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Rock Mechanics for Natural Resources and Infrastructure 
SBMR 2014 – ISRM Specialized Conference 09-13 September, Goiania, Brazil 
© CBMR/ABMS and ISRM, 2014 
 
SBMR 2014 
Mathematical Model to Quantify the Contribution of Thermal 
Stresses in Pore Pressure, Additional to the Compaction Effect 
 
Diego A. Vargas 
Corp. NATFRAC-Bucaramanga-Colombia- diegoavargass@hotmail.com diego.vargas4@correo.uis.edu.co 
 
Zuly H. Calderón 
Universidad Industrial de Santander, Escuela de Ingenieria de Petróleos- Bucaramanga-Colombia- calderon@uis.edu.co 
 
Darwin C. Mateus 
ECOPETROL- Piedecuesta – Colombia- darwin.mateus@ecopetrol.com.co 
 
Reinel Corzo 
ECOPETROL- Piedecuesta – Colombia- reinel.corzo@ecopetrol.com.co 
 
Oscar J Acevedo 
UT-PEXLAB- Piedecuesta – Colombia - osjavi1789@gmail.com 
 
SUMMARY: Pore pressure is one of the most critical variables on the geomechanical model 
design, and upon this variable depends largely the successful drilling of oil wells. This pressure is 
calculated conventionally considering the mechanical stresses, essentially the disequilibrium for 
compaction, using empirical correlations developed from information taken mainly from the Gulf of 
Mexico. The use of these equations, does not reflect real situations most of the time, giving pore 
pressure values inferior to the formation pressures, which may complicate the drilling operations, 
especially on exploratory wells, increasing the nonproductive times (NPT). The main objective of 
this investigation is to implement a mathematical model which includes the most common 
overpressure causes documented in literature, and which allow to quantify the effects of: the under 
compaction and the thermal stresses due to the water expansion, the kerogen cracking and the oil 
in shale formations. In order to include the effect of the water expansion, the geothermal gradient 
and a sedimentation history of a Colombian basin were taken into account. For the hydrocarbons 
generation, an organic material maturation model was applied, to determine the oil and gas fraction 
generated. This model was applied to a Colombian basin and providing as a result, a pressure 
profile quantifying the effect of each mechanism mentioned above.The results obtained with the 
model shows coherence with the events reported on the study area wells. Similarly, it could be 
evidenced that, despite the compaction is the main cause of overpressure in depths no deeper than 
3000 meters, at greater depths, the thermal stresses contribution may be up to 14% of the total 
overpressure. For this reason, the proposed model allows to decrease the uncertainty on the pore 
pressure model. Additionally, the pressure model was simulated with the software PetroMod with 
the objective of including the boundary conditions, along with the horizontal and vertical flow 
conditions across the permeable layer, increasing the representativeness of the pore pressure model. 
 
KEYWORDS: Pore Pressure, Thermal Stresses, Finite Differences. 
 
 
 
 
 
 
SBMR 2014 
1 INTRODUCTION 
 
Pore pressure studies on sedimentary basins 
turned out to be the last decades one of the 
most critical items on the planning of drilling 
projects in the oil and gas industry. The 
conventional methodologies used in the pore 
pressure estimation are based on the 
disequilibrium for compaction as the main 
mechanism of overpressure generation, such 
mentioned by the correlations proposed by 
Hottmann and Johnson (1965) and Eaton 
(1975). In order to consider fluids expansion , 
one of the most popular methodologies was 
proposed by Bowers (1995). 
 Since the correlations stated above may give 
values lower than the real ones , the estimation 
of the pore pressure by the mathematical model 
implementations which structure depend upon 
the analyzed variables, merge as an alternative. 
On this investigation the thermal stresses 
will be taken into account (fluids expansion and 
hydrocarbon generation Grauls, (1999)). The 
aquathermal effect is represented by the term 
proposed by Lou and Vasseur (1992). 
For the hydrocarbon generation terms, the 
maturation model proposed by Tissot, B. and D. 
H. Welte, (1984), will be considered. This 
model allows estimating the kerogen fraction 
which will become oil and then the gas kerogen 
fraction. With this fraction and the proposed 
terms for Hantschel and Kauerauf (2009), an 
estimation of the hydrocarbon generation 
influence may be done. 
The implementation of these terms, by the 
construction of a differential model used in one-
dimension basins modeling will be discussed 
further in this article 
 For the estimation of the pore pressure from 
this model, the more representative variables 
were analyzed, such as the compressibility, 
porosity, permeability and viscosity on the 
under compaction term, the temperature on the 
aquatermal effect term and the organic matter 
content with their respective mature grading, for 
the term which represents the hydrocarbon 
generation effect. The model was solved and it 
obtained estimating the overpressure 
corresponding to each one of the studied terms. 
 
2 CAUSES OF OVERPRESSURE 
 
The causes of overpressure has been studied 
and classified according to some authors 
(Osborne and Swarbrick, (1997) and Grauls, 
(1999)). 
On this investigation, the under compaction, 
water expansion and hydrocarbon generation 
effects were taken into account. 
 
2.1 Mechanical Stresses 
 
The vertical stress is the main cause of 
overpressure generation Grauls, (1999), 
although this may be dissipated, if great 
volumes are conducted along the faulting planes 
Oscorne and Swarbrick, (1997). 
 The under compaction occurs when the 
deposition rate of the sediments is highly 
superior to the fluids expulsion rate, which may 
be improved for permeability reductions, 
especially on shales. 
 
2.2 Thermal Stresses 
 
According to Barker, (1972), the water 
expansion consists on the increase on the fluids 
volume in the pore due to temperature changes. 
This author shows the importance ofthe hydro-
thermal effect by making a graphic analysis of 
water behavior in a closed system. On the other 
hand, Luo and Vasseur (1992) shows through a 
mathematical model that the contribution of this 
term is minimum, even under favorable 
geological conditions. 
 The hydrocarbon generation effect appears 
with the oil generation window at 200°F (93°C) 
and the vitrinite reflectance (Ro) higher than 
0.7% Spcencer, (1987). 
 The conditions for gas generation are a Ro 
higher than 2% and temperatures above 340°F 
(175 °C) Hedberg, (1974). The contribution of 
this mechanism, as well the oil mechanism, 
depends upon the TOC (Total Organic Carbon) 
and the kerogen type present. 
 
3. PORE PRESSURE ESTIMATION BY A 
MATHEMATICAL MODEL 
 
The Terzaghi’s (1923) theory of effective stress, 
 
 
 
 
SBMR 2014 
can be used for the developing of an 
overpressure model in one-dimension.These 
variables are calculated as a function of the 
depth and the temperature. 
Applying the Terzaghi’s law (Equation 1) 
and Darcy’s law (Equation 2), this can derive an 
equation used to model the basin (Equation 3). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Luo and Vasseur, 1992, based on Terzaghi’s 
previous work, performeda mass balance for 
the fluid and for the rock material, obtaining a 
mathematical model which allows calculating 
the pore pressure by the basins modeling 1D 
(Equation 4). 
 
 
 
 (
 
 
) (
 
 
 
 
 
) 
 
 
 (
 
 
) 
 
Based on this model, other terms were 
introduced, to take into account the causes 
undar analysis in this study 
 The previous model depends upon variables 
such as: porosity ( ), permeability ( ), 
viscosity ( ) and compressibility ( ), which 
can be calculated as a function both of depth 
and temperature. Besides, the compressibility 
depends on the compaction coefficient (α). 
 
 
 The porosity was determined by Athy’s 
law (1930) as a function of depth. 
 
 
 
 
 The permeability is a function of the 
porosity, as proposed by Terzaghi 
(1925). When the λ parameter varies 
between 10
-2
 – 10-7. The equation 
representing this formulation is detailed 
below. 
 
 
 
 For the viscosity, Mercer’s model 
(1975) was used,a correlation as a 
function of temperature for ranges 
between 0-300°C, as shown below, was 
used. 
 
 
 
 
 (7) 
 
 
 
 (8) 
 
 The compressibility of the rock was 
determined by Athy’s law (1930). 
 
 
 
 
 (9) 
 
This one depends on the compaction 
coefficient (α) and for its calculation, a 
porosity profile must be determined 
from sonic logs and afterwards the 
Athy’s law must be applied (Figure 1). 
With this law (Equation 5), an 
exponential tendency is generated and 
this parameter is calculated. 
 
 
Figure 1. Calculation of the compaction coefficient. 
 
The equation 4 was solved using finite 
differences and the models were implemented 
for each one of the variables mentioned above. 
Finally, a pore pressure profile for compaction 
effect is obtained, as shown in figure 2. 
In this figure, by comparing the results 
obtained with the compaction method with the 
mud weight used during drilling, a significantly 
low value is observed. For that reason, the other 
overpressure causes must be added y the model 
0
1000
2000
3000
4000
5000
0 0.2 0.4 0.6 0.8
POROSITY ( ) 
Porosity Athy 1930
D
EP
TH
 (
m
) 
 
 
 
 
SBMR 2014 
 
Figure 2. Pore Pressure profile for Compaction 
 
The next step was to add the thermal 
expansion effect term, studied by Luo and 
Vasseur (1992), who add a term ( 
 
 
) to the 
equation 4, as seen in equation 11. This 
additional term stands for the temperature 
variation as a function of time and a constant 
(water expansion coefficient). 
 
 
 
 
 
 
 
 
(
 
 
) (
 
 
 
 
 
) 
 
 
(
 
 
) 
 
 
 
 
In order to quantify this contribution, a 
variation model of depth as a function of time 
(sedimentation history) and a local geothermal 
gradient of 30°C/Km were used. Having the 
behaviors previously stated, the variation of 
temperature as a function of time could be 
shown (Figure 3). 
 
 
Figure 3. Variation of Temperature in function of time 
After that, the equation 11 was solved and 
the pore pressure profile including water 
expansion was obtained, as shown on figure 4. 
After quantifying the water expansion effect, 
a very irrelevant contribution can be observed 
(approximately 3%), and therefore, the pore 
pressure profile is still highly below to the mud 
weight required to prevent influx 
 
 
Figure 4. Pore Pressure Profile for Compaction and 
Water Expansion (blue color). 
 
The next term to be evaluatedwas the 
hydrocarbon generation effect. For this term, an 
organic matter maturation model proposed by 
Tissot, B. and D. H. Welte, (1984), was used, 
which determines the fraction of kerogen 
becoming oil and subsequently gas (figure 5). 
 
 
Figure 5. Fraction of hydrocarbons generated 
0
1000
2000
3000
4000
5000
0 20 40 60 80 100 120
PRESSURE (MPa) 
Lithostatic Hidrostatic
Compaction Mud Weight
D
EPD
EP
TH
 (
m
) 
0
30
60
90
120
150
0 50 100 150
TIME (Ma) 
TE
M
P
ER
A
TU
R
E 
0
1000
2000
3000
4000
5000
0 20 40 60 80 100 120
PRESSURE (MPa) 
Lithostatic Hidrostatic
Compaction Mud Weight
Water Expansión
D
EP
D
EP
TH
 (
m
) 
0
1000
2000
3000
4000
5000
0 0.2 0.4 0.6 0.8 1
HYDROCARBONS 
Oil Fraction Gas Fraction
D
EP
TH
 (
m
) 
 
 
 
 
SBMR 2014 
As seen in the previous figure, the 
generation of oil starts at shallow depths (1000 
m) because there is no need of high 
temperatures for primary cracking to occur 
(93°C) Spencer (1987). The gas generation 
coming from secondary cracking needs high 
temperatures, for that reason, their generation 
start at deeper depths. 
Hantschel and Kauerauf, (2009) in order to 
quantify the effect of the oil and gas generation, 
proposed the terms shown in equations 12 and 
13, respectively. 
 
(
 
 
 
 
 
)
 
 
 (12) 
 
(
 
 
 
 
 
)
 
 
 (13) 
 
Adding the previous terms into the equation 
11, the equation 14 is obtained. 
 
 
 
(
 
 
) (
 
 
 
 
 
) 
 
 
(
 
 
) 
 
 
 
 (
 
 
 
 
 
)
 
 
 (
 
 
 
 
 
)
 
 
 
 
The previous model was solved, and a pore 
pressure profile was obtained, which includes 
the compaction effect and the thermal effects, 
as shown in figure 6. 
 
 
Figure 6. Pore Pressure Profile for compaction and 
thermal stresses (orange color). 
 
Analyzing the previous profiles for each one 
of the mechanisms, it is observed that at 
shallow depths (3000 m), the thermal stresses 
represents only the 8% of the total pore pressure 
(Figure 7A), but below that, the contribution for 
these stresses rises significantly, reaching 
values such as 14% for higher depths (4500 m). 
(figure 7B). 
 
 
 
 
 
 Water Expansion Effect hydrocarbon 
 Compaction 
 
 
Figure 7. Contribution of each mechanism A) 2000 m y 
B) 4500 m 
 
The mathematical model used in this work was 
implemented by Vargas (2013) where Vargas 
got pore pressure values for each mechanism. 
 
4. MODELING OF PORE PRESSURE BY 
PETROMOD SOFTWARE 
 
The pore pressure calculation was performed 
using PetroMod software, used in basins 
modeling, and pressure profiles including the 
same overpressure mechanisms studied were 
obtained. Furthermore, the software takes into 
account the pressure contribution, generated for 
the lateral transferences. The obtained results 
were calibrated with well data. 
 Figure 8 shows the calibration of the model 
with geochemical properties data, such as 
vitrinite reflectance, which responds the contour 
conditions of the model (heat flow, sediment-
water interface temperature SWIT). 
The models could be calibrated with 
petrophysical data, temperature or pressure 
data, the same way with the vitrinitereflectance. An example of porosity calibration 
0
1000
2000
3000
4000
5000
0 20 40 60 80 100 120
PRESSURE (MPa) 
Lithostatic Hidrostatic
Compaction Mud Weight
Water Expansión Effect hydrocarbon
D
EP
D
EP
TH
 (
m
) 
A 
B 
92% 
3% 
5% A 
86% 
4% 
10% B 
 
 
 
 
SBMR 2014 
using the Athy’s law (1930) is shown in the 
figure 9. 
 
 
 
 
 
 
 
Figure 8. Calibration of the vitrinite reflectance 
 
 
 
Figure 9. Porosity calibration for a formation. 
 
 The pressure distribution obtained 
numericallyof the two-dimensional model, 
using the same mechanisms implemented on the 
mathematical model is shown on the figure 10, 
which shows at shallow depths (less than 2000 
meters) the increases of pressure are completely 
normal, while at higher depths, where the effect 
of thermal stresses is important, significant 
increases of pressure are observed, as well as 
their distribution along the layer. 
 
 
 
 
 
 
Figure 10. Pore Pressure obtained by the software 
 
Comparing both software and mathematical 
model profiles of pore pressure (figure 11), it is 
shown that the result obtained by PetroMod is 
higher than the obtained by mathematical 
model, approaching more to the drilling events. 
This could be explained by the lateral 
transferences between layer. 
 
 
Figure 11. Pressure Profile by mathematical model and 
software. 
 
0
1000
2000
3000
4000
5000
0 50 100 150
PRESSURE (MPa) 
Presure PetroMod Lithostatic
Hidrostatic Mud Weight
Effect hydrocarbon
D
EP
TD
EP
TH
 (
m
) 
%Ro Software 
%Ro Laboratory 
 
 
 
 
SBMR 2014 
The comparison of the numerical results with 
the drilling events (influx) is observed on figure 
12. 
 
 
Figure 12. Comparison between the software results and 
the drilling events. 
 
 
Analyzing results, it can be shown that at 
depths inferior to 1700m, the mud weight is 
enough to control the pore pressure. Between 
1700 and 2700m, influxes were observed, 
which indicates that the pore pressure is higher 
than the mud weight. This pressure excess 
could be demonstrated by the 2D basins 
modeling. Besides the overpressure 
mechanisms studied, this prediction includes 
lateral transferences between layers. Finally, at 
depths greater than 2700 m it was observed that 
the mud weight is enough to control the pore 
pressure. 
 If only undercompaction is taken into 
account to get the pore pressure profile (figure 
2) it does not fit with the events, particularly 
influxes that occurred during the drilling phase, 
this is due to thermal stresses generate a pore 
pressure increasing (Figure 6) reaching an 
additional 14% (Fig. 7B) 
 
5. CONCLUSIONS 
 
 It was confirmed that if geological 
conditions are favorable (TOC higher 
than 1%, temperatures higher than 
93°C), other mechanisms of 
overpressure may occur (thermal 
stresses), but still yet, the compaction is 
the main cause of overpressure (Figure 
7) 
 It was corroborate that aquathermal 
effect is rather irrelevant, because it 
wasshown that the excess of pressure is 
compensated with the viscosity 
reduction, as stated by Luo and Vasseur 
(1992) (Figure 4). 
 It was shown in this application, that at 
depths greater than 4500 m the effect of 
hydrocarbons generation, is important 
because, the contribution reached a 
value of 14% further to the pressure 
generated by compaction (Figure 7B). 
 It was shown that the results obtained 
with basin modeling are more reliable 
than those of the mathematical model, 
because the 2D model can include 
lateral transferences between the layers. 
 
 
 NOMENCLATURE 
 
T = Temperature 
t = Time 
 Effective stress, [M/Lt2], [psi] 
 = Vertical stress, [M/Lt2], [psi] 
P = Pore pressure, [M/Lt
2
], [psi] 
 Depth, [L], [ft] 
Cr= Compressibility of the rock [1/(M/Lt2)], [1/psi] 
 = Density [ML-3], [lb/ft3] 
 = Kerogen density [ML-3], [lb/ft3] 
 = Gas density [ML-3], [lb/ft3] 
µ = Viscosity [M/Lt
1
] 
 Porosity 
 k =Permeability [L
2
] 
 = Variation of gfas density [ML-3], [lb/ft3] 
 = Variation of oil density [ML-3], [lb/ft3] 
 = Coefficient of thermal expansion [T-1], 
 = Relationship between porosity and permeability [L2] 
α = Compaction coefficient [L-1] [ft-1] 
b = Compaction coefficient [1/(M/Lt
2
)], [1/psi] 
g = Gravity[Lt
-2
], [ft.s
-2
] 
V=Velocity [L.s
-1
] 
 Variation 
 
ACKNOWLEDGMENT 
 
I acknowledge mainly to GOD for everything, 
to the workers of Colombian Institute of Oil 
(ICP), to the Universidad Industrial de 
Santander (UIS) and to the Wellbore Stability 
research Group (GIEP) for the oportunity of 
acquire new kwnoledges. 
0
1000
2000
3000
4000
5000
0 5 10 15 20 25
PRESSURE (ppg) 
INFLUX
PRESSURE MUD OF DRILLING
PRESSURE MUD PETROMOD
 
 
 
 
SBMR 2014 
 
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