Structure maps and contouring

Quick refresher

A contour joins points of equal value and separates higher values from lower ones. A structure contour is a horizontal line on the surface of a layer. Structure contours run along the strike (the horizontal direction of the layer), and where they crowd together, the dip is steeper.
The top is plotted where the borehole met it. For a deviated well (one that leaves the vertical) this is the point calculated from the directional survey (the measured path of the well), at its true vertical depth (the straight-down depth) below sea level. It is not at the wellhead, and it is not at the measured depth.
The contour interval is the same over the whole map. We choose it not finer than the accuracy of the depths, and so that contours do not merge on the steepest limbs (the sides of a fold), while inside the trap there are enough of them to calculate volumes.
The triangle method (mechanical contouring) is reproducible, but it treats the layer as flat in each triangle. So the contours are angular, and a crest cannot come out higher than the highest well.
Inverse distance weighting draws “bull's eyes” (closed rings of contours) around wells, and far from them it flattens the surface towards the mean of the data. Minimum curvature builds the smoothest surface through the points. It can restore a crest between wells, but beyond the data it continues the slopes.
A map built from wells alone almost always simplifies the structure. The shape between wells comes from seismic data and the geological model. Every map shows the data points, the contour interval, the method and the zero of the depth reference.

Key relationships

z = H - h_{\text{rt}},\qquad z_{\text{elev}} = -z

depth of a point below sea level from the true vertical depth $H$ below the rotary table (the turntable on the rig floor) and the rotary table elevation $h_{\text{rt}}$ (its height above sea level); and the elevation of the point

s_c = L\,\frac{c - z_a}{z_b - z_a}

where the contour of value $c$ crosses a segment of length $L$ between points with values $z_a$ and $z_b$ (linear interpolation)

\hat z = \frac{\sum_i z_i/d_i^{\,p}}{\sum_i 1/d_i^{\,p}}

inverse distance weighting; usually $p = 2$

\nabla^4 z = 0

minimum curvature: between data points the surface obeys the biharmonic equation, like a bent thin plate

Common pitfalls

Plotting the top of a deviated well at the wellhead, or at the measured depth.
Mixing depths below sea level with absolute elevations, or losing the sign of the rotary table elevation where the ground is below sea level.
Taking a contour interval finer than the accuracy of the data: contours every 2 m for seismic depths with an error of tens of metres are false precision.
Taking a computer map to be objective. Its appearance depends on the method, the cell size, the search radius and the smoothing, all of which a person chooses.
Trusting contours beyond the wells: there interpolation is replaced by extrapolation.

After this lesson you can

list the data of a structure map and bring them to one reference system: the position of the top in a deviated well from the directional survey, depths below sea level, and seismic horizons that were converted to depth and tied to wells
choose a contour interval and draw contours by hand with the triangle method, which is linear interpolation between wells
tell apart mechanical, equal-spacing, parallel and interpretive contouring, and explain in which direction each one distorts the structure
explain how a program builds a grid by inverse distance weighting and by minimum curvature, and estimate the depth at a planned point by different methods
tell apart the types of maps: structure maps, thickness maps (isopach and isochore), net thickness maps and maps of layer properties
recognise “bull's eyes” around wells, crossing contours, edge effects, and errors of the zero of reference and of signs

Contours and kinds of maps

A contour is a line on a map along which a quantity has the same value everywhere. It separates the area of higher values from the area of lower ones. A map of contours turns scattered measurements into a continuous picture, from which we read the shape of a surface, trends and anomalies. The main maps of petroleum geology are structure maps. Their contours are structure contours: lines of equal depth of the top or base of a layer. The topic structure contours and sections explains how to find the dip angle and azimuth from structure contours and how the three-point problem is solved. Here we deal with how such maps are built.

Depths on structure maps are measured from sea level. In Russian-language practice, structure contours are labelled with absolute elevations: below sea level they are negative (−2410, −2420…). In English-language practice, they are labelled with depths below sea level (true vertical depth subsea, TVDSS), as positive numbers. Both ways are correct. The error is to mix them on one map, or not to say which one is used. In this lesson, as in the topic Stress, strain and structural analysis, we label depths below sea level $z$ , and the absolute elevation is $z_{\text{elev}} = -z$ .

Maps that are built with contours

Map: Structure map, of the top or the base of a layer
Contours: structure contours: equal depths or absolute elevations
What it is for: shape of the trap, crest, closure, depths of planned wells

Map: Isopach map
Contours: equal true thicknesses, measured normal to the bedding
What it is for: how the thickness of the layer changes; direction of sediment supply

Map: Isochore map
Contours: equal thicknesses in the vertical direction
What it is for: rock volume between two surfaces

Map: Map of effective (oil-saturated) thickness
Contours: equal thickness of reservoir picked by cut-offs
What it is for: estimates of hydrocarbons in place (Volumetric estimation of hydrocarbons in place)

Map: Maps of properties
Contours: equal porosity, permeability, pressure (isobars), temperature (isotherms)
What it is for: properties for volume estimates and models

Map: Lithofacies map, subcrop map (a map beneath an unconformity)
Contours: boundaries of rocks or their shares; where layers meet the unconformity surface from below
What it is for: where the reservoir and the seal are, where a layer is cut off

Map	Contours	What it is for
Structure map, of the top or the base of a layer	structure contours: equal depths or absolute elevations	shape of the trap, crest, closure, depths of planned wells
Isopach map	equal true thicknesses, measured normal to the bedding	how the thickness of the layer changes; direction of sediment supply
Isochore map	equal thicknesses in the vertical direction	rock volume between two surfaces
Map of effective (oil-saturated) thickness	equal thickness of reservoir picked by cut-offs	estimates of hydrocarbons in place (Volumetric estimation of hydrocarbons in place)
Maps of properties	equal porosity, permeability, pressure (isobars), temperature (isotherms)	properties for volume estimates and models
Lithofacies map, subcrop map (a map beneath an unconformity)	boundaries of rocks or their shares; where layers meet the unconformity surface from below	where the reservoir and the seal are, where a layer is cut off

True thickness is linked to vertical thickness through the dip angle: $h = h_\text{TVT}\cos\delta$ (thickness of a layer in a well). The topic Cross-sections from maps, map combination and faults on maps explains how thickness maps are made by subtracting two structure surfaces, and how sections are built from maps. The topic Gross rock volume from maps explains how rock volume is calculated from maps.

Contours obey a few rules. All of them follow from the fact that a contour is a line of equal value on a single-valued surface:

They do not cross or merge. At each point of the map the surface has one depth. The exception is vertical and overhanging parts (the fault plane of a reverse fault, an overturned limb). There one point of the map lies above two points of the surface, and each block is shown separately.
They run between higher and lower values. On one side of a contour all values are higher than it, and on the other side they are lower. A 2420 m contour cannot run so that a point of 2431 m ends up on the side of smaller depths.
They repeat at a turning point. Where the slope changes sign, on a crest, in a syncline (a trough-like downward fold) or in a saddle (a low pass between two crests), the same contour occurs on both sides of the turning point.
They close or run off the edge. A contour is either closed within the map, or it ends at the edge of the map or at a fault. It does not end in empty space.
The interval is constant. The contour interval is the difference between the values of neighbouring contours. It is the same over the whole map. Contours are labelled, and closed depressions are marked with hachures: short ticks that point down the slope.

Data for a structure map

All the data are plotted on a base map in one coordinate system. It is usually a rectangular one, in metres, in the Gauss–Krüger or UTM (Universal Transverse Mercator) projection (a way of flattening the curved earth onto a map; reference systems). Azimuths are converted to the grid north of the map (the north direction of its grid), and depths to one level, sea level.

A formation pick (top) is the depth at which the top of a layer is picked from logs, core or cuttings. We read it off the log as a measured depth (MD) from the rotary table (the turntable on the rig floor). The map needs three numbers: the coordinates of the point where the borehole crossed the top, and the depth of this point below sea level, $z = H - h_{\text{rt}}$ . Here $H$ is the true vertical depth below the rotary table, and $h_{\text{rt}}$ is the rotary table elevation (its height above sea level). The elevation is negative if the rotary table is below sea level, as on the Caspian coast. In a vertical well, the point lies under the wellhead, and $H$ equals the measured depth. In a deviated well, the point can be hundreds of metres from the wellhead, and $H$ is hundreds of metres less than the measured depth. We calculate them from the directional survey (the measured inclination and azimuth of the hole) by the minimum curvature method (Directional survey calculations).

Worked exampleThe top of a layer in a deviated well: where it is on the map

Given (assumed data)

Well pad of wells 8 and 11: wellhead at x = 650 m, y = 450 m; rotary table elevation $h_{\text{rt}} = -14.0$ m (the ground is 20 m below sea level, the rotary table is 6 m above the ground)
Pick of the top in deviated well 11: 2651.6 m measured depth
Directional survey at the pick (minimum curvature): inclination 31.2°, azimuth 11.9°; offset from the wellhead 863.4 m north and 202.4 m east; true vertical depth $H = 2414.6$ m
Vertical well 8 of the same pad: top at a depth of 2486.2 m below sea level

Solution

Coordinates of the top on the map: $x = 650 + 202.4 = 852.4$ m, $y = 450 + 863.4 = 1313.4$ m. The point is $\sqrt{863.4^2 + 202.4^2} = 886.8$ m from the wellhead.
Depth below sea level: $z = 2414.6 - (-14.0) = 2428.6$ m. The absolute elevation is −2428.6 m.
If by mistake we plot the measured depth, reduced to sea level, at the wellhead, we get $2651.6 + 14.0 = 2665.6$ m. This is 237.0 m deeper than the true value. If we plot the correct depth of 2428.6 m but at the wellhead, then two depths of the top end up at one point of the map: 2428.6 m from well 11 and 2486.2 m from well 8. The difference is 57.6 m.
Sensitivity to survey errors: if all the inclination readings below the kick-off point (where the well starts to deviate) were too high by 0.2°, the top would be 3.1 m higher and would shift in plan by 5.8 m. An azimuth error of 0.5° would shift it in plan by 7.7 m.

Answer: The top of well 11 is plotted at the point x = 852.4 m, y = 1313.4 m, which is 886.8 m from the wellhead, with a depth of 2428.6 m below sea level (absolute elevation −2428.6 m).

What it means. Plotting errors of hundreds of metres in plan, and tens or hundreds of metres in depth, are several times larger than the contour interval of the map (10 m). Plotted at the wellhead, the top of well 11 would make a false “hump” almost 60 m high on the pad, next to vertical well 8. The errors of the directional survey itself are an order of magnitude smaller, metres vertically and metres in plan. But they grow with the length of a deviated hole, and for deep wells with a large offset we take them into account. The data in this example are assumed for illustration.

Picks of the top of the layer: the data for the maps in this lesson

Well: 1
x, m: 2100
y, m: 1450
z, m: 2400.9

Well: 2
x, m: 2850
y, m: 2050
z, m: 2416.6

Well: 3
x, m: 1350
y, m: 1000
z, m: 2432.4

Well: 4
x, m: 2550
y, m: 750
z, m: 2503.4

Well: 5
x, m: 1700
y, m: 2450
z, m: 2451.8

Well: 6
x, m: 3550
y, m: 1300
z, m: 2504.2

Well: 7
x, m: 550
y, m: 2100
z, m: 2471.2

Well: 8
x, m: 650
y, m: 450
z, m: 2486.2

Well: 9
x, m: 3300
y, m: 2750
z, m: 2459.0

Well: 10
x, m: 3250
y, m: 300
z, m: 2529.5

Well: 11 (deviated, from the pad of well 8)
x, m: 852.4
y, m: 1313.4
z, m: 2428.6

Well	x, m	y, m	z, m
1	2100	1450	2400.9
2	2850	2050	2416.6
3	1350	1000	2432.4
4	2550	750	2503.4
5	1700	2450	2451.8
6	3550	1300	2504.2
7	550	2100	2471.2
8	650	450	2486.2
9	3300	2750	2459.0
10	3250	300	2529.5
11 (deviated, from the pad of well 8)	852.4	1313.4	2428.6

Coordinates are measured from the south-west corner of a block of 4 × 3 km, with x to the east and y to the north. $z$ is the depth of the top below sea level. Wells 1–10 are vertical. For well 11, the point where the borehole crossed the top is given (the example above). The highest pick is 2400.9 m, and the deepest is 2529.5 m. The data are assumed for illustration.

Seismic data cover the area densely: a reflecting horizon is followed at every point of the survey, tens of metres apart. But seismic data measure time: the two-way travel time of the wave to the boundary and back, in milliseconds. We convert time to depth with a velocity model (Seismic interpretation). Before the tie to wells, the error of such a depth can be a few per cent of the depth, which is tens of metres. So we tie the seismic surface to wells. At each well we find a correction, which is the difference between the pick and the seismic depth at the same point. We build a map of the corrections and add it to the seismic surface. Wells give the accuracy of depths, and seismic data give the shape of the surface between wells. The best maps combine both.

Besides depths, other data are plotted on the map. Dip angles and azimuths from dipmeter data (a log that measures the dip of layers) and borehole image logs suggest the direction of the contours at a well. Points where boreholes crossed faults bound the blocks. Fluid contacts, places where layers meet an unconformity surface from below, and the limits of the reservoir are needed for maps of an accumulation.

Hand contouring: triangles and the contour interval

We begin with the choice of the contour interval. Three considerations decide it:

Accuracy of the data. The interval is not finer than the error of the depths. For picks in wells with an error of about a metre, we can draw contours every 5–10 m. For seismic depths before the tie, we can draw them only every few tens of metres.
Legibility. On the steepest limb the contours must not merge. The distance between them is $w = \Delta z/\mathrm{tan}\,\delta$ . At a dip of 7° and an interval of 10 m, this is about 80 m, or 3 mm on a map at 1:25,000.
Purpose. To calculate the volume of a trap, we need at least 4–5 contours inside the closure. For a regional overview map, an interval of tens of metres is enough.

For the lesson's block, the depths of the picks range from 2400.9 to 2529.5 m, the error of the picks is about a metre, and the dip of the limbs is up to 7–8°. An interval of 10 m matches the accuracy of the data, gives 12 contours for the whole map, and does not merge on the limbs.

The simplest and most reproducible way to draw contours by hand is the triangle method (triangulation), or mechanical contouring. It assumes that between neighbouring wells the slope of the surface is constant:

Contouring by the triangle method

1
Join the wells into triangles
Join the nearest points so that the sides do not cross and the triangles are as close to equilateral as possible. Avoid long narrow triangles along the edge of the data.
2
Choose the contour interval
Choose it by the accuracy of the data, the steepness of the limbs and the purpose of the map.
3
Mark the sides
On each side, find the points where the contours pass, by linear interpolation between the values at its ends.
4
Join points of equal value
Inside each triangle the contours are straight parallel segments at equal distances.
5
Check
Each contour separates higher values from lower ones, contours do not cross, they are labelled, and depressions are marked.

Where a contour crosses the side of a triangle

s_c = L\,\frac{c - z_a}{z_b - z_a}

where:

$s_c$ distance from point $a$ along the side to the point where the contour of value $c$ passes, m
$L$ length of the side between points $a$ and $b$ , m
$z_a,\ z_b$ values at the ends of the side (depths of the top in two wells), m
$c$ value of the contour that lies between $z_a$ and $z_b$ , m

Units: any consistent units of length; values in one set of units

Holds when

The surface between two points is taken as flat (the slope is constant). This is an assumption of the method, not a property of the layer.
A contour crosses a side only if its value lies between $z_a$ and $z_b$ .

Contours on a side are spaced equally, $L\,\Delta z_c/(z_b - z_a)$ apart, where $\Delta z_c$ is the contour interval. Inside a triangle, the surface is the plane through the three wells, the same as in the three-point problem. So contours in it are straight and parallel. On the sides, where the planes of neighbouring triangles meet at an angle, the contours bend.

CalculatedA map of the top of the layer from 11 picks by the triangle method: a network of Delaunay triangles (a standard way of dividing the points into triangles; thin grey lines), contours every 10 m, and labels that give depths below sea level. Deviated well 11 was drilled from the pad of well 8, and the dash is the projection of its borehole on the plan. The star is planned well 12, and the dotted line A–A′ is the line of the cross-section in the next part of the lesson. The narrow triangle 7–8–11 at the western edge gives a slope of about 11° where the model has about 3.5°. Long narrow triangles along the edge of the data distort the map.

Worked exampleContours in the triangle of wells 1–2–5

Given (assumed data)

Well 1: x = 2100 m, y = 1450 m, top at 2400.9 m
Well 2: x = 2850 m, y = 2050 m, top at 2416.6 m
Well 5: x = 1700 m, y = 2450 m, top at 2451.8 m
Contour interval: 10 m

Solution

Side 1–5: the length is $\sqrt{400^2 + 1000^2} = 1077.0$ m, and the difference of depths is $2451.8 - 2400.9 = 50.9$ m. The 2410 m contour crosses it at $1077.0\cdot 9.1/50.9 = 192.5$ m from well 1. The contours 2420, 2430, 2440 and 2450 m cross it at 404.1, 615.7, 827.3 and 1038.9 m. Each next contour is $1077.0\cdot 10/50.9 = 211.6$ m farther.
Side 2–5: the length is 1217.6 m and the difference of depths is 35.2 m. The contours 2420, 2430, 2440 and 2450 m cross it at 117.6, 463.5, 809.4 and 1155.3 m from well 2 (a step of 345.9 m). Side 1–2: the length is 960.5 m and the difference is 15.7 m. Only the 2410 m contour crosses it, at 556.7 m from well 1.
We join the points of equal value: the 2410 m contour between sides 1–2 and 1–5, and the contours 2420–2450 m between sides 1–5 and 2–5. We get five parallel segments. The last one, 2450 m, is short, right by well 5 (2451.8 m).
Check with the three-point problem (structure contours): the slopes of the elevation of the top to the east and to the north are $g_x = 0.014990$ and $g_y = -0.044904$ . Then $\mathrm{tan}\,\delta = 0.04734$ , the dip angle is 2.71° and the dip azimuth is 341.5°. The contours run along the strike 71.5–251.5° at a spacing of $10/0.04734 = 211.2$ m.
In the neighbouring triangle 1–5–11 the plane dips at 2.76° towards an azimuth of 326.5°. When crossing side 1–5, the contours turn by 15°.

Answer: In triangle 1–2–5, the contours 2410–2450 m are straight segments along the strike 71.5–251.5° at a spacing of 211.2 m. On the sides of the triangle they bend.

What it means. Side 1–5 almost coincides with the dip direction, so the step of the contours along it (211.6 m) is almost equal to the distance between them measured at right angles to the contours (211.2 m). This is a handy check. The method adds nothing to the data. The surface is a set of flat faces, the breaks on the sides are a property of the method and not of the layer, and the crest cannot be higher than well 1. But the result is reproducible: from the same data, two specialists get the same map. So we take the mechanical map as the starting point for interpretation, and as a neutral one when interpretations disagree. The data in this example are assumed for illustration.

Mechanical contouring is only one of four ways to draw contours by hand. They differ in what they assume between wells, and so they distort the structure in different ways:

Four ways to draw contours by hand

Way: Mechanical (by triangles)
Assumption: the slope between neighbouring wells is constant
What we get: a reproducible angular map; crests and depressions do not go beyond the extreme values of the data, so the structure is more likely understated

Way: Equal spacing
Assumption: the dip angle is the same over the whole limb; contours at equal distances, taken from the most reliable part or from dipmeter data
What we get: crests higher and depressions deeper than in the wells, and closures outside the data are possible: the most optimistic version

Way: Parallel
Assumption: contours run parallel to one another, and the distances between them may change
What we get: a smooth map that repeats the general pattern of the structure; in boldness it lies between the mechanical and the equal-spacing way

Way: Interpretive
Assumption: all the data and ideas about the geology are taken into account: the style of folds and faults, the direction of sediment supply, seismic data
What we get: the most plausible but subjective map; different specialists will get different versions

Way	Assumption	What we get
Mechanical (by triangles)	the slope between neighbouring wells is constant	a reproducible angular map; crests and depressions do not go beyond the extreme values of the data, so the structure is more likely understated
Equal spacing	the dip angle is the same over the whole limb; contours at equal distances, taken from the most reliable part or from dipmeter data	crests higher and depressions deeper than in the wells, and closures outside the data are possible: the most optimistic version
Parallel	contours run parallel to one another, and the distances between them may change	a smooth map that repeats the general pattern of the structure; in boldness it lies between the mechanical and the equal-spacing way
Interpretive	all the data and ideas about the geology are taken into account: the style of folds and faults, the direction of sediment supply, seismic data	the most plausible but subjective map; different specialists will get different versions

It is useful to compare the ways on the same grid of wells. The difference between them is an honest measure of the uncertainty of the map where there are no seismic data.

Computer contouring: the grid and interpolation

A program does not draw contours at once. First it covers the area with a regular grid of square cells and estimates the value at each node (corner of a cell) from the nearest data. This is gridding, and the result is called a gridded model of the surface, or a grid. Then the contours are drawn on the grid, by the same linear division of the cell sides as of the triangle sides in hand contouring. Everything is decided by the way the values at the nodes are estimated.

Triangulation (triangulated irregular network, TIN) is the computer version of the triangle method. Usually we build a Delaunay triangulation: the triangles are chosen so that inside the circle through the corners of any triangle there are no other points. Of all the ways to divide the points into triangles, this one makes the smallest angle of the triangles as large as possible, so the triangles come out close to equilateral. Outside the polygon that is stretched over the outer wells, the surface is not defined.

Inverse distance weighting (IDW) estimates the value at a node as a weighted average of the values in wells: the farther a well is, the smaller its weight.

Inverse distance weighting

\hat z(x, y) = \frac{\sum_{i=1}^{n} w_i\,z_i}{\sum_{i=1}^{n} w_i},\qquad w_i = \frac{1}{d_i^{\,p}}

where:

$\hat z$ estimate of the value at the point $(x, y)$ , at a grid node or at a planned well, m
$z_i$ value at the $i$ -th data point, m
$d_i$ distance from the point $(x, y)$ to the $i$ -th data point, m
$p$ exponent; usually 2, —
$n$ number of points taken into account: all of them, or the nearest ones within the search radius, —

Units: any consistent units: distances and values in their own units; the weights enter only as a ratio

Holds when

At a data point itself, the estimate equals its value.
The weight depends only on distance, not on direction and not on one well being “shadowed” by another. So programs limit the number of points and take them by sectors.

Near points weigh more than far ones: for $p = 2$ , a well twice as near weighs four times as much. The estimate is a weighted average, so it always lies between the smallest and the largest values of the data. The method cannot raise a crest above the highest well. For $p > 1$ the slope of the surface at each data point is zero. Flat areas and closed contours, “bull's eyes”, appear around wells. Far from the data the weights even out, and the surface tends to the mean value of all the wells.

Minimum curvature gridding looks, among all the surfaces that pass through the data points, for the smoothest one: the one that bends least overall, which means that its total squared curvature is smallest. The physical analogy is a thin elastic plate that is pressed down to the data points. It bends exactly as much as the points force it to.

The condition of minimum curvature

\iint \left(z_{xx}^2 + 2\,z_{xy}^2 + z_{yy}^2\right) dx\,dy \to \min,\qquad z(x_i, y_i) = z_i

where:

$z(x, y)$ the surface we look for, m
$z_{xx},\ z_{xy},\ z_{yy}$ second derivatives of the surface with respect to the coordinates: measures of its curvature, 1/m
$(x_i, y_i),\ z_i$ data points and the values at them, m

Units: consistent units of length

Holds when

The surface is smooth and continuous. Faults are not provided for. They are set separately, or else the program “smooths” a step into a bend.
Between data points such a surface obeys the biharmonic equation $\nabla^4 z = 0$ . The program solves it on the grid by repeated steps (iterations).

The surface keeps the slopes that come to the wells and continues them between wells. So it can rise above the highest well, and restore a crest that lies between wells. The other side is “overshoot”: where close points differ sharply, and beyond the data, the surface bends higher or lower than reality. To limit overshoot, “tension” is added to the equation, which makes the surface less flexible.

There are other methods too. A trend surface is a smooth surface given by a polynomial of the coordinates (a simple formula), fitted to the points by least squares (a best-fit method). It does not pass through the points, but it shows the regional tilt, and the deviations from it (residuals) pick out local structures. Kriging weights the points taking into account the distance at which the values of the layer are still similar to each other. Together with the map, it gives an estimate of its error. It needs fairly dense data and a variogram (Spatial correlation and the variogram).

Cell size. A cell that is comparable with the distance between wells smooths the structure and does not let the surface pass through the picks. Usually it is taken several times smaller than the distance between neighbouring wells. For seismic data it is comparable with the step of the seismic grid.
Search radius and number of points. A radius that is too small leaves nodes without data, and the surface falls into pieces. A radius that is too large pulls distant wells from other parts of the structure into the estimate.
Smoothing. It makes contours prettier but takes the surface away from the data. After smoothing we check the mismatches at wells.
Faults. If a fault is not set as a line or a polygon, the program draws the surface through it continuously, and a step turns into a steep bend (Cross-sections from maps, map combination and faults on maps).

CalculatedThe same 11 picks built by three methods, and the surface of the model that they were taken from. Contours every 10 m, bold ones every 50 m. Dots are wells, and the cross is the true crest. The triangulation is defined only inside the area spanned by the wells (grey outside it). Inverse distance weighting draws “bull's eyes” and flattens the corners of the map. Minimum curvature is closest to the model, but it too falls short of the crest.

Worked exampleDepth of the top at a planned well from three maps

Given (assumed data)

Planned well 12: x = 2000 m, y = 1750 m, inside triangle 1–2–5, 316 m north-north-west of well 1
Data: the 11 picks from the table above
Inverse distance weighting: all 11 wells, $p = 2$

Solution

Triangles: this is the plane of wells 1–2–5 from the previous example. For a shift of 100 m to the west from well 1, the depth of the top grows by $100\cdot 0.014990 = 1.499$ m. For 300 m to the north it grows by $300\cdot 0.044904 = 13.471$ m. So $z = 2400.9 + 1.499 + 13.471 = 2415.9$ m.
Inverse distances: to well 1, 316.2 m, weight $1/316.2^2 = 10.002\cdot 10^{-6}$ m⁻². To wells 5, 2 and 3: 761.6, 901.4 and 992.5 m, with weights 1.724, 1.231 and 1.015 in the same units of $10^{-6}$ m⁻². The sum of the weights of all 11 wells is 17.166.
It is convenient to average the deviations from a round depth of 2400 m. The sum of the weights times the deviations is $\sum w_i(z_i - 2400) = 403.8$ . The four nearest wells give 151.6 of it, and the other seven give 252.2. The estimate is $\hat z = 2400 + 403.8/17.166 = 2423.5$ m.
Minimum curvature (calculated by a program from the same 11 points): 2398.5 m.
Well 12 was drilled: the top is at 2384.9 m.

Answer: The forecast of the top is 2415.9 m by triangles, 2423.5 m by inverse distances and 2398.5 m by minimum curvature. In fact it is at 2384.9 m.

What it means. All three maps put the top deeper than it turned out to be. The crest lies between wells, 16 m higher than the highest of them, well 1. By their design, triangles and inverse distances cannot rise above 2400.9 m. Inverse distance weighting also pulls the estimate towards the mean of the wells: the four nearest wells give 81% of the sum of the weights, but the seven distant and deep ones give most of the sum of the deviations. Minimum curvature continued the slopes of the limbs up to the crest and made the smallest error, but it too does not know what is not in the data. The shape of a crest between wells is shown by seismic data. A map from wells alone understates the closure of the trap and the rock volume. The data in this example are assumed for illustration.

CalculatedA section along the line A–A′ with bends at wells 5, 1 and 4: the top from the model and from the three methods. The triangles give a broken line only inside the area spanned by the wells. Inverse distances give flat areas at wells and flattening beyond the outer wells. Minimum curvature rises to the crest between wells 5 and 1 and continues the slopes beyond the data.

Typical errors and how to recognise them

Signs of errors on a map

What is seen: Closed contours around single wells: “bull's eyes”
Possible cause: inverse distance weighting; a wrong pick; a depth measured from a different zero level
What to do: compare the well with its neighbours and with seismic data; check the pick and the elevation; change the method

What is seen: Contours cross or merge
Possible cause: a construction error; two surfaces on one map (blocks of a reverse fault, an overturned limb)
What to do: check the values; build the blocks separately

What is seen: A crest, closure or steep slope in a corner of the map
Possible cause: an edge effect: extrapolation beyond the wells
What to do: show the limit of the data area; do not take closure and volumes from this part

What is seen: Breaks of contours along lines between wells
Possible cause: mechanical interpolation by triangles
What to do: this is a property of the method and not of the layer; seismic data give the shape between wells

What is seen: A step of tens of metres at one well
Possible cause: a deviated well plotted at the wellhead or at the measured depth; the sign of the elevation was lost
What to do: recalculate from the directional survey and check the zero levels

What is seen: A smooth pattern where seismic data see a fold
Possible cause: a cell that is too large, or strong smoothing
What to do: make the cell smaller, reduce the smoothing

What is seen	Possible cause	What to do
Closed contours around single wells: “bull's eyes”	inverse distance weighting; a wrong pick; a depth measured from a different zero level	compare the well with its neighbours and with seismic data; check the pick and the elevation; change the method
Contours cross or merge	a construction error; two surfaces on one map (blocks of a reverse fault, an overturned limb)	check the values; build the blocks separately
A crest, closure or steep slope in a corner of the map	an edge effect: extrapolation beyond the wells	show the limit of the data area; do not take closure and volumes from this part
Breaks of contours along lines between wells	mechanical interpolation by triangles	this is a property of the method and not of the layer; seismic data give the shape between wells
A step of tens of metres at one well	a deviated well plotted at the wellhead or at the measured depth; the sign of the elevation was lost	recalculate from the directional survey and check the zero levels
A smooth pattern where seismic data see a fold	a cell that is too large, or strong smoothing	make the cell smaller, reduce the smoothing

The edge effect is the disagreement of methods where there are no more data beyond the nearest wells. In the south-east corner of the lesson's block (x = 3900, y = 200 m), beyond well 10, the top of the model lies at 2535.4 m. Inverse distance weighting gives 2500.8 m, which is 35 m higher. Far from the data it is pulled to the mean of the wells (2462.2 m), and it draws a false flattening, or even a false closure. Minimum curvature gives 2545.4 m, which is 10 m deeper. It continued the slope that comes to the edge, and overshot. None of these figures is measured. This is extrapolation, and on a good map the area without data is marked.

The uncertainty of a map grows with distance from wells. At a well the depth is known with the accuracy of the pick. Halfway to the neighbouring well it is known with the accuracy of the method and of seismic data. Beyond the outer wells it is known only with the accuracy of the assumptions. If several maps fit the data equally well, we do not average them but use them as scenarios. We give the depth of a planned well as a range, and we calculate the rock volume for the low, base and high cases (Gross rock volume from maps, Probabilistic volumetrics by Monte Carlo simulation). Each new well is a check of the map. The disagreement between the forecast and the pick shows how far the map can be trusted elsewhere.

Terms and symbols

Contour, structure contour: A line of equal value on a map. A structure contour is a line of equal depth (absolute elevation) of a geological surface.
Contour interval: The difference between the values of neighbouring contours. It is constant over the whole map.
Depth below sea level and absolute elevation: The depth below sea level is $z = H - h_{\text{rt}}$ (TVDSS, positive). The absolute elevation is $z_{\text{elev}} = -z$ (negative below sea level).
Formation pick: The depth at which the top (base) of a layer is picked from logs, core or cuttings. It is read along the borehole and converted into coordinates and depth below sea level.
Isopach map and isochore map: Maps of equal true thickness (at right angles to the layers) and of equal vertical thickness.
Triangle method: Mechanical contouring: linear interpolation along the sides of triangles that join neighbouring points. On a computer it is the Delaunay triangulation (TIN).
Grid: A regular network of nodes at which the program estimates the value of the surface. Contours are drawn from it.
Inverse distance weighting: An estimate at a node as the weighted average of the values of the data points, with weights $1/d^p$ .
Minimum curvature: The smoothest surface through the data points (a model of a thin elastic plate).
“Bull's eye”: Closed contours around a single data point. They are an artefact of the method or a sign of a wrong point.
Edge effect: Distortions of the surface outside the area of data, where interpolation gives way to extrapolation.

Common mistakes and misconceptions

Wrong

“A deviated well can be plotted at the wellhead: the top is the same.”

In fact

The top is plotted at the point where the borehole crossed it, by the directional survey. In the lesson's example, the point is almost 900 m from the wellhead, and a top plotted at the wellhead would disagree with the neighbouring vertical well by 58 m.

Wrong

“A computer map is objective, and a hand-drawn map is subjective.”

In fact

A computer map depends on the method, the cell size, the search radius, the smoothing and the treatment of faults. A person chooses all of them. It is reproducible with the same settings, but it is not “true”.

Wrong

“The finer the contour interval, the more accurate the map.”

In fact

The interval cannot be more accurate than the data. Contours every 2 m for depths with an error of 10–20 m draw not the layer but the noise of interpolation.

Wrong

“The crest of a structure is where the highest well is.”

In fact

Wells rarely hit the crest exactly. Triangles and inverse distances will not raise the surface above the highest well. In the lesson's example, the true crest lies between wells, 16 m higher.

Wrong

“The zero contour of a thickness map is drawn through the wells with zero thickness.”

In fact

The zero contour is the limit of the extent of the layer. It runs between wells where the layer is present and wells where it is absent.

In depth: advanced detail

Why inverse distances give “bull's eyes” and triangles give breaks

Let the point $(x, y)$ approach well $i$ , so that $d_i \to 0$ . Subtract the value $z_i$ from the estimate, and divide the numerator and the denominator by the weight $w_i$ :

The inverse distance estimate near a well

\hat z - z_i = \frac{\sum_{j \ne i} (d_i/d_j)^p\,(z_j - z_i)}{1 + \sum_{j \ne i} (d_i/d_j)^p}

where:

$d_i,\ d_j$ distances from the point to well $i$ and to the other wells $j$ , m
$z_i,\ z_j$ values at the wells, m
$p$ exponent of the method, —

Units: consistent units

Holds when

An identity transformation of the formula of the method. All wells $j \ne i$ within the search radius are taken into account.

As $d_i \to 0$ , the deviation from the value at the well falls as $d_i^{\,p}$ . For $p > 1$ this is faster than the distance itself. The slope of the surface at the well is zero, and a flat area forms around it. For $p = 1$ the surface at the well has a break, like the tip of a cone. For $p < 1$ it has a sharp peak. The larger $p$ is, the wider the flat areas and the sharper the transitions between them. For very large $p$ the map falls apart into steps around the wells.

Triangulation has a different problem: the surface is continuous, but its slope changes abruptly at each side of a triangle, where two planes meet. This gives the breaks of the contours. The Delaunay triangulation itself is unique if no four points lie on one circle. With such a coincidence, the diagonal of the quadrilateral is chosen arbitrarily, and two programs may give slightly different maps.

Minimum curvature: the thin plate and its exact solution

For points on an unbounded plane, the minimum curvature problem has an explicit solution. The surface is written as $z = a_0 + a_1 x + a_2 y + \sum_i c_i\,U(r_i)$ , where $r_i$ is the distance to the $i$ -th point, and $U(r) = r^2\ln r$ is the solution of the biharmonic equation for a point force that is applied to a plate. The coefficients are found from $n + 3$ linear equations: the surface passes through all $n$ points, and $\sum c_i = \sum c_i x_i = \sum c_i y_i = 0$ . This is a thin-plate spline. The lesson's maps were calculated in just this way.

Mapping programs solve the same equation on a grid. At each node where there are no data, the value is adjusted by iterations so that the finite-difference analogue of $\nabla^4 z$ equals zero, and the nodes near data points are pulled to their values. Far from the data the surface tends to an inclined plane, which is why slopes continue beyond the edge. The “tension” $T$ changes the equation to $(1 - T)\nabla^4 z - T\nabla^2 z = 0$ . For $T = 0$ we get minimum curvature. For $T = 1$ we get a surface stretched over the points like a membrane, without overshoots, but with breaks at the data points.

Connections to other topics

Stress, strain and structural analysis — structure contours, the dip angle from them, the three-point problem, and sections with a distorted scale.
Directional survey calculations — how the position and depth of a point of the borehole are calculated from the directional survey.
Directional wells: applications and profile — zeros of depth reference, the rotary table elevation, absolute elevations and coordinate systems.
Well correlation — how consistent picks are obtained, from which maps are built.
Cross-sections from maps, map combination and faults on maps — sections from maps, thickness maps as the difference of surfaces, and faults on maps.
Gross rock volume from maps — rock volume of an accumulation from structure maps.
Spatial correlation and the variogram — spatial variability and kriging.
Traps and timing — crest, closure and spill point of a trap.
Seismic interpretation — following reflecting horizons and converting time to depth.

Check yourself

1
The picks of the top in an area are from 1812 to 1897 m, with an error of about a metre. The limbs dip up to 6°. The map is needed to calculate the volume of a trap about 40 m high. Which contour interval is reasonable?
1. A.50 m
2. B.5 m
3. C.1 m
4. D.25 m
Show the answer
Correct answer: B. 5 m
With an interval of 5 m, 8 contours fit inside the closure, which is enough for volumes. On a limb with a dip of 6° they run $5/\mathrm{tan}\,6° \approx 48$ m apart, which is about 2 mm on a map at 1:25,000. An interval of 1 m is almost equal to the error of the picks and will merge on the limbs. Intervals of 25 and 50 m give no more than one or two contours inside the trap.
2
The top of a layer is picked in a deviated well at a depth of 3105.0 m along the hole. By the directional survey, at this point the true vertical depth from the rotary table is 2846.3 m, the offset from the wellhead is 412 m north and 655 m east, and the rotary table elevation is +32.0 m. Find the depth of the top below sea level, its absolute elevation and its position relative to the wellhead.
Show the answer
$z = H - h_{\text{rt}} = 2846.3 - 32.0 = 2814.3$ m below sea level, and the absolute elevation is −2814.3 m. The offset is $\sqrt{412^2 + 655^2} \approx 774$ m at an azimuth of $\arctan(655/412) \approx 57.8°$ , to the north-east. Here the elevation is positive, and it is subtracted. On the Caspian coast it is negative, and the depth below sea level comes out greater than the depth from the rotary table.
3
Which way of building a map cannot raise a crest above the highest well?
1. A.Interpretive contouring
2. B.Minimum curvature gridding
3. C.Equal-spacing contouring
4. D.Triangulation and inverse distance weighting
Show the answer
Correct answer: D. Triangulation and inverse distance weighting
Linear interpolation in a triangle and the weighted average of inverse distances always lie between the values of the data. Minimum curvature continues the slopes and can rise higher. Equal-spacing and interpretive contouring directly allow a crest between wells.
4
Why do closed contours appear around wells on a map built by inverse distance weighting, and why are there more of them when the exponent is raised from 2 to 4?
Show the answer
At the well itself its weight grows without limit, and the estimate approaches its value faster than the distance grows. For $p > 1$ the slope at the well is zero, a flat area forms around it, and the contours go round this area in a closed ring. As $p$ grows, the influence of each well becomes sharper: flat areas are wider, transitions between them are steeper, and the map falls apart into steps around the wells. There is no geology in these rings. They are a property of the method.
5
On a map, the contours −2450 and −2460 m cross at two points. What can this mean?
1. A.A saddle between two crests
2. B.A normal occurrence at a small contour interval
3. C.A construction error, or two surfaces on one map, for example the blocks of a reverse fault
4. D.A very steep dip of the layer
Show the answer
Correct answer: C. A construction error, or two surfaces on one map, for example the blocks of a reverse fault
A single-valued surface cannot have two depths at one point. So contours of different values do not cross at any dip and any interval. A crossing is allowed only where the surface overhangs itself, at a reverse fault or an overturned limb. Then the blocks are mapped separately. In all other cases it is an error.
6
In a corner of the map with no wells, inverse distance weighting shows a gentle closure, and minimum curvature shows continuing descent. Which map should we believe, and what should we do?
Show the answer
Neither: both figures are extrapolation. Far from the data, inverse distance weighting tends to the mean of the wells and draws a false flattening and closures. Minimum curvature continues the slopes and can overshoot. We need data, seismic data or a well. Until then, we mark the area without data on the map and do not use it to calculate closure and volumes.

Updated: 15 October 2026