MAPPING THROW IN PLACE OF VERTICAL SEPARATION: A COSTLY SUBSURFACE MAPPING MISCONCEPTION

Daniel J. Tearpock
Subsurface Consultants & Associates
Lafayette, La.

Richard E. Bischke
Department of Geological & Geophysical Sciences
Princeton University
Princeton, N.J.

Perhaps the most costly subsurface mapping error made by interpreters is to utilize throw instead of vertical separation.

Indeed the two terms are often confused or used interchangeably between different working groups; i.e., vertical separation is often called throw, or throw is substituted for vertical separation. These two terms are, however, not identical and are a measure of two distinctly different geometric properties.

This article demonstrates that to contour throw instead of vertical separation can often result in mapping errors or 25% or greater.

TERMINOLOGY, DEFINITIONS

The term throw is commonly used in the structural literature to describe a fault displacement component, and its definition is well established (Page, 1859; Reid et al., 1913; Hill, 1947, 1959; Billings, 1972). Throw is illustrated in Fig. 1 along with the vertical separation. It is important to understand that throw, in contrast to vertical separation, is a fault slip or displacement component and is defined in the A.G.I. Glossary as the vertical component of the net slip.

As throw is directly related to fault slip, for cross sections drawn perpendicular to strike or parallel to maximum dip (Fig. 1)

Throw = AC =

AB * sin(0)

Where:

AB= fault slip displacement

0 = fault dip perpendicular to strike

Thus as the fault changes dip, the value of throw for a set amount of vertical separation (AE) must also change. It must be emphasized at this point that throw, which is related to fault dip, and displacement cannot be directly measured from electric logs, as will be shown shortly. Later methods will be presented that will enable the interpreter to calculate the amount of throw knowing the vertical separation and other properties, such as fault and bed dip. Normally however, throw should not enter into proper mapping techniques.

The vertical separation (AE) in Fig. 1 can be defined as the distance that a bed has been vertically displaced during faulting (Reid et al., 1913; Hills, 1947). This distance is of primary importance to the mapper as vertical separation is directly recorded by (or readily calculated from) electric logs. To illustrate this point consider the following hypothetical example. Assume that a structure exists that contains beds that dip uniformly to the west (Fig. 2), that two wells are drilled into these beds, and that these wells produced the SP logs shown in Fig. 2. The dashed line in Fig. 2 is a future normal fault, which will displace so that Well 1 is juxtaposed over the lower or footwall portion of Well 2 (Fig. 3).

From this geometric configuration the following observations can be made. As Well 1 is displaced above the footwall portions of Well 2, the top of bed B in the hanging wall is brought into contact with the top of bed C in the footwall. Therefore, the missing section in Well 1 of the hanging wall consists of bed B to the top of bed C; i.e., the sequence of beds A, B, and C in Fig. 2 becomes the sequence A and C in Fig. 3. Closer inspection of the electric logs reveals that the missing section in the fault cut at Well 1 (in the hangwall) is represented by the coarsening upward sequence and its lower shale present in the structurally lower hanging wall portion of Well 2.

This hypothetical example clearly demonstrates that the throw is not equal to the missing section in the hanging wall portion of Well 1. However, the missing section is present in the lower hanging wall portion of Well 2 and is equivalent to the vertical separation as defined in Fig. 1. The conclusions are that electric logs record the vertical separation and not the throw and that throw does not directly enter into proper contouring techniques.

Summary: From the exercise it can be seen that distance AE is a direct measure of the section missing from the fault cut. It is also apparent from Figs. 1 and 3 that the amount of throw (AC) is not directly recorded by electric logs nor is it simply related to AE, the vertical separation.

ONE COMMON ERROR

A common mapping mistake is to measure the vertical separation in electric logs and then apply the vertical separation as if it were throw across the fault cut (Fig. 4).

The correct method for applying the log data across the fault cut is to project the value of the vertical separation (AE) upward from point E, and then project bed A (along with its curvature) across the fault cut.

In general a bed may be nonplanar, and thus the curvature (i.e., gradient) of a bed in the downthrown block must be projected up to the equivalent bed in the up thrown block. Bed A at the correct level (Fig. 4) can then be projected downward into the upthrown block. If, however, the vertical separation (AE) is taken to be throw (A'C in Fig. 4), then bed A will be projected across the fault cut in the example at too high a level. It will be demonstrated that this procedure will result in mapping errors that are unacceptably large.

The point to remember for this case is that the bed and its inclination (i.e., dip) must be projected from a point directly above its footwall cutoff and not from the beds' hanging wall cutoff (Fig. 4).

QUANTITATIVE RELATIONSHIPS

Vertical separation can be related to throw using the relationships contained in Fig. 5.

Performing some elementary trigonometry and using the law of sins results in

 AE*sin(p/2 - 0) = AB*sin(0 - 0) (2)

where:

0 = bed dip perpendicular to strike

0 = fault dip perpendicular to strike and substituting

 AB = AC/sin(0) (3)

into equation (2) yields

 AE/AC = sin(0) - 0)/[sin-(0)*sin(p/2 - 0)] (4)

and utilizing trigonometric identities yields

 AE/AC =/

Where:

AE/AC is taken relative to the absolute value, and 0 and 0 are taken clockwise.

MAP CHECKING PROCEDURES

Normal well log determinations are capable of calculating the values of the vertical separation and the bed dip (Tearpock and Bischke, 1990). If it were possible to measure throw directly from well log data (or by independent means such as seismic), then equation (5) could be used to determine the dip of the fault at the working level; i.e.,

 0 = arctan [tan(0))/(AE/AC + 1.0)] (6)

However, the authors do not presently know of any technique to determine throw directly from well log data, although seismic data could be used. Nevertheless, the nomogram presented in Fig. 6 and derived from equation (6) can be used to check results for consistency while mapping electric log data. This nomogram is designed to cover the range of dips normally encountered during daily operations, and thus Fig. 6 can be used to check the consistency of the contour maps and ensure that the contours are on the correct or proper level.

Furthermore, in Fig. 6 the fault and bed dips can be taken in either the clockwise or counterclockwise direction. If the bed dips exceed the fault dip, and if the beds and the fault dip in the same direction, then AE/AC must be added to plus 1.0 in Fig. 6. This is the case of the repeat section in normal faulting. The appendix gives three examples on how to apply the nomogram.

ERROR ANALYSIS

The relationships in Fig. 7 can be used to calculate an error analysis of the problem; i.e., how much error is introduced into maps that assume that fault throw is vertical separation? It should be readily apparent from Figs. 3 and 4 that if throw is mapped in place of vertical separation, the errors involved in this assumption could result in a well that totally misses its target.

It is derived from the law of sins noticing that AE = A'C:

 BA*sin(0 - 0)) = A'C*sin(p/2 - 0) (7)
 As
 BA = TA/sin(0) (8)
 TA/A'C = sin(0)*cos (0)/sin(0 - 0) (9)
 where the error analysis is taken relative to a horizon that is incorrectly mapped as throw or relative to length A'C on Fig. 7; i.e., the percent distance that a correctly mapped horizon exists relative to an incorrectly mapped horizon.
 Thus Fig. 8, which is derived from equation (9), can be used to measure the error introduced by improper mapping techniques.
 An examination of Fig. 8 shows that unacceptably large errors (greater than 20%) are introduced for faults dipping at angles of less than about 30 (or greater than about 150) and for beds dipping at greater than 5 (or less than 175, recognizing that a 180 dip is lower than a 175 dip).
 For faults dipping at 45, the bed dips should not exceed about 10 (lower portion of Fig. 8) or dip at less than about 170. Notice that if the beds are dipping in the same direction as the fault, then the errors encountered are correspondingly larger (compare the upper and lower half of Fig. 8).
 If the bed dips are about equal to the fault dips and the beds dip in the same direction as the fault, as is often the case along the flanks of salt domes, then the errors involved in mapping throw as vertical separation can readily exceed several hundred percent.
 In Fig. 8 the error is relative to the depth of an incorrectly mapped horizon or a horizon mapped as throw. Thus in Fig. 8 a bed that dips at 135 (or 45 to the west) into a fault dipping at 45 (to the east) will have an error of 50%, which means that the correct level to the bed under consideration is 50% or one half the vertical distance from the incorrect level. Thus, relative to the correctly mapped bed the error is,
 (incorrect depth level - correct depth level)/correct depth level
 or
 error= (1 - 0.5)/0.5 X 100 = 100%
 relative to the level that should have been correctly mapped as vertical separation.
 The conclusion is that the errors involved in substituting throw for vertical separations are larger than interpreters may realize or be willing to accept.
STRUCTURAL CASE STUDY
 A technique has been presented for projecting established contours from one fault block across a fault into the opposing fault block to correctly contour the displacement resulting from a fault. The fault component referred to, as determined from well log correlation or seismic, is vertical separation.
 An incorrect technique has been presented for contouring displacement as throw.
 Next to be considered is a case study that illustrates the impact of preparing a map with the incorrect as opposed to correct contouring technique.
 This field example shows an anticlinal high cut by a large 600 ft down-to-the-south normal "Fault A" and a second smaller 100 ft normal compensating down-to-the-north "Fault B" (Fig. 9). The structure map (Fig. 10) was prepared with the incorrect assumption that the missing section in the wellbores is equal to throw.
 When throw is mapped the structural contours are projected through the fault gap perpendicular to the strike of the fault. Based on this interpretation, two development wells were proposed.
 Location "X," upthrown to Fault A in Reservoir A, is estimated to penetrate the reservoir updip and to the east of Well 12 at 4,960 ft subsea. The purpose of the well is to improve the drainage efficiency of this reservoir. Location "Y," upthrown to Fault B, is estimated to penetrate Reservoir B at 5,360 ft subsea and is designed to be drilled in the optimum structural position to drain the attic reserves in this reservoir.
 The structure map in Fig. 11 was constructed using the missing section in the wellbores as vertical separation and not throw. This process is best visualized by imagining that all of the well data are restored to their initial or pre faulted" position.
 If this is done then the contouring process would be trivial because no fault currently exists. Thus to map vertical separation (instead of throw) project the contours across the fault gap as if the fault were not present (i.e., pre faulted state) and then adjust the contours across the fault gap for their "post-faulted" values of vertical separation.
 The dashed contours in the fault gap showing the projection of the contours from one fault block to the next provide an example. This interpretation results in a different structural picture. First, Well "X" is estimated to penetrate the formation at 5,030 ft subsea, the depth to the oil-water contact in Reservoir A. Therefore Well "X," if drilled, would be dry.
 The penetration point for Well "Y," in Reservoir B, is estimated at - 5,415 ft, 55 ft deeper than that shown in Fig. 10.
 Based on the correctly contoured map, the proposed well is about 800 ft west of the optimum position to efficiently drain the remaining reserves in Reservoir B.
 The correctly contoured map has the following impact on these two reservoirs and proposed wells.
 Reservoir A (upthrown to Fault A)
It eliminates the drilling of a dry hole.
It improves the volumetric reserve estimate, in this case a 36% reduction in reserves.
 Reservoir B (upthrown to Fault B)
It improves the volumetric reserves estimates for the reservoir (a reduction of 11%).
It improves the configuration of the reservoir, affecting the location of the proposed development well.
 The magnitude of error created by incorrectly mapping throw instead of vertical separation is greatest near or on the crest of a structure (Figs. 8, 9). The amount of error inherent along cross section B of Fig. 10 can be determined from Fig. 8. The beds are estimated to dip at 13 in the vicinity of the 55 dipping "Fault A" (Fig. 10). Therefore the error relative to the incorrectly mapped horizon is 19%.
APPENDIX
 Nomogram Fig. 6 is an attempt to generalize Equation 6 so that it can be applied to daily mapping routines.
 As vertical separation and throw can be read directly from geologic or seismic maps (or constructed from cross section), Fig. 6 can be used to check results in order to prevent costly mistakes and errors and to establish a consistent terminology when communicating between different working groups.
 This diagram can also be used to predict fault dip if the ratio AE/AC can be determined from seismic data and the bed dip is known. However, the seismic data should image true dip and not apparent dip. As the diagram has several minor complications, three examples will be presented.
Example 1:
 This simple example of beds dipping into a fault is encountered in many fields (Fig. 12-1). The length of throw, or the vertical component of the slip (AC), is measured along with the vertical separation (AE). As the input to Fig. 6 is a ratio (AE/AC), the units do not matter, although the units should be consistent (either in depth, length, time, etc.).
 The ratio AE/AC is then calculated, and the absolute value of the quantity (AE/ AC-1.0) is determined (Table 1). In Fig. 6 the values for the quantities (AE/AC-1.0) have been plotted along the abscissa on a logarithmic scale. The S shaped lines on the diagram denote the different bed dips (0) and range from 1.0 to 80. One then projected upward from the calculated (AE/AC-1.0) value to the point on the diagram where the vertically extended line intersects the known bed dip (0) (Fig. 6). This point is then projected over to the ordinate, and the corresponding fault dip (0) is read off the diagram (Fig. 13).
Example 2:
 In this example the bed dips at about the same angle that the fault dips, and this case is commonly encountered along the flanks of salt domes (Fig. 12-2). Again the ratio AE/AC is determined, and the absolute value of (AE/AC-1.0) is calculated. This number will always be positive. The results are shown (Table 1, Fig. 13.)
Example 3:
 Example 3, which also comes from the flanks of a salt dome, is the case where the normal faulting cuts steeply dipping beds and results in a repeat section (Fig. 12-3). In this more unusual example the ratio AE/AC must be added to 1.0 (i.e. the quantity AE/AC + 1.0 is determined). The results of the calculations are shown (Fig. 13, Table 1).
REFERENCES
 Billings, M.P., 1972, Structural Geology, 3rd ed., Prentice-Hall, Englewood Cliffs, N.J., 606 p.
 Hill, M.L., 1947, Classification of Faults, American Association of Petroleum Geologists Bulletin, Vol. 31, pp. 1664-1673.
 Hill, M.L., 1959, Dual classification of faults: AAPG Bulletin, Vol. 43, No. 1, pp. 217-237.
 Page, D., 1859, Handbook of Geological Terms and Geology, William Blackwood & Sons, London.
 Reid, H.F., Davis, W.M., Lawson, A.C., Ramsome, F.L., and Committee, 1913, Report of the committee on the nomenclature of faults: Geological Society of America Bulletin, Vol. 24, pp. 163-186.
 Tearpock, D.J. and J. Harris, 1987, Subsurface Geological Mapping Techniques-A Training Manual: Tenneco Oil Co., Houston, Tex., p. 106.
 Tearpock, D. and R.E. Bischke, 1990, Applied Subsurface Geological Mapping, Prentice-Hall, 650 p., in press.
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