Estimating Reservoir Permeability Through Well Logging

М. М.  Khasanov, К. V. Тoropov, А. А. Lubnin, (NK Rosneft OJSC)

Introduction
Many major problems in the simulation of the hydrodynamic models of oil reserves are connected with profile estimations of the vertical distribution of permeability in drill-holes. Known methods of macro-scale (selective flow) permeability testing – drill-stem testing – estimation on the basis of Geophysical Drill-Hole Logging (GDHL) and detailed flow measurements provide more relative rather than absolute information [1]. These methods allow the estimation of the intervals with maximum and minimum permeability. However, quantitative estimates, obtained on the basis of these methods, are rather rough and, what is more important, they are not compatible with the mega-scale, that is average permeability estimates along the reservoir section, obtained with the help of Drill-Hole Hydrodynamic Research (DHHDR) or on the basis of the data obtained during normal operation of the drill-hole. The reasons for these problems are discussed in detail in the study [1]. When there is a more widespread source of information, such as Geophysical Drill-Hole Logging (GDHL), these problems are due to the fact that GDHL provides data for estimating the volumetric measures, such as porosity and saturation, but not the dynamic measures, such as permeability. Empirical formulae, depending on porosity and permeability factors or more complex algorithms (multiple regressions, neural networks), connecting permeability with the volumetric characteristics, and measured based on the results of GDHL, provide of low accuracy estimates.

Notwithstanding the above problems, the existing methods of selective flow permeability estimation, in particular methods based on GDHL, are extremely useful, due to their comparative function; that is a comparative instrument for the comparison and ranking of various intervals/flows on the basis of their permeability.

To construct an adequate, quantitative profile of the vertical distribution of permeability, it is necessary to work out a solution for the integration of various data, obtained from different information sources (well-core analysis, reservoir test, flow measurements and GDHL) into a single model when measuring on various (macro and mega) scales [1]. This being said, it is necessary to take into account the following factors:

Differences in the measuring conditions. Core analysis is carried out in a laboratory environment after special treatment of the samples, resulting in significant deviations between the permeability parameters measured in the laboratory and the permeability in the “in-situ” environment.   Furthermore, the absolute permeability is estimated on the basis of the core, whereas the reservoir test or the interpretation of the Drill-Hole Hydrodynamic Research (DHHDR) data provide the parameters for effective permeability, corresponding to the actual saturation of the reservoir.

Calibration test rqguirments of permeability estimates obtained from GDHL data. When integrating macro-scale (GDHL) and mega-scale (DHHDR) permeability estimates in one single model, it is necessary to ensure that for each drill-hole the average permeability through the reservoir thickness, estimated on the basis of GDHL data, is equal to the average permeability through the reservoir thickness, estimated on the basis of DHHDR results or on the basis of normal drill hole operation data. Usually, these conditions are ensured by multiplication of the permeability parameters, obtained on the basis of GDHL data, by a certain correction factor, which is the same for all levels of the reservoir in the drill-hole and for all drill-holes, located in similar geological conditions [1].

Thus, the parameters of average permeability, estimated on the basis of DHHDR data or normal operations, are acknowledged as a reference for the calibration of GDHL data.

The correction of permeability parameters by the multiplication of the correction factor achieves a close alignment of the estimated drill-hole production rate with the actual production rate. However, it is not possible to simultaneously work out an adequate description of oil production rates using the same method (e.g. a well defined dynamic analysis of the flow of water in drill-hole production).

The reasons can be understood if we examine the permeability of the reservoir levels as a random value, the distribution function of which (normally assumed as logarithmically normal) depends on two parameters – the average value (mathematical estimate) and the range of deviations from the average value (dispersion). The dispersion identifies the level of the vertical heterogeneity of the reservoir and, in particular, the water-oil displacement characteristics (e.g., according to the Dykstra-Parsons model [2-6]). The multiplication of the permeability values by the correction factor is equivalent to the shift of a cumulative distribution function along a “permeability logarithm” axis. This shift allows for the achievement of the desired value of average permeability, however, the range of the permeability distribution curve is out of control because of this, so that the desired values of heterogeneity are not achievable.

This study offers a method for the correction of permeability values, allowing the desired type of distribution function to be achieved (e.g. logarithmically normal) with the specified data – mathematical estimates and dispersion. This method is based on the results of the theory of order statistics [7, 8] and permeability recovery capability in the sample collection range upon availability of prior information in a type of distribution function. It is assumed that the permeability range of the reservoir is carried out with the help of GDHL. Mathematical estimates should be made on the basis of DHHDR or normal operation data, and the dispersion – on the basis of core sample analysis or water flooding developments.  One of the main advantages of the proposed method is a considerable reduction in the time required for the process of matching the hydrodynamic 3D models to the history of oil production on site (history matching) and the improvement of the stability of    the results due to the fact that the initial approximation to the field permeability estimate (at the stage of model initialisation) is matched as much as possible to the actual flow properties of the reservoir. This is achieved through the use of all the available laboratory and field data, including statistical data on the field production performance (oil and other fluids) at the stage of model initialisation.

Practical application of the proposed method has been reviewed using the example of one of the oilfields of NK Rosneft OJSC.

Statement of the Problem
Let us assume that a drill-hole is working on a reservoir of effective thickness H, consisting of n levels (beds and intermediate beds); with hi and ki being the thickness and absolute permeability of the i-interval, respectively, and kgi being the absolute permeability of the i-level, obtained based on GDHL data (i =1, 2, …, n).

As was mentioned before, kgi values may differ considerably from the true ki values, therefore the direct use of kgi may result in low accuracy in estimating the two most important indexes: drill-hole productivity and vertical non-homogeneity of the reservoir permeability, which determine, to a large extent, the displacement efficiency of oil by water and gas.

These two indexes are connected with the parameters of the permeability distribution function by the mathematical calculation

(1)

and the dispersion

(2)

where E[•] means the process for estimating the mathematical expectation; and σ is the root mean square (standard) deviation of permeability. Drill-hole productivity is proportional to the product KH (K being the average permeability along the thickness of the reservoir, which is determined by the influx of the reservoir fluid to the drill-hole). The value K may be estimated by processing the DHHDR results. If active measurements for DHHDR are not available, it is recommended to use normal operation data (passive measurements) for estimating K. Normal operation data are understood to be a dynamic series of well production rate and bottom-hole measurements, carried out during well operation in the pseudo-steady state (with a relatively small change in the bottom-hole pressure due to natural reasons), or during the changeover for a recently drilled well to steady production.

The value K is determined by the analysis of the fluid flowing into the well in natural reservoir conditions, therefore K is the effective permeability. Making an allowance for this, we can write:

(3)

where Kr – is relative permeability with average fluid saturation throughout the thickness of the reservoir. If during the exploration, the influx of crude oil without water is produced (which happens in most cases), Kr will represent the relative permeability for the oil in the presence of the interstitial water in the reservoir.

Permeability probability distribution along the section is normally described as a logarithmically normal function. Thus, the probability distribution function ϕ(x) of x = ln k will be determined by the formulae:

(4)

where µ is the mathematical expectation and σIn is the standard deviation of the permeability logarithm.
It is easy to the obtain permeability probability distribution function F(x) for the logarithmically normal law from the formula (4):

(5)

where f (k) – is the permeability probability distribution density, estimated by the formula

(6)

The mathematical expectation k and the mean square deviation σ of permeability are connected with the parameters of µ and σIn by the formulae [3, 4]

(7)

(8)

We assume that kp is the permeability estimated with the formula

(9)

As it appears from the mathematical meaning of the distribution function F(k), the p value determines the percentage of the intermediate strata with k < kp permeability in the effective thickness of the reservoir. For example, k50 is the median value; intermediate strata with permeability less than k50, and intermediate strata with permeability higher than k50, will each account for 50% of the effective thickness. The permeability median value is normally identified by the symbol km, that is k50 = km. Since the function ϕ(x) in the formula (4) is symmetrical to the axis x = µ, we obtain µ = ln k50 = ln km.

As a result, the formula (7) can be represented as follows:

(10)

As is known, the deviations of the argument, with normal distribution, to the left and to the right from the median value by the value of the mean square deviation, result in the following approximate values for the distribution function: 0.16 (16%) and 0.84 (84%).
Therefore (Fig. 1):

(11)

Thus, the degree of heterogeneity of the reservoir may be characterised as:

(12)

However, for this purpose, the Dykstra-Parsons coefficient is normally used in modern oil engineering practice [2-6]:

(13)

As follows from (11) and (13), the coefficient VDP for logarithmically normal distribution is connected with the parameter σIn by the formula:

(14)

By inserting formula (14) into formula (8), we obtain:

(15)

so that:

(16)

where:

(17)

Productive strata are most commonly characterised by VDP values in the range 0.7 to 0.8. Therefore, the most commonly encountered values of the coefficient A (VDP) for heterogeneous strata are in the range of 2 to 3.5, i.e. the mean permeability value is 2 to 3.5 times higher than the median value.

An alternative characteristic of the degree of non-homogeneity is the coefficient of the variation V=σ / k . For logarithmically normal distribution, based on formulae (7) and (8), we obtain:

(18)

The value VDP can be obtained based on a direct analysis of the core sample or by processing the oil field data on the water encroachment dynamics of the production wells.

In the latter case, an adequate value of VDP is determined by matching the displacement characteristics, obtained within the stratified model of the stratum [2, 5, 6] with the characteristics observed in reality.

The formulae (14) and (16) are connected by the parameters µ and σIn of the logarithmically normal permeability distribution function with the actual measured characteristics of the well – weighted mean strata thickness permeability k and coefficient VDP.

The permeability distribution function may often deviate from the logarithmically normal. However, knowing this factor, it is always possible to express the values of the distribution parameters with the values k and VDP (we will only examine two-parameter distribution functions).

It is clear from the assessment above that from the mathematical point of view, the task of identifying the profile of the vertical distribution of the permeability in a drill-hole is confined to finding the functional connection between the permeability values, obtained on the basis of GDHL and its true values

(19)

This functional connection allows for the fulfilment of the following conditions:
»    the function λ(k) is uniform;
»    the function of permeability distribution along the section ƒ(k), estimated with the help of function (19), has a predetermined form;
»    the parameters of the function ƒ(k) ensure that the estimated values of k and VDP coincide with the estimates obtained based on experimental and oil field data.

Further, to simplify the explanation, we shall assume that the function ƒ(k) is logarithmically normal, although the methods proposed in this study, are applicable for any distribution. The method of correction of kg values based on their multiplication by some correction coefficient to achieve the desired mean permeability value, often used in practice, means that the function λ(k) is assumed to be linear:

(20)

where the multiplier C is identified from the condition

(21)

resulting from the formula (3).

By taking the logarithm of (20), we obtain

(22)

Fig. 2 shows that the conversion (22) doesn’t allow us to obtain the desired distribution function (curve 3), since it doesn’t change either the shape of curve 1, or its range.

Method of Non-Reference Measurements
Let’s assume that we have a pile of stones. Is it possible to estimate the weight of each stone having mechanical beam scales without balance weights? In a more general form, this task is referred to as a task of non-reference measurement [7] and may be formulated as follows.

Let’s assume that (x1, x2, … xn) is a selection consisting of n occurrences of a random value X; xi is the occurrence number i of a random value (i = 1, 2, … n). Is it possible to estimate value xi, having only the comparing element, which allows to compare  xi  values with one another, but without having any references, which would allow us to measure the xivalues directly? (In our example with the stones, the scales are the comparing element, and missing references are the weights).

In [7] it is shown that it is possible to solve this task with the help of the theory of order statistics, provided that the function of distribution of the random value X is known.  It is easy to see that the problem of estimating the true permeability of the strata intervals ki based on GDHL data is also a task of non-reference measurement, and therefore may be resolved by the methods of the theory of order statistics.

Switching to a probablistic description, let’s review a random set of selections (x1, x2, … xn). In many practical applications [4, 7 – 9], it is useful to use organised selections within a range of values (x(1), x(2), … x(n)) obtained from the initial selections  (x1, x2, … xn) by transposition of the values xi in the increasing order  (x(1), x(2), … x(n)).

The value i in the range of values selected is said to have the ordinal number or rank i. It is clear that the values that have the same rank i in various random selections will be different due to random chance. Therefore the values x(i) are occurrences of some random value, which we will identify as x(i). Therefore, the random range of selected values can be presented as a set of random values
(x(1), x(2), … x(n))).  The element x(i) of this selection is called the i-order statistic, and the area of mathematical statistics studying the properties of ordered selections is called the Theory of Order Statistics [7, 9].

If ƒ(x) is the probability density and F(x) is the function of the distribution of the original population, then the distribution density of i-order statistics Ψi(x) in the selected range n can be estimated by the formula [7]

(23)

The mean value Ei of order statistics is:

(24)

The dispersion of order statistics is:

(25)

Fig. 3 shows graph curves for the initial random value distribution density, having a normal distribution ƒ(x), and a distribution of order statistics Ψi(x), when i = 6 and i = 20 (n = 30), as an example. Fig. 3 shows that the probability density of order statistics and represents a rather “narrow” function, therefore for the estimation of the unknown value x(i), when the number n has a high value, it is possible to use the mathematical estimate Ei: x(i) ≈ Ei

The idea of this substitution is the basis of the non-reference measurement method [7]. The tolerance error
x(i) – Ei when using this method will be reduced with the increase of the selection size n.

The estimation of E1 values is carried out using the formulae (23) and (24) and by a numerical integral evaluation (24).

Non Reference Measurement Method for the estimation of permeability distribution in drill holes
First of all, let us assume that the permeability values on the basis of GDHL kgi are obtained at equal formation intervals h1 = h2 = …= hn. This applies when the well-logging curves and the results of their interpretation are presented in an LAS-format, when the parameters of the formation are estimated evenly, at every 0.1 or 0.2 metres. This restriction allows a direct application of the idea of non-reference estimations, described above.

Let us assume that the permeability distribution is logarithmically normal. The following initial information is provided:
» values of permeability of the formation intervals kgi obtained on the basis of GDHL data (i=1, 2, … n);
» average permeability through the reservoir thickness K.
» Dykstra–Parsons coefficient of heterogeneity VDP

It is required to estimate the true permeability of the ki intervals (i = 1, 2, … n);

The task shall be resolved in the following sequence:
1. The distribution parameter σln (4) is estimated using formula (14).
2. The mathematical permeability expectation is estimated using formula (3).
3. The distribution parameter μ (4) is estimated using formula (16).
4. The formation intervals shall be arranged in increasing order by comparing the kgi values corresponding to different intervals. Thus, the initial selection (kg1, kg2, …, kgn) will be transformed to a selection in the sequence
(kg(1), kg(2), …, kg(n)).
5. The mathematical expectations of order statistics Ei will be estimated by numerical integration using formulae (23) and (24), which allows us to obtain a range of selected true permeability values (k(1), k(2), …, k(n)) using an approximate substitution k(i) ≈ Ei.
6. By comparing the range of selected values (kg(1), kg(2), …, kg(n)) and (k(1), k(2), …, k(n)), we obtain a functional connection between the true permeability and the permeability obtained on the basis of GDHL data: k = λ (kg).
The described order of calculations is used for all wells, where the required initial information is available. This being said, it is necessary to carry out a thorough comparative analysis of all the studied intervals and ratios obtained for all wells, in order to identify some general patterns, which may be also used safely for similar wells, where part of the initial information is not available (e.g. there are GDHL data available, but no data on the parameters of the true permeability distribution function – K and VDP).

Example: We assume that the average effective permeability of a formation cut open by a certain drill hole K=7.2 mD, the coefficient VDP =0.8, and the relative permeability Kr=0.8. The permeability values based on GDHL data kgi, forming a selection size of n=29, and characterising the formation intervals of 0.4 m thickness, are listed in Table 1 below in the order of increasing depth of the intervals.

1. Estimate of standard deviation using formula (14)

2. Average permeability is estimated with formula (3)

3. The parameter of logarithmically normal distribution (4) is estimated using formula (16)

4. The values kgi from Table 1 are arranged in Table 2 in increasing order.

5. The values of mathematical expectations Ei are estimated using formulae (23) and (24) and arranged in Table 2.

6. Reconstruct the permeability vertically by reverse rearrangement of intervals in the order of their increasing depth (see Table 1). Values k(i) from Table 2 to Table 1 shall be transferred so that the values of kg, located in one row with k(i) in Table 1 and Table 2 coincide.

Fig. 4 shows the relationship of k = λ(kg), reflecting the functional connection between the columns 2 and 3 of Table 1.

As you can see, this relationship is non-linear, unlike the relationship (20). It should be noted that the average value

and the coefficient of heterogeneity

coincide with the initially assigned values.

Practical application of this method
The proposed method for estimating the permeability range was tested at the time of the creation of the sector hydrodynamic model of one of the oilfields of Rosneft. Initially, for the estimation of the permeability of this oilfield, a relationship of porosity and permeability was used:

(26)

or

(27)

where m is the porosity in %, estimated on the basis of GDHL data.

As a result of this estimate, using a method of non-reference measurement, the adjusted values of permeability were obtained, the interval comparison of these with the porosity values, and following a generalisation of these estimates for all the wells, this showed that the relationship of porosity and true permeability is described by the function

(28)

The comparison of the initial (26) and the corrected (28) petro-physical functions are shown in Fig. 5. Fig. 6 shows the results of the estimates, obtained with the use of an initial hydrodynamic model and the model in which the permeability was estimated using the method of non-reference measurements. As can be seen, the proposed method allows the immediate achievement (without using the history matching procedure) of a very good match of the estimated results with the field data.

The details of the generalised estimates, assuming a random distribution of the values kgi along the depth of the formation (different hi) shall be reviewed in the next publication on this issue.

Abstract
The proposed algorithm for estimating  profiles of the vertical permeability distribution may be used for the re-interpretation of the GDHL results, with an allowance for well operation data. This method of permeability value correction allows us to achieve conformity between the design (estimated) capacity and the true well production capacity, and also for the solution of problems such as adequate descriptions of the characteristics of oil production from the reservoir (correct estimation of water encroachment dynamics).

On the basis of examples from the actual oil field, it was shown that the application of the proposed algorithm for the construction of 3D hydrodynamic models allows for a good initial match with the development history, without using a “history matching” procedure.

When deriving the key relationships, which form the basis of the algorithm, it was assumed that the thickness of the intervals were equal. This assumption complies with the presentation of the geophysical information in the LAS-file format. However, such files are not always available, and therefore the proposed method requires generalisation for cases where there are varying thicknesses in the intermediate strata.

Reference
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This article was published in the NK Rosneft Scientific and Technical Newsletter (Nauchno-tekhnicheskiy Vestnik OAO “NK “Rosneft”, No.4, 2009, pp. 14-21; ISSN 2074-2339) and won the 2-rd prize in the 2009 competition for the best publication in the newsletter.  Printed with permission from the Editorial Board.