ABSTRACT

Keywords: hydraulic fracturing, fracture gradient, foothills, Colombia, Mirador Superior, Barco, Effective Stress Method, Pseudo-Overburden Stress Method.

RESUMEN

Estimaciones del gradiente de fractura son fundamentales en la predicción de la presión requerida para fracturar una formación hidráulicamente. El principal objetivo de este trabajo consiste en proponer una nueva metodología para calcular un valor de gradiente de fractura basado en la aplicación de dos nuevos y diferentes métodos: Pseudo-Overburden Stress Method  y Effective Stress Method. Estos nuevos métodos fueron obtenidos modificando y mejorando dos métodos propuestos en la literatura, acondicionándolos en un orden lógico y sistemático,  haciendo posible su aplicación a pozos onshore, incorporando una nueva función para calcular constantes de calibración con la menor incertidumbre asociada, y ampliando su campo de aplicación para  involucrar formaciones a profundidades diferentes a las de la calibración inicial al incluir un nuevo subproceso. Adicionalmente, estos métodos incorporan parámetros de entrada como gradiente de fractura, esfuerzo vertical y presión de poro, los cuales describen las condiciones geomecánicas de la formación. Esta metodología es validada en las formaciones Mirador Superior y Barco en el Piedemonte Colombiano. Los resultados de esta metodología son comparados con los valores de obtenidos de pruebas Minifrac^TM. La aplicación de esta metodología permite la predicción de valores confiables de gradiente de fractura. 

Palabras Clave: fracturamiento hidráulico, gradiente de fractura, Piedemonte, Colombia, Mirador Superior, Barco.

RESUMEN

Estimações do gradiente de fratura são fundamentais na predição da pressão requerida para fraturar uma formação hidraulicamente. O principal objetivo deste trabalho consiste em propor uma nova metodologia para calcular um valor de gradiente de fratura baseado na aplicação de dois novos e diferentes métodos: Pseudo-Overburden Stress Method  e Effective Stress Method. Estes novos métodos foram obtidos modificando e melhorando dois métodos propostos na literatura, acondicionando-os em uma ordem lógica e sistemática,  fazendo possível a sua aplicação a poços onshore, incorporando uma nova função para calcular constantes de calibração com a menor incerteza associada, e ampliando o seu campo de aplicação para  envolver formações a profundidades diferentes às da calibração inicial ao incluir um novo sub-processo. Adicionalmente, estes métodos incorporam parâmetros de entrada como gradiente de fratura, esforço vertical e pressão de poro, os quais descrevem as condições geomecânicas da formação. Esta metodologia é validada nas formações Mirador Superior e Barco no Pé de Montanha Colombiano. Os resultados desta metodologia são comparados com os valores de obtidos de provas Minifrac^TM. A aplicação desta metodologia permite a predição de valores confiáveis de gradiente de fratura.

Palavras Chave: fraturamento hidráulico, gradiente de fratura, Pé de Montanha, Colômbia, Mirador Superior, Barco.

NOMENCLATURE

gf     Fracture gradient (psi/ft)

D     Formation depth (ft)

ø_o     Surface pseudo porosity (dimensionless)

K_ø    Declination Pseudo porosity (1/ft)

σ_pseudo Pseudo-Overburden stress (psi)

σ_ν     Vertical stress (psi)

ρ_g     Grain density (lb/ft³)

ρ_f     Formation fluid density (lb/ft)

ρ_w    Water density (lb/ft³)

D_w    Water depth (ft)

σ_f     Fracture stress (psi)

σ_p     Pore pressure (psi)

σ_v     Vertical Stress (psi)

K     Stress ratio Constant (dimensionless)

g_f     Fracture Gradient (psi/ft)

g_v     Vertical stress gradient (psi/ft)

g_p     Pore pressure gradient (psi/ft)

INTRODUCTION

Increasing the hydrocarbon production has been the essence over the last few years and, therefore, different strategies have been implemented to make reservoirs even more productive. One of the most effective strategies is the hydraulic fracturing, a stimulation method involving fracturing the productive strata by injecting a fluid at high pressure. The induced fracture is maintained open with a propping agent, and the high permeability propped fracture facilitates hydrocarbon flow to the well face. This stimulation method is mostly applied in wells that evidence near-wellbore damage (presence of a positive Skin) or in those wells in low permeability formations. Usually, the service companies executing the hydraulic fracturing operation reports fracture gradients values based on ISIP data (Instantaneous Shut-In Pressure). This means that pressure drops caused by tortuosity and perforation friction are not considered in this paper and, fracture gradient values are based on ISIP values.

This article proposes and implements a new me-thodology for fracture gradient calculation involving two new methods: the Pseudo-Overburden Stress Method and the Effective Stress Method. The objective of this methodology is to predict a fracture gradient value with the least uncertainty for the well of interest by selecting the greatest of the two fracture gradient values obtained from the two methods (this leads to a more conservative design). This methodology is thus adjusted according to the conditions prevailing during the prediction process.

The two methods involved were selected based mainly on the following criteria:

• The two methods are calibrated from real target strata characteristics, leading to predictions with less uncertainty. These facts overcome the limitations of the traditional methods.
• The fracture gradient, vertical stress, and pore pressure values of the offset wells used in calibration of the methods represent the specific geomechanical conditions of the formations of interest.
• Input data such as Minifrac^TM field data and geomechanical parameters required by the methods (which are distinctive input data of the two involved methods) are available in the field, and are generally considered to be reliable values.
• The Pseudo-Overburden Stress Method proposed here is based on the method of Rocha and Bourgoyne (1996). This new method, in contrast to the base method, is applicable to wells without water columns (onshore wells), exhibits a logic and systematic order, incorporates a new function "Selection K_ø and ø_o (1)" to calculate calibration constants with the less associated uncertainty, and includes a new subprocess "ME" which allows involving formations at depths different from the initial calibration depths.

The Effective Stress Method is based on the method of Brennan and Annis (1984). This new method, in contrast to the base method: is applied to non-over-pressured formations, exhibits a logic and systematic order, and incorporates a new sub-process "EE" which allows involving formations at depths different from the initial calibration depths.

The proposed methodology is validated in the Mirador Superior and Barco Formations in a field located in Colombian Andean Foothills. Results were compared to the values obtained from Minifrac^TM field data.

Fracture Gradient

The fracture gradient is the geomechanical parameter that determines the necessary pressure to be applied to fracture the formation (Fjaer, Holt, Horsrud, Raaen & Risnes,1996):

The fracture gradient together with the losses due to friction and hydrostatic pressure are the base to obtain the surface treatment pressure, and, therefore, the required hydraulic power.

Calculation of the Fracture Gradient by Field Tests

Different field tests such as Leak-Off (LOT) (Altun, Langlinais & Bourgoyne Jr., 2001), Extended Leak-Off (ELOT) (Addis, Yassir, Willoughby & Enever, 1998), Step-Rate Test (SRT) (Singh,  Agarwal & Krase, 1987) and Minifrac^TM are conducted in field to determine the fracture gradient.

Method of Rocha and Bourgoyne (1996)

This method is based on a compaction model that considers the depth of each well in presence of water column. The Pseudo-Overburden stress expression is calibrated with the value of the two constants (K_ø and ø_o)which characterize the target formation. Model development begins expressing the formation porosity as follows:

This method begins with the calculation of the Pseudo-Overburden stress corresponding to each well of interest. Then, a graph is drawn with these points in one axis and the fracture stress values obtained from LOT in the other axis. The objective is to find the values corresponding to the constants K_ø and ø_o for which the slope of the trend line obtained in the curve equals to 1.0. This indicates that the Pseudo-Overburden stress is equal to the fracture stress.

Method of Brennan and Annis (1984)

This method is based on the theory proposed by Hubbert and Willis (1956).The generalized form is presented as follows:

This method consists in finding a function that relates the effective fracture stress to the effective vertical stress for a set of calibration wells located in the vicinity of the well of interest. This ratio is applied to estimate the fracture gradient value in the well of interest. The fracture stress value is obtained from LOT.

If Equation 4 is divided by depth, the basic form of the equation becomes:

The stress ratio constant can also be expressed in function of gradients:

Fracture gradient with application of the Equation 5 can be predicted if a curve illustrating effective fracture gradient vs. vertical effective stress gradient with a defined slope line is observed. If no direct relation is found between the stress gradient values, a specific correlation to relate them is designed as follows:

Isolating the fracture gradient:

This method consists in finding a defined value for the stress ratio constant by graphing the effective fracture gradients vs. effective vertical stress gradients of calibration wells and obtaining the slope of the curve. Then, it is possible to apply the Equation 4. If a defined value for the curve slope cannot be found, the next step is to find a specific correlation for these two effective stress values.

PROPOSED METHODOLOGY

The proposed methodology to estimate fracture gradient involves two new methods:

• Pseudo-Overburden Stress Method
• Effective Stress Method
]]>

These new methods were developed based on the methods proposed by Rocha and Bourgoyne (1996) and Brennan and Annis (1984) respectively.

One innovative fact considered by this proposed methodology is including a security factor by selecting the greatest fracture gradient value obtained after implementing the two new methods; this allows a more conservative operation design. Underestimating the fracture gradient value leads to an inappropriate surface equipment design for the conduction of operations. This fact is reinforced by considering that many hydraulic fracturing operations have failed due to insufficient hydraulic power.

Pseudo-Overburden Stress Method

This new method, in contrast to the method proposed by Rocha and Bourgoyne (1996), is applicable to wells without water columns (onshore wells), exhibits a logic and systematic order, incorporates a new function "Sele-ction K_ø and ø_o (1)" to calculate calibration constants with the less associated uncertainty, and includes a new sub-process which will be called "ME", that allows involving formations at depths different from the initial calibration depths in the Pseudo-Overburden Stress Method.



The new function: Selection K_ø and ø_o (1), requires selecting an initial K_ø and ø_o constants value range, then, through an iterative process, it will calculate such constants with the less associated uncertainty.

The expression for the Pseudo-Overburden Stress wi-thout considering water column is presented as follows:



The calibration of this method is based on data obtained from off-set wells with the same target formation or eventually in formations with similar characteristics.

]]> The Pseudo-Overburden Stress Method is presented as follows:

1.   Data collection of σ_ν, ρ_g and ρ_f for the calibration well and for the prediction well at the depth of interest.

2.   Is either LOT or Minifrac^TM field data available?

3.   Data collection of either LOT or Minifrac^TM for calibration wells.

4.   Graph (D) vs. σ_f for the calibration wells.

5.   Is there any trend that shows that σ_f increases in depth?

6.   Feeding the Pseudo-Overburden stress equation (Equation 9) with K_ø, ø_o,  D, σ_ν and ρ_g for the interest well and conduct the prediction of σ_f.

7.   Is the value of σ_f required at other depth?

8.   Determine fracture gradient value by dividing σ_f    by depth (D).

9.   The Pseudo-Overburden Method is not applicable.

]]> 10.Finding directly the fracture gradient value is not possible by this method.

ME: Sub-process to calculate σ_f  at a different depth from the value used in the initial prediction.

A.  Graph σ_f  vs. Depth (D) for the calibration wells and the well of interest at the corresponding depth.

B.  Draw a trend line from the graph obtained on A and express fracture stress in function of depth.

C.  Feed the correlation obtained on B with the depth at which the value of σ_f f is desired.

Selection K_ø and ø_o (1)

S1.  Select a range of values for the constants ø_o and K_ø, as: (ø_oi, ø_of) and (K_øi, K_øf), respectively, as well as the increment steps for the constant  values: δK_ø and δø_o.

S2.  For K_ø, between K_øi and K_øf, with K_ø = K_ø+ δK_ø, complete S3.

S3.  For ø_o, between ø_oi and ø_of, with ø_o = ø_o + δø_o, complete from S4 to S9.

S4.  Generate a set of values where X set are the fracture stress values and the Y set are the Pseudo-Overburden stress values (Equation 9).

]]> S5.  Draw a trend line by the minimum square value method and calculate its slope for the set of values X and Y, calculated in S4.

S6.  A set of values Y’ is generated corresponding to the evaluation of the line obtained on S5. In addition, a set of values Y_ideal = X corresponding to the unitary slope line is created.

S7.  The absolute error between the set of values Y’ and Y_ideal is calculated.

S8.  Is the error calculated in S7 the least?

S9.  Save the couple of values (K_ø, ø_o).

Effective Stress Method

This new method, in contrast to the method proposed by Brennan and Annis (1984), is applied to non-over-pressured formations, exhibits a logic and systematic order, and incorporates a new subprocess which  will be called "EE", that allows involving formations at depths different from the initial calibration depths in the Effective Stress Method.

The Effective Stress Method is presented as follows:



Where:

]]> 1.   Data collection of g_p and (g_ν - g_p) for the calibration wells and for the prediction well at the depth of interest.

2.   Is either LOT or Minifrac^TM field data available?

3.   Data collection of σ_f and g_f for the calibration wells.

4.   Tabulate (g_f - g_p) and (g_ν - g_p) for each calibration well.

5.   Obtain the value of K for each calibration well (Equation 6).

6.   Graph K vs. depth (D).

7.   Is there uniform trend for the K value in depth

8.   (K = constant)?

9.   Graph (g_f - g_p)  vs. (g_ν - g_p)   and determine the slope which corresponds to the K value.

10.    Apply the Equation 5.

]]> 11.    It is not possible to apply the Equation 5.

12.    Graph (g_f - g_p) vs. (g_v - g_p)   and draw a trend line.

13.    Express the trend line as it is presented in the Equation 7.  

14.    Solve for fracture gradient (Equation 8).

15.    Feed the equation obtained in the step 13 (Equation 8) with g_p and g_ν values for the interest well and find g_f.

16.    Is the g_f value required at a different depth?

17.    It is not possible to obtain the fracture gradient value directly by this method.

EE: Sub-process to calculate g_f  at a different depth from the value used in the initial prediction.

A.  Graph σ_f vs. Depth (D) for the calibration wells and the well of interest at the corresponding depth.

B.  Draw a trend line from the graph obtained on A and express fracture stress in function of depth.

]]> C.  Feed the correlation obtained on B with the depth at which the σ_f  value is desired. 

D.  Divide the fracture stress value obtained on C by the new depth value to obtain g_f.

APPLICATION OF THE METHODOLOGY AND RESULTS

The proposed methodology is applied to predict the fracture gradient for Mirador Superior and Barco formations in FHC 7 and FHC 9L wells respectively.

Calculation of the Fracture Gradient value from the Pseudo-Overburden Stress Method

To calculate the fracture gradient for Mirador Superior formation in FHC 7 well, 6 fracture gradient values will be used from offset wells. Likewise, to calculate this property for Barco formation in FHC 9L well, 4 fracture gradient values will be used. Fracture gradient calculation for each formation will be conducted simultaneously following the numbering system proposed:

1.   Formation grain density, vertical stress and fluid density values for the calibration and prediction wells on Mirador Superior and Barco formations are presented as follow:





2.   Minifrac^TM field data are available. Go to 3.

]]> 3.   The Fracture Stress Values from Minifrac^TM field data for calibration wells are collected.

4.   Depth vs. fracture stress graphs for calibration wells on Mirador Superior and Barco formations are presented as follow:



5.   From 4, it is possible to see an increasing trend of fracture stress in depth. This indicates the Pseudo-Overburden Stress Method is applicable. Go to apply Selection function K_ø and ø_o (1) for each formation.

Selection K_ø and ø_o (1)

The objective of the Selection K_ø and ø_o (1) function is to determine the values of the K_ø and ø_o constants for which the Pseudo-Overburden stress value equals to the fracture stress value. This can be obtained by graphing Pseudo-Overburden Stress vs. fracture stress and finding a trend line with a slope of 1.0.

• Mirador Superior formation

S1. The value ranges for constants K_ø and ø_o to be applied are the next:

For K_ø:  (0,00001; 0,0015)

For ø_o : (0,9; 1,0)

]]> With an increment step of:

δ K_ø= 0,00005

δ ø_o = 0,001

Once the program was run (from S2 to S9), the obtained values are presented as follow:

K_ø = 0,00011

ø_o  = 0,9

• Barco formation

S1.  The value ranges for constants K_ø  and ø_o to be applied are the next:

For K_ø : (0,000002; 0,0001)

For ø_o : (0,8; 1,0)

]]> With an increment step of:

δ K_ø= 0,000005

δ ø_o = 0,001

Once the program was run (from S2 to S9), the obtained values are presented as follow:

K_ø= 0,000097

ø_o = 0,8

Go to 6.

6. Once the calibration constants are obtained, it is possible to estimate the Pseudo-Overburden Stress Value (Equation 3) which equals to the fracture stress value. The results are presented as follow:

• Mirador Superior formation

σ_pseudo = σ_f =  10936,41 psi

]]> • Barco formation

σ_pseudo = σ_f  =  12096,22 psi

7. The fracture stress value is not required at a different depth at which the test was conducted. Go to 8.



8. If the fracture stress values obtained on 6 are divided by their corresponding depth, fracture gradients are obtained:

• Mirador Superior formation



• Barco formation



Comparison of fracture gradients obtained from the Pseudo-Overburden Stress Method and from Minifrac^TM field data

]]> The fracture gradient for the target formations in FHC 7 and FHC 9L wells are known from Minifrac^TM field data. Therefore, a comparison between these values and the values obtained by the Pseudo-Overburden Stress Method at the same depth can be accomplished to validate the results. Table 3 presents the comparison.

Table 3 reveals that fracture gradient values obtained from the Pseudo-Overburden Stress Method are very close to that obtained from Minifrac^TM field data. This reflects the effectiveness of this method.

Calculation of fracture gradient from the Effective Stress Method

To calculate the fracture gradient for Mirador Superior Formation in FHC 7 well, 6 fracture gradient values will be used from offset wells. Likewise, to calculate this property for Barco Formation in   FHC 9L well, 4 fracture gradient values will be used. Fracture gradient calculation for each formation will be conducted simultaneously following the numbering system proposed:

1.   Pore pressure gradient and vertical stress gradient for calibration and prediction wells on Mirador Superior and Barco are presented as follow:





2.   Minifrac^TM field data are available. Go to 3.

3.   The Fracture Stress Values from Minifrac^TM field data for calibration wells are collected.

4.   Effective fracture gradient values and effective vertical stress gradient values for each well are calculated.

]]> 5.   The value corresponding to the stress constant ratio value (Equation 6) for calibration wells in each formation is determined.

6.   Stress constant ratio in depth for calibration wells on Mirador Superior and Barco formations.



7.   From 6, it is possible to observe that the stress constant ratio (K) is not a constant value. Go to 10.

10.  Since the stress constant ratio (K) is not a constant,  equation 5 cannot be applied. Go to 11.

11.  After graphing effective fracture gradient vs. effective vertical stress gradient, it is possible to obtain the trends lines (which will be expressed mathematically on 12). Go to 12.

12. The relationship between the effective fracture gradient and effective vertical stress gradient (based on trend lines) for each formation is presented as follow:

• Mirador Superior Formation



• Barco formation

]]>

13. Solving the expressions obtained on12 for fracture gradient:

• Mirador Superior Formation



• Barco formation



14. Solving the equations obtained on 13 with the values of pore pressure gradient and effective vertical stress gradient of the prediction wells for each formation:

• Mirador Superior formation



• Barco formation

]]>

15. The fracture gradient value is not required at a different depth at which the test was conducted.

Comparison of fracture gradients obtained from the Effective Stress Method and from Minifrac^TM field data

The fracture gradients for FHC 7 and FHC 9L wells are known from Minifrac^TM field data. Therefore, a comparison between these values and the values obtained by the Effective Stress Method at the same depth can be accomplished to validate the results. Table 6 presents the comparison:



Table 6 reveals that fracture gradient values obtained from the Effective Stress Method are very close to that obtained from Minifrac^TM field data. This reflects the effectiveness of this method.

Selection of fracture gradient values for the prediction wells

As the final stage of this methodology, and it was former mentioned, the greatest fracture gradient values corresponding to each formation are selected and presented as follow:



RESULT ANALYSIS

]]> For the Mirador Superior Fomation, the fracture gra-dient value obtained from the Pseudo-Overburden Stress Method and the Effective Stress Method was the same (i.e., 0,74 psi/ft), and this was the value selected as definitive. For the Barco Formation, the definitive fracture gradient value selected by the methodology corresponds to the value obtained by the Pseudo-Overburden Stress Method (i.e., 0,77 psi/ft). Due to this, it was greater than that obtained by the Effective Stress Method.

The quantitative comparison between the fracture gradient values obtained from the proposed methodolo-gy and that obtained from Minifrac^TM field data, reveals error percentages of 4,22% and 1,31% respectively. This demonstrates the effectiveness of the methodology.

The fracture gradient values obtained from this new methodology overestimate the values obtained from Minifrac^TM tests. For practical purposes, the fact of overestimating the fracture gradient value leads to ensure the success of the treatment in a higher measure, since this value conditions the operational design (hydraulic power requirements) to accomplish the treatment.

CONCLUSIONS

• This paper proposes a new methodology to calculate a fracture gradient value involving two new methods: Pseudo-Overburden Stress Method and Effective Stress Method. A reliable fracture gradient value with low associated uncertainty was obtained from this methodology.
• Despite the different considerations which each method is based on, the fracture gradient values obtained from them for the Barco formation are considerably closed (i.e., 0,73 and 0,77 psi/ft) and for Mirador Superior formation are the same (i.e., 0,74 psi/ft).
• The methodology proposed in this paper can be applied in formations with different characteristics to that presented in this work, this due to the fact that two new methods that compose the methodology are calibrated from real characteristics of the target formation, leading to predictions based on real formation conditions.
• The fact that the fracture gradient values obtained from this methodology are not very high at all despite the issue that Colombian Andean Foothills experiments strike-slip Faulting Regime is a subjective matter. This because it has been proved in some geomechanical studies in Colombian Fields (Mateus, Corzo, García & Marín, 2008) that even in fields which experiment Normal Faulting, fracture gradient values reach values up to 0,9 psi/ft.
• The K_ø  and ø_o values obtained from the Pseudo-Overburden Stress Method and the ratios between the vertical and horizontal stress gradients obtained from the Effective Stress Method for Mirador Superior and Barco formations can be extrapolated to the same formations through Colombian Andean Foothills, ensuring no important variations in the faulting regime. [/body]
]]> [back]ACKNOWLEDGMENTS

The authors express their most sincere feelings of gratitude to Universidad Industrial de Santander (UIS), ECOPETROL S.A. -Instituto Colombiano del Petróleo (ICP) and to the Wellbore-Stability Research Team (UIS - ICP Agreement) for their unconditional support during the development of this work.

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REFERENCES

Addis, M. A., Yassir, N., Willoughby, D. R. & Enever, J. A. (1998). Comparison off leak-off test and extended leak-off test data for stress estimation. SPE/ISRM Eurock’ 98, Trondheim, Norway. SPE 47235.         [ Links ]

Altun, G., Langlinais, U. J. & Bourgoyne Jr., A.T. (2001). Application of a new model to analyze leak-off test. 1999Annual Technical Conference and Exhibition, Houston. SPE 72061.        [ Links ]

Anderson, R. A., Ingram, D.S. & Zanier, A. M. (1973). Determining fracture pressure gradients from well logs. Journal of Petroleum Technology, 25, (11).        [ Links ]

Brennan, R. M. & Annis, M. R. (1984). A new fracture gradient prediction technique that shows good results in Gulf of Mexico abnormal pressure. 59^th Annual Technical Conference and Exhibition, Houston, Texas. SPE 13210.         [ Links ]

Eaton, B. A. (1969). Fracture gradient prediction and its applications in oilfield operations. Journal of petroleum technology, 21 (10).        [ Links ]

Fjaer, E., Holt, R. M., Horsrud, P., Raaen, A. & Risnes, R. (1996). Petroleum related rock mechanics. Developments in petroleum science, 33, (1). 269-282.        [ Links ]

Hubbert, M. K. & Willis, D. G. (1956). Mechanics of hydraulic fracturing. Petroleum Branch Fall Meeting, Los Angeles. AIME 210: 153-66.        [ Links ]

Mateus, D., Corzo, R., García, R. & Marín, M. (2008). Estudio Geomecánico Prospecto Petrolea Sur. Informe Técnico, Ecopetrol S.A.- Instituto Colombiano del Petróleo (ICP).        [ Links ]

Rocha, L. A. & Bourgoyne, A. T. (1996).  A new simple method to estimate fracture pressure gradient. Paper first presented at the 1994 SPE Intl. Petroleum Conference and Exhibition of Mexico , in Veracruz, Mexico .SPE 28710.         [ Links ]

Salz, L. B. (1979). Relationship between fracture propagation pressure and pore pressure. 52nd Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, Denver, Colorado. SPE 6870.        [ Links ]

Singh, P. K., Agarwal, R. G. & Krase, L. D. (1987). Systematic design and analysis of step-rate tests to determine formation parting pressure.  62nd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers Dallas, TX, SPE 16798-MS.        [ Links ]

Zoback, M. D. (2007). Reservoir geomechanics. (First edition). Cambridge: Cambridge University Press.        [ Links ] ]]> 1998 98 Trondheim 2001 1999 Houston 1973 25 11 11 1984 Houston Texas 1969 21 10 10 1996 33 1 1 269-282 1956 210 153-66 Ecopetrol S.A.- Instituto Colombiano del Petróleo (ICP) 2008 1996 1994 Veracruz 1979 Denver Colorado 1987 Dallas TX First