- Email: [email protected]
The diffusivity equation for geopressure prediction using well logs
Author's Accepted Manuscript
The Diffusivity Equation for Geopressure Prediction using Well Logs Alberto López Manríquez, Kamy Sepehrnoori
www.elsevier.com/locate/petrol
PII: DOI: Reference:
S0920-4105(14)00332-5 http://dx.doi.org/10.1016/j.petrol.2014.10.009 PETROL2825
To appear in:
Journal of Petroleum Science and Engineering
Received date: 12 December 2013 Revised date: 27 September 2014 Accepted date: 12 October 2014 Cite this article as: Alberto López Manríquez, Kamy Sepehrnoori, The Diffusivity Equation for Geopressure Prediction using Well Logs, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2014.10.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The Diffusivity Equation for Geopressure Prediction using Well Logs Alberto López Manríquez and Kamy Sepehrnoori
Corresponding Author: Alberto Lopez Manriquez Phone number: 1 956 414 3505 e-mail: [email protected]
Abstract
Common practice in industry is to obtain geopressures from indirect methods applied to data obtained from seismic or electrical logs. Because the diffusivity equation describes the pore pressure behavior of a saturated porous media, in this work, it is being used as the basis to attempt a solution to estimate pore pressure in producing formations and above them, in the overburden. Formation properties captured through well logging in offset wells are introduced to our solution of the diffusivity equation as “normalized values” to estimate pore pressures. For a traditional pore pressure assessment, as it is proposed in this work, the problem is set to a time frame. The resulting boundary value problem is solved by applying a “non-strict inverse approach” until a satisfying solution is found. Pressure data from offset wells are used as calibration parameters to obtain the pore pressure distribution along depth of investigation. To prove effectiveness and accuracy of our novel technique, results from real data in conventional and complex geological basins are discussed.
Research highlights
•
This novel method follows accepted theoretical criteria of diffusion transfer phenomena.
•
An analytic approach of solution is being applied rather than empirical.
•
The model is applicable for complex lithological scenarios others than Tertiary formations.
•
Because no Normal Trend of Compaction identification is needed, no graphical method is required.
•
The method is useful for identifying additional sources of abnormal pressures other than disequilibrium compaction.
Keywords: Geopressure prediction; diffusion; diffusivity equation; pore pressure; fracture pressure; consolidation; abnormal pressure; secondary compaction; well logging
1. Introduction
Common practice in industry is to obtain geopressures from methodologies applied to data obtained either prior to drilling from seismics, during drilling from operating parameters, or post drilling from well logs. A vast review covering years’ worth of research using these methodologies is contained in different specialized books, thesis, technical papers, journals, and all literature dedicated to the study in depth of geopressure prediction and estimation. Predicting geopressures from seismics is a subject that has been approached by geophysicists and the theory behind it is covered in specialized literature. Lopez (2013) proposed the prediction (predrill) of geopressures using data from passive magnetotellurics as a complementary technique to reduce uncertainty.
This work is focused on the estimation of geopressures using well log information (post-drill). Harrold (2001) provides an excellent review of literature dedicated to this topic. He proposed a methodology for pore pressure estimation from void ratios and mean effective stress. He suggests that his approach is more consistent with available data in comparison with results obtained from previous methods based on porosity and vertical effective stress. In this sense, it is remarkable to point out that even enormous efforts have been dedicated to estimate pore pressure from well logs; one, if not the most popular approach used in current days is Eaton’s approach. Based on the theory of consolidation proposed by Terzaghi, Eaton (1975) proposed a series of empirical equations to estimate geopressures from well logs. The simplicity of these equations has been the key factor to make Eaton’s equations one of the most recurred methods. To the best of our knowledge, this set of equations proposed by Eaton is still the basis for software used in petroleum industry to perform geopressures assessments when well logs are available.
In this work, a new set of equations is proposed to estimate geopressures from well logs. The diffusivity equation is used as a means to attempt a solution to estimate geopressures. The significance of these equations rests on the premise that pore pressure distribution in a porous medium obeys diffusivity principles. In this sense, the behavior of pore pressure all along the depth of investigation can be represented by the solution of the diffusivity equation. Formation properties captured through well logging are introduced to the solution of the diffusivity equation to estimate formation pore and fracture pressures. Time-dependent behavior of pressure can be modeled using this approach, which can be a valuable tool when identifying lateral drainage or pressuring, or identifying natural depletion of pay zones. Formation fracture and overburden pressures are then estimated using Eaton’s relationships. A simple but reliable equation proposed by Christman (1973) is used to estimate fracture pressure gradients for deep water wells in offshore scenarios. The results of this new proposed methodology are valuable for research, interpretation, correlation, and identification of abnormal pressure zones for exploration, drilling, completion, and production purposes.
2. Geopressures detection techniques
There exists a series of methods to estimate geopressures before, while, and after drilling a well. Regional geology and geophysical methods are widely used to predict geopressures before drilling. Seismics methods to attempt to reconstruct the behavior of compaction trend during burial are commonly used in industry and permit the prediction of formation pressure. To evaluate pressure quantitatively during drilling, Mouchet and Mitchell (1989) and Fertl (1976), among other authors, provide a list of methods for this purpose, such as formation tests, seismic interval velocities, drilling rate (“d” exponent), shale density, gas shows, kicks/mud losses, and wireline logs (resistivity/conductivity, sonic, density). The following section highlights the basics of Eaton’s technique, a postdrill method to estimate geopressures.
2.1 Effective stress transformation model
Based on Terzaghi’s stress relationship principle, expressed in Equation 1, Eaton (1975) developed a set of empirical equations to estimate geopressures from well logs data. = −
(1)
The empirical equations to estimate formation pore pressure from sonic and resistivity well logs formulated by Eaton represent an effective stress transformation model expressed in the following form:
= − −
= − −
∆
∆
(2a)
(2b)
In his work, Eaton concluded that the accuracy of the results from these equations depends on the following main aspects: the quality of the input data, the interpreter who establishes the normal trend of compaction (NTC), and the correct value of the exponent (α) on the ratio term, which for resistivity data varied from 1.2 to 1.5 and for sonic data is close to 3.0 to the applied data Eaton used for his study. In this respect, Lopez-Solis et al. (2006) found and concluded that Eaton’s exponent, α, is a factor of calibration which can be adjusted according to particular conditions to match results. They found that for a specific area offshore Gulf of Mexico fields, Eaton’s exponent (α) takes a value of 0.4 to match calculated to observed values in the field.
2.2 Normal Trend of Compaction
The normal trend of compaction (NTC) represents the optimum fitted linear trend of measured data (velocity or resistivity) in the low permeable formations within the transition zone. Shaker (2007) states that pore pressure profile is divided into three segments as illustrated in Figure 1. They are as follows: 1) the shallow section is unconfined and liberating water. 2) the middle hydrodynamic section, named the transition zone, where compaction disequilibrium is active between the lower confined pressure zones and the unconfined upper sections. A dewatering process from sediments due to pressure gradient occurs in this transition zone. Finally, 3) in the lowest segment, below the top of geopressured zone, dewatering process does not respond to normal compaction, creating pore pressure to deviate from normal hydrostatic pressure.
Figure 1. Three generic subsurface zones for establishing a NTC trend. (from Shaker 2007).
Many authors, Lopez-Solis et al. (2006), Shaker (2007), Young and Lepley (2005), and Swarbrick (2002), among others, have written about the significance of the correct fitting of the NTC on the final results of pressure analysis. Young and Lepley (2005) states that geopressures analyses based on multiple NTC should be a matter of discussion; even in areas where drilling penetrates rocks of different geological Eras. Additional complications arise where shale/sand sequences are encountered. Besides, he states that NTC should be fitted to resistivity values obtained only from intervals containing shale.
2.3 Other approaches
The benefits of Eaton’s approach have been recognized, but at the same time the limitations of this approach have been highlighted. Since Eaton’s approach comes from the theory of consolidation, it is assumed that pore pressure estimation does not reflect sources of abnormal pressures, such as fluid expansion, hydrocarbon generation, dynamic lateral transfer, thermal, chemical/ secondary compaction, or diagenesis other than disequilibrium compaction. Based on this concept, researchers have been focusing efforts in finding ways to introduce all possible sources of pore pressure and overburden changes to modern models. Harrold (2001) proposed a methodology for pore pressure estimation from void ratios and mean effective stress. His equation has the following form:
≈ ( + − − )
(3)
He suggests that this approach is more consistent with his available data than with previous methods based on porosity and vertical effective stress. Additionally, he claims that his method can detect overpressure generated by mechanisms other than disequilibrium compaction.
Miller proposed an alternative way of computing effective stress in Terzaghi’s equation 1. His equation is based on the properties of sound when transmitted through a porous medium. He states that the vertical effective stress can be obtained from:
= − ( − ! )" # % &'((
(4)
Replacing the effective stress, σeff , into Terzaghi’s equation 1, pore pressure can be computed as follows:
= − ,-
− ! . − ./
(5)
Bowers’ approximation is another equation widely used in pore pressure prediction and is useful when sonic properties of formations are available. This particular equation is recognized to be useful when loading/unloading pore pressure mechanisms are anticipated.
Even Eaton’s approach, based on an empirical formula to estimate pore pressure, is still widely used until present days in geopressure assessments when wireline logs are available.
3. The Diffusivity Equation
Literature review on the subject leads to interesting solutions of the diffusivity equation using several techniques for particular conditions. It is not the intention in this work to elaborate on the subject, but to present solutions obtained by the method of separation of variables related to this topic.
Charlez (1991 & 1997) describes that diffusivity comes from three different sources: hydraulic, mass, and thermal sources. An expression of diffusivity is obtained when Darcy’s law is introduced to a mass balance. Assuming the system is under equilibrium and mass diffusion through a porous medium and temperature remaining at equilibrium allow simplifying the diffusivity equation to the following partial differential equation:
0 2
1 2
−3
2455 2
6
= ∇9 7
(6)
Specific loading paths allow rearranging equation (6) to other expressions more commonly used in reservoir engineering applications such as
:
2 2
=
6
;<=( >=? @
∇9
(7)
where ∇9 is the Laplacian operator of P, and for a Cartesian coordinate system, we have
∇9 =
2A 2B A
+
2A 2 A
+
2A 2C A
(8)
In a one-dimensional medium, the diffusivity equation can be expressed as follows:
2 2
=D
EA 2C A
(9)
Solving equation (9) by separation of variables allows a special solution that can be written as a linear combination of the product of functions of the individual variables. The general solution of equation (9) is expressed as follows:
(F, H) = I " #J K cos FOD: ⁄P 9 + R ST- FOD: ⁄P 9
(10) (Solution found in Kreyszig 1999 and Olver 2013)
where FOD: ⁄P 9 = -U and D = (-UP ⁄F)9 /:. P9 =
6
;<=( >=? @
and µ coefficients represent rock and fluid formation properties. Cf and Cr are formation fluid and rock
compressibilities respectively, Cr being stress path dependent.
This solution represents the pore pressure behavior of a saturated porous media, A, B, and Cn being constants depending on the boundary and initial conditions existent in the pore pressure medium.
Geopressures estimation analysis requires knowledge of the boundary and initial conditions. However, in a problem like this, either at a well location or at basin level, boundary conditions and initial condition are hardly known.
In this respect, particular solutions to the diffusion equation (9), when different boundary conditions (BC) and initial condition (IC) are applied, can be found in literature. A brief description of them is presented below. A more detailed explanation can be found in Kreyszig (1999) and Olver (2013) as many other references.
3.1 Solution of PDE (9) with Dirichlet non-homogeneous BC
P(0,t)=P1 ,
P(L,t)=P2 ,
P(z,0)=f(z)
Imposing these BC implies fixed predefined pressure P values at z=0 and z=L
The particular solution of (9) is as follows: (F, H) = 0 +
A #V W
F + ∑^0 Y "Z[ −
\A ]A WA
ST-
]C W
where 9
W
Y = _a (F)STW
]C W
`F
(11)
3.2 Solution of PDE (9) with Dirichlet homogeneous BC
P(0,t)=0 ,
P(L,t)=0 ,
P(z,0)=f(z)
Dirichlet homogeneous BC represent fixed pressures equal to zero at both ends. This is a particular condition of equation (11) resulting in:
(F, H) = b Y "Z[ c− ^0
d-9 U 9 H -UF f ST-
e9 e
where Y =
2 W -UF h (F)ST`F e a e (12)
It is interesting to note that from these particular solutions, where Dirichlet conditions are imposed, P(z,t) depends only on the sin (z) term. The cos (z) portion of the solution vanishes.
Aside, an interesting particular solution to a heat transfer problem where the interval z from 0 to L is assumed as a semi-infinite interval was found in Olver (2013). By analogy, we replace pressure instead of temperature. In this way, it is assumed pressure inside the earth depends only on depth and time. It is also assumed that pressure at surface conditions z=0 behaves according to the following arbitrary expression:
(13)
P(0,t)= A cos wt
At large depth, P(L,t) is assumed unvarying, and hence tends to a mean pressure, P(L,t)→i .
The resulting solution to these BC is as follows:
(F, H) = K "
#C(
j ) Ak
cos clH − Fml.2df
(14)
This equation (14) resembles the solution presented by Francesco (2008) to Terzaghi’s consolidation equation. It is interesting to point out that these solutions (14 and 15) were obtained assuming different approaches. Francesco’s expression is as follows:
∆(F, H) = ∆" #6C cos (lH − nF)
(15)
∆ defined as the variation of pore pressure.
3.3 Solution of PDE (9) with Neumann homogeneous BC
Px(0,t)=0 ,
Px(L,t)=0 ,
P(z,0)=f(z)
When Neumann BC are applied, it is assumed that no pressure exchange occurs at the boundaries z=0 and z=L.
The solution of equation (9) is given by
(F, H) = b o "Z[ c− ^0
d-9 U 9 H -UF f PpS
e9 e
where o =
2 W -UF h (F)PpS `F e a e (16)
It is interesting to note that the particular BC imposed to the problem defines P(z,t) depending only on the cos(z) term while the sin (z) part of the solution vanishes.
3.4 Solution of PDE (9) with Robin BC at one end
Px(0,t)=0 , P(z,0)=f(z)
Px(L,t)=-δ P(L,t) ,
When a Robin BC is imposed at z=L, δ being positive; then it states that pressure at this boundary is being taken out of the system at the same amount pressure is maintained. Dirichlet BC is imposed at z=0.
The solution of (9) is as follows:
(F, H) = ∑^0 "Z[(−D H) ST-(l F)
(17)
4. Modeling Pore Pressure
As stated in the previous section, in a geopressure estimation problem, boundary and initial conditions are hardly known. Therefore, a “non-strict inverse approach” to the solution of the problem is proposed using equation (10). Data obtained from well logs capture and contain rock and fluid formation properties, data from sonic, resistivity, and/or density logs are introduced into the equation as “normalized values”. For instance, sonic properties are introduced as follows:
∆Hq /∆H
st
= OD: ⁄P 9
(18)
As stated previously, FOD: ⁄P 9 = -U and D = (-UP ⁄F)9 /:; being P 9 =
6
;<=( >=? @
and µ coefficients representing rock and fluid
formation properties. Cf and Cr are formation fluid and rock compressibilities respectively.
Applying (18) into equation (10), the following expression is obtained:
F ∆Hq F ∆Hq (F, H) = I " #J cK cos c ∗ f + R ST- c ∗ ff e ∆H st e ∆H st (19)
This solution represents the pore pressure behavior of a saturated porous media for the boundary conditions existent in the porous medium analyzed. Similar expressions can be obtained when resistivity and/or density well log information is analyzed. Accuracy of results from using different source of formation data will vary according to the resolution and the behavior of data applied.
Initially, a reasonable first approximation of the values of constants A and B associated with the general solution is introduced to equation (19). For a traditional pore pressure assessment, as proposed in this work, the problem is set to a time frame t=0. When
pressure data from offset wells are available, they are used as calibration parameters to find A and B values satisfying a P(z,t) solution accordingly. In this way, the particular solution of P(z,t), equation (19), can be computed all along the depth of investigation (depth of the well).
The proposed equation (19) captures the cosine and sine behavior of the solution of the diffusivity equation and the boundary conditions result in a periodic eigenvalue problem. Therefore, it is suggested that weights of A and B values on the solution could help identify boundary effects taking place. These A and B values have been encountered to be between 0 and 1.
o =
2 W -UF h (F)PpS `F e a e
Y =
2 W -UF h (F)ST`F e a e
5. Modeling Fracture Pressure
Fertl (1976) and Mouchet and Mitchell (1989), among others, provide a detailed description of the different prevailing methods for the prediction of fracture pressure gradients. Once pore pressure has been estimated using our proposed methodology, fracture pressure gradient determination implemented in this model is based on the general relationship between pore pressure and stress given by equation (20). This relationship allows taking into account variable pore pressure and variable vertical effective stress related to depth by introducing Poisson’s ratio into the calculations, assuming formations to be elastic (Eaton’s 1969). This set of conditions provides the connection between pore pressure and fracture pressure estimation. When including these parameters, existing in-situ stresses are being taken into account in calculations. For tectonically relaxed basins, normal faulting, the minimum horizontal stress is horizontal; therefore, overburden represents vertical stress. In this sense, a variable overburden stress gradient is taken into account. This particular characteristic of modeling constrains even more finding a particular solution to P(z,t) since this must satisfy both: fracture and pore pressure calibration data available at the same time.
vw =
x
0#x
(yw − w) + w
(20)
When density log information is available, the model allows for the inclusion of these data representing the overburden gradient.
Geopressures prediction in an offshore scenario requires inclusion of the effect of water depth on the overburden stress. Fracture gradient determination assumes that overburden comes from rock matrix and formation fluid stresses. Offshore, sea water does not have rock matrix; therefore, because fracture gradients depend on overburden, fracture gradients offshore are lower than those computed onshore compared at equivalent depths. This effect becomes significant as water depth increases. To overcome this condition, the concept of stress ratio, introduced by Christman, is implemented in this model to properly estimate overburden pressure gradient offshore using a relationship in the form:
ywz{ ≈ Kw| + }w{z + yw/W
(21)
6. Discussion of Results
This section presents the discussion of results of five different case studies. These discussions are focused first on the comparison of estimated geopressures (pore and fracture pressures) with the calibrated values of pressure obtained from the field (when available). Second, when no calibration data are available, these discussions are established based on comparison of results obtained from the proposed methodology in this work versus results obtained from Eaton’s approach. The cases discussed here are not confined to a specific area of interest; instead, well log data correspond to different geological basins. It is important to note that without data to calibrate or compare, results obtained from any geopressure analysis are only estimates. Harrold (2001) provides a table containing hierarchy of data sources according to its reliability. Table 1 is taken from this reference.
Data Sources Repeat Formation Tester (RFT, MDT) Formation Interval Tester (FIT) Drill Stem Test (DST) Well or pressure kick Connection gas Measuring/Logging While Drilling (MWD/LWD) Mudweight Equivalent depth Drilling exponent (D, Dc)
Hierarchy high
low
Table 1. Sources of pressure data organized by reliability (From Harrold 2001)
6.1 Case Study 1
Wire logs from an offshore well were obtained; sonic and resistivity profiles along depth are shown in Figures 2a and 2b. Figure Fig 2a shows the normal trend of compaction (NTC) line adjusted to the resistivity profile according to Eaton’s method. A smooth slope was imposed based on the known geology of the area, which consists of sedimentary y packages of the same geologic Era (Cenozoic) made up of a silisiclastic sequence alternating with shale (0 (0-418 m) Pleistocene, (418-1924 m) upper er Pliocene, (1924-2229 m) middle Pliocene, (2229-3111 m) lower Pliocene and (3111 (3111-3388 m) upper pper Miocene, without reaching the top of the middle Miocene.
Figure 2a. Resistivity log from wireline showing NTC
Figure 2b. Sonic log from wireline
Figure 3 illustrates the results of geopressures estimated using the proposed equation. Pore pressure is represented by the blue dots dot profile while fracture pressure by the red. Validity of these results is established during superimposition of pore pressure values valu measured in the well using Modular Formation Dynamics Testing (MDT), a direct pore pressure measurement tool. Validity of fracture pressure profile is achieved while superimposing estimated fracture pressure obtained from Leak Off Test (LOT) at 1665 16 m.
Because fracture pressure calculation is pore pressure dependent, this particular approach of modeling imposes additional constraints. The green dots in this figure represent values of pore pressure measured at seven different points along depth. It can be seen that these pore pressure values match along the calculated pore pressure profile except one of them at 2790 m.
Figure 3. Geopressures profile obtained from the proposed equation
Figure 4 illustrates pore pressure values along depth expressed in equivalent density (ED). The light blue straight line in this figure represents the mud density (MW) used to drill the well for the interval. According to the drilling record of the well, water flow was detected from interval 3000 - 3350 m (bottom). Mud density had to be increased to reestablish control of the well. Figure 4 also shows the predicted fracture pressure (red) and overburden (green) profiles. It is remarkable that the predicted values of fracture gradient match the only fracture pressure control point obtained at 1665 m from a LOT, represented for the orange dot. The effect of water depth on fracture gradient and overburden stress is included in modeling. The agreement found between predicted geopressures (pore and fracture pressures) with available measured values in the field provides confidence on the validity of applied methodology.
Figure 4. Geopressures expressed in equivalent density (ED) profile obtained from the proposed equation
Figure 5 illustrates a comparison of results of estimated pore pressures between the proposed equation and Eaton’s equation. Matching is excellent all along depth of interest since the two curves collapse practically to one single behavior. Eaton’s pore pressure is represented by the blue dots profile, while pore pressure obtained using the proposed model by the red. MDT data are included in this figure to express the excellent match of results.
Figure 5. Pore pressure comparison between Eaton’s and Lopez and Sepehrnoori’s equations
6.2 Case Study 2
Figure 6 illustrates the comparison of results between estimated pore pressures using both equations. Again, matching between results is excellent in the lower part of the interval. Some deviations can be seen above 1800 m. The well was drilled through sedimentary rocks of Cenozoic; mainly sand sequences alternating with shale. Dispersion of results by Eaton’s curve above 1800 m is attributed to the shale-sand sequences found in these Tertiary formations drilled in this well. Dispersion of original ∆to data from the NTC imposed during calculation creates this behavior. Young and Lepley (2005) recognize this condition when he states that if an empirical pressure model is used, sands will create pressure regression. He suggests not taking into account these “sand-influences” anomalies in the assessment.
Calibration of these results is given by the green dots in this figure; they represent values of average reservoir pressure (P avg) reported at different horizons (2000, 2200 and 2700 m). It can be seen that these pressure values match the calculated pore pressure
profile along depth giving reliability to the obtained results. Additional information to calibrate results from horizons above 1800 m should be valuable to fully complete this assessment.
Figure 6. Pore pressure comparison between Eaton’s and Lopez and Sepehrnoori’s equations
Figure 7 shows pore (blue) and fracture (red) pressures expressed in ED format profile. The light blue line in this figure represents the mud densities (MW) used to drill a well in the area while the purple dots show reservoir pressures. Drilling mud densities all along depth are also in agreement with the so-called operational window between pore and fracture pressure gradients.
Figure 7. Geopressures expressed in equivalent density (ED); profile obtained from the proposed equation
6.3 Case Study 3
Figure 8, like in the past two cases, shows the comparison of results between estimated pore pressures. Matching between results is excellent along depth of interest except for the bottom part of the section below 3500 m. Validity of results for this case is based on superimposing pore pressure values measured in the well using MDT and fracture pressures estimates from LOT. The green dots in this figure represent values of pore pressure measured at nine different points along depth. No MDT data was taken below 3500 m to validate the curve below this depth.
A possible explanation for the mismatch between curves below 3500 m can be given by describing the stratigraphic column in the area. Above 3500 m, lithology is described as an upper sedimentary package of the same geologic Era from recent Pleistocene to middle Miocene. This section is similar to the geological areas in the Gulf coast, terrigenous tertiary Era sediments, as those where Eaton and Hottman and Johnson based their studies. The lower Miocene (below 3000m) is described by lithic sandstones, limolites (turbiditic fans), and limestone-clay matrix. Below 3200 m rock is characterized with primary and secondary porosity. This dual-
porosity characteristic below 3200 m may be one of the reasons for the difference between the two curves. This material will be discussed in a separate publication.
Figure 8. Pore pressure comparison between Eaton’s and Lopez and Sepehrnoori’s equations
Figure 9 illustrates pore and fracture pressure values along depth expressed in equivalent density. As before, the light blue straight line in this figure represents mud densities (MW). Computed values of fracture gradient (red curve) match at two (1900 and 3500 m) of the four fracture gradient pressure points obtained at 1400, 1900, 2660, and 3500m from LOT represented by the orange dots. Since the well was drilled offshore, the effect of deep water depth on fracture gradient and overburden stress is included in the modeling. No better match could be obtained without compromising the principles of theory previously given in this same work. Assuming the quality of all data from LOT is reliable for this deep water case, certainly it suggests a better fracture gradient estimation at intermediate and deep depths than at shallow depths. At the same time, it is recognized that to qualify the accuracy of the deep water fracture gradient equations used in this modeling, more deep water case studies are needed.
Figure 9. Geopressures expressed in equivalent density (ED) profile obtained from the proposed equation
6.4 Case Study 4
This case study involves a different lithological column. A general description of the area of this case study is given as follows: Upper Cretaceous formations (0-1100 m) composed of alternating sandstones units with presence of mudstones and shales. From 1100-2600 m upper and lower cretaceous horizons consisting of calcareous mudstone with slight presence of shale are encountered. From 2600-2800 m Limestone is found predominantly. This lithology differs from conditions used by Eaton (1975) and Mathews and Kelly (1967) to obtain their equations.
Figure 10 shows geopressure profiles obtained from analyzing the data. It is clearly seen that although both curves follow a similar trend, mismatch between them exists. The green dots represent the value of pressure measured at the depths of 1555 and 1575 m in pay zones in the only near offset well, serving as the only calibration points. The abnormal pressure detected from 2600 to 2700 m could not be corroborated with calibration.
Figure 10. Pore pressure comparison between Eaton’s (One-NTC) and Lopez andSepehrnoori’s equations
It is not the intention of this work to make strict conclusions about findings for this single case, but to open a discussion about the usefulness of the equation proposed to identify sources of overpressure related to the geological environment. An explanation for the disagreement between results can be obtained when reviewing literature. It has been pointed out for some authors (Young and Lepley (2005), Swarbrick (2002), and Shaker (2007)) that in more complex scenarios like this, where rock formations belonging to Eras other than Tertiary or more than one geological Era are drilled, definition of a NTC line is complicated. This condition causes in many occasions that more than one NTC is used by interpreters to match results. When doing that, the principle of the NTC is compromised and the resulting geopressure assessment could be not reliable (Young and Lepley 2005).
To help the discussion, analysis using Eaton’s equation was repeated imposing two different NTC: the first above 1100 m and the second below these depths. Adjusting these two NTC lines, pore pressure profile calculated using Eaton’s equation matched much better the estimated values with our proposed equation (19). Figure 11 depicts this behavior. It can be seen from this figure that the two curves match better than what presented in Figure 10.
To make further conclusions, much more information needs to be collected from the studied field. Particularly, reliable pore and fracture pressures measurements below 2500 m are needed to corroborate the existence of a presumably high pressure zone there. Fracture pressure at other levels than pay zones will help make further conclusions. Otherwise, the discussion is merely speculation. By now, it is important to note that in the case studies 1, 2, and 3, where lithology resembles the conditions covered by Eaton’s equation, the proposed equation 19 outstandingly matches with Eaton’s. On the other hand, when complex lithologies different from those studied by Eaton are encountered, results from both equations deviate. Two NTC had to be imposed to find better matching between results.
Figure 11. Pore pressure comparison between Eaton’s (2NTC) and Lopez and Sepehrnoori’s equations
Figure 12 shows the equivalent density profile obtained. The purple, the light blue, and the orange lines (MW1, MW2, and MW3) represent drilling mud densities for wells drilled in the area expressed as equivalent density. As it can be seen, mud density MW1 (purple line) is mostly in agreement with the pore pressure profile. From 2000 m to the bottom, MW1 increments of density obey pore pressure behavior computed in the confined zone. A mud density (up to 1.8 g/cm3) was used to drill interval 2350 - 2900 m, as illustrated by the orange line (MW3). Based on last observation, and in order to provide additional information to compare and discuss results, it was seen from daily drilling reports that no manifestation of gas or liquids were detected on surface when drilling through interval 2000 to 2900 m in wells (MW1 and MW3) drilled in the area. The well represented by MW2 (light blue line) reached only
1500 m approximately to an upper target. The abnormal pressure detected from 2600 to 2700 m could nnot ot be corroborated at the time of writing this work since no more information from deep wells in the area was available.
Figure 12. Geopressures expressed as equivalent density from resistivity logs
6.5 Case Study 5
An onshore well was drilled to 6480 m planning to explore for hydrocarbons in Cretaceous formations. This area is recognized as a tectonically active region; faulting has also been identified. Tertiary formations were drilled to 5700 m, mainly sequences of o shales and sands. Cretaceous formations ormations belonging to Mesozoic were the target of this well. These formations were encountered from 5700 through the bottom of the well and are characterized as naturally fractured carbonated rocks and dolomite.
Perez et al. (2010) reviewed previous geop geopressure ressure assessments and identified that in the interval 2700 to 3800 m, where a sand/shale sequence is encountered; abnormal low pressured zones were erroneously interpreted and identified. This misinterpretation misinterpr resulted in erroneous pore pressure estimation ion at this stratigraphic level.
Further review using the proposed equation shows that although not mentioned by the authors, between 2000 and 2700 m a high abnormal zone was also originally misinterpreted. Figure 13 (obtained from Perez et al.) shows the comparison of geopressure modeling before and after they reinterpreted lithology and additional information. The profile in the left shows the initial interpretation while the one in the right shows Perez et al. improved interpretation. It can be seen how at the two intervals mentioned, pore pressure (PP) purple curve flattens, no longer showing abnormal high or low pore pressures at these levels.
Figure 13. Comparison of geopressure models before and after reinterpretation (from Perez at al. 2010)
Modeling of data for a well in the area was conducted. A high abnormal pressure section is identified below 4500 m, as shown in Figure 14. No abnormal pressures are estimated above this level. This corroborates Perez et al. findings related to the misinterpretation of results originated by the effect created by the sand and shaly interval drilled (2700 – 3800m). This same sand/shaly effect can be the reason of disparity between results for this particular interval. Eaton’s equation (one NTC) underestimates pore pressure at this interval when comparing against results from the proposed equation. The high pressure zone below 4500 m is perfectly identified by the proposed equation while Eaton’s fails to match the calibration point at 6000 m.
The target formation (6000 m approximately) is located below the high pressured zone and presumably has been naturally depleted from producing wells in the area. It is interesting to remark at this point that tectonics events in the region may cause lateral drainage along faults. It is also important to mention that below 5700 m Cretaceous naturally fractured carbonated and dolomitized formations have been identified. This particular condition could be one of the reasons why in this segment results from the two models deviate. Average reservoir pressure has been reported for some wells in the area. For an offset well, the original reservoir pressure reported was 11800 psi. Since reservoir pressure could have experienced natural depletion, reported data have to be timely according to the date wireline logs were taken to make comparison time-frame agreeable. Average reservoir pressure reported (Perez et al. 2010) is 11000 psi at an average depth level of 6000m. Figure 14 shows this data with a green dot which serves as a calibration point.
Figure 14. Comparison of pore pressure profiles between Eaton’s and the proposed equation
Figure 15 illustrates pore (red dots) and fracture pressure (blue dots) values along depth expressed in equivalent density (ED) (E format profile. These results come from the proposed equation. The light blue and purple lines represent mud density (MW1 and MW2). 2). Both computed geopressures gradients (pore and fracture) are in agreement with the geopressure “reinterpreted” model presented by Perez et al. To visualize this statement, results in Figure 15 should be compared with those seen in Figure 13 (right). ( On the other hand, Eaton’s equation using a single NTC could not match results obtained either by the proposed equation or by Perez Pe et al. reinterpreted. Based on these results, the proposed equation could be a valuable tool to analyze complex geological scenarios sce others than Tertiary formations.
Figure 15. Geopressures expressed as equivalent density
It is important to mention that modeling using Eaton’s equation with multiple NTC could cause misinterpretation of pore pressure. press
nesses of the proposed equation 7. Strengths and weaknesses
Strengths •
The proposed methodology is simple; it requires conventional information delivered by either sonic or resistivity logs.
•
The proposed methodology follows accepted theoretical criteria of diffusion transfer phenomen phenomenaa and an analytic method of solution rather than an empirical approach.
•
It is useful for application in complex lithological scenarios others than Tertiary formations (e.g. cases 3, 4 and 5). In these th cases, the proposed equation showed better agreement w with existent data.
•
It is well known that to fit results in complex scenarios, Eaton’s method requires more than one NTC lines imposed arbitrarily arbitraril by the interpreter. This practice has been identified by some authors as an action that compromises the effective effect stress transformation model proposed by Eaton.
•
Deviation between results, when comparing Eaton’s versus the proposed pore pressure solution, has been identified in formations formatio recognized with dual porosity. It is suggested that the proposed equation sho should uld be used for pore pressure estimation in geological scenarios under these conditions. It could be a valuable tool to identify sources of abnormal pressures such as secondary compaction or diagenesis encountered in naturally fractured formations.
•
It was shown that particular BCs applied define the weight of cosine and sine functions in the general solution equation. Particular solutions shown in the review section of this work have either cosine or sine functions but not both. The proposed equation (19) captures the cosine and sine behavior of the solution of the diffusivity equation. Therefore, it is suggested that weights of A and B values on the solution could help identify boundary effects taking place. For the presented cases, A values varied from 0.1 to 0.2 and B values from 0.6 to 0.9. There is no need for NTC identification, which eliminates the issue of defining the NTC’s slope by interpreter, who in many cases uses more than one arbitrary NTC lines to match results with calibrating data. This common practice is against the premise stated by consolidation theory briefly discussed in this work.
•
Because no NTC is needed, no graphic method is required.
Weaknesses •
Only few data for case studies were available at the time of this work.
•
More data from the field are needed to find trends of A and B calibration parameters according to the geological conditions of studied basins.
•
Like any indirect method, accuracy of results highly depends on the quality of data and the expertise of interpreter.
•
Much more analysis is required to further conclude some of the observations discussed here.
8. Conclusions
This work describes a self-consistent methodology for calculating pore and fracture pressures using information from well logs. As with any indirect method, the results of this methodology are interpretive. As is with wireline logs or seismic data, they do not measure pressure directly independent of how data are processed.
As it was described in the “strengths and weaknesses” section, the proposed equation satisfies conventional and unconventional geological scenarios for estimating pore and fracture pressures. For this reason, it is considered as a valuable tool to identify sources of abnormal pressures, such as secondary compaction or diagenesis encountered in naturally fractured formations.
The proposed set of equations presented in this work could be valuable for the researcher who is looking for a way to interpret the different mechanisms involved in abnormal geopressure origins and sources. It is also useful for the explorer looking for a general description of pressure profile in areas other than Tertiary and for the engineer in charge of planning drilling and completion.
9. Nomenclature
A, B, Cn
= Constants
AGRF
= Air gradient from rig floor, psi/ft
Cf
= fluid compressibility, 1/psi
Cr
= pore volume compressibility, 1/psi
D
= total depth, m, (ft)
exp (e)
= base of natural logarithm
FPG
= fracture pressure gradient
f
= function
fM
= Miller’s fitting parameter
h
= representative depth, m, (ft)
k
= formation permeability, mDarcies
L
= total depth of investigation, m
OG
= overburden gradient, psi/ft
OGDW
= overburden gradient deep water, psi,ft
OGML
= overburden gradient mud line, psi,ft
PG
= pore pressure gradient, psi/ft
P
= Formation pore pressure, psi
Phyd
= Hydrostatic pressure, psi
Ro
= resistivity observed, ohm m
Rn
= resistivity on normal trend line, ohm m
S
= Overburden total stress, psi
t
= time, sec (min)
V
= speed of sound in shale, ft/µsec
Vma
= speed of sound in the rock matrix, ft/µsec
Vml
= mud line velocity (speed of sound in liquid), ft/µsec
WGWD
=Water gradient water depth, psi/ft
z
= depth of interest, m, (ft)
α
= Eaton’s exponent, dimensionless
β
= thermal expansion coefficient, dimensionless
∆P
= Pore pressure variation, psi
∆tavg
= Sonic Average Interval Travel Time, µsec/ft
∆tn
= Sonic ITT on normal trend line (µsec/ft)
∆to
= Sonic Interval Travel Time (ITT),µsec/ft
~ 9
= Lapacian operator
εkk
= Volumetric strain, dimensionless
φ
= porosity, fraction
η
= Biot's modulus, psi
λ
= eigenvalue
µ
= fluid viscosity, cP
σeff
= rock matrix effective stress, psi
σm
= mean effective stress, psi
υ
= Poisson’s ratio, dimensionless
10. References
Charlez, P.A. 1991, Rock Mechanics. Vol. 1 Theoretical Fundamentals, Editions Technip, Paris. Charlez, P.A. 1997, Rock Mechanics. Vol. 2 Petroleum Applications, Editions Technip, Paris. Christman, S.A. 1973, Offshore Fracture Gradients, JPT 4133-PA, SPE Vol 25, Number 8, Pag. 910-914. Eaton, B.A. 1975, The Equation for Geopressure Prediction from well logs. paper SPE 5544, 50th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, Dallas, TX, Sept 28 – Oct 1, 1975. Fertl, W.H. 1976. Abnormal Formation Pressures, Elsevier Scientific Publishing Company. Francesco, R.D. 2008, Exact Solution to Terzaghi’s Consolidation Equation, GEO&GEO Instruments, research & development, Teramo, Italy. Harrold, T. 2001. Porosity and effective stress relationships in mudrocks, Durham theses, Durham University, http://etheses.dur.ac.uk/4338 (accessed April 23, 2013). Kreyszig, E. 1999. Advanced Engineering Mathematics, 8th edition, John Wiley & Sons, Inc. USA. Lopez, A. 2013. Geopressure prediction from Passive Magnetotellurics. Paper SPE 166356-MS, presented at the ATCE 2013 held in New Orleans, Louisiana, USA 30 September – 2 October 2013. Lopez-Solis, V., Velazquez-Cruz, D., Jardinez-Tena, A., and Castaneda, G. 2006. Normal Resistivity Trends for Geopressure Analysis in Mexican Offshore Wells. Technical paper OTC 18192, May (2006). Miller, T (unknown year). Miller Pore Pressure algorithm, Knowledge Systems publication (accessed May 6, 2013). Mouchet, J.P. and Mitchell, A. 1989. Abnormal Pressures while Drilling. Societe Nationale Elf Aquitaine.
Olver, P.J. 2013, Introduction to Partial Differential Equations, Chapter 4, Lecture notes. Perez, C., Mandujano, H., Puentes, F., Isacci J., Marino, L., and Obediente, R. 2010. Drilling Optimization of Wells from Crater Field: Decreasing Drilling Times and Operational Risks. Paper SPE 139035 presented at the SPE Latin American & Caribbean Petroleum Engineering Conference held in Lima, Peru, 1-3 December 2010. Shaker, S. 2007, Calibration of Geopressure Predictions using the Normal Compaction Trend: Perception and Pitfall, Technical article: CSEG Recorder, P. 29-35, (January 2007). Swarbrick, R.E. 2002, Challenges of Porosity-based Pore Pressure Prediction. Technical article: CSEG Recorder, P. 77-77, (September 2002). Young, R.A. and Lepley. T. 2005. Five Things Your Pore Pressure Analyst Won’t Tell You. Technical paper AADE-05-NTCE-12, April (2005).
The Diffusivity Equation for Geopressure Prediction using Well Logs Alberto López Manríquez, Kamy Sepehrnoori
www.elsevier.com/locate/petrol
PII: DOI: Reference:
S0920-4105(14)00332-5 http://dx.doi.org/10.1016/j.petrol.2014.10.009 PETROL2825
To appear in:
Journal of Petroleum Science and Engineering
Received date: 12 December 2013 Revised date: 27 September 2014 Accepted date: 12 October 2014 Cite this article as: Alberto López Manríquez, Kamy Sepehrnoori, The Diffusivity Equation for Geopressure Prediction using Well Logs, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2014.10.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The Diffusivity Equation for Geopressure Prediction using Well Logs Alberto López Manríquez and Kamy Sepehrnoori
Corresponding Author: Alberto Lopez Manriquez Phone number: 1 956 414 3505 e-mail: [email protected]
Abstract
Common practice in industry is to obtain geopressures from indirect methods applied to data obtained from seismic or electrical logs. Because the diffusivity equation describes the pore pressure behavior of a saturated porous media, in this work, it is being used as the basis to attempt a solution to estimate pore pressure in producing formations and above them, in the overburden. Formation properties captured through well logging in offset wells are introduced to our solution of the diffusivity equation as “normalized values” to estimate pore pressures. For a traditional pore pressure assessment, as it is proposed in this work, the problem is set to a time frame. The resulting boundary value problem is solved by applying a “non-strict inverse approach” until a satisfying solution is found. Pressure data from offset wells are used as calibration parameters to obtain the pore pressure distribution along depth of investigation. To prove effectiveness and accuracy of our novel technique, results from real data in conventional and complex geological basins are discussed.
Research highlights
•
This novel method follows accepted theoretical criteria of diffusion transfer phenomena.
•
An analytic approach of solution is being applied rather than empirical.
•
The model is applicable for complex lithological scenarios others than Tertiary formations.
•
Because no Normal Trend of Compaction identification is needed, no graphical method is required.
•
The method is useful for identifying additional sources of abnormal pressures other than disequilibrium compaction.
Keywords: Geopressure prediction; diffusion; diffusivity equation; pore pressure; fracture pressure; consolidation; abnormal pressure; secondary compaction; well logging
1. Introduction
Common practice in industry is to obtain geopressures from methodologies applied to data obtained either prior to drilling from seismics, during drilling from operating parameters, or post drilling from well logs. A vast review covering years’ worth of research using these methodologies is contained in different specialized books, thesis, technical papers, journals, and all literature dedicated to the study in depth of geopressure prediction and estimation. Predicting geopressures from seismics is a subject that has been approached by geophysicists and the theory behind it is covered in specialized literature. Lopez (2013) proposed the prediction (predrill) of geopressures using data from passive magnetotellurics as a complementary technique to reduce uncertainty.
This work is focused on the estimation of geopressures using well log information (post-drill). Harrold (2001) provides an excellent review of literature dedicated to this topic. He proposed a methodology for pore pressure estimation from void ratios and mean effective stress. He suggests that his approach is more consistent with available data in comparison with results obtained from previous methods based on porosity and vertical effective stress. In this sense, it is remarkable to point out that even enormous efforts have been dedicated to estimate pore pressure from well logs; one, if not the most popular approach used in current days is Eaton’s approach. Based on the theory of consolidation proposed by Terzaghi, Eaton (1975) proposed a series of empirical equations to estimate geopressures from well logs. The simplicity of these equations has been the key factor to make Eaton’s equations one of the most recurred methods. To the best of our knowledge, this set of equations proposed by Eaton is still the basis for software used in petroleum industry to perform geopressures assessments when well logs are available.
In this work, a new set of equations is proposed to estimate geopressures from well logs. The diffusivity equation is used as a means to attempt a solution to estimate geopressures. The significance of these equations rests on the premise that pore pressure distribution in a porous medium obeys diffusivity principles. In this sense, the behavior of pore pressure all along the depth of investigation can be represented by the solution of the diffusivity equation. Formation properties captured through well logging are introduced to the solution of the diffusivity equation to estimate formation pore and fracture pressures. Time-dependent behavior of pressure can be modeled using this approach, which can be a valuable tool when identifying lateral drainage or pressuring, or identifying natural depletion of pay zones. Formation fracture and overburden pressures are then estimated using Eaton’s relationships. A simple but reliable equation proposed by Christman (1973) is used to estimate fracture pressure gradients for deep water wells in offshore scenarios. The results of this new proposed methodology are valuable for research, interpretation, correlation, and identification of abnormal pressure zones for exploration, drilling, completion, and production purposes.
2. Geopressures detection techniques
There exists a series of methods to estimate geopressures before, while, and after drilling a well. Regional geology and geophysical methods are widely used to predict geopressures before drilling. Seismics methods to attempt to reconstruct the behavior of compaction trend during burial are commonly used in industry and permit the prediction of formation pressure. To evaluate pressure quantitatively during drilling, Mouchet and Mitchell (1989) and Fertl (1976), among other authors, provide a list of methods for this purpose, such as formation tests, seismic interval velocities, drilling rate (“d” exponent), shale density, gas shows, kicks/mud losses, and wireline logs (resistivity/conductivity, sonic, density). The following section highlights the basics of Eaton’s technique, a postdrill method to estimate geopressures.
2.1 Effective stress transformation model
Based on Terzaghi’s stress relationship principle, expressed in Equation 1, Eaton (1975) developed a set of empirical equations to estimate geopressures from well logs data. = −
(1)
The empirical equations to estimate formation pore pressure from sonic and resistivity well logs formulated by Eaton represent an effective stress transformation model expressed in the following form:
= − −
= − −
∆
∆
(2a)
(2b)
In his work, Eaton concluded that the accuracy of the results from these equations depends on the following main aspects: the quality of the input data, the interpreter who establishes the normal trend of compaction (NTC), and the correct value of the exponent (α) on the ratio term, which for resistivity data varied from 1.2 to 1.5 and for sonic data is close to 3.0 to the applied data Eaton used for his study. In this respect, Lopez-Solis et al. (2006) found and concluded that Eaton’s exponent, α, is a factor of calibration which can be adjusted according to particular conditions to match results. They found that for a specific area offshore Gulf of Mexico fields, Eaton’s exponent (α) takes a value of 0.4 to match calculated to observed values in the field.
2.2 Normal Trend of Compaction
The normal trend of compaction (NTC) represents the optimum fitted linear trend of measured data (velocity or resistivity) in the low permeable formations within the transition zone. Shaker (2007) states that pore pressure profile is divided into three segments as illustrated in Figure 1. They are as follows: 1) the shallow section is unconfined and liberating water. 2) the middle hydrodynamic section, named the transition zone, where compaction disequilibrium is active between the lower confined pressure zones and the unconfined upper sections. A dewatering process from sediments due to pressure gradient occurs in this transition zone. Finally, 3) in the lowest segment, below the top of geopressured zone, dewatering process does not respond to normal compaction, creating pore pressure to deviate from normal hydrostatic pressure.
Figure 1. Three generic subsurface zones for establishing a NTC trend. (from Shaker 2007).
Many authors, Lopez-Solis et al. (2006), Shaker (2007), Young and Lepley (2005), and Swarbrick (2002), among others, have written about the significance of the correct fitting of the NTC on the final results of pressure analysis. Young and Lepley (2005) states that geopressures analyses based on multiple NTC should be a matter of discussion; even in areas where drilling penetrates rocks of different geological Eras. Additional complications arise where shale/sand sequences are encountered. Besides, he states that NTC should be fitted to resistivity values obtained only from intervals containing shale.
2.3 Other approaches
The benefits of Eaton’s approach have been recognized, but at the same time the limitations of this approach have been highlighted. Since Eaton’s approach comes from the theory of consolidation, it is assumed that pore pressure estimation does not reflect sources of abnormal pressures, such as fluid expansion, hydrocarbon generation, dynamic lateral transfer, thermal, chemical/ secondary compaction, or diagenesis other than disequilibrium compaction. Based on this concept, researchers have been focusing efforts in finding ways to introduce all possible sources of pore pressure and overburden changes to modern models. Harrold (2001) proposed a methodology for pore pressure estimation from void ratios and mean effective stress. His equation has the following form:
≈ ( + − − )
(3)
He suggests that this approach is more consistent with his available data than with previous methods based on porosity and vertical effective stress. Additionally, he claims that his method can detect overpressure generated by mechanisms other than disequilibrium compaction.
Miller proposed an alternative way of computing effective stress in Terzaghi’s equation 1. His equation is based on the properties of sound when transmitted through a porous medium. He states that the vertical effective stress can be obtained from:
= − ( − ! )" # % &'((
(4)
Replacing the effective stress, σeff , into Terzaghi’s equation 1, pore pressure can be computed as follows:
= − ,-
− ! . − ./
(5)
Bowers’ approximation is another equation widely used in pore pressure prediction and is useful when sonic properties of formations are available. This particular equation is recognized to be useful when loading/unloading pore pressure mechanisms are anticipated.
Even Eaton’s approach, based on an empirical formula to estimate pore pressure, is still widely used until present days in geopressure assessments when wireline logs are available.
3. The Diffusivity Equation
Literature review on the subject leads to interesting solutions of the diffusivity equation using several techniques for particular conditions. It is not the intention in this work to elaborate on the subject, but to present solutions obtained by the method of separation of variables related to this topic.
Charlez (1991 & 1997) describes that diffusivity comes from three different sources: hydraulic, mass, and thermal sources. An expression of diffusivity is obtained when Darcy’s law is introduced to a mass balance. Assuming the system is under equilibrium and mass diffusion through a porous medium and temperature remaining at equilibrium allow simplifying the diffusivity equation to the following partial differential equation:
0 2
1 2
−3
2455 2
6
= ∇9 7
(6)
Specific loading paths allow rearranging equation (6) to other expressions more commonly used in reservoir engineering applications such as
:
2 2
=
6
;<=( >=? @
∇9
(7)
where ∇9 is the Laplacian operator of P, and for a Cartesian coordinate system, we have
∇9 =
2A 2B A
+
2A 2 A
+
2A 2C A
(8)
In a one-dimensional medium, the diffusivity equation can be expressed as follows:
2 2
=D
EA 2C A
(9)
Solving equation (9) by separation of variables allows a special solution that can be written as a linear combination of the product of functions of the individual variables. The general solution of equation (9) is expressed as follows:
(F, H) = I " #J K cos FOD: ⁄P 9 + R ST- FOD: ⁄P 9
(10) (Solution found in Kreyszig 1999 and Olver 2013)
where FOD: ⁄P 9 = -U and D = (-UP ⁄F)9 /:. P9 =
6
;<=( >=? @
and µ coefficients represent rock and fluid formation properties. Cf and Cr are formation fluid and rock
compressibilities respectively, Cr being stress path dependent.
This solution represents the pore pressure behavior of a saturated porous media, A, B, and Cn being constants depending on the boundary and initial conditions existent in the pore pressure medium.
Geopressures estimation analysis requires knowledge of the boundary and initial conditions. However, in a problem like this, either at a well location or at basin level, boundary conditions and initial condition are hardly known.
In this respect, particular solutions to the diffusion equation (9), when different boundary conditions (BC) and initial condition (IC) are applied, can be found in literature. A brief description of them is presented below. A more detailed explanation can be found in Kreyszig (1999) and Olver (2013) as many other references.
3.1 Solution of PDE (9) with Dirichlet non-homogeneous BC
P(0,t)=P1 ,
P(L,t)=P2 ,
P(z,0)=f(z)
Imposing these BC implies fixed predefined pressure P values at z=0 and z=L
The particular solution of (9) is as follows: (F, H) = 0 +
A #V W
F + ∑^0 Y "Z[ −
\A ]A WA
ST-
]C W
where 9
W
Y = _a (F)STW
]C W
`F
(11)
3.2 Solution of PDE (9) with Dirichlet homogeneous BC
P(0,t)=0 ,
P(L,t)=0 ,
P(z,0)=f(z)
Dirichlet homogeneous BC represent fixed pressures equal to zero at both ends. This is a particular condition of equation (11) resulting in:
(F, H) = b Y "Z[ c− ^0
d-9 U 9 H -UF f ST-
e9 e
where Y =
2 W -UF h (F)ST`F e a e (12)
It is interesting to note that from these particular solutions, where Dirichlet conditions are imposed, P(z,t) depends only on the sin (z) term. The cos (z) portion of the solution vanishes.
Aside, an interesting particular solution to a heat transfer problem where the interval z from 0 to L is assumed as a semi-infinite interval was found in Olver (2013). By analogy, we replace pressure instead of temperature. In this way, it is assumed pressure inside the earth depends only on depth and time. It is also assumed that pressure at surface conditions z=0 behaves according to the following arbitrary expression:
(13)
P(0,t)= A cos wt
At large depth, P(L,t) is assumed unvarying, and hence tends to a mean pressure, P(L,t)→i .
The resulting solution to these BC is as follows:
(F, H) = K "
#C(
j ) Ak
cos clH − Fml.2df
(14)
This equation (14) resembles the solution presented by Francesco (2008) to Terzaghi’s consolidation equation. It is interesting to point out that these solutions (14 and 15) were obtained assuming different approaches. Francesco’s expression is as follows:
∆(F, H) = ∆" #6C cos (lH − nF)
(15)
∆ defined as the variation of pore pressure.
3.3 Solution of PDE (9) with Neumann homogeneous BC
Px(0,t)=0 ,
Px(L,t)=0 ,
P(z,0)=f(z)
When Neumann BC are applied, it is assumed that no pressure exchange occurs at the boundaries z=0 and z=L.
The solution of equation (9) is given by
(F, H) = b o "Z[ c− ^0
d-9 U 9 H -UF f PpS
e9 e
where o =
2 W -UF h (F)PpS `F e a e (16)
It is interesting to note that the particular BC imposed to the problem defines P(z,t) depending only on the cos(z) term while the sin (z) part of the solution vanishes.
3.4 Solution of PDE (9) with Robin BC at one end
Px(0,t)=0 , P(z,0)=f(z)
Px(L,t)=-δ P(L,t) ,
When a Robin BC is imposed at z=L, δ being positive; then it states that pressure at this boundary is being taken out of the system at the same amount pressure is maintained. Dirichlet BC is imposed at z=0.
The solution of (9) is as follows:
(F, H) = ∑^0 "Z[(−D H) ST-(l F)
(17)
4. Modeling Pore Pressure
As stated in the previous section, in a geopressure estimation problem, boundary and initial conditions are hardly known. Therefore, a “non-strict inverse approach” to the solution of the problem is proposed using equation (10). Data obtained from well logs capture and contain rock and fluid formation properties, data from sonic, resistivity, and/or density logs are introduced into the equation as “normalized values”. For instance, sonic properties are introduced as follows:
∆Hq /∆H
st
= OD: ⁄P 9
(18)
As stated previously, FOD: ⁄P 9 = -U and D = (-UP ⁄F)9 /:; being P 9 =
6
;<=( >=? @
and µ coefficients representing rock and fluid
formation properties. Cf and Cr are formation fluid and rock compressibilities respectively.
Applying (18) into equation (10), the following expression is obtained:
F ∆Hq F ∆Hq (F, H) = I " #J cK cos c ∗ f + R ST- c ∗ ff e ∆H st e ∆H st (19)
This solution represents the pore pressure behavior of a saturated porous media for the boundary conditions existent in the porous medium analyzed. Similar expressions can be obtained when resistivity and/or density well log information is analyzed. Accuracy of results from using different source of formation data will vary according to the resolution and the behavior of data applied.
Initially, a reasonable first approximation of the values of constants A and B associated with the general solution is introduced to equation (19). For a traditional pore pressure assessment, as proposed in this work, the problem is set to a time frame t=0. When
pressure data from offset wells are available, they are used as calibration parameters to find A and B values satisfying a P(z,t) solution accordingly. In this way, the particular solution of P(z,t), equation (19), can be computed all along the depth of investigation (depth of the well).
The proposed equation (19) captures the cosine and sine behavior of the solution of the diffusivity equation and the boundary conditions result in a periodic eigenvalue problem. Therefore, it is suggested that weights of A and B values on the solution could help identify boundary effects taking place. These A and B values have been encountered to be between 0 and 1.
o =
2 W -UF h (F)PpS `F e a e
Y =
2 W -UF h (F)ST`F e a e
5. Modeling Fracture Pressure
Fertl (1976) and Mouchet and Mitchell (1989), among others, provide a detailed description of the different prevailing methods for the prediction of fracture pressure gradients. Once pore pressure has been estimated using our proposed methodology, fracture pressure gradient determination implemented in this model is based on the general relationship between pore pressure and stress given by equation (20). This relationship allows taking into account variable pore pressure and variable vertical effective stress related to depth by introducing Poisson’s ratio into the calculations, assuming formations to be elastic (Eaton’s 1969). This set of conditions provides the connection between pore pressure and fracture pressure estimation. When including these parameters, existing in-situ stresses are being taken into account in calculations. For tectonically relaxed basins, normal faulting, the minimum horizontal stress is horizontal; therefore, overburden represents vertical stress. In this sense, a variable overburden stress gradient is taken into account. This particular characteristic of modeling constrains even more finding a particular solution to P(z,t) since this must satisfy both: fracture and pore pressure calibration data available at the same time.
vw =
x
0#x
(yw − w) + w
(20)
When density log information is available, the model allows for the inclusion of these data representing the overburden gradient.
Geopressures prediction in an offshore scenario requires inclusion of the effect of water depth on the overburden stress. Fracture gradient determination assumes that overburden comes from rock matrix and formation fluid stresses. Offshore, sea water does not have rock matrix; therefore, because fracture gradients depend on overburden, fracture gradients offshore are lower than those computed onshore compared at equivalent depths. This effect becomes significant as water depth increases. To overcome this condition, the concept of stress ratio, introduced by Christman, is implemented in this model to properly estimate overburden pressure gradient offshore using a relationship in the form:
ywz{ ≈ Kw| + }w{z + yw/W
(21)
6. Discussion of Results
This section presents the discussion of results of five different case studies. These discussions are focused first on the comparison of estimated geopressures (pore and fracture pressures) with the calibrated values of pressure obtained from the field (when available). Second, when no calibration data are available, these discussions are established based on comparison of results obtained from the proposed methodology in this work versus results obtained from Eaton’s approach. The cases discussed here are not confined to a specific area of interest; instead, well log data correspond to different geological basins. It is important to note that without data to calibrate or compare, results obtained from any geopressure analysis are only estimates. Harrold (2001) provides a table containing hierarchy of data sources according to its reliability. Table 1 is taken from this reference.
Data Sources Repeat Formation Tester (RFT, MDT) Formation Interval Tester (FIT) Drill Stem Test (DST) Well or pressure kick Connection gas Measuring/Logging While Drilling (MWD/LWD) Mudweight Equivalent depth Drilling exponent (D, Dc)
Hierarchy high
low
Table 1. Sources of pressure data organized by reliability (From Harrold 2001)
6.1 Case Study 1
Wire logs from an offshore well were obtained; sonic and resistivity profiles along depth are shown in Figures 2a and 2b. Figure Fig 2a shows the normal trend of compaction (NTC) line adjusted to the resistivity profile according to Eaton’s method. A smooth slope was imposed based on the known geology of the area, which consists of sedimentary y packages of the same geologic Era (Cenozoic) made up of a silisiclastic sequence alternating with shale (0 (0-418 m) Pleistocene, (418-1924 m) upper er Pliocene, (1924-2229 m) middle Pliocene, (2229-3111 m) lower Pliocene and (3111 (3111-3388 m) upper pper Miocene, without reaching the top of the middle Miocene.
Figure 2a. Resistivity log from wireline showing NTC
Figure 2b. Sonic log from wireline
Figure 3 illustrates the results of geopressures estimated using the proposed equation. Pore pressure is represented by the blue dots dot profile while fracture pressure by the red. Validity of these results is established during superimposition of pore pressure values valu measured in the well using Modular Formation Dynamics Testing (MDT), a direct pore pressure measurement tool. Validity of fracture pressure profile is achieved while superimposing estimated fracture pressure obtained from Leak Off Test (LOT) at 1665 16 m.
Because fracture pressure calculation is pore pressure dependent, this particular approach of modeling imposes additional constraints. The green dots in this figure represent values of pore pressure measured at seven different points along depth. It can be seen that these pore pressure values match along the calculated pore pressure profile except one of them at 2790 m.
Figure 3. Geopressures profile obtained from the proposed equation
Figure 4 illustrates pore pressure values along depth expressed in equivalent density (ED). The light blue straight line in this figure represents the mud density (MW) used to drill the well for the interval. According to the drilling record of the well, water flow was detected from interval 3000 - 3350 m (bottom). Mud density had to be increased to reestablish control of the well. Figure 4 also shows the predicted fracture pressure (red) and overburden (green) profiles. It is remarkable that the predicted values of fracture gradient match the only fracture pressure control point obtained at 1665 m from a LOT, represented for the orange dot. The effect of water depth on fracture gradient and overburden stress is included in modeling. The agreement found between predicted geopressures (pore and fracture pressures) with available measured values in the field provides confidence on the validity of applied methodology.
Figure 4. Geopressures expressed in equivalent density (ED) profile obtained from the proposed equation
Figure 5 illustrates a comparison of results of estimated pore pressures between the proposed equation and Eaton’s equation. Matching is excellent all along depth of interest since the two curves collapse practically to one single behavior. Eaton’s pore pressure is represented by the blue dots profile, while pore pressure obtained using the proposed model by the red. MDT data are included in this figure to express the excellent match of results.
Figure 5. Pore pressure comparison between Eaton’s and Lopez and Sepehrnoori’s equations
6.2 Case Study 2
Figure 6 illustrates the comparison of results between estimated pore pressures using both equations. Again, matching between results is excellent in the lower part of the interval. Some deviations can be seen above 1800 m. The well was drilled through sedimentary rocks of Cenozoic; mainly sand sequences alternating with shale. Dispersion of results by Eaton’s curve above 1800 m is attributed to the shale-sand sequences found in these Tertiary formations drilled in this well. Dispersion of original ∆to data from the NTC imposed during calculation creates this behavior. Young and Lepley (2005) recognize this condition when he states that if an empirical pressure model is used, sands will create pressure regression. He suggests not taking into account these “sand-influences” anomalies in the assessment.
Calibration of these results is given by the green dots in this figure; they represent values of average reservoir pressure (P avg) reported at different horizons (2000, 2200 and 2700 m). It can be seen that these pressure values match the calculated pore pressure
profile along depth giving reliability to the obtained results. Additional information to calibrate results from horizons above 1800 m should be valuable to fully complete this assessment.
Figure 6. Pore pressure comparison between Eaton’s and Lopez and Sepehrnoori’s equations
Figure 7 shows pore (blue) and fracture (red) pressures expressed in ED format profile. The light blue line in this figure represents the mud densities (MW) used to drill a well in the area while the purple dots show reservoir pressures. Drilling mud densities all along depth are also in agreement with the so-called operational window between pore and fracture pressure gradients.
Figure 7. Geopressures expressed in equivalent density (ED); profile obtained from the proposed equation
6.3 Case Study 3
Figure 8, like in the past two cases, shows the comparison of results between estimated pore pressures. Matching between results is excellent along depth of interest except for the bottom part of the section below 3500 m. Validity of results for this case is based on superimposing pore pressure values measured in the well using MDT and fracture pressures estimates from LOT. The green dots in this figure represent values of pore pressure measured at nine different points along depth. No MDT data was taken below 3500 m to validate the curve below this depth.
A possible explanation for the mismatch between curves below 3500 m can be given by describing the stratigraphic column in the area. Above 3500 m, lithology is described as an upper sedimentary package of the same geologic Era from recent Pleistocene to middle Miocene. This section is similar to the geological areas in the Gulf coast, terrigenous tertiary Era sediments, as those where Eaton and Hottman and Johnson based their studies. The lower Miocene (below 3000m) is described by lithic sandstones, limolites (turbiditic fans), and limestone-clay matrix. Below 3200 m rock is characterized with primary and secondary porosity. This dual-
porosity characteristic below 3200 m may be one of the reasons for the difference between the two curves. This material will be discussed in a separate publication.
Figure 8. Pore pressure comparison between Eaton’s and Lopez and Sepehrnoori’s equations
Figure 9 illustrates pore and fracture pressure values along depth expressed in equivalent density. As before, the light blue straight line in this figure represents mud densities (MW). Computed values of fracture gradient (red curve) match at two (1900 and 3500 m) of the four fracture gradient pressure points obtained at 1400, 1900, 2660, and 3500m from LOT represented by the orange dots. Since the well was drilled offshore, the effect of deep water depth on fracture gradient and overburden stress is included in the modeling. No better match could be obtained without compromising the principles of theory previously given in this same work. Assuming the quality of all data from LOT is reliable for this deep water case, certainly it suggests a better fracture gradient estimation at intermediate and deep depths than at shallow depths. At the same time, it is recognized that to qualify the accuracy of the deep water fracture gradient equations used in this modeling, more deep water case studies are needed.
Figure 9. Geopressures expressed in equivalent density (ED) profile obtained from the proposed equation
6.4 Case Study 4
This case study involves a different lithological column. A general description of the area of this case study is given as follows: Upper Cretaceous formations (0-1100 m) composed of alternating sandstones units with presence of mudstones and shales. From 1100-2600 m upper and lower cretaceous horizons consisting of calcareous mudstone with slight presence of shale are encountered. From 2600-2800 m Limestone is found predominantly. This lithology differs from conditions used by Eaton (1975) and Mathews and Kelly (1967) to obtain their equations.
Figure 10 shows geopressure profiles obtained from analyzing the data. It is clearly seen that although both curves follow a similar trend, mismatch between them exists. The green dots represent the value of pressure measured at the depths of 1555 and 1575 m in pay zones in the only near offset well, serving as the only calibration points. The abnormal pressure detected from 2600 to 2700 m could not be corroborated with calibration.
Figure 10. Pore pressure comparison between Eaton’s (One-NTC) and Lopez andSepehrnoori’s equations
It is not the intention of this work to make strict conclusions about findings for this single case, but to open a discussion about the usefulness of the equation proposed to identify sources of overpressure related to the geological environment. An explanation for the disagreement between results can be obtained when reviewing literature. It has been pointed out for some authors (Young and Lepley (2005), Swarbrick (2002), and Shaker (2007)) that in more complex scenarios like this, where rock formations belonging to Eras other than Tertiary or more than one geological Era are drilled, definition of a NTC line is complicated. This condition causes in many occasions that more than one NTC is used by interpreters to match results. When doing that, the principle of the NTC is compromised and the resulting geopressure assessment could be not reliable (Young and Lepley 2005).
To help the discussion, analysis using Eaton’s equation was repeated imposing two different NTC: the first above 1100 m and the second below these depths. Adjusting these two NTC lines, pore pressure profile calculated using Eaton’s equation matched much better the estimated values with our proposed equation (19). Figure 11 depicts this behavior. It can be seen from this figure that the two curves match better than what presented in Figure 10.
To make further conclusions, much more information needs to be collected from the studied field. Particularly, reliable pore and fracture pressures measurements below 2500 m are needed to corroborate the existence of a presumably high pressure zone there. Fracture pressure at other levels than pay zones will help make further conclusions. Otherwise, the discussion is merely speculation. By now, it is important to note that in the case studies 1, 2, and 3, where lithology resembles the conditions covered by Eaton’s equation, the proposed equation 19 outstandingly matches with Eaton’s. On the other hand, when complex lithologies different from those studied by Eaton are encountered, results from both equations deviate. Two NTC had to be imposed to find better matching between results.
Figure 11. Pore pressure comparison between Eaton’s (2NTC) and Lopez and Sepehrnoori’s equations
Figure 12 shows the equivalent density profile obtained. The purple, the light blue, and the orange lines (MW1, MW2, and MW3) represent drilling mud densities for wells drilled in the area expressed as equivalent density. As it can be seen, mud density MW1 (purple line) is mostly in agreement with the pore pressure profile. From 2000 m to the bottom, MW1 increments of density obey pore pressure behavior computed in the confined zone. A mud density (up to 1.8 g/cm3) was used to drill interval 2350 - 2900 m, as illustrated by the orange line (MW3). Based on last observation, and in order to provide additional information to compare and discuss results, it was seen from daily drilling reports that no manifestation of gas or liquids were detected on surface when drilling through interval 2000 to 2900 m in wells (MW1 and MW3) drilled in the area. The well represented by MW2 (light blue line) reached only
1500 m approximately to an upper target. The abnormal pressure detected from 2600 to 2700 m could nnot ot be corroborated at the time of writing this work since no more information from deep wells in the area was available.
Figure 12. Geopressures expressed as equivalent density from resistivity logs
6.5 Case Study 5
An onshore well was drilled to 6480 m planning to explore for hydrocarbons in Cretaceous formations. This area is recognized as a tectonically active region; faulting has also been identified. Tertiary formations were drilled to 5700 m, mainly sequences of o shales and sands. Cretaceous formations ormations belonging to Mesozoic were the target of this well. These formations were encountered from 5700 through the bottom of the well and are characterized as naturally fractured carbonated rocks and dolomite.
Perez et al. (2010) reviewed previous geop geopressure ressure assessments and identified that in the interval 2700 to 3800 m, where a sand/shale sequence is encountered; abnormal low pressured zones were erroneously interpreted and identified. This misinterpretation misinterpr resulted in erroneous pore pressure estimation ion at this stratigraphic level.
Further review using the proposed equation shows that although not mentioned by the authors, between 2000 and 2700 m a high abnormal zone was also originally misinterpreted. Figure 13 (obtained from Perez et al.) shows the comparison of geopressure modeling before and after they reinterpreted lithology and additional information. The profile in the left shows the initial interpretation while the one in the right shows Perez et al. improved interpretation. It can be seen how at the two intervals mentioned, pore pressure (PP) purple curve flattens, no longer showing abnormal high or low pore pressures at these levels.
Figure 13. Comparison of geopressure models before and after reinterpretation (from Perez at al. 2010)
Modeling of data for a well in the area was conducted. A high abnormal pressure section is identified below 4500 m, as shown in Figure 14. No abnormal pressures are estimated above this level. This corroborates Perez et al. findings related to the misinterpretation of results originated by the effect created by the sand and shaly interval drilled (2700 – 3800m). This same sand/shaly effect can be the reason of disparity between results for this particular interval. Eaton’s equation (one NTC) underestimates pore pressure at this interval when comparing against results from the proposed equation. The high pressure zone below 4500 m is perfectly identified by the proposed equation while Eaton’s fails to match the calibration point at 6000 m.
The target formation (6000 m approximately) is located below the high pressured zone and presumably has been naturally depleted from producing wells in the area. It is interesting to remark at this point that tectonics events in the region may cause lateral drainage along faults. It is also important to mention that below 5700 m Cretaceous naturally fractured carbonated and dolomitized formations have been identified. This particular condition could be one of the reasons why in this segment results from the two models deviate. Average reservoir pressure has been reported for some wells in the area. For an offset well, the original reservoir pressure reported was 11800 psi. Since reservoir pressure could have experienced natural depletion, reported data have to be timely according to the date wireline logs were taken to make comparison time-frame agreeable. Average reservoir pressure reported (Perez et al. 2010) is 11000 psi at an average depth level of 6000m. Figure 14 shows this data with a green dot which serves as a calibration point.
Figure 14. Comparison of pore pressure profiles between Eaton’s and the proposed equation
Figure 15 illustrates pore (red dots) and fracture pressure (blue dots) values along depth expressed in equivalent density (ED) (E format profile. These results come from the proposed equation. The light blue and purple lines represent mud density (MW1 and MW2). 2). Both computed geopressures gradients (pore and fracture) are in agreement with the geopressure “reinterpreted” model presented by Perez et al. To visualize this statement, results in Figure 15 should be compared with those seen in Figure 13 (right). ( On the other hand, Eaton’s equation using a single NTC could not match results obtained either by the proposed equation or by Perez Pe et al. reinterpreted. Based on these results, the proposed equation could be a valuable tool to analyze complex geological scenarios sce others than Tertiary formations.
Figure 15. Geopressures expressed as equivalent density
It is important to mention that modeling using Eaton’s equation with multiple NTC could cause misinterpretation of pore pressure. press
nesses of the proposed equation 7. Strengths and weaknesses
Strengths •
The proposed methodology is simple; it requires conventional information delivered by either sonic or resistivity logs.
•
The proposed methodology follows accepted theoretical criteria of diffusion transfer phenomen phenomenaa and an analytic method of solution rather than an empirical approach.
•
It is useful for application in complex lithological scenarios others than Tertiary formations (e.g. cases 3, 4 and 5). In these th cases, the proposed equation showed better agreement w with existent data.
•
It is well known that to fit results in complex scenarios, Eaton’s method requires more than one NTC lines imposed arbitrarily arbitraril by the interpreter. This practice has been identified by some authors as an action that compromises the effective effect stress transformation model proposed by Eaton.
•
Deviation between results, when comparing Eaton’s versus the proposed pore pressure solution, has been identified in formations formatio recognized with dual porosity. It is suggested that the proposed equation sho should uld be used for pore pressure estimation in geological scenarios under these conditions. It could be a valuable tool to identify sources of abnormal pressures such as secondary compaction or diagenesis encountered in naturally fractured formations.
•
It was shown that particular BCs applied define the weight of cosine and sine functions in the general solution equation. Particular solutions shown in the review section of this work have either cosine or sine functions but not both. The proposed equation (19) captures the cosine and sine behavior of the solution of the diffusivity equation. Therefore, it is suggested that weights of A and B values on the solution could help identify boundary effects taking place. For the presented cases, A values varied from 0.1 to 0.2 and B values from 0.6 to 0.9. There is no need for NTC identification, which eliminates the issue of defining the NTC’s slope by interpreter, who in many cases uses more than one arbitrary NTC lines to match results with calibrating data. This common practice is against the premise stated by consolidation theory briefly discussed in this work.
•
Because no NTC is needed, no graphic method is required.
Weaknesses •
Only few data for case studies were available at the time of this work.
•
More data from the field are needed to find trends of A and B calibration parameters according to the geological conditions of studied basins.
•
Like any indirect method, accuracy of results highly depends on the quality of data and the expertise of interpreter.
•
Much more analysis is required to further conclude some of the observations discussed here.
8. Conclusions
This work describes a self-consistent methodology for calculating pore and fracture pressures using information from well logs. As with any indirect method, the results of this methodology are interpretive. As is with wireline logs or seismic data, they do not measure pressure directly independent of how data are processed.
As it was described in the “strengths and weaknesses” section, the proposed equation satisfies conventional and unconventional geological scenarios for estimating pore and fracture pressures. For this reason, it is considered as a valuable tool to identify sources of abnormal pressures, such as secondary compaction or diagenesis encountered in naturally fractured formations.
The proposed set of equations presented in this work could be valuable for the researcher who is looking for a way to interpret the different mechanisms involved in abnormal geopressure origins and sources. It is also useful for the explorer looking for a general description of pressure profile in areas other than Tertiary and for the engineer in charge of planning drilling and completion.
9. Nomenclature
A, B, Cn
= Constants
AGRF
= Air gradient from rig floor, psi/ft
Cf
= fluid compressibility, 1/psi
Cr
= pore volume compressibility, 1/psi
D
= total depth, m, (ft)
exp (e)
= base of natural logarithm
FPG
= fracture pressure gradient
f
= function
fM
= Miller’s fitting parameter
h
= representative depth, m, (ft)
k
= formation permeability, mDarcies
L
= total depth of investigation, m
OG
= overburden gradient, psi/ft
OGDW
= overburden gradient deep water, psi,ft
OGML
= overburden gradient mud line, psi,ft
PG
= pore pressure gradient, psi/ft
P
= Formation pore pressure, psi
Phyd
= Hydrostatic pressure, psi
Ro
= resistivity observed, ohm m
Rn
= resistivity on normal trend line, ohm m
S
= Overburden total stress, psi
t
= time, sec (min)
V
= speed of sound in shale, ft/µsec
Vma
= speed of sound in the rock matrix, ft/µsec
Vml
= mud line velocity (speed of sound in liquid), ft/µsec
WGWD
=Water gradient water depth, psi/ft
z
= depth of interest, m, (ft)
α
= Eaton’s exponent, dimensionless
β
= thermal expansion coefficient, dimensionless
∆P
= Pore pressure variation, psi
∆tavg
= Sonic Average Interval Travel Time, µsec/ft
∆tn
= Sonic ITT on normal trend line (µsec/ft)
∆to
= Sonic Interval Travel Time (ITT),µsec/ft
~ 9
= Lapacian operator
εkk
= Volumetric strain, dimensionless
φ
= porosity, fraction
η
= Biot's modulus, psi
λ
= eigenvalue
µ
= fluid viscosity, cP
σeff
= rock matrix effective stress, psi
σm
= mean effective stress, psi
υ
= Poisson’s ratio, dimensionless
10. References
Charlez, P.A. 1991, Rock Mechanics. Vol. 1 Theoretical Fundamentals, Editions Technip, Paris. Charlez, P.A. 1997, Rock Mechanics. Vol. 2 Petroleum Applications, Editions Technip, Paris. Christman, S.A. 1973, Offshore Fracture Gradients, JPT 4133-PA, SPE Vol 25, Number 8, Pag. 910-914. Eaton, B.A. 1975, The Equation for Geopressure Prediction from well logs. paper SPE 5544, 50th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, Dallas, TX, Sept 28 – Oct 1, 1975. Fertl, W.H. 1976. Abnormal Formation Pressures, Elsevier Scientific Publishing Company. Francesco, R.D. 2008, Exact Solution to Terzaghi’s Consolidation Equation, GEO&GEO Instruments, research & development, Teramo, Italy. Harrold, T. 2001. Porosity and effective stress relationships in mudrocks, Durham theses, Durham University, http://etheses.dur.ac.uk/4338 (accessed April 23, 2013). Kreyszig, E. 1999. Advanced Engineering Mathematics, 8th edition, John Wiley & Sons, Inc. USA. Lopez, A. 2013. Geopressure prediction from Passive Magnetotellurics. Paper SPE 166356-MS, presented at the ATCE 2013 held in New Orleans, Louisiana, USA 30 September – 2 October 2013. Lopez-Solis, V., Velazquez-Cruz, D., Jardinez-Tena, A., and Castaneda, G. 2006. Normal Resistivity Trends for Geopressure Analysis in Mexican Offshore Wells. Technical paper OTC 18192, May (2006). Miller, T (unknown year). Miller Pore Pressure algorithm, Knowledge Systems publication (accessed May 6, 2013). Mouchet, J.P. and Mitchell, A. 1989. Abnormal Pressures while Drilling. Societe Nationale Elf Aquitaine.
Olver, P.J. 2013, Introduction to Partial Differential Equations, Chapter 4, Lecture notes. Perez, C., Mandujano, H., Puentes, F., Isacci J., Marino, L., and Obediente, R. 2010. Drilling Optimization of Wells from Crater Field: Decreasing Drilling Times and Operational Risks. Paper SPE 139035 presented at the SPE Latin American & Caribbean Petroleum Engineering Conference held in Lima, Peru, 1-3 December 2010. Shaker, S. 2007, Calibration of Geopressure Predictions using the Normal Compaction Trend: Perception and Pitfall, Technical article: CSEG Recorder, P. 29-35, (January 2007). Swarbrick, R.E. 2002, Challenges of Porosity-based Pore Pressure Prediction. Technical article: CSEG Recorder, P. 77-77, (September 2002). Young, R.A. and Lepley. T. 2005. Five Things Your Pore Pressure Analyst Won’t Tell You. Technical paper AADE-05-NTCE-12, April (2005).