Methods to determine pore pressure

Methods and complications in determining pore pressures are discussed below

Methods to determine pore pressure

Equivalent depth methods

One example of analysis using a trend line is the equivalent depth method illustrated in Fig. 1. This method first assumes that there is a depth section over which the pore pressure is hydrostatic, and the sediments are normally compacted because of the systematic increase in effective stress with depth. When the log of a measured value is plotted as a function of depth, NCTs can be displayed as straight lines fitted to the data over the normally compacted interval. Because the value of the measured physical property is a unique function of effective stress, the pore pressure at any depth where the measured value is not on the NCT can be computed from

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  Fig. 1—Illustration of the equivalent depth method using sonic ΔT. The normal compaction trend (NCT) is a straight line in log-linear space that has been fitted to the decrease in slowness as a function of depth where sediments are normally compacting. The effective stress at depth Z is equal to the effective stress at depth A, and thus, the pore pressure at depth Z is simply P_z = P_a + (S_z–S_a).

....................(1)

where P_a,z and S_a,z are the pore pressure and the stress at z, the depth of interest and a, the depth along the normal compaction trend at which the measured parameter is the same as it is at the depth of interest. The only unique assumption required by equivalent depth methods is that effective stress is a linear function of depth.

The ratio method

In the ratio method, pore pressure is calculated using the assumption that, for sonic delta-t, density, and resistivity, respectively, the pore pressure is the product of the normal pressure multiplied (or divided by) the ratio of the measured value to the normal value for the same depth.

and

....................(2)

where the subscripts n and log refer to the normal and measured values of density, resistivity, or sonic delta-t; P_p is the actual pore pressure, and P_hyd is the normal hydrostatic pore pressure. Calibration of this method requires knowing the appropriate normal value of each parameter. It is important to realize that, in contrast to trend-line methods, the ratio method does not use overburden or effective stress explicitly and thus is not an effective stress method. This can lead to unphysical situations, such as calculated pore pressures that are higher than the overburden. The ratio method is also applied to analyses of pore pressure from the drilling exponent (Fig. 2).

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  Fig. 2—Illustration of the ratio method. Here, d_cn is the expected value of the drilling exponent based on extrapolating a normal trend, and d_co is its measured value (modified after Mouchet and Mitchell).

Eaton's method

Perhaps the most widely publicized pore-pressure-estimation technique is Eaton’s method, shown graphically in Fig. 3. Here, stress is used explicitly in the equations

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  Fig. 3 - Lines for computing pore pressure expressed as an equivalent density, calculated using Eaton's method and the drilling exponent. Notice that these "lines" are not linear in semilog space (modified after Mouchet and Mitchell).

and

....................(3)

where P_p is pore pressure; S is the stress (typically, S_v); P_hyd is hydrostatic pore pressure; and the subscripts n and log refer to the normal and measured values of resistivity (R) and sonic delta-t (ΔT) at each depth. The exponents shown in Eq. 3 are typical values that are often changed for different regions so that the predictions better match pore pressures inferred from other data.

The major problem with all trend-line methods is that the user must pick the correct normal compaction trend. Sometimes are too few data to define the NCT. Unfortunately, if the NCT is defined over an interval with elevated pore pressure, the method will give the wrong (too low) pore pressure, leading to severe risks for drilling.

Resistivity method with depth-dependent normal compaction trendline (NCT)

In Eaton’s original equation, it is difficult to determine the normal shale resistivity or the shale resistivity in the condition of hydrostatic pore pressure. One approach is to assume that the normal shale resistivity is a constant. However, the normal resistivity (Rn) is not a constant in most cases, but a function of the burial depth. Thus normal compaction trendline needs to be determined for pore pressure prediction. Based on the relationship of measured resistivity and burial depth in the formations with normal pressures, the following equation of the normal compaction trend of resistivity can be used (Zhang, 2011):

Rn =R0 exp(bZ)

where Rn is the shale resistivity in the normal compaction condition; R0 is the shale resistivity in the mudline; b is the constant; Z is the depth below the mudline.

Substituting this Rn into the Eaton’s resistivity equation, pore pressure can be calculated.

Sonic method with depth-dependent normal compaction trendline (NCT)

Based on the data of the measured sonic transit time in the formations with normal pore pressures, the following general relationship of the normal compaction trend of the transit time is proposed (refer to Zhang, 2011):

DTn =DTm+(DTml-DTm)exp(-cZ)

where DTm is the compressional transit time in the shale matrix (with zero porosity); DTml is the mudline transit time; c is the compaction constant; Z is the depth below the mudline.

Substituting this DTn into the Eaton’s velocity/transit time equation, pore pressure can be calculated.

Effective stress methods

Methods that treat the problem correctly are often referred to as effective stress methods. The basis of the approach is summarized in Fig. 4. In this example, the top set of plots shows data recorded over a normally compacted section. The mean stress and pore pressure are shown as a function of depth at the left. Because the effective stress increases with depth, the porosity decreases with depth, as shown in the middle. If the porosity-stress function is a power law, it will plot as a straight line in linear-log space. There is, in fact, no restriction on the functional form of the porosity-stress function. The lower set of plots in Fig. 4 show the effect of an increase in pore pressure below a certain depth, as represented by the dashed line that diverges from the hydrostatic line in the lower left plot. In this case, the pore pressure in the overpressured zone increases at the same rate as the mean stress, such that the effective stress is constant. The plot of log σ vs. Φ follows the compaction trend only until it reaches the depth of overpressure, after which there is no change in either porosity or effective stress (lower right plot). This type of profile is typical of regions in which a pore-pressure increase is caused by the inability of the pore fluids to escape during burial and compaction.

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  Fig. 4—Illustration of the effective stress method for pore pressure prediction (modified after Swarbrick).

In general, effective stress methods must be calibrated, preferably using log data. However, they can also be calibrated empirically using approaches similar to those used to select trend lines, and they account explicitly for local changes in overburden and other stresses. The equation plotted in Fig. 4 is an example of relationships of the form

....................(4)

Athy first proposed this type of relationship in 1930, also proposing values for the parameters. Use of Athy’s original parameters is not recommended because they were based on analysis of overconsolidated shales from Oklahoma and thus are not applicable to young sediments. An appropriate algorithm for Athy’s method is to solve the following set of equations.

where

....................(5)

and f is the acoustic formation factor and is derived by calibration; Δtma is the matrix transit time. This type of relationship allows extension to account for effects such as cementation and thermal transformations by modifying the functional form of the exponent.

Complications

Ultimately, all pore-pressure methods must be calibrated; this is done empirically in most cases. Typical approaches rely on drilling experience to provide calibration points. These calibration points are based either on the occurrence of kicks, in which case the pore pressure in the sand producing the kick must be higher than the equivalent mud weight and lower than the kill mud weight, or on observations of instabilities in shales. In the former case, it is assumed that the pore pressures in shales and the adjacent sands are the same. In the latter case, the assumption is that instabilities occur when the mud weight has fallen below the pore pressure. In fact, wellbore instabilities that are due to compressive breakouts can occur at pressures that are higher or lower than the pore pressure. Therefore, the assumption that collapse begins to occur at a mud weight equal to the pore pressure can result either in an overestimate or an underestimate of P_p. If neither occurs, P_p is assumed (sometimes incorrectly) to be less than P_mud.

A further complication is that all of these methods require that the rock obeys a single, monotonic, compaction-induced trend, and that no other effects are operating. In reality, active chemical processes can increase cementation, leading to increased stiffness (higher velocities), which can mask high pore pressure, and increased resistance to further compaction, which can lead to erroneous prediction of the onset of pore pressure. Elevated temperatures lead to a transformation of the predominant shale mineral. For example, an increase in temperature transforms a water-bearing smectite to a relatively water-free (and more dense) illite. This transformation occurs over a range of temperatures near 110°C, but they can vary with fluid chemistry; furthermore, the depth at which this temperature is exceeded varies from basin to basin. Dutta developed a method that expands the argument of the exponential relationship of Eq. 4 to account for temperature effects and diagenesis (cementation and other changes that occur over time).

Pore fluid properties can also have a significant effect on pore-pressure predictions. This is because resistivity and velocity are both affected by the type and properties of the pore fluid. Changes in the salinity of brines will change resistivity, because pore fluid conductivity increases with salinity; thus a salinity increase (for example, adjacent to or beneath a salt dome) could be misinterpreted as an increase in pore pressure. Fluid conductivity is also a function of temperature.

Substitution of hydrocarbons for brine will increase resistivity, because hydrocarbons do not conduct electricity; this can mask increases in pore pressure that often accompany the presence of hydrocarbons. Because hydrocarbons are more compliant and less dense than brines, compression-wave velocity will decrease and shear-wave velocity will increase as hydrocarbon saturation increases. High gas saturation or API index will amplify this affect. Because a change from water to hydrocarbon affects resistivity and compressional velocity in opposite ways, simultaneous pore-pressure analyses using both measurements can sometimes identify such zones. It is more difficult to identify and deal with changes in fluid salinity.

Undercompaction

Most shale properties are, fortunately, characterized by fairly simple and single-valued functions of effective stress while on the compaction trend. When unloading occurs and the material becomes overcompacted, they do not follow the same relationship. This is because when the effective stress decreases, porosity and other properties are less sensitive to effective stress (see Fig. 5). Fortunately, relationships between porosity (or density) and other properties are different for overcompacted sediments than they are for the same sediment when it is normally compacted or undercompacted, as shown in laboratory data (Fig. 6). This provides a way to differentiate between undercompacted and overcompacted shales, using plots of velocity vs. density (Fig. 7). Once the domains have been separated, independent calibrations can be used to determine the pore pressure.

In highly lithified, older sediments, as in the case of overcompacted sediments, it is very difficult to use trend-line analyses to determine pore pressure. This is because, in these sediments, the sensitivity of porosity to effective stress is small. Even in such cases, however, it is sometimes possible (with accurate models derived from laboratory measurements and calibrated against in-situ direct measurements) to utilize resistivity or velocity measurements to estimate pore pressure.

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  Fig. 5—This figure shows the loading path and the confining pressure as a function of strain recorded during compaction experiments conducted using two samples of a poorly consolidated, shaley turbiditic sand of Miocene age. Sample 1 was maintained at its saturated condition; Sample 2 was cleaned and dried before testing (modified after Moos and Chang).

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  Fig. 6—This figure shows laboratory measurements of porosity vs. pressure (a) and porosity vs. slowness (b) along compaction trends and during reductions and subsequent increases in effective confining pressure in a poorly consolidated, shaley turbiditic sand of Miocene age. The separation of overcompacted from normally compacted or undercompacted sediments in plots of porosity vs. slowness makes it possible to use combined measurement of these parameters both to determine pore pressure and to identify the overpressure mechanism in both undercompacted and overcompacted domains (after Moos and Zwart).

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  Fig. 7—Relationships between density and sonic ΔT can be used to distinguish overcompacted from normally compacted or undercompacted sediments, as shown in this figure.

Centroid and buoyancy effects

The previous discussion of pore-pressure-prediction algorithms applies exclusively to shales and other low-permeability materials that undergo large amounts of shear-enhanced compaction. Because these algorithms do not work very well in sands, it is often assumed that pore pressures in sands are similar to those in adjacent shales. In reality, this is often not the case because the low permeability of shale makes it possible for it to maintain a pore pressure that is quite different from that in the adjacent sand. Two active processes, both of which can lead to very much higher (or lower) sand pore pressure than shale pore pressure that can be maintained over geological time because of low shale permeability, are the centroid and buoyancy effects.

The classical centroid effect occurs when an initially flat reservoir surrounded by and in equilibrium with overpressured shale is loaded asymmetrically and tilted, leading to a hydrostatic gradient in the sand that is in equilibrium with the original pore pressure at the depth of the sand prior to tilting. At the same time, pore pressure in the shale, which has extremely low permeability after it has been compacted, changes in such a way as to maintain a constant effective stress equal to the original effective stress at the depth of the sand prior to tilting. At the depth of the centroid (usually taken to be the mean elevation of the sand), the shale and sand pore pressures are equal. This effect is shown diagrammatically in Fig. 8. Because the pressure in the shale decreases upward at the same rate as the overburden (that is, proportional to the density of the shale itself), it is much lower at the top of the reservoir than is the pore pressure within the reservoir, which decreases at a slower rate that is proportional to the fluid density in the reservoir. Below the centroid, pore pressure in the sand is less than that in the adjacent shale.

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  Fig. 8—This figure shows diagrammatically a typical centroid geometry (left) and pore pressure profiles (right) in a reservoir sand and in the surrounding shale that develop because of the centroid effect. Pressure at the top of the sand is higher than in the adjacent shale, whereas pressure at the base of the sand is lower (modified after Bruce and Bowers).

Buoyancy effects occur when hydrocarbons begin to fill a tilted reservoir. The lighter hydrocarbons migrate to the top of the structure. Pressure at depth within the reservoir still follows a hydrostatic gradient (Fig. 9). The pressure in the gas at the top of the reservoir decreases upward more slowly, at a rate proportional to the density of the gas, which can be less than one-fourth the density of water. This leads to elevated pressure at the top of the structure. The same process occurs when oil fills a reservoir, but since the density difference is not as large for oil, the effect is less pronounced.

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  Fig. 9—This figure illustrates the buoyancy effect caused by systematic filling of a reservoir with lowdensity hydrocarbons. As filling progresses from Stage 1 to 2 to 3, the gas column grows, but the pressure is always in equilibrium with the centroid pressure at the gas-water contact, and so the pressure at the top of the reservoir increases (courtesy GeoMechanics Intl. Inc.).

The reservoir can continue to fill until the structure’s sealing capacity is exceeded. In the example, seal capacity is exceeded when the pressure at the top of the reservoir is high enough to cause the sealing fault to slip. However, in extreme cases, the reservoir pressure can be close to the least principal stress in the adjacent shale.

Nomenclature

e	= base of a natural logarithm, e = 2.718281828 …
f	= acoustic formation factor, used in Eq. 5
Pa	= Pore pressure at deptha
Phyd	= hydrostatic pressure, MPa, psi, lbm/gal
Pp	= pore pressure, MPa, psi, lbm/gal
Pz	= Pore pressure at depth z, MPa, psi, lbm/gal
Rlog	= measured value of resistivity, ohm-m
Rn	= normal value of resistivity, ohm-m
S	= total stress, MPa, psi
Sa	= stress at depth a, MPa, psi, lbm/gal
Sz	= axial stress along a wellbore, MPa, psi
Δtma	= matrix transit time, μs/ft
ΔT	= temperature difference between the fluid in a well and the adjacent rock
ΔTlog	= measured value of sonic transit-time at a given depth, μs/ft
ΔTn	= normal value of sonic transit-time at a given depth, μs/ft
σ	= Terzaghi effective stress, MPa, psi

Superscripts

β	= coefficient multiplying the effective vertical stress in Athy’ s relationship, Eq. 4
σv	= effective vertical stress in Athy’ s relationship, Eq. 4

See also

PEH:Geomechanics_Applied_to_Drilling_Engineering

Noteworthy papers in OnePetro

Glenn L. B. 2001. Determining an Appropriate Pore-Pressure Estimation Strategy, Offshore Technology Conference, 30 April-3 May. 13042-MS. http://dx.doi.org/10.4043/13042-MS

Muthukumarappan R., Douglas M. M., and Scott B. D. 2002. Diagnostic Fracture Injection Test in Coals to Determine Pore Pressure and Permeability, SPE Gas Technology Symposium, 30 April-2 May. 75701-MS. http://dx.doi.org/10.2118/75701-MS

Page champions

Fersheed Mody, Ph.D., P.E.