Abnormal pore pressures are encountered worldwide,often resulting in drilling problems such as kicks, blow-outs, borehole instability, stuck pipe, and lost circulation.Because of this, a quantitative predrill prediction of porepressure is required for the safe and economic drilling ofwells in overpressured formations. In this paper, the basicconcepts used in pore-pressure prediction are defined, andthe way in which pore pressure can be estimated usingvelocity measurements is explained. A predrill estimate ofpore pressure can be obtained from seismic velocities usinga velocity-to-pore-pressure transform calibrated with offsetwell data. However, velocities obtained from processingseismic reflection data often lack the spatial resolutionneeded for accurate pore-pressure prediction, due toassumptions such as layered media and hyperbolic move-out. In the following paper in this section, Chopra andHuffman review the available methods of velocity modelbuilding and discuss the advantages and disadvantages ofeach. Once suitable velocities are obtained, a velocity-to-pore-pressure transform is required. This article providesthe rock physics basis underlying such transforms.

Sediment compaction. Following deposition in the marineenvironment, sediments are initially unconsolidated andhave a high porosity and permeability. As a result, the waterin the pore space is in pressure communication with the sur-face, and the weight of the solid phase is supported at thegrain contacts and has no influence on the pressure in thefluid (Bourgoyne et al., 1986). The pore pressure p in the fluidis then given by the hydrostatic pressure of a column of for-mation water extending to the surface (Figure 1). Sedimentsin which the pore pressure is approximately equal to thehydrostatic pressure are said to be normally pressured, the nor-mal pressure at depth h below the sea surface being givenby

where g is the acceleration due to gravity, and ρfluid(z) is thefluid density at depth z.

As the sediment is buried to greater and greater depth,the weight of the overlying rocks increases, and the increas-ing stress acting at the grain contacts leads to rearrangementof the grains, resulting in lower porosity and permeability.If the rate of sedimentation exceeds the rate at which fluidcan be expelled from the pore space, or if dewatering is inhib-ited by the formation of seals during burial, the pore fluidbecomes overpressured and thus supports part of the over-burden load. Overpressure generated in this way is said toresult from disequilibrium compaction or undercompaction,this being the most common mechanism for generatingoverpressure in deepwater sediments.

For the purposes of illustration, the total vertical stressS is assumed to be given by the combined weight of the rockmatrix and the fluids in the pore space overlying the inter-val of interest:

where ρ(z) is the density at depth z below the surface, andg is the acceleration due to gravity. Part of this load is sup-

ported by the fluid pressure p, while the remainder is sup-ported by the rock matrix and is referred to as the effectivestress σ defined by:

Uniaxial strain measurements on unconsolidated sedi-ments have shown that the void ratio e defined by e=φ/(1-φ) decreases linearly with increase in the logarithm of thevertical effective stress:

where e* is the void ratio at a chosen effective stress σ*, andβ is the compression coefficient (Aplin et al., 1995). This equa-tion may be rewritten as

An introduction to velocity-based pore-pressure estimationCOLIN M. SAYERS, Schlumberger Data and Consulting Services, Houston, USA

1496 THE LEADING EDGE DECEMBER 2006

Figure 2. Porosity versus depth curves obtained using the relation ofAplin et al. (1995) assuming a grain density of 2.65 g/cc and a fluiddensity of 1.05 g/cc for φ* = 0.5, σ* = 100 kPa and various values ofthe compaction coefficient β.

Figure 1. In normallypressured sediments, thepore pressure equals thehydrostatic pressure of acolumn of formation waterextending to the surface.

(1)

This equation allows the vertical effective stress σ to beestimated given a measurement of porosity or density. Thepore pressure is then given by the difference between thetotal vertical stress and the vertical effective stress.

Starting from Equation 1, Aplin et al. (1995) have deriveda relation describing porosity/depth trends in a horizontallystratified sequence of normally pressured, gravitationallycompacted sediments. Figure 2 shows the prediction of thisrelation for the case in which β is assumed to be indepen-dent of depth. The vertical effective stress at any depth isthen given by Equation 1.

In drilling applications, the pressure and overburden gra-dient, defined as pressure and vertical stress divided by truevertical depth (TVD) below the kelly bushing (KB), are morefrequently used than are pressure and vertical stress, andare often expressed in pounds per gallon (ppg) where 1psi/ft = 19.25 ppg =2.31 g/cc. Thus the overburden gradi-ent (OBG) can be written as (Traugott, 1997)

where W is water depth, D is true vertical depth, A is theair gap (vertical distance between KB and sea surface), ρseais the seawater density, and ρavg is the average sediment den-sity between the sea bottom and the depth of interest.

Various empirical relations for the overburden gradienthave been used. An example is the expression given byTraugott (1997) in which the average sediment densitybetween the sea bottom and the depth of interest is approx-imated by the expression

where ρavg has units of ppg, and D, W, and A have units offt.

Since

it follows that this equation corresponds to a sediment den-sity that varies with depth z below sea bottom as

where ρ0 is the density at the mudline. Assuming a graindensity ρs and fluid density ρf the porosity may be calcu-lated from this equation using the relation

(2)

Figure 3 plots porosity as a function of effective stressusing Traugott’s equation and a grain density of 2.65 g/ccand fluid density of 1.05 g/cc together with a best fit of aporosity/effective stress relation of the form

(3)

where φc is a critical porosity above which the effectivestress is zero, and σmax is the effective stress correspondingto φ=0 in Equation 3. It is seen that the power law given byEquation 3 is in good agreement with the porosity/effec-tive stress relation implied by Traugott’s relation.

Figure 4 compares the variation of density and porosityversus depth below mudline given by Traugott’s relationwith the prediction of Equation 3. Good agreement is seen.

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Figure 3. Porosity/effective stress relation implied by Traugott’s relationcompared with the prediction of the power law given by Equation 3.

Figure 4. Density and porosity versus depth below the mudline impliedby Traugott’s relation compared with the prediction of the power lawgiven by Equation 3.

Figure 5. Velocity versus porosity obtained from Gardner’s relation com-pared with the data presented by Issler (1992) for offshore wells in theBeaufort-Mackenzie Basin, Northern Canada.

Pore pressure estimation from velocity. A relation betweenvelocity and effective stress can be obtained from the pre-vious equations relating porosity to effective stress given arelation between velocity and porosity. A suitable relationis Gardner’s relation (1974) between density and velocity:

which allows velocity to be calculated from porosity usingEquation 2. Figure 5 compares the velocity versus porosityprediction from Gardner’s relation, using the parametersgiven by Castagna et al. (1993) for shales, with the data pre-sented by Issler (1992) for offshore wells in the Beaufort-Mackenzie Basin, Northern Canada.

Figure 6 shows the velocity versus effective stress impliedby Gardner’s equation, with the same parameters as before,and the porosity/effective stress relation implied by

Traugott’s expression shown in Figure 4. It is seen that thevelocity versus effective stress relation that results agreeswell with the relation

(4)

proposed by Bowers (1995).Equation 4 allows the effective stress σ to be determined

from velocity measurements using the relation:

The pore pressure p can then be determined as the dif-ference between the total vertical stress S and the verticaleffective stress σ.

Several other velocity/effective stress relations exist inthe literature, and several of these are discussed by Gutierrezet al. in this issue of TLE. The most widely used approachin the industry is the method of Eaton (1975) which esti-mates the vertical component of the effective stress σ fromthe seismic velocity v, via the relation:

σNormal and vNormal in this equation are the vertical effectivestress and seismic velocity expected if the sediment is nor-mally pressured, while n is an exponent that describes thesensitivity of velocity to effective stress. The pore pressureis then given by:

To use Eaton’s method, the deviation of the measuredvelocity from that of normally pressured sediments vNormalmust be estimated. Usually, a suitable parameterized expres-sion for vNormal is chosen, and the parameters are obtainedby fitting to the shallow velocities assuming that the shal-low sediments are normally pressured. However, over-pressure often begins at shallow depths thus invalidatingthis approach. The assumption that shallow sediments arenormally pressured is also unnecessary, because the para-meters defining vNormal may be determined by an inversionof existing pressure data, as pointed out by Sayers et al.(2002). Consider, for purposes of illustration, the simple lin-ear variation of vNormal with depth given by:

where z is depth measured from the seafloor and v0 is thevelocity of sediments at the seafloor. Typical values of thevertical velocity gradient k lie in the range 0.6 to 1 s-1 (Xu etal., 1993). To predict pore pressure the parameters v0, k, andEaton exponent n must be determined. Using a 3D grid-based method (Sayers et al., 2002), the numeric ranges ofparameters v0, k, and n may be determined via root-mean-square (rms) analysis of the residuals ∆p = pmeas - ppred:

where ppred is the predicted pore pressure, pmeas is the mea-sured pore pressure, and N is the number of pore pressuremeasurements. Figure 7, for example, shows a slice at a con-stant value of n=3 through the 3D space with axes v0, k, andn. Given an estimate of the error in the pressure measurementsused to calibrate the transform, the region of parameter spaceconsistent with the data can be identified, thus allowing a pre-

1498 THE LEADING EDGE DECEMBER 2006

Figure 6. Velocity versus effective stress relation obtained by Traugott’srelation with Gardner’s expression compared with the Bowers’ relationgiven by Equation 4.

Figure 7. Constant n slice through the 3D space with axes v0, k, and nshowing contours of ∆prms defined in the text for n=3.

diction of pore pressure with uncertainty to be made. A sim-ilar approach may be used to determine the parameters v0, A,and B in Bowers’ relation.

Discussion. While disequilibrium compaction or undercom-paction is the most common mechanism for generating over-pressure in deepwater sediments, several other overpressuremechanisms may also occur. Several of these are describedbelow.

Clay diagenesis. The smectite-to-illite transformation thatoccurs as the mixed-layer clay systems in the Gulf of Mexicoundergo burial depends strongly on the time-temperaturehistory of the sediment and causes reordering of the clayplatelets and redistribution of effective stress (Dutta, 2002).Dutta (1988) incorporates the time-dependent temperaturehistory by use of an effective stress/porosity relation of theform:

where σ0 is a constant, e is the void ratiodefined earlier, A(T) is a polynomial intemperature T, and B(t) is a diageneticintegral depending on time (and tem-perature) that describes the smectite toillite conversion. Given a relationbetween velocity and porosity, thisequation leads to a velocity/effectivestress relation that depends on the time-dependent temperature history of thesediment.

Unloading. Unloading refers to adecrease in effective stress acting on therock frame, as may occur if the porepressure increases in a sediment at afixed depth. Such an increase in porepressure may occur, for example, due totemperature increase, clay dewateringor conversion of kerogen to lower mol-ecular weight hydrocarbons. As shownin Figure 8, sediments follow a differ-ent path on unloading than on loading(Bowers, 1995), and this difference inthe velocity/effective stress relationneeds to be taken into account if unload-ing occurs. As illustrated in Figure 8b,the velocity may drop significantlyupon unloading, while the densitychanges by only a small amount. Avelocity/density crossplot may there-fore help to distinguish between load-ing and unloading.

Lateral transfer. While the pore pres-sure within an impermeable shale canincrease with increasing depth at a ratethat is greater than the hydrostatic gra-dient, pore pressures within a dippingpermeable layer follow the hydrostaticgradient (Figure 9). As a result, flowoccurs from the deeper overpressuredshales into the sand and from the sandinto the shallower shales. As a result ofthis lateral transfer the sand transmitspore pressure updip (Yardley andSwarbrick, 2000), and this can lead tokicks when drilling into the crest of adipping sand. A method for calculatingthe pressure within a dipping sand has

been presented by Stump et al. (1998), and is discussed fur-ther by Dutta and Khazanehdari (2006) in this issue of TLE.

Conclusion. Several commonly used terms in the pore-pres-sure-prediction literature have been defined, and variousmechanisms that result in overpressure have been discussed.Disequilibrium compaction or undercompaction is the mostcommon source of overpressure in young, rapidly buriedsediments, and it is shown how velocity versus effectivestress methods arise naturally by coupling the change inporosity with increasing effective stress with the porositydependence of velocity. In any particular application, addi-tional mechanisms such as clay diagenesis, unloading, andlateral transport may also play a role in causing overpres-sure to occur, and an appropriate velocity-to-pore pressuretransform based on a careful analysis of offset well datatogether with fit-for-purpose seismic velocities at the pro-posed drilling location is required for a reliable estimate of

DECEMBER 2006 THE LEADING EDGE 1499

pore pressure. Uncertainty in the prediction can be reducedby updating the velocity-to-pore pressure transform usingdata acquired while drilling as described by Malinverno etal. (2004).

Suggested reading. “Assessment of β, the compression coeffi-cient of mudstones and its relationship with detailed lithology”by Aplin et al. (Marine and Petroleum Geology, 1995). AppliedDrilling Engineering by Bourgoyne et al. (SPE, 1986). “Pore pres-sure estimation from velocity data: Accounting for pore pres-sure mechanisms besides undercompaction” by Bowers (SPEDrilling and Completion, 1995). “Rock physics—The linkbetween rock properties and AVO response” by Castagna et al.(in J. P. Castagna and M. M. Backus, eds., Offset-dependent reflec-tivity — Theory and practice of AVO analysis: Investigations inGeophysics, SEG). “Velocity determination for pore pressure

prediction” by Chopra and Huffman (TLE, 2006). “Fluid flowin low permeable porous media” by Dutta (in Migration ofHydrocarbons in Sedimentary Basins, Editions Technip, 1988).“Geopressure prediction using seismic data: current status andthe road ahead” by Dutta (GEOPHYSICS, 2002). “Estimation offormation fluid pressure using high-resolution velocity frominversion of seismic data and rock physics model based on com-paction and burial diagenesis of shales” by Dutta andKhazanehdari (TLE, 2006). “The equation for geopressure pre-diction from well logs” by Eaton (SPE paper 5544, 1975).“Formation velocity and density—the diagnostic basis for strati-graphic traps” by Gardner et al. (GEOPHYSICS, 1974). “Calibrationand ranking of pore pressure prediction models” by Gutierrezet al. (TLE, 2006). “A new approach to shale compaction andstratigraphic restoration, Beaufort-Mackenzie Basin andMackenzie Corridor, Northern Canada” by Issler (AAPGBulletin, 1992). “Integrating Diverse Measurements to PredictPore Pressure With Uncertainties While Drilling” by Malinvernoet al. (SPE paper 90001, 2004). “Predrill pore pressure predic-tion using seismic data” by Sayers et al. (GEOPHYSICS, 2002).“Pressure differences between overpressured sands and bound-ing shales of the Eugene Island 330 field (Offshore Louisiana,USA) with implications for fluid flow induced by sediment load-ing: Overpressures in Petroleum Exploration” by Stump et al.(Bull. Centre Rech, Elf Explor. Prod., Mem, 1998). “Pore/frac-ture pressure determinations in deep water” by Traugott (WorldOil, Deepwater Technology Special Supplement, 1997). “Some effectsof velocity variation on AVO and its interpretation” by Xu etal. (GEOPHYSICS, 1993). “Lateral transfer: a source of additionaloverpressure?” by Yardley and Swarbrick (Marine and PetroleumGeology, 2000). TLE

Corresponding author: csayers@houston.oilfield.slb.com

1500 THE LEADING EDGE DECEMBER 2006

Figure 8. Variation of porosity (a) and velocity (b) with effective stressupon loading and unloading.

Figure 9. Overpressure resulting from lateral transfer.