Changes

:<math>\mbox{TVD}  = \sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \cos \alpha_{i}</math>

:<math>\mbox{TVD}  = \sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \cos \alpha_{i}</math>

:<math>\mbox{NSD} &= \sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \sin \alpha_{i}\cos \beta_{i}\\\mbox{EWD} &= \sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \sin \alpha_{i} \sin \beta_{i}</math>

:<math>\mbox{NSD} = \sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \sin \alpha_{i}\cos \beta_{i}</math>

:<math>\mbox{EWD} = \sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \sin \alpha_{i} \sin \beta_{i}</math>

[[file:conversion-of-well-log-data-to-subsurface-stratigraphic-and-structural-information_fig3.png|thumb|{{figure number|3}}Segment of a curved well path showing angular and dimensional relationships between the top and bottom of the interval.]]

[[file:conversion-of-well-log-data-to-subsurface-stratigraphic-and-structural-information_fig3.png|thumb|{{figure number|3}}Segment of a curved well path showing angular and dimensional relationships between the top and bottom of the interval.]]

where

where

{{quotation|NSD = north-south displacement

* NSD = north-south displacement

 

* EWD = east-west displacement

EWD = east-west displacement

* α = inclination angle, in degrees from the vertical, from the survey

 

* β = compass bearing, in degrees clockwise from north, from the survey.

α = inclination angle, in degrees from the vertical, from the survey

* ''i'' = survey point number (i = 0 at surface)

 

β = compass bearing, in degrees clockwise from north, from the survey.

 

''i'' = survey point number (i = 0 at surface)

The intervals can be defined in several ways depending on the accuracy and simplicity of calculation required. The ''tangential'' or ''terminal angle method'' (Figure 4a) assumes a constant deviation for the entire interval from one survey point to the next. Thus, the measured depths (MD_''i'', MD_''i''–1) for each interval coincide with the depth at the survey points, and the angle used would be for the lower survey point. Although easy to calculate, this method is likely to be substantially in error and is generally not recommended<ref name=pt06r20>Craig, J. T. Jr., Randall, B. V., 1976, Directional survey calculation: Petroleum Engineer, March, p. 38–54.</ref><ref name=pt06r57>Inglis, T. A., 1987, Directional Drilling, Petroleum Engineering and Development Studies, Volume 2: London, Graham and Trorman, chap. 9, p. 155–171.</ref>. It is mentioned here for historical reasons, as it has been widely used. Alternatively, the ''angle averaging method'' (Figure 4b) uses the average for the two survey points at either end of the segment.

The intervals can be defined in several ways depending on the accuracy and simplicity of calculation required. The ''tangential'' or ''terminal angle method'' (Figure 4a) assumes a constant deviation for the entire interval from one survey point to the next. Thus, the measured depths (MD_''i'', MD_''i''–1) for each interval coincide with the depth at the survey points, and the angle used would be for the lower survey point. Although easy to calculate, this method is likely to be substantially in error and is generally not recommended<ref name=pt06r20>Craig, J. T. Jr., Randall, B. V., 1976, Directional survey calculation: Petroleum Engineer, March, p. 38–54.</ref><ref name=pt06r57>Inglis, T. A., 1987, Directional Drilling, Petroleum Engineering and Development Studies, Volume 2: London, Graham and Trorman, chap. 9, p. 155–171.</ref>. It is mentioned here for historical reasons, as it has been widely used. Alternatively, the ''angle averaging method'' (Figure 4b) uses the average for the two survey points at either end of the segment.

These formulas are derived for calculation of well positions from data at specific survey points. Calculation of the location of a stratigraphic top or other depth of interest on the well log may require interpolation of the inclination and bearing angles between survey points. The simplest approach is to interpolate linearly between the survey points above and below the point of interest:

These formulas are derived for calculation of well positions from data at specific survey points. Calculation of the location of a stratigraphic top or other depth of interest on the well log may require interpolation of the inclination and bearing angles between survey points. The simplest approach is to interpolate linearly between the survey points above and below the point of interest:

:<math>\alpha_{m}  = \alpha_{i-1} + [(\alpha_{i} - \alpha_{i-1}) \times (\mbox{MD}_{m} - \mbox{MD}_{i-1})</math>

:<math>\alpha_{m}  = \alpha_{i-1} + [(\alpha_{i} - \alpha_{i-1}) \times (\mbox{MD}_{m} - \mbox{MD}_{i-1}) / (\mbox{MD}_{i} - \mbox{MD}_{i-1})]</math>

:<math>&\qquad /(\mbox{MD}_{i} - \mbox{MD}_{i-1})]\\\beta_{m} &= \beta_{i-1} + [(\beta_{i} - \beta_{i-1}) \times (\mbox{MD}_{m} - \mbox{MD}_{i-1})\\&\qquad /(\mbox{MD}_{i} - \mbox{MD}_{i-1})]</math>

:<math>\beta_{m} = \beta_{i-1} + [(\beta_{i} - \beta_{i-1}) \times (\mbox{MD}_{m} - \mbox{MD}_{i-1}) / (\mbox{MD}_{i} - \mbox{MD}_{i-1})]</math>

where

where

{{quotation|''m'' = marker depth of interest

[[file:conversion-of-well-log-data-to-subsurface-stratigraphic-and-structural-information_fig5.png|thumb|{{figure number|5}}Circular approximations of a curved well course showing angles used for the approximations. (a) Radius of curvature method showing chords of horizontal and vertical circles. This method assumes a constant radius of curvature (constant increase or decrease in deviation between survey points). (b) Minimum curvature method showing chord of single circle and the angle ϕ, which describes the chord.]]

 

''i'' = next survey point below the marker depth

More sophisticated approaches to well-depth correction are the radius of curvature method (Figure 5a) and the minimum curvature method (Figure 5b). The ''radius of curvature method'' approximates the well path as a circular arc in the vertical plane, which is then wrapped around a vertical cylinder. The equations in the method (from <ref name=pt06r20 />) are as follows:

More sophisticated approaches to well-depth correction are the radius of curvature method (Figure 5a) and the minimum curvature method (Figure 5b). The ''radius of curvature method'' approximates the well path as a circular arc in the vertical plane, which is then wrapped around a vertical cylinder. The equations in the method (from <ref name=pt06r20 />) are as follows:

:<math>\mbox{TVD}  = (180/\pi)\sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times (\sin \alpha_{i} - \sin \alpha_{i-1})</math>

:<math>\mbox{TVD}  = (180/\pi)\sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times (\sin \alpha_{i} - \sin \alpha_{i-1}) /(\alpha_{i} - \alpha_{i-1})</math>

:<math>&\quad \ /(\alpha_{i} - \alpha_{i-1})\\\mbox{NSD} &= (180/\pi)^{2}\sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times (\cos \alpha_{i-1} - \cos \alpha_{i})\\&\quad \ {\times}\, (\sin \beta_{i} - \sin \beta_{i-1})/[(\alpha_{i} - \alpha_{i-1}) \times (\beta_{i} - \beta_{i-1})]\\\mbox{EWD} &= (180/\pi)^{2}\sum \{(\mbox{MD}_{i} - \mbox{MD}_{i-1})/[(\alpha_{i} - \alpha_{i-1})\\&\quad \ {\times}\, (\beta_{i} - \beta_{i-1})]\} \times (\sin \alpha_{i} - \sin \alpha_{i-1})\\&\quad \ {\times}\, (\cos \beta_{i} - \cos \beta_{i-1})\\\mbox{HD} &= (\mbox{NSD}_{b} - \mbox{NSD}_{t})^{2} + (\mbox{EWD}_{b} - \mbox{EWD}_{t})^{2})^{1/2}</math>

:<math>\mbox{NSD} = (180/\pi)^{2}\sum (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times (\cos \alpha_{i-1} - \cos \alpha_{i}) {\times}\, (\sin \beta_{i} - \sin \beta_{i-1})/[(\alpha_{i} - \alpha_{i-1}) \times (\beta_{i} - \beta_{i-1})]</math>

 

:<math>\mbox{EWD} = (180/\pi)^{2}\sum \{(\mbox{MD}_{i} - \mbox{MD}_{i-1})/[(\alpha_{i} - \alpha_{i-1}) \times \, (\beta_{i} - \beta_{i-1})]\} \times (\sin \alpha_{i} - \sin \alpha_{i-1}) \times \, (\cos \beta_{i} - \cos \beta_{i-1})</math>

:<math>\mbox{HD} = (\mbox{NSD}_{b} - \mbox{NSD}_{t})^{2} + (\mbox{EWD}_{b} - \mbox{EWD}_{t})^{2})^{1/2}</math>

where

where

 

* HD = net horizontal displacement

{{quotation|HD = net horizontal displacement

* ''b'' = bottom of interval of interest

 

* ''t'' = top of interval of interest

''b'' = bottom of interval of interest

 

''t'' = top of interval of interest

The ''minimum curvature method'' approximates the well path as a single circular arc. The equations in this method (from <ref name=pt06r20 /><ref name=pt06r57 />) are as follows:

The ''minimum curvature method'' approximates the well path as a single circular arc. The equations in this method (from <ref name=pt06r20 /><ref name=pt06r57 />) are as follows:

:<math>\phi  = \cos^{-1}[\cos \alpha_{i-1} \cos \alpha_{\rm i} + \sin \alpha_{\rm i} \sin \alpha_{i-1} \cos (\beta_{i} - \beta_{i-1})]</math>

:<math>\phi  = \cos^{-1}[\cos \alpha_{i-1} \cos \alpha_{\rm i} + \sin \alpha_{\rm i} \sin \alpha_{i-1} \cos (\beta_{i} - \beta_{i-1})]</math>

:<math>\mbox{TVD} &= 180/\pi \sum [\tan (\phi/2) \times (\mbox{MD}_{i} - \mbox{MD}_{i-1})\\&\quad \ {\times}\, (\cos \alpha_{i-1} + \cos \alpha_{i})]/\phi\\\mbox{NSD} &= 180/\pi \sum [\tan (\phi/2) \times (\mbox{MD}_{i} - \mbox{MD}_{i-1})\\&\quad \ {\times}\, (\sin \alpha_{i-1} \cos \beta_{i-1} + \sin \alpha_{i} \cos \beta_{i})]/\phi\\\mbox{EWD} &= 180/\pi \sum [\tan (\phi/2) \times (\mbox{MD}_{i} - \mbox{MD}_{i-1})\\&\quad \ {\times}\, (\sin \alpha_{i-1} \sin \beta_{i-1} + \sin \alpha_{i} \sin \beta_{i})]/\phi</math>

:<math>\mbox{TVD} = 180/\pi \sum [\tan (\phi/2) \times (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \, (\cos \alpha_{i-1} + \cos \alpha_{i})]/\phi</math>

:<math>\mbox{NSD} = 180/\pi \sum [\tan (\phi/2) \times (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \, (\sin \alpha_{i-1} \cos \beta_{i-1} + \sin \alpha_{i} \cos \beta_{i})]/\phi</math>

:<math>\mbox{EWD} = 180/\pi \sum [\tan (\phi/2) \times (\mbox{MD}_{i} - \mbox{MD}_{i-1}) \times \, (\sin \alpha_{i-1} \sin \beta_{i-1} + \sin \alpha_{i} \sin \beta_{i})]/\phi</math>

These methods are especially useful when the deviation angle is built or decreased rapidly with respect to the survey interval.

These methods are especially useful when the deviation angle is built or decreased rapidly with respect to the survey interval.

''True stratigraphic thickness'' (TST) is the thickness of a stratigraphic unit measured in the direction perpendicular to the bedding planes of the unit (Figure 2). It is a critical measure for understanding both the structural and stratigraphic development of a field. The true stratigraphic thickness is derived from the true vertical depths by the following equation:

''True stratigraphic thickness'' (TST) is the thickness of a stratigraphic unit measured in the direction perpendicular to the bedding planes of the unit (Figure 2). It is a critical measure for understanding both the structural and stratigraphic development of a field. The true stratigraphic thickness is derived from the true vertical depths by the following equation:

:<math>\mbox{TST}  = (\mbox{TVD}_{b} - \mbox{TVD}_{t}) \times (\cos \delta') - [(\mbox{NSD}_{b} - \mbox{NSD}_{t})^{2}</math>

:<math>\mbox{TST}  = (\mbox{TVD}_{b} - \mbox{TVD}_{t}) \times (\cos \delta') - [(\mbox{NSD}_{b} - \mbox{NSD}_{t})^{2} + \, (\mbox{EWD}_{b} - \mbox{EWD}_{t})^{2}]^{1/2} \times \sin \delta')</math>

:<math>&\quad \ {+}\, (\mbox{EWD}_{b} - \mbox{EWD}_{t})^{2}]^{1/2} \times \sin \delta')</math>

 

In this equation, δ′ indicates the apparent dip of the bed in the direction of the horizontal displacement (Figure 6), which is written as

In this equation, δ′ indicates the apparent dip of the bed in the direction of the horizontal displacement (Figure 6), which is written as

:<math>\delta' = \tan^{-1} [\tan \delta \cos (\beta - \varepsilon)]</math>

:<math>\delta' = \tan^{-1} [\tan \delta \cos (\beta - \varepsilon)]</math>

where

where

 

* δ = true dip

{{quotation|δ = true dip

* β = bearing of horizontal displacement between well penetration of top and bottom of unit, or

 

:= tan^–1HEWD_b– EWD_''t'')/(NSD_''b''- NSD_''t'')

β = bearing of horizontal displacement between well penetration of top and bottom of unit, or

* ε = bearing of dip vector

 

= tan^–1HEWD_b– EWD_''t'')/(NSD_''b''- NSD_''t'')

 

ε = bearing of dip vector

If the well is straight (no change in deviation) for the length of the interval of interest, this formula reduces to

If the well is straight (no change in deviation) for the length of the interval of interest, this formula reduces to

:<math>\mbox{TST} = (\mbox{MD}_{b} - \mbox{MD}_{t}) \times \cos (\alpha + \delta')</math>

:<math>\mbox{TST} = (\mbox{MD}_{b} - \mbox{MD}_{t}) \times \cos (\alpha + \delta')</math>

where MD = measured depth

where MD = measured depth.

It is important to note the sign convention for the two angles α and δ′. The deviation is measured from the vertical and is positive, whereas the dip is measured from the horizontal and is positive if it is in the same direction as the deviation and negative if the dip is opposite to the deviation.

It is important to note the sign convention for the two angles α and δ′. The deviation is measured from the vertical and is positive, whereas the dip is measured from the horizontal and is positive if it is in the same direction as the deviation and negative if the dip is opposite to the deviation.

''True vertical thickness'' (TVT) is the thickness of a geological unit in a well measured in the vertical direction (Figure 2). It is a valuable measure for volumetric calculations because it is unaffected by variations in the dip of the unit and can be derived by subtracting computer-gridded structural horizons. In a deviated well with a nonhorizontal unit, the TVT is difficult to calculate because, as the well steps out horizontally, it no longer cuts the bottom of the unit vertically below the point where it penetrated the top of the unit (Figure 5). If the dip is in the same direction as the deviation, the unit will appear thicker than it actually is, whereas if the dip is in the opposite direction, the unit will be shortened. The TVT is calculated according to the following formula:

''True vertical thickness'' (TVT) is the thickness of a geological unit in a well measured in the vertical direction (Figure 2). It is a valuable measure for volumetric calculations because it is unaffected by variations in the dip of the unit and can be derived by subtracting computer-gridded structural horizons. In a deviated well with a nonhorizontal unit, the TVT is difficult to calculate because, as the well steps out horizontally, it no longer cuts the bottom of the unit vertically below the point where it penetrated the top of the unit (Figure 5). If the dip is in the same direction as the deviation, the unit will appear thicker than it actually is, whereas if the dip is in the opposite direction, the unit will be shortened. The TVT is calculated according to the following formula:

:<math>\mbox{TVT}  = (\mbox{TVD}_{b} - \mbox{TVD}_{t}) - [(\mbox{NSD}_{b} - \mbox{NSD}_{t})^{2}</math>

:<math>\mbox{TVT}  = (\mbox{TVD}_{b} - \mbox{TVD}_{t}) - [(\mbox{NSD}_{b} - \mbox{NSD}_{t})^{2} + \, (\mbox{EWD}_{b} - \mbox{EWD}_{t})^{2}]^{1/2} \times \tan \delta'</math>

:<math>&\quad \ {+}\, (\mbox{EWD}_{b} - \mbox{EWD}_{t})^{2}]^{1/2} \times \tan \delta'</math>

==See also==

==See also==

* [[Introduction to geological methods]]

* [[Introduction to geological methods]]

* [[Subsurface maps]]

* [[Subsurface maps]]

* [[Effective pay determination]]

* [[Effective pay determination]]

* [[Multivariate data analysis]]

* [[Multivariate data analysis]]

* [[Geological cross sections]]

* [[Geological cross sections]]

* [[Fluid contacts]]

* [[Fluid contacts]]

* [[Geological heterogeneities]]

* [[Geological heterogeneities]]

==References==

==References==