Gravity Methods PDF

Shell Intensive Training Programme Gravity Methods 4.1 Theory of Gravitation The theory of all gravitational studies is Newton’s Law of Gravity, which quantifies the mutual attraction between two masses (m1 and m2) separated by a distance r given by:-...

Shell Intensive Training Programme Gravity Methods 4.1 Theory of Gravitation The theory of all gravitational studies is Newton’s Law of Gravity, which quantifies the mutual attraction between two masses (m1 and m2) separated by a distance r given by:- F=Gm1 m2/R2 Newton (4.1) Where G = universal gravitational constant and is equal to 6.67 x 10-11 N m2 kg-2. The attraction that is exerted by the Earth (mass, M) on any mass (m) outside of it on the surface of the earth is always directed towards the Earth’s centre with an acceleration (denoted by g ) which is roughly equal to 9.8 ms-2. This is because a point mass and a spherical symmetrical body of the same total mass have exactly the same gravitational field. Thus, to a first approximation, the earth’s gravitational field always looks as though all the mass is concentrated at the centre of the earth. The force on a mass (m0 falling towards the Earth under the influence of gravity alone may also be written as :- F = mass 5 acceleration = mg (4.2) And so, combining this with equation 4.1 F = mg = GmM/R2 Where R = distance to the centre of the Earth (Radius of the earth) and M = mass of the Earth. So from equations 4.2 and 4.3 g = GM/R2 (4.4) and this provides a value of g at all points on the Earth’s surface. The general relationship between gravity and the Earth’s average density (p) is given by: P = mass/volume = M/(4πR3/3) = 3M/4π R3 (4.5) (assuming that the Earth is a perfect sphere) and from equation 4.4, M = gR2/G (4.6) Thus: Page 1 of 19 © Univation Shell Intensive Training Programme P = 3gR2/4πR3G = 3g/4πRG (4.7) This equation shows that if g, R and G are known or measured, the earth’s average density (p) can be calculated. Using modern measurements, ρ has been found to be 5520 kgm-3. The fact that this value is much greater than the densities of rocks (2500-3000 kgm-3, table 2 ) indicates that the earth’s density increases towards the centre of the Earth. Using the known values of some of the parameters in equation 4.4, the acceleration due to gravity is calculated as follows:- g = Gm/R2 = 6.67510-1156.151024 6.456.451012 g from above is approximately 10m/s2 or 9.8m/s In reality, the acceleration due to gravity is not the same at all points on the Earth’s surface. The value of g = 9.8m/s2 is an average value and so are both the mass and density of the earth. Hence there are points of the earth, where there are excess mass/density and other points where there are deficit mass/density due to the inhomogenities in the earth’s density. Geophysical interest in gravity centres particularly on the measurement and interpretation of these variations of g in terms of the Earth’s subsurface structure. The general form of equation 4 which is regularly used in the calculation of gravity effects of subsurface rock is given by:- g = 4 pRG/3 ------(4.8) This equation shows that g depends on two variables, R and p. If the Earth were perfectly spherical (R = constant) and perfectly uniform (p = constant), g would have the same value everywhere on the Earth’s surface. However, if at some time the earth departs from sphericity, the value of g at that time will depart from the value. Again, if there is a local variation/anomaly in density at a point, g will at that point vary. In principle, therefore, it should be possible to use measured variations in g at the Earth’s surface to deduce variations of R and p, hence deduce variations in the density of rocks beneath the earth’s surface. Page 2 of 19 © Univation Shell Intensive Training Programme The Earth is not spherical but corresponds to an ellipsoid of revolution; that is, it is flattened at the poles. The ellipticity of the earth can be measured very accurately from satellite; the equatorial radius of 6378 km is about 21.5 km longer than the polar one. Since, to a first approximation, g is inversely proportional to R2, g must be greater at the pole than at the equator. There is another important force, the centrifugal force, due to the Earth’s rotation, which also reduces the resultant force of gravity at all places on the surface other than the pole. The centrifugal force at the Earth’s surface acts at right angles to the rotational axis of the Earth and reaches a maximum at the equator. The net gravitational force at any point on the surface is the resultant of forces due to internal mass and centrifugal action. The spheroid is an ideal mean sea-level surface whose net gravitational force varies, but whose surface is everywhere at right angles to the force. Both the geometrical flattening of the Earth and the centrifugal force are taken into account in the International Gravity Formula (IGF), which describe the variation of gravity at sea-level with latitude (λ), where 9.780318 m s-2 is the value of g at the equator: IGF = 9.7803185 (1 + 0.005278895 sin2 λ - 0.000023462 sin4 λ) m s-2 (4.9) The IGF is the best-fitting value at the sea-level surface for an idealised homogenous Earth that is distorted only by the effects of rotation. The Earth’s surface, however, is affected by inhomogeneities of density beneath the surface and therefore it departs from the theoretical IGF surface. The theoretical IGF surface is known as the spheroid. The value of gravity varies systematically around the smooth surface of the spheroid in accordance with equation 4.9. Units of Gravity The mean value of gravity at the Earth’s surface is about 9.80 m s-2. Variations in gravity caused by sub-surface density variations are of the order of 100 µm s-2. This unit of micrometre per second is referred to as the Gravity Unit (gu), while the c.g.s unit of gravity is the milligal (1 mgal = 10-3 cm s -2), equivalent to 10 gu. Page 3 of 19 © Univation Shell Intensive Training Programme The milligal The Gal: The unit of measurement in gravity surveying is the milligal which is a smaller unit of the Gal. This unit was named after Galileo Galilei, an Italian Physicist who first proposed that the Earth was round and not flat. 1 gal = 1 cm/s2 = 10-2m/s2 1 milligal = 10-3 gal = 10-5m/s2 = 10510-6m/s2 1 milligal (mgal) = 10µm/s2 The acceleration due to gravity g is then given by g = 9.8m/s2 = 980 gals Therefore g = 980000 mgal The earth’s rotation and flattening cause gravity to increase by roughly 5300 mgal from equator to the pole and yet this is only about 0.5% variation. In gravity surveys, it is necessary to measure accurately the small changes in gravity caused by subsurface rocks. These require an instrumental sensitivity of the order of 0.01mgal. 4.2 Instrumentation The measurement of the absolute value of gravity requires complex apparatus and a long period of observation. However, the determination of relative values of gravity, that is, the differences of gravity between different sites, is relatively simple. The instrument used to rapidly measure gravity is called a gravimeter. It records the direct effects of the pull of gravity on a weight suspended by a delicate spring. The extension of the spring is proportional to the pull according to Hooke’s law:- δs = δg(m/k) where δs = extension of the length of the spring, δg = increase in gravity , m = weight of suspended mass and k = elastic spring constant. Having a large mass and a weak spring could increase the sensitivity of the instrument, but this is not practical. This type of gravimeter, in which the spring functions to support the mass and also as the measuring device, is known as a stable or static gravimeter and is not very sensitive. Hence, Optical, mechanical or Page 4 of 19 © Univation Shell Intensive Training Programme electronic amplification of the extension is used. Modern instrument such as the LaCoste and Romberg gravimeter are constructed so that an additional force acts in the same direction as gravity and opposes the restoring force of the spring. The spring is pretensioned during manufacture so that its natural tendency is to uncoil. An increase in gravity streches the spring against its restoring force and the extension is augmented by the built-in pretension. These are called unstable instruments and are very sensitive, providing an accuracy of 10-8 of the normal field of gravity (fig. 4.1) The high sensitivity of the modern gravimeter makes them easily susceptible to undergo small changes in their own properties. To reduce these to a minimum, the critical elements are enclosed in an evacuated chamber and additional precautions, such as a thermostatting system or some system of compensating for temperature changes, are normally incorporated in the instrument. In some cases, the Locoste and Romberg gravimeter is enclosed in a thermos bottle Housing in order to reduce Page 5 of 19 © Univation Shell Intensive Training Programme or remove the effects of temperature variations on the spring of the gravimeter. Despite this, gravimeters are subject to drift i.e., there is a gradual change in gravity reading with time, which is observed even when an instrument is left at a fixed location. This is mainly the result of the imperfect elasticity of the springs, which undergo creep with time. The effect is compensated for by a drift correction, which is obtained by repeated observations at some stations during the day. Readings at other stations can then be adjusted by comparison with the drift curve obtained. Gravimeters are also affected by tidal attraction so that tidal corrections must be applied and this means that if the time of each measurement is known, this correction is implicit in the drift correction. While normal gravimeters can be used for marine surveying by lowering them in a waterproof housing to the sea-bed, their use is limited. However, continuos gravity measurements can be made at sea using specially modified instruments, although the effects of waves and the ship’s motion considerably reduce the accuracy of the measurements. Some of these problems can be removed by mounting the meter on a gyroscopically stabilised platform. Rock densities Gravity anomalies result from the difference in density or density contrast between a body of rock and its surroundings. A body of rock with a density of ρ1 surrounded by material of density ρ2 the density contrast ∆ρ is given by: ∆ρ = ρ1 - ρ2 The sign of the density contrast determines the sign of the gravity anomaly. A knowledge of rock density is necessary both for the application of the Bouguer correction and for the interpretation of gravity data. Rock densities are dependent on composition and porosity. The main cause of density variation in sedimentary rocks is variation in porosity. In general terms, sedimentary rocks tend to become denser with depth, due to compaction, and with age, due to cementation. In igneous and metamorphic rocks the main control of density is composition. The density ranges of common rock types are shown in Table 1. Page 6 of 19 © Univation Shell Intensive Training Programme Table 1. Ranges and averages of densities of common rocks Rock Density (kgm-3) Range Average Sediments and sedimentary rocks Sandstone 1600-2760 2350 Shale 1560-3200 2400 Limestone 2040-2900 2550 halite 2100-2600 2220 gypsum 2200-2600 2350 coal 1340-1800 1500 oil 600-900 igneous rocks sandesite 2400-2800 2610 granite 2500-2810 2640 basalt 2700-3300 2990 gabbro 2700-3500 3030 peridotite 2780-3370 3150 Page 7 of 19 © Univation Shell Intensive Training Programme metamorphic rocks quartzite 2500-2700 2600 schist 2390-2900 2640 granulite 2520-2730 2650 marble 2600-2900 2750 slate 2700-2900 2790 gneiss 2590-3000 2800 4.3. Gravity Field Measurements. In a gravity survey, gravity measurements are made at a number of sites. The spacing of the stations will vary depending on the intended purpose of the survey. In regional reconnaissance surveys the distance between locations may be several kilometres while in localised surveys for detailed mineral or geotechnical work, distances would involve sites that may be only a few metres apart. The density of sites should ideally be highest where the gravity field is changing most rapidly in order to obtain maximum precision. Measurements may be in a straight-line profile arrangement or in the pattern of a grid of locations. Whichever is used, the inter- location distance would depend on the magnitude of the survey (regional or localised). A record is made at each site of the location, time, elevation (or water depth) and gravimeter reading. This data should be as accurate as possible. At a suitable base station, gravity readings are repeated at regular time intervals throughout the survey period. This extra data would usually be used in the calculation of the drift and tidal corrections (Fig 4.2) In gravity surveying the measurement of the differences in gravity between selected sites is usually tied to a reference point. Ideally, this reference point should be one of the known network of stations at which absolute gravity measurements have been determined (the International Gravity Standardization Network (IGSN)). The Page 8 of 19 © Univation Shell Intensive Training Programme differences are known as ∆g values. If it is necessary to have absolute gravity values, then a measurement at a base station where the absolute value of gravity is known will be required. The absolute values for each survey location can then be calculated by adding the ∆g values to the absolute value at the reference station. However, for a localised gravity prospecting surveys, a base station so selected to which all ∆g values are referenced to should be on a known elevation value location such as a bench mark or on a road junction locatable on a topographic map.. The height differences of all the locations with reference to the base station must be known or measured. This is usually done with a surveyor’s level. Note that elevation (height) difference of about 10cm can generate up to 0.1 mgal variation or more. Where a surveyor’s level is not available, barometric leveling may suffice. 4.4. Gravity Data Analysis The gravity difference between two points (a site and a reference point) depends upon a number of factors other than the presence of density variations. The values Page 9 of 19 © Univation Shell Intensive Training Programme of ∆g therefore must be corrected in order to eliminate these other effects before they can be interpreted; i.e. it is necessary to work out what the ∆g value would be if: The sites were all at the same latitude. The sites were all at the same topographic height. Material between the sites was taken into consideration. The topography of the surrounding region was taken into account. This process is known as gravity reduction which is the first and important stage of gravity data analysis. 4.4.1 Latitude correction Between any station (s) and the reference point (p) differing in latitude, there will be a value of ∆g only due to difference in latitude given by:- ∆g = gs - gp. This latitudinal effect can be removed by applying the IGF (equation 4.9) which shows that there is a nonth-south effect gravity gradient at latitude λ being 8.12 sin 2λ gµkm-1. If the site is at a higher latitude than the reference point (i.e. where, because of the latitude effect, gs is larger than gp ), then the latitude correction must be subtracted from ∆g to give ∆gL, the gravity difference corrected for latitude. Conversely, if the site is at a lower latitude than the reference point, then the latitude correction must be added to ∆g. In small scale (localised) surveys where λ is less than 1o, latitude effects are very negligible and are usually not corrected for. 4.4.2 Elevation corrections Correction for the differing elevations of gravity stations is made in three parts (fig 4.3) Page 10 of 19 © Univation Shell Intensive Training Programme a. The free-air correction (∆gFA) corrects for the decrease in gravity with height in free air resulting from increase in distance from the Earth’s centre, according to Newton’s Law. To reduce an observation taken at height h (in metres) to datum : ∆gFA = 3.086h gµ ------------ (4.11) The free-air correction will be positive for an observation point above the reference station to correct for the decrease in gravity with elevation and negative for an observation point below the reference station. The free-air correction accounts solely for variation in the distance of the observation point from the centre of the Earth; but does not account for the gravitational effect of the rock present between the observation site and the reference station. The plotted data after the free-air correction is called Free-air anomaly. b. This effect is corrected for by the Bouguer correction, ∆gBP. The size of the Bouguer correction depends on the density of the rock column below the observation site to an infinite horizontal slab with a thickness equal to the Page 11 of 19 © Univation Shell Intensive Training Programme elevation of the site above the reference station. If ρ is the density of the rock in Kgm-3 and h is the height difference in metres, then: ∆gB = 0.4191 ρh gµ (4.12) The Bouguer correction ∆gB is subtracted at measurement points where ∆gFA is positive and is added where ∆gFA is negative. The plot (either in profile or contour grid) of the gravity anomalies against the measurement points after all the corrections up to this stage is called Bouguer Anomaly. This is usually the stage at which the plotted data can now be separated into regional and residual anomalies for interpretation. However, for surveys in vary rugged terrains, where measurement points may be very close to high topographic reliefs or valleys, terrain correction is carried out before plotting the data. Even then, this final plot would still be Bougoer anomaly. c. The Bougeur correction assumes that topography around the observation site is flat. This is rarely the case and hence a further correction, the terrain or topographic correction (∆gr) is done. This correction is always positive since the Bouguer correction will have overcompensated for any valleys and hills, which would have been excluded from the Bouguer correction. will Terrain corrections are made using a cumbersome and time consuming standard table, which takes into account height, density and decreasing effect on g with distance from the observational site. Terrain effects are minimal in areas of subdued topography and can be ignored, but in areas of rugged relief terrain effects will be considerably greater. In the latter case considerable time can be spent on this. Earth tides are caused by the gravitational effects of the Sun and the Moon. Both are due to the attraction between them and the Earth , although the effect of the Moon is much greater than the Sun. During data acquisition if observations are made over intervals shorter than the minimum Earth tide, then the tidal variations will be automatically removed during drift correction. Page 12 of 19 © Univation Shell Intensive Training Programme The Eotvos correction is applied to gravity measurements taken on moving vehicles, such as a ship or aircraft. Depending on the direction of travel, motion will generate a centripetal acceleration that either reinforces or opposes gravity. The correction required is: Eotvos correction = 75.03V sin α cos λ + 0.04154V2 gµ where V = velocity in knots, α = the heading and λ = latitude of the observation. Once the latitude, elevation, tidal and, if necessary, Eotvos corrections have been determined, the final corrected gravity difference between an observational site (s) and the reference point (p) can be calculated. (gS - gp ) corrected = ∆g corrected = ∆g -∆gL + ∆FA- ∆gBA + ∆gr -- 4.13 The adjusted figures reveal variations called gravity anomalies, which are due to density variations in the subsurface. A gravity anomaly may be positive or negative according to which of the two values is the greater. As explained earlier, the gravity data at this stage is called Bouguer gravity anomaly and are displayed as profiles (fig. 4.4 ) or as Isogal contour maps (fig. 4.5) Page 13 of 19 © Univation Shell Intensive Training Programme 3.5 Interpretation of Gravity data (Anomalies) The interpretation of gravity anomaly at this stage is generally qualitative as there must be the separation of anomalies before any quantitative interpretation. 3.5.1 Qualitative interpretation From the display of the Bouguer gravity anomaly as a profile, locations along the profile with positive or negative anomalies (now referred to as gravity highs or gravity lows respectively) can be identified, marked out and described (gravity highs would indicate the presence of highly dense rocks lying not too deep from the Page 14 of 19 © Univation Shell Intensive Training Programme surface within the point of measurement. Gravity lows would conversely point to the preponderance of very light rocks(fig.4.4).. When the display is in Isogal contour format (fig. 4.5) areas of high or low contour gradients and closures are identifiable and discernible trends and their directions can be described. 3.5.2 Quantitative interpretation (I) Separation of Anomalies Bouguer anomaly fields are usually characterized by a broad, gentle varying, regional anomaly upon which are superimposed much sharper and more compact local anomalies. In gravity prospecting , it is the latter which are usually of prime importance and the initial step in interpretation is the removal of the regional field to isolate the residual anomalies. This can be done graphically by sketching in a linear or curvilinear field manually using available geological knowledge of the area and keen visual observation (fig. 4.6). However, several analytical methods of separation of regional and residual anomalies are available such as: a. Least squares fitting b. Polynomial curve c. Second derivative d. 5-point centre ring e. Downward and upward continuation All of the above are now run with the computer. These must be done with caution because to avoid the introduction of spurious residual anomalies due to the mathematical procedures adopted. Page 15 of 19 © Univation Shell Intensive Training Programme (ii). Residual Anomaly Interpretation After obtaining a Residual Anomaly, the depth of burial and dimension of the causative subsurface body can be calculated. This may not be very accurate unless some other information are available The method of Half-width is frequently used for estimation of depth of burial (Ζ )and excess mass (m) of assumed shapes for causative bodies (fig. 4.7). Page 16 of 19 © Univation Shell Intensive Training Programme This is a method in which half of the width of the anomaly (X1/2 ) at the point of half the maximum amplitude of the anomaly (1/2∆gmax) is used as follows:- For an assumed spherically shaped causative body: Ζ = 1.3 X1/2 metres --------- 4.14 Radius of the sphere R = √∆g2max Ζ /12.8ρ -------------- 4.15 m = 26∆gmax X1/2 tons ----------- 4.16 The Bouguer anomalies have been observed to show some defined patterns for various regular geometrical shapes. These anomalies can actually be calculated from the known shapes including hypothetical faults. This is the basis for mathematical modelling where shapes are assumed together with dimensions and depth of burial and calculated anomalies obtained as a model. The model is then compared with the observed residual or regional bouguer anomaly. When they match, there is an interpretation and when they do not match, modifications are made in the original values of either depth or in dimensions until a match or a fit is achieved. However, it has been observed in the mathematical modelling of potential fields that more than one set of parameters present good fit with the field data (fig. 4.8). This means that there may be no unique solution usually referred to as the INVERSE problem or Non-uniqueness problem and this is common with potential methods of gravity and magnetics. Page 17 of 19 © Univation Shell Intensive Training Programme After obtaining excess mass and depth of burial, it is possible to make pronouncements on the availability or absence of specific types of rocks, ores or minerals or structures of interest. 3.6. Applications Gravity surveys are applicable in many areas of study and have been very useful in:- a. Defining the size and extent of ore bodies (positive anomalies) b. Delineating intrusive bodies (strong +ve anomalies but -ve for granites) c. Defining buried river channels d. Demarcating faults (changing of anomaly shape) e. Reconnaissance for hydrocarbon in terms of delimitation of extent of sedimentary basin (+ve on basin platforms and -ve within the Basin). d. Direct detection of salt dome and salt deposits and reefs (-ve anomalies). e. Concept of isostacy (-ve anomalies over mountain ranges and +ve anomalies over oceans basins) Page 18 of 19 © Univation Shell Intensive Training Programme Page 19 of 19 © Univation