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# Directional survey for 3D reservoir modeling

With the growth in drilling deviated, extended-reach, and horizontal wells, the location of the wellbore is increasingly a 3D problem. It is encountered in one of two situations:

- To direct and define the trajectory of the well during the drilling process (geosteering)
- To characterize the well path after drilling

The former has contributed to huge increases in well productivity. The latter is a vital element of integrated reservoir studies in which the aim is to generate a 3D model of the reservoir based on correct well locations. This discussion is set within the context of the latter. Geosteering is a benefit of logging-while-drilling technology.

## Principles

In directional-survey terminology, the azimuth is the orientation of the wellbore to "north," which has been variously defined as

- Magnetic north
- Geographic north (latitude and longitude)
- Grid north [e.g., the Universal Transverse Mercator (UTM) geographic coordinate system]

Azimuth is typically measured clockwise from north. The inclination is the deviation of the wellbore from the vertical. The azimuth and inclination define the coordinates of the wellbore along its length, usually relative to the location of the wellhead. Tool face quantifies the direction in which the tool is pointing. It is the angle between a reference direction on a downhole tool and a fixed reference in space. For near-vertical wells, the fixed reference is magnetic north and the (magnetic) tool face is the angle between magnetic north and the projection of the tool’s reference direction onto a horizontal plane. For more deviated wells, the fixed reference is the top of the hole and the (gravity or high side) tool face is the tool orientation with respect to the top of the hole. Directional surveys allow azimuth and inclination to be determined so that the coordinates of well location can be computed at each survey station.

## Measurement

There are two types of directional surveys: magnetic and gyroscope surveys. Traditionally, each of these has been run using either single- or multishot tools. The tools are self-contained and can be powered from the surface or with downhole batteries. For a detailed description, see Bourgoyne *et al.*^{[1]}

Magnetic surveys can be run on the drillstring while tripping or on a wireline (conducting cable) after drilling. They are openhole measurements. In the simplest form, a magnetic-survey tool comprises a downhole inclinometer and a compass unit. More advanced tools comprise arrays of three-axis accelerometers (inclinometers) and magnetometers. A three-axis accelerometer measures three orthogonal components of gravity, which combine vectorially to give the direction of the Earth’s gravitational field relative to the axis of the tool and a reference position on its circumference. A three-axis magnetometer measures the direction of the Earth’s magnetic field relative to the tool axis. As **Fig.1** illustrates, by combining the accelerometer and magnetometer data, it is possible to calculate the inclination, *I*, and azimuth, *A*, of the tool. Through the tool face, it is then possible to calculate the inclination, *α*, and azimuth, *β*, of the wellbore itself. The magnetic measurements are impacted by short-term variations in the Earth’s magnetic field and by proximity to magnetic materials such as drillpipe and casing or to magnetic minerals such as pyrite and/or hematite, the latter also occurring as a mud additive. These problems can be mitigated through the use of nonmagnetic drill collars and repeat measurements with different alignments of the tool in the wellbore. It should be noted that even where magnetic-survey tools are run in long lengths of nonmagnetic drill collars, there can still be a significant effect from steel drillstring tools.^{[2]}

One of the problems with gyroscope surveys is that the instrumentation is very sensitive and cannot withstand downhole vibrations and stresses (see Directional survey errors for more information. For this reason, gyroscope surveys have to be run on wireline, usually while entering the hole with a few additional checkshots on coming out. They interrupt the drilling process, and this adds to cost. However, they can be run in cased hole. In the simplest form, a gyro tool comprises a downhole inclinometer and gyroscope unit. Gyroscope surveys are usually mechanically driven, but other methods, which use a rotating beam of light, have been developed on the basis of fiber optics and laser technology. Mechanical gyroscopes are grouped in terms of their freedom of movement and the number of flywheels (between one and three). Three-axis systems (i.e., three orthogonal accelerometers and gyroscopes mounted on an inertial platform) have proved superior, especially in highly deviated and horizontal wells, where, for example, the axial component of the Earth’s gravity field is small. Here, the tool position is calculated from the accelerometer and gyroscope data through the tool’s inertial navigation system.

**Fig. 2** shows the reporting format of a typical directional survey. A synopsis of the errors associated with these measurements and their application has been provided by Theys.^{[3]}

**Fig. 2 – Example of the reporting format of a directional survey with depths in feet. Vertical scale on the Well Path Plot is true vertical depth (TVD). Depth markers on the Plan View trace are measured depths. The Tabular Listing links the two depth scales at measurement stations and contains the wellbore deviation, azimuth, and coordinates at the points of measurement. (Courtesy of Baker Atlas.)**

## Processing

The results of directional surveying usually take the form of inclination, *α*, and azimuth, *β*, of a borehole at a sequence of survey stations. The only other information available is the difference in measured depths for two adjacent stations, but this does not describe the shape of the well path. Starting with the coordinates of the surface reference point, the actual distance between adjacent survey stations needs to be calculated so that the coordinates of any station can be found by addition and those of any intermediate point can be found by interpolation. There are several ways of doing this interpolation. The following account is based on Inglis’ work.^{[4]}

The balanced tangential method assumes that the actual wellpath between two adjacent measurement stations can be approximated by two straight lines of equal length, *L*/2, shown as AX and BX in **Fig.3**. This leads to the following expressions for the incremental distances between adjacent survey stations in the vertical direction (Δ*V*), in the direction of the northing (Δ*N*), and in the direction of the easting (Δ*E*):

where the subscripts 1 and 2 denote the upper and lower survey stations, respectively.

An improvement on the balanced tangential method is the minimum curvature method, which replaces the two straight lines by an arc. The position of the arc is based on the amount of bending in the wellpath between the two survey stations. The amount of bending is described by a ratio factor, *F*_{r}, and quantified by a dogleg angle, *ψ* (**Fig. 4**), so that:

where

The ratio factor is applied to each of the quantities Δ*V*, Δ*N*, and Δ*E* as calculated by the balanced tangential method (**Eqs. 1** through **3**). Thus, for example, **Eq. 1** becomes

The minimum curvature method is the most widely used for computing the coordinate deliverables of directional surveys. Inglis^{[4]} provided a detailed description of the calculations.

## Applications

The uses of directional surveys include monitoring the actual wellpath to ensure that the drilling target has been reached, defining the X–Y coordinates of points in the wellbore, and determining the true vertical depth (TVD) for geological mapping. Other important operational objectives are to ensure that well paths do not collide (a noteworthy risk given that several production wells are drilled from the same platform) and that there are no changes in angularity (e.g., doglegs) that might impede tool deployment or production efficiency.

An important exercise for integrated reservoir studies is to use the TVD to evaluate the true stratigraphic thickness, *h*_{ts}, and the true vertical thickness, *h*_{tv}, of constituent reservoir beds. **Fig. 5a** shows a well with inclination or deviation, *α*, penetrating a bed with dip, *φ*, over a measured bed thickness, *h*_{m}, for the particular case in which the azimuths of the well and the dip are the same. Here, the values of *h*_{ts} and *h*_{tv} can be calculated as

In the more general case, for which the azimuth of the well, *β*, is not the same as the azimuth of the dip, *β*_{d} (**Fig. 6b**), the equations are more complex:

As an illustration of the derivation of these expressions, **Eq. 10** can be derived from **Eq. 8** by projecting the true angle of dip, *φ*, onto the vertical plane that contains the wellbore azimuth within the layer of interest (**Fig. 5b**). This introduces an apparent dip, *φ*_{a}, which is related to the true dip by the relationship:

If the apparent dip is used instead of the true dip in **Eq. 8**, we have

**Eq. 12** will give the correct answer. If we now use **Eq. 11** to substitute for *φ*_{a} in **Eq. 12**, we obtain **Eq. 10**. Other derivations of **Eq. 10** are available (e.g., those that rotate the dip or deviation to zero to simplify the equations). **Eqs. 9** and **10** can also be written in terms of wellbore coordinates.^{[5]}

**Fig. 5 – Basis for the evaluation of true stratigraphic thickness***h*, and true vertical thickness,_{ts}*h*, from processed directional-survey data using Eqs. 9 and 10, respectively. (a) The azimuths of the well and true dip are the same. (b) The azimuths of the well and true dip are not the same: the true dip is projected into the plane of the wellbore to become an apparent dip, so that the azimuths of the well and apparent dip are the same._{tv}

## Nomenclature

## Subscripts

1 | = | upper directional survey station |

2 | = | lower directional survey station |

## References

- ↑ Bourgoyne, A.T., Millheim, K.K., Chenevert , M.E. et al. 1986. Applied Drilling Engineering, 2. Richardson, Texas: Textbook Series, SPE.
- ↑ Thorogood, J.L. 1989. Directional Survey Operations Management. J Pet Technol 41 (12): 1250–1252. SPE-19462-PA. http://dx.doi.org/10.2118/19462-PA.
- ↑
^{3.0}^{3.1}Theys, P. 1999. Log Data Acquisition and Quality Control. Paris: Editions Technip. - ↑
^{4.0}^{4.1}^{4.2}Inglis, T. 1987. Directional Drilling. London: Graham & Trotman. - ↑ Boak, J.M. 1992. Conversion of Well Log Data to Subsurface Stratigraphic and Structural Information. Part 6. Geological Methods. In Development Geology Reference Manual, ME 10, 289–293. Tulsa, Oklahoma: AAPG Methods in Exploration Series, AAPG.

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## See also

Directional survey instruments

PEH:Specialized_Well-Logging_Topics