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RESISTIVITY CONCEPTS -- ARCHIE'S "LAWS"

RESISTIVITY BASICS
This Section covers resistivity concepts for real rock, including water filled, hydrocarbon saturated, and shaly rocks. All resistivity logs respond to real rocks as described here, and all petrophysical analysis procedures to determine water saturation, and hence hydrocarbon saturation, are based on the theories developed below.

Since most sedimentary rock minerals are very poor conductors (or good insulators), how does electric current flow through a rock? Many investigations have shown that most of the current flows through the water in the pores and not through the rock material. Therefore, the manner in which currents flow through water must be examined.

Pure water is also a very poor conductor. However, if salt is added to water, the solution becomes more conductive. Current is conducted through water by ions formed from the salt in solution in the water. The more ions present in the solution, the more conductive the solution will be. Since most natural waters in rocks contain salts of various kinds, the majority of natural waters are conductive.

To develop an understanding of how logging systems respond to various types of rocks the manner in which pores are interconnected must be visualized. The simplest system to imagine is an unconsolidated sand as shown below. In such a rock, the sand grains are piled on top of each other and the pore system is the space remaining between grains.


Various pore geometries have different effective path length

The grains may or may not be of uniform shape and size and packing. Cementing material may bind the particles together. All of these conditions would affect the pore system. Some other types of pores are also shown.

Thus, the distribution of pores in various rock types are not uniform, but dependent upon the genetic origin of the rock and the subsequent geologic changes to which it has been subjected.

Using the methods already developed, the equation for the resistance of a one meter cube to current flow through two parallel faces can be written:
   1. Resistance of cube of water = resistivity of water * length / area

If length and area both equal 1, then:
    2: Resistance (ohm) = RW (ohm - meter)

WHERE:
  RW = resistivity of water (ohm-m)

The resistance, in ohms, of a one meter cube of water is numerically equal to the resistivity of the water in ohm-meters. This is true for any material or combination of materials and is not restricted only to water.


 Effective path length

Now consider a one meter cube of rock that is 100% water saturated, that is, all the pores are filled with water. Resistivity of the cube may be written in terms of the current path length, area and resistivity.

The resistivity of the current path is RW and the path length (Le1) is at least one meter, but probably longer. The area is proportional to porosity. Thus:
    3: Ro = RW * Le1 / PHIe

WHERE:
  Le1 = effective path length (meters)
  PHIe = porosity (fractional)
  Ro = resistivity of rock filled with water (ohm-m)
  RW = resistivity of water (ohm-m)

The concept of formation resistivity factor is one of the most important in petrophysical analysis. It can be described by the expressions just developed. Formation resistivity factor is the ratio of the resistivity of a 100% water saturated rock to the resistivity of the water with which it is saturated.
    1: F = Ro / RW

In terms of the cubes developed, F becomes:
   4: F = Ro / RW = RW * Le1 / PHIe * 1 / RW = Le1 / PHIe

WHERE:
  F = formation factor (fractional)
  Le1 = effective path length (meters)
  PHIe = porosity (fractional)
  Ro = resistivity of rock filled with water (ohm-m)
  RW = resistivity of water (ohm-m)

From laboratory measurements made on rock samples it has been found that formation factor remains constant for a wide range of water resistivity values, as shown in the top half of the illustration below.


Formation Factor vs porosity (top) and Resistivity Index vs water saturation (bottom)
for a number of rock samples as measured in the laboratory.

Consider a cube of rock containing water and a hydrocarbon. Then:
    5: Rt = RW * Le2 / (PHIe * Sw)

WHERE:
  Le2 = effective path length (meters)
  PHIe = porosity (fractional)
  Rt = resistivity of the rock filled with water and oil (ohm-m)
  Rw = resistivity of water (ohm-m)
  Sw = water saturation (fractional)

Resistivity index is the ratio of the resistivity of the rock, to the resistivity of 100% water saturated rock, which was derived earlier. Thus:
    6: RI = Rt / Ro

Using the cubes already defined, the resistivity index becomes:
    7: RI = RW * Le2 / (PHIe * Sw) * PHIe / (RW * Le1) = Le2 / (Sw * Le1)

WHERE:
  Le1 = effective path length of water saturated rock (meters)
  Le2 = effective path length of oil and water saturated rock (meters)
  PHIe = porosity (fractional)
  RI = resistivity index (ohm-m)
  RW = resistivity of water (ohm-m)
  Sw = water saturation (fractional)

This indicates that the resistivity index varies with a number of rock properties. The graphical representation of this is shown in the bottom of the illustration shown above.

 

ARCHIE'S LAWS IN CLEAN ROCKS
Relationships just developed for cubes are not practical working equations. It is impossible to measure or assign a value to Le1 and Le2, the effective current path lengths for the two different saturation conditions.

Much work has been done to develop empirical relationships between water resistivity, porosity, and water saturation. G.E. Archie published a paper in the 1942 transactions of AIME entitled "The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics". The empirical relationships which he presented are still the most widely used today and are referred to as the Archie Equations and sometimes as Archie's Laws, although they are not as rigorous as laws are supposed to be..

First, he showed with core samples, as we have previously defined, that formation resistivity factor, water resistivity and rock resistivity are related by the following expression over wide ranges of porosity:
  8: F = Ro / RW

WHERE:
  F = formation factor (unitless)
  Ro = resistivity of rock filled with water (ohm-m)
  RW = resistivity of water (ohm-m)

Second, he showed that measured formation factor and porosity are related by the general expression:
    9: F = 1 / (PHIe ^ M)

WHERE:
  F = formation factor (unitless)
  M = cementation exponent (unitless)
  PHIe = porosity (fractional)

Archie hoped that A = 1.0 and M = 2.0, which would have made life easier in the pre-computer era, but these parameters vary in real rocks.

The formula applies for wide ranges of porosity of the same rock type. The exponent M is referred to as the cementation factor or the cementation exponent. Depending on the rock type, the value for M varies from approximately 1.3 to 2.6. The exponent M seems to be a function of the degree of cementation in clastic rocks.

The parameter A is called the tortuosity factor and clearly is related to the path length of the current flow, Le1 (or Le2). The value lies in the range 0.5 to 1.5.

Finally, Archie stated that water saturation is related to the rock resistivity by the expression:
    10: Sw = (Ro / Rt) ^ (1 / N)

WHERE:
  N = saturation exponent (unitless)
  Ro = resistivity of rock filled with water (ohm-m)
  Rt = resistivity of rock filled with water and oil (ohm-m)
  Sw = water saturation (fractional)

The exponent N is referred to as the saturation exponent, and is generally considered to have a value near 2.0. The range is similar to M, 1.3 to 2.6. Some petrophysicists treat N = M, but there is no experimental evidence for this.

These empirical relationships eliminate the need to know the effective path lengths described earlier. Note that Ro / Rt in the above equation is the inverse of the resistivity index (RI).

Archie's equations are summarized as follows:
    11: F = Ro / RW
    12: F = A / (PHIe ^ M)
    13: RI = Rt / Ro
    14: Sw = (1 / RI) ^ (1 / N) = (F * RW / Rt) ^ (1 / N)

WHERE:
  A = 1.0 = tortuosity factor
  F = formation factor (unitless)
  M = cementation exponent (unitless)
  N = saturation exponent (unitless)
  PHIe = porosity (fractional)
  RI = resistivity index
  Ro = resistivity of a rock filled with water (ohm-m)
  Rt = resistivity of a rock filled with water and oil (ohm-m)
  RW = resistivity of water (ohm-m)
  Sw = water saturation (fractional)

These equations are the basis for most resistivity log interpretation methods in use today, with the exception of very shaly sandstones, which are discussed later in thus Section..

The Winsauer porosity equation was developed as a result of a study on the formation factor versus porosity relationship of many different sandstones by W.O. Winsauer, H.M. Shearin, Jr., P.H. Masson, and M. Williams. It was published in 1952 in the Bulletin of the AAPG, in a paper entitled "Resistivity of Brine Saturated Sands in Relation of Pore Geometry". They concluded that, for the rocks studied, an expression of the following form described their experimental data:
    15: F = A / (PHIe ^ M)

WHERE:
  A = tortuosity exponent (unitless)
  F = formation factor (unitless)
  M = cementation exponent (unitless)
  PHIe = porosity (fractional)

They evaluated A and M in terms of their data with the following result:
    16: F = 0.62 / (PHIe ^ 2.15)

WHERE:
  F = formation factor (unitless)
  PHIe = porosity (fractional)

The Winsauer equation is widely used today for the evaluation of sandstones from resistivity logs. The unmodified Archie formula with A = 1.0 and M = 2.0 is often used in carbonates. Most analysts prefer to find values of A and M for the particular rocks they are evaluating. .

It is often necessary to compute porosity from the formation factor, by inverting the Archie or Winsauer equation:
    17: PHIe = (A / F) ^ (1 / M)

WHERE:
  A = tortuosity exponent (unitless)
  F = formation factor (unitless)
  M = cementation exponent (unitless)
  PHIe = porosity (fractional)

Numerical Example:
  1. Formation Factor from Resistivity:
      Ro = 10 ohm-m
      RW = 0.1 ohm-m
      F = 10 / 0.1 = 100

  2. Formation Factor from Porosity (Archie):
      PHIe = 0.20
      M = 2.0
      F = 1 / (0.20 ^ 2) = 25

  3. Porosity from Formation Factor (Archie):
      F = 100
      M = 2.0
      PHIe = (1 / 100) ^ (1 / 2) = 0.10

  4. Water Saturation (Archie):
      Ro = 10 ohm-m
      Rt = 100 ohm-m
      RW = 0.1 ohm-m
      N = 2.0
      Sw = (10 / 100) ^ (1 / 2) = 0.32

  OR F = 10 / 0.1 = 100
        Sw = (100 * 0.1 / 100) ^ (1 / 2) = 0.32


  5. Formation Factor from Porosity (Winsauer):
      PHIe = 0.20
      M = 2.15
      A = 0.62
      F = 0.62 / (0.20 ^ 2.15) = 19.7

  6. Porosity from Formation Factor (Winsauer):
      F = 18.7
      M = 2.15
      A = 0.62
      PHIe = (0.62 / 18.7) ^ (1 / 2.15) = 0.198

 

ARCHIE'S LAWS EXTENDED TO SHALY ROCKS
The extensions of the Archie resistivity equations for formation factor and water saturation to the case of shaly sands has been undertaken by many investigators over the last 45 years. More than 30 different water saturation models have been proposed over that period, of which three or four are more popular than the others.

The following material, which summarizes the scene very effectively, has been condensed from "The Evaluation of Shaly-Sand Concepts in Reservoir Evaluation", by P. E. Worthington, published by SPWLA in the Log Analyst for Jan-Feb, 1985.

"The emergence of the shaly-sand problem as it affects resistivity data can be more readily traced by considering only conditions of full water saturation in the first instance. A convenient starting point is the definition of formation factor F which was first of three equations proposed by Archie, namely:
    18: F = Ro / RW = CW / Co

WHERE:
  F = formation factor (unitless)
  Ro = resistivity of a rock filled with water (ohm-m)
  RW = resistivity of water (fractional)
  Co = conductivity of a rock filled with water (ohm-m)
  CW = conductivity of water (fractional)

A plot of Co vs CW for a given sample should furnish a straight line of gradient 1/F provided that Archie's experimental conditions of a clean reservoir rock fully saturated with brine are completely satisfied. Subject to these conditions the formation factor is precisely what the name implies; it is a parameter of the formation, more specifically one that describes the pore geometry. It is independent of CW so that a plot of CW/Co vs CW for a given sample should furnish a straight line parallel to the CW axis, as in the left hand illustration below.


 Effect of conductive minerals on resistivity - clean sand (left) and shaly sand (right).

However, around 1950 there was increasing evidence from various formations to suggest that the ratio CW/Co is not always a constant for a given sample but can actually decrease as CW decreases. The relative decrease in CW/Co at a given level of CW appeared to be more pronounced for shalier specimens. Since CW was presumed to be known, the only possible explanation for this phenomenon lay in the effect of the shale component of the reservoir rock upon Co. This effect was essentially to under reduce Co as CW decreased or, to put it another way, to impart an extra conductivity to the system at lower values of CW. For this reason the electrical manifestation of shale effects has been described in terms of an "excess of conductivity." It became advisable to regard the ratio CW/Co as an apparent formation factor Fa which is equal to the intrinsic formation factor F only when Archie's assumptions are satisfied.

Since the Archie equation was not found to be valid for all formations, a more general relationship between Co and Cw was sought in order to accommodate the excess conductivity. By rewriting the Archie equation and incorporating the excess conductivity within a composite shale-conductivity term X, it was proposed that an expression of the following form is valid for all granular reservoirs that are fully water saturated.
    20: Co = CW / F + X

WHERE:
  Co = conductivity of a rock filled with water (mS/m)
  CW = conductivity of water (mS/m)
  F = formation factor (unitless)
  X = excess conductivity term (mS/m)

For a clean sand, X approaches 0.0 and the equation reduces to Archie's. If CW is very large, X has comparatively little influence on Co and again it effectively reduces to the Archie definition. Conversely, the ratio CW/Co is effectively equal to the intrinsic formation factor F only if X is sufficiently small and/or CW is sufficiently large. Thus, although the absolute value of X can be seen as an electrical parameter of shaliness, the manifestation of shale effects from an electrical standpoint is also controlled by the value of X relative to the term CW/F.

During the period 1950-1955 evidence began to accumulate that the absolute value of the quantity X is not always a constant for a given sample over the experimentally attainable range of CW but can vary with electrolyte conductivity. The most widely accepted behavioral pattern, which has continued to be supported, was that for a given sample, the absolute value of X increases with CW to some plateau level and then remains constant as CW is increased still further. This pattern is illustrated for hypothetical data in the right hand illustration shown above. Here the terms "non-linear zone" and "linear zone" have been adopted for the regions of variable X and constant X, respectively.

It was often the practice to estimate porosity from the ratio CW/Co using a standard version of Archie's second equation in conjunction with resistivity logging data from nearby water zones. In so doing it was essential to have sufficiently clean conditions for there to be a well defined relationship between porosity and CW/Co. Where this condition was satisfied it was still possible to proceed even if the ratio CW/Co actually represented an apparent formation factor Fa instead of the intrinsic formation factor F. In the former case A and M would be pseudo-parameters which would compensate for any departure of Fa from F when calculating porosity.

The advent of porosity tools has resulted in a change of usage of Archie's second equation. It is current practice to infer porosity from porosity tool response(s) and then to calculate F using pre-determined values of A and M. In this case the resulting value of F will be wrong if the parameters A and M do not themselves relate specifically to effectively clean conditions, but have inadvertently been established on the basis of a correlation of Fa with porosity. This error can be readily transmitted to subsequent estimates of water saturation.

The shaly-sand models introduced since about 1960 have been divided into two groups:

1. Concepts based on the shale volume fraction, Vsh. These models have the disadvantage of being scientifically inexact with the result that they are open to misunderstanding and misuse. On the other hand they are applicable to logging data without the encumbrance of a core sample calibration of the shale related parameter.

2. Concepts based on the ionic double-layer phenomenon. These models have a more attractive scientific pedigree. If strictly applied, they require core-sample calibration of the shale related parameter against some log derivable petrophysical quantity. Otherwise their field application might involve approximations which effectively reduce the shale term to one in Vsh.

Although Vsh models are being progressively displaced by models of the second group, this process will not be complete until there exists an established procedure for the downhole measurement of X.

Vsh models gained credence because of earlier experimental work which showed potentially useful relationships between the amount of "conductive solids" present within a saturated granular system, such as a clay slurry, and the conductivity of the solid phase. These early data did not relate to typical reservoir rocks and they have subsequently been extrapolated far beyond their original limits. Attempts to explain the physical significance of the parameter X in terms of Vsh have had either a conceptual or an empirical basis.

Apart from the unavailability of a "universal" Vsh equation there is one other major disadvantage of Vsh models; the Vsh parameter does not take account of the mode of distribution or the composition of constituent shales. Since variations in these factors can give rise to markedly different shale effects for the same numerical shale fraction, improved models were sought which did take account of the geometry and electrochemistry of mineral-electrolyte interfaces.

The term "double layer model" is used here to describe any conceptual model which draws directly or indirectly upon the ionic double-layer phenomenon.

Waxman and Smits explained the physical significance of the quantity X in terms of the composite term BQv/F*, where Qv is the cation exchange capacity per unit pore volume, B is the equivalent conductance of sodium clay exchange cations (expressed as a function of CW at 25 degrees C) and F* is the intrinsic formation factor for a shaly sand. Thus, the Waxman-Smits model assumes that the conducting paths through the free pore water and the counterions within the ionic double layer are subject to the same geometric factor F*. The dependence of B upon CW allowed X to vary with CW so that both the non-linear and the linear zones of thr CW/Co graph could be represented through one parallel-resistor equation.

Clavier et al. sought to modify the Waxman-Smits equation to take account of experimental evidence for the exclusion of anions from the double layer. This was done in terms of a "dual water" model of free (formation) water and bound (clay) water. It was argued that a shaly formation behaves as though it were clean, but with an electrolyte of conductivity CWE that is a mixture of these two constituents. Thus the Archie definition was rewritten:
    21: Co = CWE / Fo

WHERE:
  Co = conductivity of a rock filled with water (mS/m)
  CWE = effective conductivity of water and shale (mS/m)
  Fo = apparent total formation factor (unitless)

This equation forms the basis of the dual water model.

It can be inferred by rearranging the dual water equation that the geometric factors associated with the two "parallel" conducting paths are not equal. Furthermore, the presence of the variable parameter in the shale term allowed X to vary with CW at low salinities. This meant that both the non-linear and the linear zone could be represented through this one equation.

The extension of these shaly-sand equations to take account of the hydrocarbon zone requires that terms in water saturation be inserted into the relationships. This has generally resulted in one of two distinct outcomes, a change in only the clean sand term of the corresponding water-zone equation or a modification of both the clean and shaly terms.

Changes to the clean term have been based upon Archie's third equation for clean sands If only the clean term is changed, the general water-zone equation can be transformed to:
    22: Ct = CW / F * (SW ^ N) + X

WHERE:
  Ct = conductivity of a rock filled with water and oil (mS/m)
  CW = conductivity of water (mS/m)
  F = formation factor (unitless)
  N = saturation exponent (unitless)
  SW = water saturation (fractional)
  X = excess conductivity term (mS/m)

so that when X is very small or CW is very large, this equation reduces to the clean sand equation.

Changes to the shale term have usually had the effect of introducing a factor SW ^ S where S can be loosely regarded for our purposes as a "shale-term saturation exponent." Thus, if both the clean and the shale terms are changed, the general water-zone equation can be transformed to:
    23: Ct = CW / F * (Sw ^ N) + X * (Sw ^ S)

WHERE:
  Ct = conductivity of a rock filled with water and oil (mS/m)
  CW = conductivity of water (mS/m)
  F = formation factor (unitless)
  N = saturation exponent (unitless)
  S = saturation exponent for shale term
  SW = water saturation (fractional)
  X = excess conductivity term (mS/m)

Again, this reduces to Archie in clean formations.

Specific examples of these equations are grouped into four categories in Worthington's paper, and are reproduced below. These are not "solution " equations - you would have to solve for Sw using quadratic or iterative methods.  Go to CPH Indes to find Sw solutions in Chapter Thirteen.



Paul Worthington's annotated list of water saturation equations for conductive minerals
 

Copyright E. R. (Ross) Crain, P.Eng.  email
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