CHAPTER TEN: PERMEABILITY, PRODUCTIVITY, RESERVOIR VOLUME, AND CASH FLOW

Table Of Contents
10.00 Introduction to This Chapter
10.01 Permeability of Fractures
10.02 Visual Indications
10.03 Irreducible Water Saturation
10.04 Permeability from the Wylie-Rose Method
10.05 Permeability from Formation Factor
10.06 Permeability from Porosity
10.07 Permeability from the Dumanoir and Coates Method
10.08 Permeability Examples
10.09 Relative Permeability and Water Cut
10.10 Net Pay and Cutoffs
10.11 Cumulative and Average Reservoir Properties
10.12 Reservoir Properties for Laminated Shaly Sands
10.13 Oil and Gas Reserves
10.14 Tar Assay Reserves (Weight)
10.15 Gas Hydrate Reserves
10.16 Productivity Index
10.17 Decline Rate and Economic Life
10.18 Production Profile and Cash Flow
10.19 Permeability and Productivity Routines
10.20 Calibrating Permeability to Core and Test Data <<<< NEW
10.21 In Conclusion
10.22 Exercises For Chapter Ten
10.23 Bibliography For Chapter Ten

Continue to Next Chapter

Publication History: This Chapter was originally published by Pennwell Books, Tulsa OK, in 1986 as Chapter Ten of "The Log Analysis Handbook". Section 10.12 updated for laminated shaly sands, Section 10.02 updated for NMR, permeability definitions (Section 10.00) updated and fracture permeability added (Section 10.01) August 2002. Discussion on selecting cutoffs added to Section 10, Apr 2004.

CHAPTER TEN: PERMEABILITY, PRODUCTIVITY, RESERVOIR VOLUME, AND CASH FLOW

10.00 Introduction to This Chapter
Most quantitative log analysis is aimed at defining shale content, porosity, and water saturation. These terms define the oil or gas in place in the reservoir at initial conditions. What we would really like to know is: "Is the well any good?" That is, will it produce anything, and if so, how much per day. To know this, we must determine values for permeability and productivity.

Permeability refers to the ease with which fluids flow through any substance. It is not sufficient to have oil or gas in a formation; the hydrocarbons must be able to flow from the reservoir into the well bore in order to be recovered at the surface. Absolute permeability is a physical characteristic of the rock. Permeability of a rock for oil, gas, or water is a function of the absolute permeability and the viscosity of the fluid.

Productivity describes the flow rate of oil or gas into the well bore. These two terms are obviously related. This Chapter explains how to determine these values from open hole log data.

Reservoir volume is a term used, with other adjectives, to describe the volume of hydrocarbon in the reservoir. It is also called the oil in place or gas in place. The phrases, reserves or recoverable reserves, refer to the amount of hydrocarbon in place that can actually be produced under the existing (or a proposed) recovery mechanism.

Recoverable reserves and productivity define, along with product prices and production costs, whether or not a well will make money; that is, "Is the well any good?"

Permeability is measured by flowing fluids through the rock under known conditions. This can be done on rock samples in the laboratory, or by flowing a well (in -situ measurement). It depends on the size and shape of the pores, the properties of the fluids, the pressure exerted on the fluid, and the amount of the fluid flow. Darcy's fluid flow equation relates these properties to permeability.

For linear horizontal flow:
1: Q = 1.127 * A * (K / MU) * (P1 - P2) / L

Where:
Q = quantity of fluid (bbl/day)
A = area fluid flows through (sq feet)
K = permeability (Darcies)
MU = viscosity of fluid (centipoise)
P1 - P2 = pressure differential (psi)
L = length of flow path (feet)

For non-horizontal flow (eg. up-dip in a reservoir) a gravity term must be included.

More importantly, fluid flow from a reservoir into a wellbore is not linear but radial, so the equation becomes:
2: Q = 3.07 * H * (K / MU) * (Pr - Pb) / log(Rr/Rb)

Where:
Q = quantity of fluid (bbl/day)
H = thickness of reservoir that fluid flows through (feet)
K = permeability (Darcies)
MU = viscosity of fluid (centipoise)
Pr - Pb = pressure differential from reservoir to wellbore(psi)
Rr = radius of reservoir = length of flow path (feet)
Rb = radius of wellbore (feet)

The unit used in measuring permeability is the Darcy. Permeabilties normally encountered in reservoir rocks range from less than one millidarcy in low porosity sandstones, to about fifty Darcies in fractured rock.

Although there are some general trends of increasing permeability with increasing porosity, these do not necessarily hold for any given situation. If the sand grains are large, then the pore throat diameters are large and the permeability is high. If the size of the sand grains is reduced by a factor of l00, the permeability is considerably smaller, but the porosity will be the same. Smaller pores mean larger surface areas around them, and therefore more resistance to flow (lower permeability). Various authors have proposed equations that account for these effects:
Slichner: K = 10.2 * D^2 / CK1
Terzaghi: K = CK2 * D^2 * ((PHIe - 0.13) / (1 - PHIe)^0.33)^2
Uren: K = CK3 * D^2 * PHIe^3.31
Kozeny K = CK4 * PHIe^3 / ((1 - PHIe^2) / (Sv^2)
Kozeny-Carmen K = CK5 * PHIe^2 * Rp^2 * Cpur

Where:
CK1 = 10 to 100 depending on porosity and grain packing
CK2 = constant to be determined by calibration
CK3 = constant to be determined by calibration
CK4 = 0.20
CK5 = 0.04444
Cpur = 0.216
D = average grain diameter (cm)
K = permeability (Darcy)
PHIe = porosity ((fractional)
Rp = pore throat radius (microns)
Sv = specific surface area of grains (sq cm)

Since D, Sv. and Rp are not easily measured, these are not very practical equations for use in log analysis. However, they do resemble the form of practical equations presented later in this Chapter.

Permeability measured with only one fluid in the pores is equal to the absolute permeability, because Darcy's equation accounts for the viscosity and pressures involved in the measurement process.

The effective permeability refers to permeability with more than one fluid present. Effective permeability is less than absolute permeability because the presence of a second fluid reduces the size of holes available for flow of the first fluid. If no fluid flows, the effective permeability of the rock to that fluid is zero.

Relative permeability is the ratio of effective permeability of a specific fluid to absolute permeability. Graphs of relative permeability curves, as in Figure 10.01 and 10.02, reflect the capacity of the rock to produce fluids by showing the permeability of those fluids as a function of saturation.

FIGURE 10.01: Typical relative permeability curves for oil and water

FIGURE 10.02: Relative permeability and capillary pressure curve compared

The amount of fluid flowing is not a direct result of the relative permeability, as different fluids have different viscosities. For example, if gas and oil have equal relative permeabilities, more gas than oil will flow within the rock because of the dramatic difference in viscosity.

Most core analysis reports provide three permeability measurements labeled Kmax, K90, and Kvert. Kmax is the permeability in the horizontal direction through the core with the highest permeability. This direction is determined by measuring the pressure drop across the core and then rotating the core until the minimum pressure drop is located. This permeability is sometimes labeled Khor, Khoriz, or Kh.

After rotating the core 90 degrees from the direction of Kmax, K90 is measured. K90 must be less than or equal to Kmax.

Kvert is then measured by flowing fluid through the vertical direction on the core. Kvert is often less than Kmax in sandstones and shaly sandstones. It may be higher than Kmax if semi-vertical fractures exist, as in many carbonate reservoirs.

The permeability derived from log analysis is the absolute permeability, if it is calibrated to the absolute permeability from core data. Usually, log results are calibrated to the maximum permeability from core (Kmax). Effective permeability can usually be derived from absolute permeability, using empirical relationships.

The irregular, narrow connections between pores are called capillaries. They can be likened to thin tubes connecting any two points in the reservoir. Capillary pressure is the phenomenon by which water or any wetting liquid is drawn up into a vertical capillary. The smaller the capillary, the higher the liquid rises.

Due to the variety of capillary diameters, the water saturation within a rock varies above the hydrocarbon-water contact. The water saturation caused by capillary pressure in the hydrocarbon zone is called the irreducible water saturation. A zone at irreducible water saturation will not produce water. A zone that is very wet AND at its irreducible water saturation will not flow water. A zone could have such poor permeability that an apparently wet zone can occur anywhere in an oil or gas column. This problem is solved by calling the apparently wet zone "tight", or impermeable", or "non-pay". It is not a “water zone”.

Between the water and hydrocarbon zones, is a layer of rock filled with both water and oil, with the water at a saturation higher than the irreducible saturation. It is considered the region in which both water and oil (or gas) will flow. It is termed the transition zone. This should not be confused with the same term used to describe the invasion of mud filtrate into the formation.

The more small capillaries there are, the higher the water saturation will be. Also the transition from irreducible water saturation to 100% water throughout the transition zone will be longer. The fraction of water flowing with the oil (or gas) is referred to as the water cut.

These definitions suggest that it is important to know the top depth and base of the transition zone, so that a well can be produced from above the top of the transition zone to minimize water production. This may not be obvious from specific values of water saturation, because the porosity is seldom constant, but it may be more apparent by observing the porosity times water saturation product (PHIxSW) through the transition zone. Compare the illustration on the right side of Figure 10.03 with that on the left.

FIGURE 10.03: Identifying transition zone from PHI*SW product

10.01 Permeability of Fractures
The permeability of fractures is a function of the width of the fracture. A rough relationship for permeability versus width (in inches) of a fracture is:

Where:
KF1 = number of main fracture directions
= 1 for sub-horizontal or sub-vertical
= 2 for orthogonal sub-vertical
= 3 for chaotic or brecciated
PHIfrac = fracture porosity (fractional)
Df = fracture frequency (fractures per meter)
Wf = fracture aperture (millimeters)
Kfrac = fracture permeability (millidarcies)

Note: Equations 2, 3, and 4 give identical results.

Therefore, a fracture 1 millimeter thick has a permeability of 83 darcies (or 83,000 millidarcies).

10.02 Visual Indications of Permeability on Logs
While shale content and porosity are directly related to specific log readings, the formation permeability is not easily correlated with any single log value. Indications of permeability can be found from some of the following:

1. Relatively low shale content as seen on the gamma ray log or the SP log, combined with some porosity on the sonic, density or neutron logs.

2. Mud cake buildup as seen on the caliper log. (See top left illustration in Figure 10.04).

3. Separation between the deep induction (or any deep resistivity device) and the shallow resistivity device as in the top right illustration of Figure 10.04. Separation is seen when two logs do not read roughly the same resistivity value, because fluid from the mud has invaded the formation. This causes a different resistivity to occur close to the borehole wall compared to deeper in the formation. This method is not applicable in high resistivity due to borehole effect.

4. Positive separation on a microlog, if the log is available, is another indicator of permeability. Positive separation means that the dotted curve (RES2) reads higher resistivity than the solid curve (RES1). Figure 10.04 (bottom) illustrates a modern microlog with positive separation.

5. Porosity of any significant amount usually indicates permeability. However, the amount of permeability cannot be directly related to the porosity without some outside knowledge, such as core analysis data. In the low porosity - permeability range the logarithm of permeability is often proportional to porosity, and useful crossplots of this data transformation can be made.

6. The length of the transition zone, if it can be identified, is an indicator of permeability. The longer the transition, the lower the permeability.

FIGURE 10.04: Visual indicators of permeability

10.03 Calculating Irreducible Water Saturation
Most quantitative permeability calculations require a value for the irreducible water saturation at each level in a zone even if the zone is not hydrocarbon bearing.

In a hydrocarbon bearing zone that has never been produced, the actual water saturation (SW) as calculated in Chapter Eight IS the irreducible water saturation (SWir). CAUTION: If logs are run in a reservoir that has been produced, the actual water saturation is the current saturation, not necessarily the initial saturation (SWinit) at time of discovery. The only way to estimate initial water saturation in a depleted zone from log analysis is to estimate SWir from methods described below and assume that SWinit = SWir.

SOME nuclear magnetic logs present a curve that may approximate the volume of irreducible water, also called volume of bound fluid, abbreviated as VBF or BFV. Depending on the T1 and T2 cutoffs used to process the log, this may include the bound water in the shale (Vsh * BVWSH - see Chapter Seven). If a zone is shaly AND if BFV is roughly equal to Vsh * BVWSH or somewhat larger than Vsh * BVWSH, then BFV includes bound water in the shale and VBF cannot be used directly to calculate SWir. If BFV is smaller than Vsh * BVWSH and the zone is shaly, then BFV does not include shale bound water. If zone is not shaly, shale bound water is zero and BVF includes the irreducible water volume regardless of the processing parameters.

1: BVWSH = (PHINSH + PHIDSH) / 2
2: IF VBF >= Vsh * BVWSH
3: THEN VBFc = VBF - Vsh * BVWSH
4: OTHERWISE VBFc = VBF
5: IF PHIe > 0.0
6: THEN SWir = VBFc / PHIe
7: OTHERWISE Swir = 1.0
8: IF SWir > 1.0
9: THEN SWir = 1.0

WHERE:
BVWSH = bound water in shale (fractional)
PHIe = effective porosity
PHIDSH = density log porosity in 100% shale (fractional)
PHINSH = neutron log porosity in 100% shale (fractional)
SWir = irreducible water saturation (fractional)
VBF = volume of bound fluid from NMR log (fractional)
VBFc = volume of bound fluid from NMR log corrected for bound water in shale (fractional)
Vsh = shale volume (fractional)

COMMENTS:
It may be difficult to judge whether VBF needs shale correction or not. The use of the term bound water or bound fluid instead of irreducible water should be discouraged, but it pervades the NMR literature. Bound water has traditionally meant only the water bound in the shale. If bound water must be used, distinguish between clay or shale bound water and capillary bound water.

RECOMMENDED PARAMETERS:
None

Some NMR logs present only the free fluid index (FFI) and do not provide a VBF curve. Since FFI contains only moveable fluid, shale bound water is not included and no shale correction is required.
1: IF PHIe > 0.0
2: AND IF FFI < PHIe
3: THEN SWir = (PHIe - FFI ) / PHIe
4: OTHERWISE SWir = 1.0
5: IF SWir > 1.0
6: THEN SWir = 1.0

WHERE:
FFI = free fluid index from NMR log (fractional)
PHIe = effective porosity
SWir = irreducible water saturation (fractional)

COMMENTS:
The NMR log sees a very tiny piece of rock, so the results may be a bit noisy.

RECOMMENDED PARAMETERS:
None

In the vast majority of well files, the NMR is not available or is not useful. No other log attempts to measure irreducible water volume or saturation. A very useful approach is to determine the product of porosity times water saturation (PHIxSW) from hydrocarbon zones (above the transition zone) and use this parameter in the water and transition zones to find irreducible water saturation. The product PHIxSW can also be found from capillary pressure data. Here, we take the porosity and minimum wetting phase saturation from a number of core plugs and fit a hyperbolic line to the data.

In hydrocarbon zones above the transition zone, SWir = Sw from any shale corrected method. Below the hydrocarbon zone, we use the PHIxSW method as follows:

1: IF PHIe > 0.0
2: THEN SWir = PHIxSW / PHIe
3: OTHERWISE SWir = 1.0
4: IF SWir > 1.0
5: THEN SWir = 1.0
6: SWir = MIN (Sw, SWir)

WHERE:
PHIe = effective porosity
PHIxSW = porosity water saturation product (fractional)
SWir = irreducible water saturation (fractional)

COMMENTS:
The PHIxSW product can be found by plotting PHIe vs Sw from log data and finding the best fit hyperbola to the data. Log data from the transition zone or the water zone should be excluded.

FIGURE 10.05A: Porosity vs water saturation to find PHIxSW (between 0.0200 and 0.400 in this example

The MIN command in equation 6 attempts to put the actual Sw into SWir in pay zones. However, if PHIxSW is too low, SWir will be less than Sw leading too a possible interpretation that the zone might produce with a water cut.

Once a PHIxSW product has been chosen, the irreducible water saturation for any zone can be found by entering effective porosity on the chart of Figure 10.05B, or by using the equations given above.

FIGURE 10.05B: Irreducible water saturation from PHIxSW product

Because Buckles number (PHIxSW) increases with finer grained rock, the above equation can be modified to account for the usual decrease in grain size as a zone becomes shalier. Two different formulae have been tried:
2A: SWir = PHIxSW / PHIe / (1 - Vsh)
OR 2B: SWir = PHIxSW / PHIe / (1 - Vsh^2)

RECOMMENDED PARAMETERS:
Sandstones PHIxSW
Shaly sands (shallow) 0.15 to 0.18
Clean sands (shallow) 0.12 to 0.15
Shaly sands (medium) 0.10 to 0.12
Clean sands (medium) 0.08 to 0.12
Shaly sands (deep) 0.06 to 0.08
Clean sands (deep) 0.04 to 0.06

Carbonates
Chalky/shaly limestone 0.06 to 0.08
Mississippian (LS & DOL) 0.02 to 0.04
Devonian (LS & DOL shallow) 0.15 to 0.04
Devonian (LS & DOL deep) 0.007 to 0.015
Very vuggy (LS & DOL) 0.003 to 0.007

NUMERICAL EXAMPLE:
See Section 10.08.

10.04 Permeability from the Wylie-Rose Method
This is one of the oldest permeability methods available, and is reliable when calibrated to core data.

1: PERMw = CPERM * (PHIe ^ DPERM) / (SWir ^ EPERM)

WHERE:
CPERM = permeability constant (fractional)
DPERM = porosity exponent (fractional)
EPERM = irreducible saturation exponent (fractional)
PERMw = permeability (millidarcies)
PHIe = effective porosity from any method (fractional)
SWir = irreducible water saturation (fractional)

COMMENTS:
Different researchers have found a variety of exponents based on core analysis studies, and the analyst may find many more in the literature or company files.

RECOMMENDED PARAMETERS:
RESEARCHER CPERM DPERM EPERM
OIL GAS
Morris-Biggs 65000 6500 6.0 2.0
Timur 3400 340 4.4 2.0

The resulting permeability relationship for the Morris-Biggs parameters is given in Figure 10.06 and for Timur parameters in Figure 10.07.

FIGURE 10.06: Permeability from Morris - Biggs parameters

FIGURE 10.07: Permeability from Timur parameters

NUMERICAL EXAMPLE:
See Section 10.08.

10.05 Permeability from Formation Factor
This method is based on a curve fit to core data which works well in low porosities.

1: F = A / (PHIe ^ M)
2: PERMf = FPERM / (F ^ GPERM)

WHERE:
A = tortuosity exponent (fractional)
FPERM = permeability constant (fractional)
GPERM = porosity exponent (fractional)
F = formation factor (fractional)
PERMf = calculated permeability (millidarcies)
PHIe = effective porosity (fractional)

COMMENTS:
This formula has not been as successful as the Wylie - Rose approach.

RECOMMENDED PARAMETERS:
ROCK TYPE FPERM GPERM
Sandstone 7.0 * 10 ^ 6 4.5
Limestone 4.0 * 10 ^ 6 3.5

Data from individual core studies varies considerably from these average values.

NUMERICAL EXAMPLE:
See Section 10.08.

10.06 Permeability from Porosity

A number of studies have shown a linear relationship between logarithm of permeability and linear porosity. The equation has the form: log (PERMp) = HPERM * PHIe + JPERM.

1: PERMp = 10 ^ (HPERM * PHIe + JPERM)
2: IF PERMp > 20000
3: THEN PERMp = 20000

WHERE:
JPERM = permeability constant (fractional)
HPERM = porosity exponent (fractional)
PERMp = calculated permeability (millidarcies)
PHIe = effective porosity (fractional)

COMMENTS:
Because a poor choice of HPERM could create very large values of permeability, the computation is usually constrained to prevent unreasonable answers. Results have been very good in low and high porosity when calibrated to core.

RECOMMENDED PARAMETERS:
ROCK TYPE HPERM JPERM
Carbonates
chalky 10 - 20 -2.5
fine sucrosic or
intercrystalline 20 - 30 -2.5
coarse sucrosic or
intercrystalline 25 - 50 -2.5
small vuggy 30 - 100 -2.5
large vuggy 50 - 200 -2.5
fractures 200 - 300 -3.0

Sandstones

very fine grains 10 - 20 -3.0
fine grains 15 - 25 -3.0
medium grains 20 - 30 -3.0
large grains 25 - 50 -3.0
conglomerate 20 - 50 -3.0
unconsolidated 20 - 50 -3.5
fractures 200 - 300 -3.0

Graphs of these functions can be constructed easily in spreadsheet software, as shown below

FIGURE 10.08: Permeability from Porosity

NUMERICAL EXAMPLE:
See Section 10.08.

10.07 Permeability from the Dumanoir and Coates Method
A more complex formula has recently been derived which accounts for the different effect of gas and oil.

First find the resistivity log reading in the irreducible water zone:
1: IF Sw > 0.70
2: THEN SWir = PHIxSW / PHIe
3: AND RESir = A * RW@FT / (PHIe ^ M) / (SWir ^ N)
4: OTHERWISE RESir = RESD

Then calculate the porosity exponent (LPERM)
5: LPERM = (3.75 - PHIe + ((log (RW@FT / RESir) + 2.2) ^ 2) / 2) ^ 0.5

And finally:
6: PERMd = KPERM * ( 0.077 + 1.55 * DENShy - 0.627 * DENShy ^ 2)
* (PHIe ^ (2 * LPERM)) / ((LPERM ^ 4) * (RW@FT / RESir))) ^ 2

WHERE:
A = tortuosity exponent (fractional)
KPERM = permeability constant (fractional)
LPERM = porosity exponent (fractional)
DENShy = hydrocarbon density (gm/cc)
M = cementation exponent (fractional)
N = saturation exponent (fractional)
PERMd = calculated permeability (millidarcies)
PHIe = effective porosity (fractional)
PHIxSW = porosity saturation product (fractional)
RESD = resistivity log reading (ohm-m)
RESir = resistivity at irreducible water saturation (ohm-m)
RW@FT = water resistivity at formation temperature (ohm-m)
Sw = water saturation (fractional)
SWir = irreducible water saturation (fractional)

COMMENTS:
This complex formula does not lend itself to graphical solution, although some charts are provided in the original technical paper. A simpler version of this algorithm can be found below.

RECOMMENDED PARAMETERS:
KPERM is usually set at 90,000, but can be modified with local experience.

NUMERICAL EXAMPLE:
See Section 10.08.

This is a simplification of an earlier method proposed by Dumanoir and Coates. It is more optimistic than other methods in low porosity.
1: PERMc = GPERM * PHIe ^ 4 * ((PHIt - PHIe * SWir) / (PHIe * SWir)) ^ 2

OR in clean zones:
2: PERMc = GPERM * PHIe ^ 4 * ((1 - SWir) / SWir) ^ 2

WHERE:
GPERM = permeability scale factor (fractional)
PERMc = calculated permeability (millidarcies)
PHIe = effective porosity (fractional)
PHIt = total porosity (fractional)
SWir = irreducible water saturation (fractional)

COMMENTS: This method works well in shaly sands. As the difference between PHIt and PHIe increases, Permc decreases.

RECOMMENDED PARAMETERS:
GPERM = 6500 to 10000 for oil, 650 to 1000 for gas

10.08 Permeability Examples
Assume data from Classic Example Sand B.
PHIe = 0.30
RW@FT = 0.02 ohm-m
A = 0.62
M = 2.15
SWir = Sw = 0.25
RESD = 20 ohm-m
N = 2.00
DENShy = 0.8 gm/cc

1. Wylie-Rose formula / Morris-Biggs parameters:
CPERM = 62500
DPERM = 6.0
EPERM = 2.0
PERMw = 62500 * (0.30 ^ 6) / (0.25 ^ 2) = 730 md

2. Wylie-Rose formula / Timur parameters:
CPERM = 3400
DPERM = 4.4
EPERM = 2.0
PERMw = 3400 * (0.30 ^ 4.4) / (0.25 ^ 2) = 272 md

3. Formation factor method:
FPERM = 7.0 * 10 ^ 6
GPERM = 4.50
F = 0.62 / (0.30 ^ 2.15) = 8.25
PERMf = 7.0 * 10 ^ 6 / (8.25 ^ 4.5) = 526 md

4. Porosity method:
JPERM = -3.0
HPERM = 20
PERMp = 10 ^ (20 * 0.30 - 3.0) = 1000 md

5. Dumanoir and Coates method:
KPERM = 90,000
LPERM = (3.75 - 0.30 + ((log (0.2 / 20) + 2.2) ^ 2) / 2) ^ 0.5 = 1.86
PERMd = 90,000*(0.077+1.55*0.8-0.627*0.8*0.8)*(0.3^(2*1.86)) / ((1.86^4)*(0.2/20)))^2
PERMd = 90,000 * (0.915 * 0.0113 / (11.9 * 0.01)) ^ 2 = 679 md

10.09 Relative Permeability and Water Cut
Since we are interested in how much oil or gas will flow from a well, and would rather not have any water flow, it is necessary to calculate relative permeability and water cut to determine how good a well might be.

Relative permeability for water.
1: Krw = (Sw - SWir) / (1 - SWir)

Relative permeability for oil.
2: Kro = (0.9 - Sw) / (0.9 - SWir)

Relative permeability for gas.
3: Krg = (1 - (Sw - SWir) / (1 - SWir)) * (1 - ((Sw - SWir) / (1 - Sw)) ^ 0.25 * Sw ^ 0.25)^0.5

Water oil ratio.
4: KRwo = Krw / Kro
5: WOr = BO * (VISO / VISW) * (KRwo)

Water gas ratio.
6: KRwg = Krw / Krg
7: WGr = BG * (VISG / VISW) * (KRwg)

Water cut for oil wells.
8: WCo = WOR / (1 + WOR)

Water cut for gas wells:
9: WCg = WGR / (1 + WGR)

WHERE:
BG = volume factor for gas (fractional)
BO = volume factor for oil (fractional)
Krg = relative permeability to gas (fractional)
Kro = relative permeability to oil (fractional)
Krw = relative permeability to water (fractional)
KRwg = water to gas permeability ratio (fractional)
KRwo = water to oil permeability ratio (fractional)
Sw = water saturation (fractional)
SWir = irreducible water saturation (fractional)
VISG = viscosity of gas (cp)
VISO = viscosity of oil (cp)
VISW = viscosity of water (cp)
WCg = water cut for gas well (fractional)
WCo = water cut for oil well (fractional)
WGr = water gas ratio (fractional)
WOr = water oil ratio (fractional)

COMMENTS:
Charts for solving the relative permeability relationships are provided in Figure 10.08, and for water cut in Figure 10.09. Viscosity data required for water-oil and water-gas ratio are given in Figure 10.10 for oil and gas, and in the lower right graph on Figure 10.11 for water.

Relative permeability from log analysis without core data as a control is probably pointless.

RECOMMENDED PARAMETERS:
See Figures 10.10 and 10.11.

FIGURE 10.08: Relative permeability

FIGURE 10.09: Water cut

FIGURE 10.10: Viscosity

FIGURE 10.10: Viscosity of water and pseudo critical gas properties

NUMERICAL EXAMPLE:
1. Assume data from Sand C
PHIe = 0.30
Sw = 0.50
PHIxSW = 0.06
SWir = 0.06 / 0.30 = 0.20
Krw = (0.50 - 0.20) / (1 - 0.20) = 0.375
Kro = (0.9 - 0.50) / (0.9 - 0.20) = 0.571
Krg = (1 - (0.5 - 0.3) / (1 - 0.2)) * (1 - ((0.5 - 0.3) / (1 - 0.5)) ^ 0.25 * 0.5 ^ 0.25) ^ 0.5
= (0.75 * (1 - 0.4 ^ 0.25 * 0.5 ^ 0.25) ^ 0.5 = 0.431
KRwo = 0.375 / 0.571 = 0.657
KRwg = 0.375 / 0.431 = 0.870

2. Assume oil well data:
BO = 0.8
oil gravity = 40 degrees API
VISO = 1.2 cp
FT = 160 degrees F
VISW = 0.5 cp
| salinity = 120,000 ppm
WOr = 0.8 * (1.2 / 0.5) * 0.657 = 1.26

3. Assume gas well data:
BG = 0.8
VISG = 0.015 cp
VISW = 0.5 cp
dry gas
pressure = 1800 psi
salinity = 120,000 ppm
WGr = 0.8 * (0.015 / 0.5) * 0.870 = 0.021

4. Water Cut:
WCo = 1.26 / (1 + 1.26) = 0.557
WCg = 0.021 / (1 + 0.021) = 0.021

Thus, if this is an oil zone, it will produce over 55% water on initial completion. If it is a gas well, it will produce only 2% water.

10.10 Calculating Net Pay with Cutoffs
Cumulative reservoir properties, after appropriate cut offs are applied, provide information about the pore volume (PV), hydrocarbon pore volume (HPV), and flow capacity (KH) of a potential pay zone. These values are used to calculate hydrocarbon in place, recoverable reserves, and productivity of wells. The following algorithm is a highly simplified one pass approach, which would need considerable adjustment to run on a real computer. However it is suitable for discussion purposes.

Each layer is tested against a series of cutoffs to determine if the layer can contribute to production from the well:
1: IF PHIe < PHIcut THEN PAYFLAG$ = "TIGHT"
2: OR IF Sw > SWcot THEN PAYFLAG$ = "WET"
3: OR IF Perm < PERMcut THEN PAYFLAG$ = "LOPERM"
4: OR IF Vsh > VSHcut THEN PAYFLAG$ = "SHALY"
5: OTHERWISE PAYFLAG$ = "ON"
6: IF PHIdc >= PHInc + TOLER THEN PRODFLAG$ = "GAS"
6: IF PHIdc < PHInc + TOLER THEN PRODFLAG$ = "OIL"
7: IF PHIe * Sw > PHIxSWcut THEN PRODFLAG$ = "H2O"
8: Hnet = SUM (PAYFLAG$ = "ON" * THICKi)

WHERE:
Hnet = sum of layer thicknesses which passed all cutoffs (ft or m)
Perm = permeability (md)
PERMcut = permeability cutoff (md)
PHIcut = porosity cutoff (fractional)
PHIe = effective porosity (fractional)
PHIdc = shale corrected density log porosity (fractional)
PHInc = shale corrected neutron log porosity (fractional)
PHIxSWcut = porosity saturation product cutoff (fractional)
Sw = water saturation (fractional)
SWcut = saturation cutoff (fractional)
TOLER = crossover tolerance for "GAS" flag (fractional)
Vsh = volume of shale (fractional)
VSHcut = shale volume cutoff (fractional)

COMMENTS:
The pay flag may be very sensitive to small changes in cutoffs. Any one of the four primary cutoffs can create a "FAIL" situation. This is enough to fail the layer even if other cutoffs do not fail the zone. The PRODFLAG indicates the most likely production, with "H2O" suggesting water cut with the hydrocarbon.

Some cutoffs may be set high enough or low enough so as not to be effective. For example, if PERMcut = 0, then no value of Perm could be less than PERMcut, so permeability could not fail to pass a layer.

A sonic neutron crossover can also be used to test for "GAS" flag.

More than one set of cutoffs are normally run and the results compared to find the set that appears to give reasonable results when compared to production profiles in the area.

The cutoff algorithm given above is called a Net Pay algorithm. In reservoir simulation work, the Net Reservoir is also needed. In this case, set SWcut = 1.00. To map Net Sand, set PHIcut = 0.0 and SWcut = 1.0.

RECOMMENDED PARAMETERS:
1. High porosity set:
PHIcut SWcut PERMcut VSHcut PHIxSWC
0.00 1.0 0.0 0.0 1.00
0.15 0.5 5.0 0.3 0.07
0.20 0.4 10.0 0.3 0.07
0.25 0.3 15.0 0.3 0.07

2. Low porosity set:
0.00 1.00 0.00 0.0 1.00
0.01 0.90 0.005 0.7 0.10
0.03 0.85 0.01 0.6 0.09
0.05 0.80 0.05 0.5 0.08
0.07 0.70 0.10 0.4 0.07
0.09 0.50 0.50 0.4 0.06
0.12 0.40 1.00 0.4 0.05

The only real way to determine cutoffs is to run a flowmeter in the well over the reservoir. If an interval flows oil or gas, it is above cutoffs - if it doesn't flow oil or gas, it's below cutoffs. If it flows water with oil or gas, it has failed the water saturation cutoff. This procedure requires that you perforate poor quality rock to see if it will flow. Some managers will resist this added expense as they "know" what produces and what doesn't.

Close spaced DST's can also be used in open or cased hole to simulate a flowmeter profile.

You can mimic this in the lab with flow tests in core plugs using reservoir fluids under simulated formation pressure and temperature. However, hardly anyone actually does either flowmeter or core flow analysis because it is expensive and often means completing or coring poor quality rock to find out how low the cutoffs can be set.

The pragmatic approach is much more widely used.

1. Plot core porosity vs logarithm of core permeability. Fit a semi-log line through the data points (exclude fractured plugs). For gas use a perm cutoff of 0.1 to 1.0 md, for oil use 1.0 to 5.0 md. Find the equivalent porosity on this graph corresponding to the selected perm cutoff. This is your porosity cutoff.

2. Plot porosity vs water saturation in the oil or gas leg above the transition zone. This can be log analysis data or values from capillary pressure curves. Fit a hyperbolic line through the data. Enter with porosity cutoff and find corresponding SW. This is the SW cutoff.

3. In shaly sands, plot porosity vs shale volume. Enter graph with porosity cutoff and pick corresponding shale volume. This is Vsh cutoff.

This is called a coordinated cutoff set.

Reservoir engineers sometimes plot cumulative porosity or permeability or both (sort data into ascending order first). They then place the cutoff at the 5 or 10% accumulation. This is exceedingly arbitrary but was a widespread method. It was only vaguely useful if the core contained no poor quality rock or if there was no spread in the rock properties.

The levels that pass these four tests can be checked for continuity against two thickness criteria. One is the minimum zone thickness needed to be considered as a pay zone. The second is a maximum non-pay interval allowed in the overall pay zone before the zone is broken into more than one pay zone. These two criteria are called the acceptance thickness and rejection thickness respectively.

To find the beginning of a possible pay zone, search from the top of the computed interval for the first data set with a "pass" in its cutoff field.

Then find the first deeper level with a "fail" in its cutoff field. The depths of these two points define the top (ZONETOP) and bottom (ZONEBOTTOM) of a zone. This interval thickness is tested against the acceptance criteria.

The depth of the next pay zone top is then found and the interval between pay zones tested against the rejection criteria.

1: IF PAYFLAG$ ="OIL"
2: OR IF PAYFLAG$ ="GAS"
3: OR IF PAYFLAG$ ="H2O"
4: THEN CUTOFF$ = "PASS"
5: OTHERWISE CUTOFF$ = "FAIL"
6: IF CUTOFF$ = "PASS"
7: AND TOPFLAG$ = "START"
8: THEN ZONETOP = Depth
9: AND TOPFLAG$ = "TOPFOUND"
10: AND BOTTOMFLAG$ = "START"
11: IF CUTOFF$ = "FAIL"
12: AND BOTTOMFLAG$ = "START"
13: THEN ZONEBOTTOM = Depth
14: AND BOTTOMFLAG$ = "BOTTOMFOUND"
15: AND TOPFLAG$ = "START"
16: AND NEXTTOPFLAG$ = "START"
17: IF CUTOFF$ = "PASS"
18: AND NEXTTOPFLAG$ = "START"
19: THEN NEXTTOP = Depth
20: AND NEXTTOPFLAG$ = "TOPFOUND"
21: IF ZONEBOTTOM - ZONETOP > HACCEPT
22: THEN FOR Depth = ZONETOP
23: TO Depth = ZONEBOTTOM
24: PAYFLAG$ = "PAY"
25: OTHERWISE PAYFLAG$ = "BARREN"
26: END LOOP 2
27: IF NEXTTOP - ZONEBOTTOM < HREJECT
28: THEN FOR Depth = ZONEBOTTOM
29: TO Depth = NEXTTOP
30: PAYFLAG$ = "PAY"
31: OTHERWISE PAYFLAG$ = "BARREN"
32: END LOOP 3
33; END LOOP 1

Repeat these steps until the bottom of the computation interval is reached.

At this time each level computed will have two flags set - one to indicate whether it passed cutoffs (CUTOFF$) and whether the layer is considered part of a pay zone (PAYFLAG$), even if it failed its cutoffs. To find net pay thickness, count the number of levels with the pay flag equal to "pay" and multiply by the depth increment between the data points.

32: Hnet = Sum ((IF PAYFLAG$ = "PAY") * THICKi)
33: Hgross = ZONETOP - ZONEBOTTOM
34: NETratio = Hnet / Hgross

WHERE:
Depth = current depth (ft or m)
HACCEPT = minimum pay zone thickness cutoff (ft or m)
Hgross = gross interval (ft or m)
Hnet = sum of layer thicknesses which passed all cutoffs (ft or m)
HREJECT = maximum reject zone thickness cutoff (ft or m)
NETratio = ratio of net to gross pay (fractional)
NEXTTOP = top depth of next pay zone (ft or m)
THICKi = individual layer thickness (ft or m)
ZONEBOT = bottom of computation interval (ft or m)
ZONETOP = top of computation interval (ft or m)

COMMENTS:
Some analysts prefer not to count layers which failed cutoffs but are included in net interval because they are too thin to be rejected. To accomplish this, change Step 39 to read:

39: Hnet = Sum ((IF PAYFLAG$ = "PAY" AND IF CUTOFF$ = "PASS") * THICKi)

NUMERICAL EXAMPLE
Assume data as shown in Figure 10.12. Note that data listing is in percent, not fractional, units.

FIGURE 10.12: Data for net pay example

1. If cutoffs are:
PHIcut = 3.0 (%)
VSHcut = 40.0 (%)
PERMcut = 0.0 (md)
SWcut = 90.0 (%)
PHIxSWcut = 1.0
HACCEPT = 1.0 (m)
HREJECT = 0.0 (m)

Then net pay extends from 2054.1 to include 2063.1.

2. If SWcut is lowered to 50.0, then net pay covers 2054.1 to 2059.5 and 2061.9 to 2062.2 in two zones.

3. If HREJECT = 3 m, then these two pay zones combine to form one zone because the rejected interval is less than 3.0 m.

4. If HACCEPT = 3.0 m, then the second zone is not pay because it is not thick enough.

Rejected intervals are included in the zone for Case 3 and contribute to net pay thickness.

10.11 Calculating Cumulative and Average Reservoir Properties
The reservoir volume and flow capacity per unit area are steps toward finding total reservoir volume. Average values for comparing the quality of reservoirs are also useful results from log analysis. Pore volume (per unit area), hydrocarbon pore volume, flow capacity, and the averages of shale volume, porosity, water saturation, permeability, net pay, net reservoir, net sand, and gross sand are called mappable properties, petrophysical properties, or reservoir properties.

Do not use the following algorithm in thinly laminated shaly sands - see alternate method shown below.

Pore volume (PV).
1: PhiH = SUM (PHIe * THICKi * (IF PAYFLAG$ = "PAY"))
Hydrocarbon pore volume (HPV).
2: HydH = SUM (PHIe * (1 - Sw) * THICKi * (IF PAYFLAG$i = "PAY"))
Flow capacity (KH).
3: Kh = SUM (Perm * THICKi * (IF PAYFLAG$ = "PAY"))
Average porosity.
4: PHIavg = PhiH / Hnet
Average water saturation.
5: SWavg = 1 - (HydH / PhiH)
Average permeability.
a. Arithmetic average:
6: Kavg = Kh / Hnet
b. Geometric average:
7: Kgeo = (PROD (Perm * THICKi)) ^ (1 / Ns)
c. Harmonic average:
8: Khar = Hnet / (SUM (1 / (Perm * THICKi)))

WHERE:
Hnet = net pay thickness (ft or m)
HydH = hydrocarbon volume (ft or m per unit area)
Kavg = arithmetic average permeability (md)
Kgeo = geometric average permeability (md)
Kh = flow capacity (md-ft or md-m per unit area)
Khar = harmonic average permeability (md)
Ns = number of samples in product
Perm = permeability (md)
PHIavg = average porosity (fractional)
PHIe = effective porosity (fractional)
PhiH = pore volume (ft or m per unit area)
SWavg = average water saturation (fractional)
THICKi = individual layer thickness (ft or m)

COMMENTS:
PhiH is often called PV and HydH is often called HPV.

Some analysts prefer not to count layers which failed cutoffs but are included in net interval because they are too thin to be rejected. To accomplish this, change each IF statement to read: ((IF PAYFLAG$ = "PAY" AND IF CUTOFF$ = "PASS") * THICKi) .

The harmonic average most closely reflects radial flow into a borehole. If equal sample intervals are used, this geometric formula becomes:
Kgeo = (PROD (Perm * H)) ^ (1 / H).

It does not give the same result as the previous version if layer thicknesses are unequal.

RECOMMENDED PARAMETERS:
None.

NUMERICAL EXAMPLE:
1. Assume three layers as follows:
Layer PHIe Sw Perm THICK (ft)
1 0.10 0.60 10 2
2 0.20 0.50 100 4
3 0.30 0.40 1000 6

Assume all layers pass all cutoffs:
PhiH = 0.10 * 2 + 0.20 * 4 + 0.30 * 6 = 2.8 ft
HydH = 0.10 * (1 - 0.60) * 2 + 0.20 * (1 - 0.50) * 4 + 0.30 * (1 - 0.40) * 6 = 1.56 ft
Kh = 10 * 2 + 100 * 4 + 1000 * 6 = 6420 md-ft
Hnet = 2 + 4 + 6 = 12 ft
PHIavg = 2.8 / 12 = 0.233
SWavg = 1 - 1.56 / 2.8 = 0.443
Kavg = 6420 / 12 = 535 md
Kgeo = (10 * 2 * 100 * 4 * 1000 * 6) ^ (1 / 3) = 363 md
Khar = 12 / (1 / (10 * 2) + 1 / (100 * 4) + 1 / (1000 * 6)) = 228 md

If equal sample intervals are used, (with H = 1.0),
Kgeo = 215 md

10.12 Reservoir Properties for Laminated Shaly Sands
For most reservoirs, the above equations give rational results. For laminated shaly sands, these results are inappropriate. This topic is covered in more detail in Chapter Seventeen.

Instead of treating each layer of rock individually, we must work with average values over the gross sand interval. This is necessary because most laminated shaly sands (where the laminations are usually much thinner than the logging tool's vertical resolution) will fail one or more of the standard net pay cutoffs. Therefore, we have to find a way to allow the sand fraction to pass cutoffs and exclude the shale laminations.

Such sands are often called "low resistivity pay zones". The resistivity is low because of the way resistivity logs average the two different rock types. Assume a VSHavg of 0.50 with the shale layers having a resistivity of 2.0 ohm-m and the hydrocarbon bearing sand fraction with a resistivity of 200 ohm-m. The average resistivity appears to be 101 ohm-m. BUT resistivity logs actually average the conductivity values - not the resistivities.

The shale conductivity is 500 mSeimens and the oil sand fraction is 5 mS. The average conductivity is 252 mS, which when converted back to resistivity gibes 1000 / 252 or slightly less than 4 ohm-m, giving a very high (and wrong) apparent water saturation. So the average of 2 and 200 is 4 ohm-m - at least that is what a typical resistivity log thinks!

The standard porosity equations would also give a porosity approximately one half the correct value for this example. The zone thus appears very shaly, very low porosity, very wet, and very low permeability even though the sand fraction (50% of the gross interval) is a very good pay zone.

To solve this, we assume the sand porosity is similar to the maximum clean sand porosity (PHIMAX) in nearby non-laminated sandstones. We also assume the average water saturation (SWavg) equals the average SWir in these non-laminated sands.

1: Hgross = BOTDepth - TOPDepth
2: Hnet = Hgross * (1 - VSHavg)
3: PhiH = PHIMAX * Hnet
4: HydH = PhiH * (1 - SWavg)

WHERE:
Hgross = gross pay thickness (ft or m)
Hnet = net pay thickness (ft or m)
HydH = hydrocarbon volume (ft or m per unit area)
PHIMAX = average porosity of nearby clean sands (fractional)
PhiH = pore volume (ft or m per unit area)
SWavg = average water saturation in nearby clean sands (fractional)
Vshavg = Average shale volume inside gross interval

COMMENTS:
In very thin sand lenses, porosity may be lower than PHIMAX because the grains are finer or a bit shaly. This can be approximated by replacing equations 3 and 4 with:
3a: PhiH = PHIMAX * Hgross * (1- VSHavg^2)
4a: HydH = PhiH * (1 - PHIxSW / (PHIMAX * (1 - VSHavg^2))

This reduces sand fraction porosity and raises water saturation as the sand layers get thinner relative to the shale layers.

Permeability is calculated from the same formula as the non-laminated sands nearby using PHIMAX and SWavg instead of PHIe and SWir. PHIMAX and SWavg should be mapped over the field area to see if they vary.

RECOMMENDED PARAMETERS:
None.

NUMERICAL EXAMPLE:
1. Assume VSHavg = 0.50
PHIMAX = 0.30
SWavg = 0.20
Hgross = 20 meters
Hnet = 20 * (1 - 0.50) = 10 meters
PhiH = 0.30 * 10 = 3.0 meters
HydH = 3.0 * (1 - 0.20) = 2.4 meters
Perm = 65000 * (0.3^4) /(0.2^2) = 13200 md
Kh = 13200 * 10 = 132000 md-m

If zone was treated as a dispersed shaly sand, Vsh = 0.50, PHIe would be 0.15, and Sw would be 0.80 or higher. The entire zone would fail cutoffs. If treated as pay because it actually produces oil:
Hnet = Hgross = 20 meters
PhiH = 0.15 * 20 = 3.0 meters
HydH = 3.0 * (1 - 0.80) = 0.6 meters
Perm = 65000 * (0.15^4) / (0.8^2) = 51 md
Kh = 51 * 20 = 1020 md-m (Quite a difference!)

Assuming the zone is counted as pay, it is severely underestimated as to hydrocarbon volume and permeability appears even more disappointing when a laminated shaly sand is treated as a dispersed shaly sand.

10.13 Calculating Oil and Gas Reserves
To translate pore and hydrocarbon volume per unit area, calculated from logs, into hydrocarbon in place or recoverable reserves, we need to know something about the fluids in the zone, as well as the environment under which they are kept in the reservoir.

Oil shrinks and gas expands when brought to the surface, so reservoir volumes are usually multiplied by a volume factor to account for this, to obtain the equivalent reservoir volume at surface conditions (sometimes called stock tank or standard temperature and pressure conditions). The multiplier is called the shrinkage factor (Sf), but its inverse, the formation volume factor (B) is usually used.

1: PF = KP1 * DEPTH
2: PS = KP2

Where:
KP1 = 0.46 psi/foot for English units
KP1 = 10.4 KPa/meter for Metric units
KP2 = 14.7 psi for English units
KP2 = 101 KPa for Metric units

NOTE: KP1 varies with location and depth. KP2 assumes well has no backpressure and flows to atmospheric pressure. A typical back pressure is 200 to 600 psi (200 to 3600 KPa). Use actual values for PF and PS where possible.

3: DENShy = 8829.6 / (131.5 + API) / 62.4 * KD2

Where:
KD2 = 1.00 for English units
KD2 = 1000 for Metric units

4: GOR = KG1 * API * (PF ^ KG2) * EXP (KG3 * (API / (FT + KT2)))

Where:
KG1 = 0.0362 for API < 30
KG1 = 0.0178 for API > 30
KG2 = 1.0937 for API < 30
KG2 = 1.1870 for API > 30
KG3 = 25.7242 for API < 30
KG3 = 23.9318 for API > 30
KT2 = 460'F

NOTE: PF is in psi and FT in Fahrenheit for Equation 4. Use measured GOR where available

5: Bo = 1 + KC1 * GOR * (KC2 + KC3 * GOR) * (FT - 60) * (API / DENShy)

Where:
KC1 = 0.0004677 for API < 30
KC1 = 0.0004670 for API > 30
KC2 = 0.00001751 for API < 30
KC2 = 0.00001100 for API > 30
KC3 = 0.000000018 for API < 30
KC3 = 0.0000000013 for API > 30

NOTE: GOR is in scf/bbl, DENShy is in gm/cc, and FT in Fahrenheit for Equation 5. Use measured values where possible.

6: Bg = PF * (TS + KT1) / (PS * (TF + KT1)) / ZF

Where:
KT2 = 460 for English units
KT2 = 273 for Metric units

7: VISd = 10 ^ (10 ^ (3.0324 - 0.02023 * API) / (FT ^ 1.163)) - 1
8: VISo = (10.715 / (GOR +100) ^ 0.515) * VISd ^ (5.44 / (GOR + 150) ^ 0.338)

WHERE:
Bg = gas formation volume factor (fractional)
Bo = oil formation volume factor (fractional(
DEPTH = formation depth (ft or m)
GOR = gas oil ratio (ocf/bbl or m3/m3)
PF = formation pressure (psi or KPa)
PS = standard surface pressure (psi or KPa)
TF = formation temperature (degrees C or degrees F)
TS = standard surface temperature (degrees C or degrees F)

COMMENTS:
These fluid properties equations are derived from a US Department of Energy publication. Measured values should be used if available.

Calculate oil or gas in place
1: OOIP = KV3 * HydH * AREA / Bo
2: OGIP = KV4 * HydH * AREA * Bg

Where:
KV3 = 7758 for English units
KV3 = 1 for Metric units
KV4 = 43.56 for English units
KV4 = 1 for Metric units

NOTE: AREA is in acres

Calculate reserves
3: RF = (Sxo - Sw) / (1 - Sw)
4: Roil = RF * OOIP
5: Rgas = RF * OGIP

WHERE:
AREA = reservoir area (acres or m2)
Bg = gas formation volume factor (fractional)
Bo = oil formation volume factor (fractional(
OGIP = original gas in place (mcf or m3)
OOIP = original oil in place (bbl or m3)
RF = recovery factor (fractional)
Rgas = recoverable reserves of gas (mcf or m3)
Roil = recoverable reserves of oil (bbl or m3)
Sw = water saturation in un-invaded zone (fractional)
Sxo = water saturation in invaded zone (fractional)

COMMENTS:
Recovery factor is difficult to estimate and is often known only after the pool, or an analogous pool, is depleted.

RECOMMENDED PARAMETERS:
Recovery factor can have a broad range for oil (0.01 to 0.95) and a narrower range for gas (0.50 to 0.95).

NUMERICAL EXAMPLE:
Using the data from the example in Section 10.11:
Sxo = 0.5
Sw = 0.2
HydH = 1.56 ft
GOR = 500 scf/bbl
AREA = 640 acres
gas gravity = 0.85

TF = 460 + 160 = 620 degrees R
TS = 460 + 60 = 520 degrees R
PF = 0.46 * 3000 = 1380 psi
PS = 14.7 psi
Bo = 1.05 + 0.0005 * 500 = 1.30
RF = (0.5 - 0.2) / (1 - 0.2) = 0.37
If an oil well: OOIP = 7758 * 1.56 * 640 / 1.30 = 5.958 * 10^6 bbl/section
Roil = 6 * 10^6 * 0.31 = 2.2 * 10^6 bbl/section
If gas: OGIP = 43.56 * 1.56 * 640 * 99.78 = 4.34 Bcf/section
Rgas = 4.34 * 0.37 = 1.60 Bcf/section

Remember to round your answers to two or three significant digits, which you started with.

10.14 Tar Assay Reserves (Weight)
Tar or bitumen, and sometimes heavy oil, is measured by weight of tar in place as opposed to volume of oil in place. Some Former Soviet Union countries record conventional oil reserves in tonnes.

The following formulas are for use in areas where reserves are measured in metric tonnes, or as weight fraction (or weight percent).

Tar weight.
1: WTtar = PHIe * (1 - Sw) * DENShy / 1000

Shale weight.
2: WTsh = Vsh * DENSSH / 1000

Sand weight.
3: WTsnd = (1 - Vsh - PHIe) * DENSMA / 1000

Water weight.
4: WTwtr = PHIe * Sw * DENSW / 1000

Total rock weight.
5: WTrock = WTtar + WTsh + WTsnd + WTwtr

Tar mass fraction.
6: TARfrac = WTtar / WTrock

Tar weight percent.
7: TARwt% = 100 * WTtar / WTrock

Tar in place.
8: Tar = SUM (WTtar * THICKi) * AREA

WHERE:
AREA = reservoir area (m2)
DENShy = hydrocarbon density (Kg/m3)
DENSMA = matrix density (Kg/m3)
DENSSH = shale density (Kg/m3)
DENSW = water density (Kg/m3)
THICKi = rock thickness (meters)
Tar = tar reserves in place (tonnes)
PHIe = porosity (fractional)
Sw = water saturation (fractional)
TARfrac = tar weight fraction (fractional)
TARwt% = tar weight percent (percent)
Vsh = volume of shale (fractional)
WTrock = total rock weight (tonne/m2)
WTsh = shale weight (tonne/m2)
WTsnd = sand weight (tonne/m2)
WTtar = tar weight (tonne/m2)
WTwtr = water weight (tonne/m2)

COMMENTS:
All densities are in Kg/m3 in these formulae.

Results are in tonnes/m2 except where noted. To obtain tonnes in place, multiply by area in square meters. To obtain reserves, multiply this result by a recovery factor.

CAUTION: Some core analysis results do not include water (dry basis analysis). To compare log analysis results to core, eliminate the water term from WTrock.

When using this method for oil, replace the word “tar” with “oil” to prevent confusion.

RECOMMENDED PARAMETERS:
None.

NUMERICAL EXAMPLE:
1. Assume data as follows:
PHIe = 0.30
Sw = 0.10
Vsh = 0.10
AREA = 640 acres or AREA = 3980 m2
DENSW = 1000 Kg/m3
DENShy = 800 Kg/m3
DENSSH = 2300 Kg/m3
DENSMA = 2650 Kg/m3
THICK = 10 meters

WTtar - 0.30 * (1 - 0.10) * 800 / 1000 = 0.216 tonnes/m2
WTsh = 0.10 * 2300 / 100 = 0.230 tonnes/m2
WTsnd = (1 - 0.10 - 0.30) * 265- / 1000 = 1.590 tonnes/m2
WTwtr = 0.30 * 0.10 * 1000 / 1000 = 0.030 tonnes/m2
WTrock = 0.216 + 0.230 + 1.590 + 0.030 = 2.066 tonnes/m2
TARfrac = 0.216 / 2.066 = 0.1045
TARwt% = 100 * 0.1045 = 10.45%
Tar = 0.216 * 10 * 3980 = 8600 tonnes/section

10.15 Gas Hydrate Reserves
The occurrence of light hydrocarbons at shallow depths in the Arctic and offshore East Coast USA results in the possibility of gas hydrates existing in significant quantities. Natural gas hydrates are a solid formed by the interaction of light hydrocarbons and water. The largest molecule known to form a hydrate is cyclopentane. The crystal lattice structure of the hydrate forms in water molecules with hydrocarbon occupying the void spaces in the network. Empirically, the ratio of water to gas necessary to form a hydrate is as follows:

Excess Hydrogen
1. Methane CH4.6H20 4/12 = 33%
2. Ethane C2H6.8H20 6/16 = 37%
3. Propane C3H8.17H20 8/34 = 23%

The excess hydrogen should make the neutron porosity log read too high and the density porosity log too low. However, melting near the borehole will reduce this effect and it is seldom seen. Instead the zone often looks like a normal water-invaded gas sand.

An increase in temperature increases the pressure required to form hydrates, while small percentages of ethane or propane lower the hydrating pressure considerably. Hydrogen sulfide and carbon dioxide also decrease the required pressure.

The volume of hydrocarbons are thus a function of the hydrocarbon type and the porosity only. Water saturation is meaningless. The ratio of gas to water would range from 433 SCF/bbl. for propane to 1230 SCF/bbl. for methane.

This is equivalent to 170 cubic feet (for methane) per cubic foot of pore space (or 170 m3 per cubic meter of pore space) at standard temperature and pressure, for methane, and 60 cubic feet of propane per cubic foot of pore space.

Convert pore volume to gas volume.
1: HydH = (PhiH) * BG
Calculate gas in place.
2: Vgas = KV3 * HydH * AREA

Where:
KV3 = 43.56 for English units
KV3 = 1 for Metric units

WHERE:
AREA = reservoir area (acres or m2)
BG = equivalent gas hydrate volume factor (fractional)
HydH = hydrocarbon volume (feet or meters)
PhiH = pore volume (feet or meters)
Vgas = gas in place anhydrates (mcf or m3)

COMMENTS: None

RECOMMENDED PARAMETERS:
Usual values for BG are 170 for methane, and 60 for propane.

NUMERICAL EXAMPLE:
1. Assume the following data:
PHIe = 0.20
hydrate is methane
H = 30 feet
BG = 170 scf/scf
AREA = 640 acres
HydH = (0.20 * 30) * 170 = 1020 ft
Vgas = 43.56 * 1020 * 640 = 28.4 MMcf

10.16 Productivity Index
Productivity index is an indicator of production capacity, and can be used to compare zones or wells as to quality of production to expect.

Formation and surface temperatures.
1: TFa = (KT1 + SUFT + ((BHT - SUFT) / BHTDEP) * DEPTH

Where:
KT1 = 460 'F for English units
KT1 = 273 'C for Metric units

Formation and surface pressures:
2: PFa = KP1 * DEPTH

Where:
KP1 = 0.46 psi for English units
KP1 = 1.04 KPa for Metric units

3: PSa = KP2

Where:
KP2 = 14.7 psi for English units
KP2 = 100 KPa for Metric units

Productivity index for oil.
4: Qo = KV1 * Kh * (PFa - PSa) / VISO

Where:
KV1 = 0.0001 for English units
KV1 = 0.0000756 for Metric units

Productivity index for gas.
6: Qg = KV2 * Kh * ((PFa - PSa) ^ 2) / TFa

Where:
KV2 = 0.0001 for English units
KV2 = 0.0000061 for Metric units

WHERE:
BHT = bottom hole temperature (degrees F or degrees C)
BHTDEP = depth of bottom hole temperature (ft or m)
DEPTH = average formation depth (ft or m)
Kh = flow capacity or permeability-thickness product (md-ft or md-m)
PFa = average formation pressure (psi or KPa)
PSa = average surface or pipeline delivery pressure (psi or KPa)
SUFT = actual surface (degrees F or degrees C)
TFa = formation temperature, absolute (degrees F or degrees C)
Qg = productivity gas (mcf/day or m3/day)
Qo = productivity oil (bbl/day or m3/day)
VISO = oil viscosity (cp)

COMMENTS:
Viscosity data for oil is found from Figure 10.10.

These formulae are "log analysis" versions of more complex formulae should be used as guides to well quali