On Determining the True Thickness of a Bed from the Apparent Thickness
Abstract Generated abstract
The paper addresses the calculation of a bed’s true thickness from its apparent thickness as measured in a borehole, a problem complicated in earlier treatments by the use of multiple formulas and intermediate angular quantities. It derives a single trigonometric expression using the apparent thickness, borehole zenith angle, bed dip angle, and azimuthal difference between borehole inclination and bed dip. The resulting formula is presented as simpler and more convenient for exploration drilling than prior formulations, while remaining applicable when bed attitude and borehole curvature parameters are known.
Full Text
ON THE ISSUE OF DETERMINING THE TRUE THICKNESS OF A BED FROM THE APPARENT THICKNESS
L. A. Pukhlyakov
(Presented by Professor A. V. Aksarin)
Some authors, for example N. F. Frolov et al. [2, p. 103], recommend using three formulas to determine the true thickness of a bed \((h_i)\) from the measured or apparent thickness \((h_v)\); these formulas involve the zenith angle of the borehole \((\Theta)\), the true, or full, dip angle of the bed \((\alpha)\), and the dip angle of the bed in the plane of curvature of the borehole \((\alpha_k)\). The drawback of these formulas is, first of all, that there are several of them rather than one, which complicates the work. In addition, it is a great inconvenience that each time it is necessary to calculate the dip angle of the bed in the plane of inclination of the borehole \((\alpha_k)\), whose expression also includes another quantity—the difference between the azimuths of the bed dip and the borehole inclination \((\varphi)\).
In view of the above, a new formula is proposed for determining the true thickness of a bed from the apparent thickness; by using it, the calculation can be carried out without resorting to intermediate quantities. To derive this formula, let us construct a section of the bed in the plane of its dip and, through the point where the borehole meets its roof, draw a perpendicular \(BD\), which will represent the segment expressing the true thickness of the given bed—\(h_i\). The interval of the borehole that intersects the given bed is projected onto this drawing as the segment \(BC'\) (Fig. 1).
Fig. 1. Relationship between the true thickness of a bed and the apparent thickness
It will be related to the segment \(BD\) by the relation
\[ BD = BC' \cdot \cos(\angle C'BF + \angle FBD), \tag{1} \]
but the angle \(\angle FBD\) is equal to the angle \(\angle DC'F\), since both belong to right triangles in which two angles are formed by intersecting straight lines. On the other hand, the angle \(\angle DC'F\) is the dip angle of the bed.
the bed \(\alpha\), and the segment \(BD\) expresses the true thickness of the bed; hence expression (1) will take the form
\[ h_u = BC' \cdot \cos(\angle C'BF + \alpha) \]
or
\[ h_u = BC'(\cos \angle C'BF \cdot \cos \alpha - \sin \angle C'BF \cdot \sin \alpha). \tag{2} \]
Next, the sine and cosine of the angle \(\angle C'BF\) can be expressed from the fact that the triangle \(C'BF\) is right-angled:
\[ \sin \angle C'BF = \frac{x}{BC'}, \]
\[ \cos \angle C'BF = \frac{z}{BC'}. \]
On the other hand, the legs of this triangle can be expressed through the apparent thickness of the bed \(h_v\), the zenith angle of the borehole \(\Theta\), and the difference between the azimuths of the borehole inclination and the bed dip \(\varphi\):
\[ x = h_v \cdot \sin \Theta \cdot \cos \varphi, \]
\[ z = h_v \cdot \cos \Theta, \]
whence expression (2) will take the form
\[ h_u = BC'\left(\frac{h_v \cdot \cos \Theta}{BC'} \cdot \cos \alpha - \frac{h_v \cdot \sin \Theta \cdot \cos \varphi}{BC'} \sin \alpha\right) \]
or, after transformation,
\[ h_u = h_v(\cos \Theta \cdot \cos \alpha - \sin \Theta \cdot \sin \alpha \cdot \cos \varphi). \tag{3} \]
This formula is very similar to the formula of P. M. Leontovsky [1]. The difference between them consists only in the fact that, instead of the zenith angle of the borehole \(\Theta\), P. M. Leontovsky uses the inclination angle of the survey line \(\delta\), which is related to it by the relation
\[ \delta = 90^\circ - \Theta. \]
P. M. Leontovsky’s formula is convenient for use in geological surveys, that is, at small values of the inclination angle of the survey line. The formula (3) recommended here is more convenient for use in exploration drilling, that is, at small values of the zenith angles of boreholes. In addition, the formula (3) recommended here is easier to remember than P. M. Leontovsky’s formula. This formula is universal, that is, suitable for determining the true thickness from the apparent thickness in all cases when the elements of the bed attitude and the borehole curvature are known.
REFERENCES
-
P. M. Leontovsky. Mine-Surveying Problems, part 5. Elements of the occurrence of beds (mining geometry). Izvestiya of the Ekaterinoslav Higher Mining School, issue 2, 1905 (1906).
-
N. F. Frolov, E. F. Frolov. Geological Observations and Constructions in Drilling Curved Boreholes. Gostoptekhizdat, 1957.