Equations: Correlation and maps

The equations of the “Correlation and maps” lessons in “Petroleum geoscience”, with their conditions and units, and what each symbol means in them.

Normalisation of the gamma ray log to a reference well

GR_{\text{norm}} = GR_{cl}^{\text{ref}} + I_{GR}\left(GR_{sh}^{\text{ref}} - GR_{cl}^{\text{ref}}\right),\qquad I_{GR} = \frac{GR - GR_{cl}}{GR_{sh} - GR_{cl}}

API units; the index is dimensionless

Holds when: The reference beds are present in all wells and have the same composition. We check this against core and regional data. The differences between the scales are linear (a shift and a stretch). We correct for non-linear distortions, such as washouts and barite (a heavy mineral added to the drilling fluid), before normalisation.

Well correlation →

Thicknesses of an interval in a deviated well

h = \Delta l\cos\gamma,\qquad h_{\text{TVT}} = \frac{h}{\cos\delta},\qquad \Delta H = \Delta l\cos\theta

lengths in the same units; angles in degrees

Holds when: The layer is planar and its top and base are parallel. The hole is straight over the interval; otherwise we divide the interval into parts. The azimuths of the layers and of the hole are measured from the same north. The formula for $\cos\gamma$ comes from the topic vector calculation.

Well correlation →

Least-squares line of correlation

z_{\text{B}} = \bar z_{\text{B}} + b\,(z_{\text{A}} - \bar z_{\text{A}}),\qquad b = \frac{\sum_k (z_{\text{A},k} - \bar z_{\text{A}})(z_{\text{B},k} - \bar z_{\text{B}})}{\sum_k (z_{\text{A},k} - \bar z_{\text{A}})^2}

metres; the same depth reference in both wells (true vertical)

Holds when: The events are of the same age in both wells. The fit uses reliable events from one stretch without breaks. The ratio of accumulation rates is constant over the stretch. Where it changes or there is a hiatus, we build the line in parts. The errors in the depths of events in B are larger than in A (for example, cuttings against core), so we fit $z_{\text{B}}$ against $z_{\text{A}}$ .

Well correlation →

Probability that a body reaches the neighbouring well

P = 1 - \frac{d\,\sin\alpha}{W}\quad (P \ge 0)

any consistent units of length

Holds when: The body is straight and of constant width, and its position relative to the wells is random. The formula becomes crude at small $\alpha$ , when the body is elongated almost along the line of the wells. Then the sinuosity (how much the body winds) and the length of the body decide, not its width. When $d\sin\alpha > W$ , the probability is zero, because the body is narrower than the distance between the wells measured across its axis.

Well correlation →

Macro-heterogeneity coefficients

K_{\text{sand}} = \frac{\sum h_{\text{eff}}}{\sum h_{\text{t}}},\qquad K_{\text{dis}} = \frac{1}{N}\sum_{i=1}^{N} n_i,\qquad K_{\text{mer}} = \frac{S_{\text{mer}}}{S},\qquad K_{\text{cont}} = \frac{S_{\text{res}}}{S}

dimensionless; thicknesses and areas in the same units

Holds when: The coefficients are calculated for a layer or horizon that was picked by correlation. $K_{\text{sand}}$ depends on the cut-offs by which the reservoir is picked (cut-offs are the limit values of porosity and similar properties below which rock does not count as reservoir). Here it is weighted by thickness. $K_{\text{mer}}$ and $K_{\text{cont}}$ are taken from maps that were built from wells, so their accuracy is limited by the density of the grid. In some methods the merging area is related not to the whole area of the accumulation but to the area where the reservoirs of the layer are present. When comparing objects, check how $K_{\text{mer}}$ was calculated.

Reservoir architecture and performance →

Measures of connectivity

C = \frac{V_{\max}}{V_{\text{res}}},\qquad C_w = \frac{V_w}{V_{\text{res}}}

fractions

Holds when: Bodies are connected if they touch with sand across a surface. Touching along an edge or at a point does not count. Static connectivity takes into account the geometry of the bodies, but not the permeability of contacts or the baffles inside bodies.

Reservoir architecture and performance →

Time constant of a compartment behind a baffle

\tau = \frac{V_p\,c_t}{T},\qquad T = \frac{k_b\,A}{\mu\,b}

Holds when: The pressure inside the compartment is the same everywhere, because the compartment is much more permeable than the baffle. The pressure on the other side is constant: the neighbouring layer is much larger than the compartment, or is supported by injection. If the compartments are comparable in size, replace $V_p$ with $V_1V_2/(V_1 + V_2)$ . Flow through the baffle is single-phase (only one fluid flows) and nearly steady. We ignore the compressibility of the baffle itself.

Reservoir architecture and performance →

Where a contour crosses the side of a triangle

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

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 $z_b$ .

Structure maps and contouring →

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}}

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.

Structure maps and contouring →

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

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).

Structure maps and contouring →

Isochore as the difference of surfaces, and isopach

h_\text{TVT}(x, y) = z_\text{base}(x, y) - z_\text{top}(x, y),\qquad h = h_\text{TVT}\cos\delta_b

lengths in the same units; angles in degrees

Holds when: The top and the base at the point belong to one block: depths are not subtracted across a fault (section on fault polygons). For $h$ , the top and the base are nearly parallel over a distance of the order of the layer thickness. Otherwise the thickness is measured at right angles to the mean surface. Depths are measured from one level and are positive downwards.

Cross-sections from maps, map combination and faults on maps →

Throw, heave and the dip of the fault plane from a fault polygon

w = \Delta_\perp,\qquad t = z_\text{hw} - z_\text{fw},\qquad \mathrm{tan}\,\delta = \frac{t}{w}

lengths in metres (on the ground, not on the map); angles in degrees

Holds when: The displacement is along the dip of the fault plane. For an oblique displacement, the “paired” points on the two edges are offset along the fault by the strike-slip component, and these paired points are the ones to compare (components of displacement). The fault plane is planar (not curved) over the width of the strip. The dip of the layers does not affect the relations: at the edges of the strip we compare one and the same point of the layer before and after the displacement (the derivation is in the advanced section).

Cross-sections from maps, map combination and faults on maps →

Shift of the polygon of the base relative to the polygon of the top

\Delta q = \frac{h_\text{TVT}}{\mathrm{tan}\,\delta \pm \mathrm{tan}\,\delta_b}

m; angles in degrees

Holds when: “+” if the layer dips towards the fault plane, and “−” if it dips the same way, as in the formula for the vertical separation of a layer in the topic on faults. The layer and the fault plane are planar (not curved) over the distance $\Delta q$ , and the thickness is constant. The same holds for both walls.

Cross-sections from maps, map combination and faults on maps →

Offset of like-valued contours along a fault

\Delta s = \frac{t}{\mathrm{tan}\,\delta_\parallel}

Holds when: The displacement is along the dip: a point of the hanging wall and its “pair” in the footwall lie opposite each other across the fault, along the line at right angles to it, and differ in depth by $t$ . The layer is inclined along the fault. If it strikes parallel to the fault ( $\delta_\parallel = 0$ ), the contours are not offset but run along the strip, and we read the throw from the number of contours that are “missed” across it.

Cross-sections from maps, map combination and faults on maps →

Rock volume above the contact from the area–depth curve

\mathrm{GRV} = \int_{z_\text{crest}}^{z_\text{OWC}} A_\text{top}(z)\,dz

SI: area in m², depth in m, volume in m³; convenient: km² × m = million m³

Holds when: The layer is thicker than the accumulation: under the whole accumulation the base is below the contact, and all the rock between the top and the contact is the layer (bottom water everywhere). The thin layer is in the next section. The contact is horizontal. Areas are taken only inside the closing contour: parts of the map beyond the spill point do not belong to the trap. The volume includes all the rock of the layer, non-reservoir rock too. The share of reservoir is taken into account separately (gross and net thickness).

Gross rock volume from maps →

Area of a polygon from the coordinates of its vertices

A = \frac12\left|\sum_{i=1}^{n} \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right|,\qquad (x_{n+1}, y_{n+1}) = (x_1, y_1)

coordinates in metres, area in m²

Holds when: The contour is closed and does not cross itself. The vertices are frequent enough that the straight lines between them follow the contour. If there is a “hole” (a closed depression) inside the contour, its area is subtracted.

Gross rock volume from maps →

Volume of a slice between neighbouring contours: the trapezoidal and pyramidal rules

\Delta V_\text{trap} = \frac{\Delta z}{2}\,(A_i + A_{i+1}),\qquad \Delta V_\text{pyr} = \frac{\Delta z}{3}\left(A_i + A_{i+1} + \sqrt{A_i\,A_{i+1}}\right)

areas and interval in consistent units

Holds when: The trapezoidal rule is exact if the area in the slice grows linearly with depth. A dome close to a paraboloid of revolution behaves this way. The pyramidal rule (the formula for a truncated pyramid) is exact if the horizontal sections are similar and the area grows as the square of depth below the apex: a cone or a pyramid. A common practical rule: use pyramids when the ratio of the smaller area to the larger one is less than 0.5 (usually near the crest), and trapezoids in the other slices.

Gross rock volume from maps →

The cap above the first contour, and Simpson's rule

V_\text{cap} = \frac{A_0 h_\text{cap}}{3}\ \text{(cone)}\ \ldots\ \frac{A_0 h_\text{cap}}{2}\ \text{(paraboloid)},\qquad V = \frac{\Delta z}{3}\left(A_0 + 4A_1 + 2A_2 + \ldots + 4A_{n-1} + A_n\right)

consistent units

Holds when: A cone cap is a sharp crest (the area grows as the square of depth), and a paraboloid cap is a smooth crest (the area grows linearly). A real cap is usually between them. Simpson's rule is exact for curves up to a cubic. It needs a smooth curve and an equal interval.

Gross rock volume from maps →

Rock volume of a thin layer from the curves of the top and the base

\mathrm{GRV} = \int_{z_\text{crest}}^{z_\text{OWC}} \left[A_\text{top}(z) - A_\text{base}(z)\right] dz

m², m, m³

Holds when: The area at the base is counted from the same crest. Above the base, its curve is zero. If the vertical thickness of the layer is constant and equal to $h_\text{TVT}$ , the base is the top lowered by $h_\text{TVT}$ , and $A_\text{base}(z) = A_\text{top}(z - h_\text{TVT})$ . If the thickness changes, we need a separate map of the base.

Gross rock volume from maps →

Geometric correction factor

\mathrm{GCF} = \frac{\mathrm{GRV}}{A_\text{OWC}\,h_\text{TVT}};\qquad \text{paraboloid: } \mathrm{GCF} = 1 - \frac{h_\text{TVT}}{2H}\ \ (h_\text{TVT} \le H),\qquad \frac{H}{2h_\text{TVT}}\ \ (h_\text{TVT} \ge H)

dimensionless

Holds when: The formulas on the right are for a paraboloid dome, whose area grows linearly with depth. For a cone, $\mathrm{GCF} = \dfrac{1 - (1 - h_\text{TVT}/H)^3}{3h_\text{TVT}/H}$ for $h_\text{TVT} \le H$ , and $H/(3h_\text{TVT})$ for $h_\text{TVT} \ge H$ . The thickness is constant and the contact is horizontal.

Gross rock volume from maps →

Symbols

Symbol	Meaning and context
$\Delta l$	length of hole between the top and the base of the interval (difference in measured depth), mWell correlation
$\Delta H$	difference in true vertical depth between the top and the base in this well, mWell correlation
$\hat z$	estimate of the value at the point $(x, y)$ , at a grid node or at a planned well, mStructure maps and contouring
$\Delta q$	horizontal shift of the cutoff of the base relative to the cutoff of the top, in the direction of the dip of the fault plane, mCross-sections from maps, map combination and faults on maps
$\Delta s$	distance along the fault between the points where the same contour of the layer ends in the footwall and in the hanging wall, mCross-sections from maps, map combination and faults on maps
$\Delta z$	contour interval, the thickness of the slice, mGross rock volume from maps
$\Delta V$	volume of rock in the slice, m³ (million m³ for km²)Gross rock volume from maps
$(x_i, y_i),\ z_i$	data points and the values at them, mStructure maps and contouring
$A$	area of the baffle across the flow: the length of the boundary times its height, m²Reservoir architecture and performance area bounded by the contour, m²Gross rock volume from maps
$\bar z_{\text{A}},\ \bar z_{\text{B}}$	mean depths of the selected events, mWell correlation
$A_0 \ldots A_n$	areas of the contours at an equal interval $\Delta z$ , from the first to the last, m²Gross rock volume from maps
$A_0,\ h_\text{cap}$	area of the first contour below the crest, and the height of the crest above it, m², mGross rock volume from maps
$A_i,\ A_{i+1}$	areas inside neighbouring contours (the smaller one belongs to the upper contour), m² or km²Gross rock volume from maps
$A_\text{OWC}$	area inside the outer pool outline, m²Gross rock volume from maps
$A_\text{top}(z)$	area inside the contour of the top at depth $z$ , within the closure of the trap, m²Gross rock volume from maps
$A_\text{top}(z),\ A_\text{base}(z)$	areas inside the contour $z$ on the maps of the top and the base, m²Gross rock volume from maps
$b$	slope of the line — the ratio of accumulation rates in B and in A, —Well correlation width (thickness) of the baffle in the direction of flow, mReservoir architecture and performance
$c$	value of the contour that lies between $z_a$ and $z_b$ , mStructure maps and contouring
$C$	share of the reservoir in the largest connected body, fractionReservoir architecture and performance
$c_t$	total compressibility of the rocks and fluids of the compartment, 1/PaReservoir architecture and performance
$C_w$	share of the reservoir that is connected to at least one well, fractionReservoir architecture and performance
$d$	distance between the wells, mWell correlation
$d_i$	distance from the point $(x, y)$ to the $i$ -th data point, mStructure maps and contouring
$GR$	GR reading in the well being normalised, APIWell correlation
$GR_{cl},\ GR_{sh}$	readings against the reference clean bed and the reference shale member in the well being normalised, APIWell correlation
$GR_{cl}^{\text{ref}},\ GR_{sh}^{\text{ref}}$	readings against the same beds in the reference well, APIWell correlation
$GR_{\text{norm}}$	the same reading on the scale of the reference well, APIWell correlation
$\mathrm{GRV}$	gross rock volume between the top and the contact, m³Gross rock volume from maps
$h$	true thickness (isopach), at right angles to the bedding, mCross-sections from maps, map combination and faults on maps
$H$	height of the accumulation, from the crest to the contact, mGross rock volume from maps
$h_{\text{eff}},\ h_{\text{t}}$	effective thickness (the sum of the reservoir layers) and total thickness of the horizon in a well, mReservoir architecture and performance
$h_\text{TVT}$	thickness of the layer in the vertical direction (isochore) at a point of the map, mCross-sections from maps, map combination and faults on maps vertical thickness of the layer at the fault, mCross-sections from maps, map combination and faults on maps vertical thickness of the layer, mGross rock volume from maps
$h,\ h_{\text{TVT}}$	true stratigraphic thickness (TST) and true vertical thickness (TVT), mWell correlation
$I_{GR}$	gamma ray index (what we find from the GR), —Well correlation
$k_b$	permeability of the baffle, m²Reservoir architecture and performance
$K_{\text{cont}}$	lithological continuity coefficient: the share of the area where the layer is present as reservoir, —Reservoir architecture and performance
$K_{\text{dis}}$	dissection coefficient, the average number of reservoir layers per well, —Reservoir architecture and performance
$K_{\text{mer}}$	lithological connectivity (merging) coefficient: the share of the area where neighbouring layers merge, with no non-reservoir rock between them, —Reservoir architecture and performance
$K_{\text{sand}}$	sandiness coefficient, the share of reservoir rock in the section (net-to-gross ratio), —Reservoir architecture and performance
$L$	length of the side between points $a$ and $b$ , mStructure maps and contouring
$n$	number of points taken into account: all of them, or the nearest ones within the search radius, —Structure maps and contouring number of vertices; we go round them in order, clockwise or anticlockwise, —Gross rock volume from maps number of slices; it must be even, —Gross rock volume from maps
$n_i,\ N$	number of reservoir layers in the $i$ -th well; number of wells, —Reservoir architecture and performance
$p$	exponent; usually 2, —Structure maps and contouring
$P$	probability that a body penetrated by one well is also penetrated by the other, —Well correlation
$s_c$	distance from point $a$ along the side to the point where the contour of value $c$ passes, mStructure maps and contouring
$S_{\text{mer}},\ S_{\text{res}},\ S$	area of the merging zones, area where the reservoir is present, area of the accumulation, km²Reservoir architecture and performance
$t$	throw (vertical amplitude), mCross-sections from maps, map combination and faults on maps throw at the place where the contour ends in the hanging wall, mCross-sections from maps, map combination and faults on maps
$T$	transmissibility of the baffle: the flow rate through it per unit of pressure difference, m³/(s·Pa)Reservoir architecture and performance
$\mathrm{tan}\,\delta_\parallel$	average slope of the cutoff line in the footwall along the strike of the fault over the stretch $\Delta s$ ; in practice, the apparent slope of the layer along the fault, —Cross-sections from maps, map combination and faults on maps
$V_{\max}$	volume of the largest connected body, m³Reservoir architecture and performance
$V_p$	pore volume of the compartment, m³Reservoir architecture and performance
$V_{\text{res}}$	volume of all the reservoir, m³Reservoir architecture and performance
$V_w$	volume of the bodies that wells penetrate or that are connected to them, m³Reservoir architecture and performance
$w$	heave (horizontal amplitude) of the fault at this place, mCross-sections from maps, map combination and faults on maps
$W$	width of the body (channel, channel belt, sand layer) across its axis, mWell correlation
$x_i,\ y_i$	coordinates of the $i$ -th vertex of the contour on the map, in metres on the ground, mGross rock volume from maps
$z_{\text{A},k},\ z_{\text{B},k}$	depths of the $k$ -th event in the reference well A and in the well being matched, B, mWell correlation
$z_a,\ z_b$	values at the ends of the side (depths of the top in two wells), mStructure maps and contouring
$z_\text{crest}$	depth of the crest, the highest point of the top, mGross rock volume from maps
$z_\text{crest},\ z_\text{OWC}$	depths of the crest and the contact, mGross rock volume from maps
$z_\text{fw},\ z_\text{hw}$	depths of the layer at the edges of the polygon in the footwall and the hanging wall, at points that lie opposite each other across the fault, along the line at right angles to it, mCross-sections from maps, map combination and faults on maps
$z_i$	value at the $i$ -th data point, mStructure maps and contouring
$z_\text{OWC}$	depth of the contact (the oil–water contact, OWC, or the gas–water or gas–oil contact), not deeper than the spill point, mGross rock volume from maps
$z_\text{top},\ z_\text{base}$	depths of the top and the base below sea level at the same point of the map, mCross-sections from maps, map combination and faults on maps
$z_{xx},\ z_{xy},\ z_{yy}$	second derivatives of the surface with respect to the coordinates: measures of its curvature, 1/mStructure maps and contouring
$z(x, y)$	the surface we look for, mStructure maps and contouring
$\alpha$	angle between the axis of the body and the line joining the wells, °Well correlation
$\gamma$	angle between the hole and the normal to the bedding: $\cos\gamma = \|\sin\delta\,\sin\theta\,\cos(a - \varphi) - \cos\delta\,\cos\theta\|$ , °Well correlation
$\delta$	dip angle of the fault plane, °Cross-sections from maps, map combination and faults on maps
$\delta,\ a$	dip and dip azimuth of the layers, °Well correlation
$\Delta_\perp$	width of the polygon, measured at right angles to the strike of the fault, mCross-sections from maps, map combination and faults on maps
$\delta_b$	dip angle of the layer at this point, °Cross-sections from maps, map combination and faults on maps dip angle of the layer in a section across the strike of the fault, °Cross-sections from maps, map combination and faults on maps
$\theta,\ \varphi$	inclination and azimuth of the hole over the interval, °Well correlation
$\mu$	viscosity of the fluid that flows through the baffle, Pa·sReservoir architecture and performance
$\tau$	time constant: in this time the pressure difference between the compartment and the neighbouring layer falls by a factor of $e \approx 2.72$ , sReservoir architecture and performance