimodal porosity distribution

Modelling Results - Case 2: Bimodal Porosity

Figure 8 shows the modeling results for a reservoir with bimodal porosity distribution. Significant hydrocarbons are indicated using conventional log analysis methods.

	Higher Porosity Average Porosity: 13 percent Standard Deviation: 2 percent Lower Porosity Average Porosity: 7 percent Standard Deviation: 1 percent Hydrocarbon Pore Volume Thickness: 0.857
Figure 8: Deep and shallow apparent resistivity: Bimodal porosity distribution

Modelling Results - Case 3: Tight Sand With 'Sweet Spots'

Figure 9 shows a portion of the modeling results for a tight reservoir with higher porosity 'sweet spots'. The model does not contain any hydrocarbons but significant hydrocarbons are indicated using conventional log analysis methods. This results from the influence of the dominant low porosity, high resistivity rock investigated by the deep reading logging device.

	Annotated values are porosity Yellow cells: "High" porosity cells Purple cells: "Low" porosity cells Speckled shading: Cells measured by Rt log Hatched shading: Cells measured by Ro log
Figure 9: Example of reservoir model grid with mostly tight sands

Figure 10 shows a small portion of the modelled deep and shallow resistivity. In this example the low porosity population is dominant resulting in baseline resistivity values that are far higher that for the earlier examples where the reservoir is dominated by the higher porosity population. Reservoir properties close to the well are used to predict wet resistivity (Ro).

	Higher Porosity Average Porosity: 13 percent Standard Deviation: 2 percent Lower Porosity Average Porosity: 7 percent Standard Deviation: 1 percent Hydrocarbon Pore Volume Thickness: 0.524
Figure 10: Modelled deep and shallow resistivity - dominantly low porosity with 'sweet spots'

Lessons From Stochastic Modelling of Heterogeneous Reservoirs

Serious errors in hydrocarbon reserves estimates can result when combining data with different volumetric response functions. This happens primarily in heterogeneous reservoirs when high resolution log data (measurement close to the wellbore) 'sees' more optimistic reservoir properties than the deeper reading logs (Induction logs for example).

Some common causes of reservoir heterogeneity are;

• Normal variance of reservoir properties,
• Normal reservoirs with diagenetic cemented zones,
• 'Tight' sands (low porosity) with higher porosity 'sweet spots'

Close visual examination of the log data is essential for a robust Petrophysical interpretation. Stochastic modelling clearly shows that wireline log response to reservoir heterogeneity results in very 'noisy' interpretations. This is a warning signal that the digital log interpretation may not be robust.

Here are some indications of a robust log analysis;

• Ro(predicted) and Rt(measured) curves should overly in water saturated reservoirs and also in non-reservoir sections,
• The Ro and Rt curves should have similar vertical resolution. Looking at Figures 6, 7 and 10, for example, it is obvious that the Ro log has higher resolution than the Rt (deep reading) log. Our modelling suggests that this log pattern is most likely due to reservoir heterogeneity.
• There should be a coherent hydrocarbon response. This has two components. Ro will decrease when porosity values increase (Ro value is dominated by value of porosity raised to the power of about -2). If there is significant hydrocarbon saturation then the Rt value should increase as Ro decreases. Looking at figure 8, visual examination of the log data shows that there are really no intervals that would confidently be interpreted as hydrocarbon-bearing. The thin zone at about 1152 has a good hydrocarbon signature, but this zone is pretty isolated and does not form a component in a more wide-spread hydrocarbon response.

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