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Biot’s Coefficient: A Highly Undervalued Geomechanical Parameter

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Biot in apartment at 300 Central Park West, New York, September 1964 (Source:
Recently, I reviewed about 90 geomechanical lab reports (different labs, different formations, different companies) containing extensive measurements of all different sorts of mechanical rock properties and I could not find even a trace of Biot’s coefficient measurement. Due to its role in definition of effective stresses, we know that it is unlikely to solve any problem in geomechanics without direct use of poroelasticity theory. Nevertheless, in comparison to other geomechanical parameters (i.e., pore pressure, stresses, elastic and failure properties), Biot’s coefficient seems to be the most undervalued parameter in our studies as we leave it for our back-calculations or lucky guestimations. This does not seem scientifically just as the role of this parameter is not less than many of the other parameters.  So, why don’t we put more effort in measurement of this parameter? Is this because we have a great knowledge of this parameter and we know how to do accurate estimation??? Or only because it is complicated and expensive to measure? Or maybe because it is not in the routine procedure of commercial labs? Is any or all of these reasons good enough to neglect this important parameter and its influence on the results? What do you think?
Unfortunately, this is not our only issue with Biot’s coefficient. Probably a more fundamental one is the eligibility of poroelastic models for modeling many of the rocks especially the unconventional plays.

Two PetroGem Presentations on Geomechanics at Geoconvention 2017

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PetroGem Inc. will present two talks on geomechanics at the coming Geoconvention 2017 next week.

One of the talks (Tuesday May 16th, 10:40-11:05am, Room:Telus 104-106) discusses a potential mechanism that might be responsible for the observed seismicity induced by hydraulic fracturing and water disposal.

Induced Seismicity

The other talk (Monday May 15th, 3:00-3:25pm, Room:Telus 103) tries to show the necessity of geoethical dialogues for application of geomechanics in the oil and gas industry.


Reservoir Containment Assessment and Its Dynamic Nature

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Synopsis: In many cases, reservoir containment and caprock integrity assessment is considered as a one-time requirement for proving the safety of a subsurface project while it is important to remember that such assessments are dynamic processes that should continue during the entire life of the project and, in some cases, even after its cessation. This article presents a workflow for dynamic reservoir containment assessment (DCAP) that accounts for the dynamic nature of this process. 

Spills from Heavy Oil Projects
Some cases of oil spills close to heavy oil projects in Alberta, Canada (source: AER).

Underground activities are growing very fast while the technologies used in these activities become more and more aggressive and risky. For instance, these days, industry is using high temperature steam, it is injecting chemical fluids in the rocks and it is intentionally fracturing the rocks. On the other hand, public sensitivity towards environment has been increasing on a daily basis. Economic reasons also play an important role as by controlling the containment of the reservoir we can prevent unwelcomed problems such as wellbore damage or surficial leakage that can be quite costly for any project. Major consequences of containment loss are:

  • Leakage of reservoir fluids
  • Ground deformation, in general, and ground surface subsidence/heave, specifically
  • Well integrity issues
  • Induced seismicity (due to induced fracturing and reactivation of faults and existing fractures)
  • Inflow of outside fluids into the reservoir
  • Heat Loss

All the containment-related geomechanical hazards must be studied under a comprehensive program called caprock integrity or reservoir containment assessment. In many cases, reservoir containment and caprock integrity assessment is considered as a one-time requirement for proving the safety of a project (and receiving operations approval from authorities) while it is important to note that reservoir containment assessment should be considered as a dynamic process for the entire life of the project and that may continue even after ceasing the underground operations. The rest of this article presents Dynamic Containment Assessment Program (DCAP), a generalized workflow with different modules required for containment assessment and caprock integrity analysis. Examples of operations that such workflow can be applied to are:

  • Conventional production/water flooding
  • Gas sequestration/storage
  • Thermal operations
  • Unconventional shale gas/oil
  • Nuclear waste deposits
  • Compressed air storage
  • Underground water production
Different mechanisms that can lead to loss of hydraulic integrity and containment of reservoirs (source: Soltanzadeh, 2009)

Dynamic Containment Assessment Program (DCAP)

This workflow implements data, tools, and techniques from different disciplines such as geology, petrophysics, geophysics, reservoir engineering, well engineering, hydrogeology, geochemistry, etc. Ideally, all these information resources are integrated in a comprehensive dynamic process that can even continue after cessation of underground operations. Different steps of this workflow are:

  • Appraisal data acquisition
  • Site characterization
  • Data interpretation and modeling
  • Feasibility assessment
  • Operational criteria and recommendations
  • Field monitoring
  • Real-time data updating


Diagram of Dynamic Caprock Integrity Program (DCAP)

Appraisal Data Acquisition

In the appraisal phase of geomechanical assessment of reservoir containment, data are collected from several different sources including:

  • Geological studies
  • Geophysical surveys
  • Hydrogeological and geochemical characterizations
  • Petrophysical studies
  • Production/injection rate histories
  • Pressure and temperature histories
  • Wireline logs
  • Geomechanical lab and field tests
  • Leakage evidence
  • Well drilling, completion, fracturing and treatment experience
  • Ground deformation data

and other sources that may either directly or indirectly help to build an accurate earth model that includes all the sedimentary succession from below the reservoir up to the ground surface. Special attention must be paid to the reservoir and its primary caprock(s) in this process.

Site Characterization

After data acquisition, the collected data are used to characterize different properties of rocks in the study area and sedimentary succession of interest. Ideally, a Mechanical Earth Model (MEM) should be constructed based on integrated processing of these data. Such a model usually includes:

  • Geological structure
  • Sealing mechanisms
  • Hydrogeological and fluid flow characteristics
  • Petrophysical characteristics
  • Geomechanical properties

Site characterization must be seen as an ongoing process during the entire life of the project and, ideally, it should include the monitoring period after cessation of operations.

Data Uncertainty and Ongoing Variations: Data uncertainty is always a main characteristic of data for subsurface studies that usually is addressed in the developed mechanical earth model using geostatistical methods. It is important to note that site characterization is a dynamic process and any additional data that becomes available during the life of the project can improve the quality of characterization. On the other hand, the character of a site may significantly vary with time due to different operations such as hydrocarbon production and fluid/steam injection. Such processes change the fluid content, as well as the pressure and temperature within the reservoir and its surrounding rock and, consequently, can affect the petrophysical and geomechanical properties of the rock.

Characterizing Sealing Mechanisms: Initial sealing mechanisms can be identified by studying the geological structure of the field and its constituent faults and fractures, their mechanical and hydraulic properties, hydrogeological information, and in-situ pore pressure, temperature and stresses. The pressure history of the reservoir and the records of well testing are also very important in this process. Any evidence of reservoir fluid leakage is also very useful to identify sealing mechanisms and their potential alteration during the production life of the reservoir.

A Mechanical Earth Model created for reservoir integrity assessment showing static Young’s modulus variations (source: Soltanzadeh and Hawkes, 2012)

Data Interpretation and Modeling

Geomechanical models are constructed based on the collected data and site characterization. These models are calibrated using historical data and utilized to identify the potential geomechanical issues in the past history and the future life of the reservoir. Different types of geomechanical modeling tools may be used to studying these issues. These tools cover a broad band from simpler analytical and semi-analytical models to more complicated numerical models.

Different Geomechanical Models: Analytical and semi-analytical models are usually constrained by simplifying assumptions regarding the geometry, mechanical properties, and fluid flow characteristics of the system. To consider more details for the problem (e.g., more realistic geometry and material properties), using numerical models is essential. However, more detailed data are required for more complex models. In an ideal case, the geomechanical models are fully coupled with fluid flow models but, in reality, the degree of coupling might be looser due to different issues such as time, cost, and computational power.

Diagram of coupling between fluid flow/heat transfer simulator, geomechanics model and fracture model (Source: Soltanzadeh, 2015).

Modeling Process: The developed models, along with historical production and injection rates, pressure and temperature history of the field can be used to study the geomechanical response of the reservoir during its production life. These studies are capable of identifying induced fractures and reactivation of existing fractures and faults, and their effect on the sealing mechanisms of the field during this period. The validity of the results of modelling can be evaluated using historical geomechanical data such as recorded wellbore instabilities, seismic activities, and ground deformations. History of reservoir treatment activities such as hydraulic fracturing may have significant effects on the hydraulic integrity and must be considered during these studies. In addition, the developed models are used to predict the geomechanical response of the field to future developments such as injection and production during operations.

It should be noted that besides numerical modeling, it is also necessary to experimentally test caprock sealing properties such as capillary entry pressure, to ensure the capability of the caprock for preventing capillary leakage. Another important issue in modeling of these operations is accounting for the hysteresis behaviour of the reservoir and its surrounding rock when they become subjects of repeating cycles of injection and production during operations. After starting the operations, the developed models must be updated during the operations and calibrated with the real-time data.

A numerical model showing heave induced by injection in a reservoir (source: Soltanzadeh and Hawkes, 2012)

Feasibility Assessment

Geomechanical feasibility is evaluated based on the collected data and modeling results. The major issues considered for this assessment include minimizing the potential for leakage, wellbore stability concerns, induced seismicity, and ground deformation. This process may also include geomechanical assessment of injectivity enhancement potential such as hydraulic fracturing. The feasibility assessment must be studied in the context of provincial/state and federal regulations. Feasibility assessments may lead to different conclusions: In cases which potential risks are not tolerable and cannot be mitigated or controlled, the project may be disqualified. In other cases, limitations and operational criteria may be defined and recommended to minimize the potential risks for the project.

Operational Criteria and Recommendations

If the reservoir is qualified for underground operations, some criteria are usually defined to ensure the safety of operation. These criteria are applied to limit the injection rates, fluid pressures, fluid temperatures, ground deformations, etc. In addition, instructions are given for wellbore (re-)design and treatment. These criteria may be a direct result of modeling or imposed by regulations and standards. The initial criteria and limitations may change during the life of the project when the new data and observations become available for updating the feasibility assessment results.

Draft of reservoir containment requirements for application of oil sands projects in Alberta (source: AER)

Field Monitoring

Field monitoring is an important part of any subsurface project that is designed to record the potential changes induced by the field operations. The results from monitoring are employed to evaluate the field performance and identify the changes in the field condition during and after operations. Some of the monitoring techniques used for this purpose are: seismic surveys, microseismic monitoring, well logging and monitoring, groundwater sampling, soil contamination measurement, tilt meters, satellite monitoring of surficial deformation.

The acquired data from the reverse analysis of monitoring results can be very valuable for understanding and predicting the geomechanical behaviour of the field. Such analyses provide information about fluid flow within the reservoir, potential leakage, ground deformation, and location and characteristics of faults and fractures and rock properties.

microseismic monitpring
Sequence of three snapshots in time showing microseismic events during cyclic steam stimulation operation in a heavy oil field in Alberta, Canada (source: McGillivray, 2005)

Real-time Data Updating

The collected initial data mentioned in the appraisal stage must be updated and modified by using the newly acquired information from different sources that become available during the reservoir’s life. As mentioned, one important source for such data is field monitoring. Other sources include new geological, geophysical, petrophysical, geochemical, and hydrogeological studies. In addition, new logs, lab tests, and field tests and data from wellbore stability studies can be very useful. These real-time data will be implemented to update the site characterization for the field and, subsequently, for updating the geomechanical, fluid flow and other models. The results of such analyses are used to re-define and modify the operational criteria and recommendations for continuation of the operation.


Ethics of Geomechanics: A Thriving Discipline and Its Growing Responsibility

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Bitumen Spill Canada
Figure 1. An incident of bitumen spill close to a thermal operation site in Alberta, Canada (source:

Access the pdf version of the article here.

During the last two decades, reservoir geomechanics has been showered with attention from petroleum industry, academia and regulatory institutions mainly because modern technologies, new perspectives and economic opportunities have led to exponential growth of aggressive underground operations such as massive hydraulic fracturing, waste disposal, underground storage of greenhouse gases and in-situ thermal projects, all calling for geomechanics not just to help them with increasing their efficiency but also to answer some crucial questions on their safety and potential risks such as excessive ground deformation, fluid leakage, air, soil and water contamination and induced seismicity. In fact, none of these concerns are quite new to the world but they have never been operated in a scale as large as today’s plus that, in the current sensitive social platform, their economic, sociopolitical and environmental importance can hardly be overlooked. This popularity has come with a huge load of professional and ethical responsibility for geomechanics as a discipline that is primarily responsible for assessment of these risks. When it comes to the application of geosciences, relevant ethical issues will fall under the umbrella of ‘geoethics’, a developing branch of ethics that is much younger and less famous than its celebrity cousin, bioethics. While growing to adolescence, theoretical and practical aspects of geoethics seem to receive less attention from the technical community (including geomechanics experts) in comparison to the environmental activists, ethics philosophers, politicians and business managers. Nevertheless, with its crucial role in assessment of risks and concerns, joining the discourse of geoethics is an excellent opportunity for geomechanics to prove its commitment to the welfare of the society and environment. To accomplish this task, geomechanics community (that includes regulatory agencies, academia, and industry) along with other parties need to think of establishing a comprehensive framework that, at the very least, will include the following elements:

  • Ethical Platform: Developing or adopting an ethical platform on how to treat problems that are imposing risks on the environment and society and how to define a balance between economic development, preservation, and social prosperity is the first step. Professional integrity and scientific honesty are obviously inseparable parts of such a platform but it will definitely need to be much more comprehensive than a general code of ethics for a specific profession.
  • Acknowledging Uncertainty: Open and clear recognition of the existing uncertainties in different processes of data acquisition, modeling, design, operation and monitoring is critical. All the decisions made by geomechanics experts involve a (remarkable) level of uncertainty and, consequently, all the relevant risks must be assessed by bringing the uncertainty into account. Any analysis needs to clearly acknowledge and address all the different potential scenarios that may put the society and environment and at risk and provide the best possible estimation of their probability to the decision makers and public. Different obstacles that may make this process difficult are scientific prejudice and overconfidence, technical ignorance, communication inefficiency, and lack of professional integrity.
  • Regulations: Standard design, operational, and monitoring codes need to be developed by regulatory institutes in collaboration with the scientific community and industry to ensure the minimum requirements for safety and preservation are fulfilled. Similar to other disciplines (take the field of ‘construction’ as an example), coming up with such regulatory guidelines will need investment from all the parties especially the governments and intergovernmental agencies. These investments are used to form specialized research institutes with the duty of providing the best-practice guidelines. Enforcing ultra-conservative advices backed up with justifications such as ‘lack of knowledge’ or ‘immaturity of science’ usually is not a smart long-term move. With such lame excuses in effect, several of the currently existing developments in the world would never have had a chance to happen. The main role of regulatory institutes is taking the lead on developing knowledge, science and technology whenever necessary.
  • Education: Training on environmental, social and economic aspects of relevant risks and their potential impacts is crucial. Such training should be a part of a systemic education in academia and industry for geomechanics practitioners. Different elements of ethics, especially geoethics must be a part of such educational system. It is important to ensure that all the practitioners are familiar with the codes of conduct through proper education. Also, professional associations who are regulating the practice of the discipline need to show more profession-specific attention to education and qualification of their members.
  • Scientific Freedom: Importance of freedom of research and science cannot be emphasized enough. All the involved sectors need to ensure the circulation of knowledge is not bottlenecked for any unnecessary reason such as politics or higher profit. Practitioners need to feel ‘free’ in expressing their opinion on the matters concerning the society and environment regardless of the outcomes. It is important that proper whistleblower policies will be in effect in all the areas with potential georisks.
  • Transparency: Without a minimum level of transparency in providing details on different processes of design, execution, monitoring and observation, preventing undesired situation will be very difficult. Along with respecting the interests of the investors, industry needs to ensure that confidentiality does not act as a barrier for sharing crucial information with public.
  • Public Communication: Communicating with the society and media can be quite a challenge for the technical communities including reservoir geomechanics due to their complex physical nature. Nevertheless, this cannot be used as an excuse for not providing understandable explanation for the issues related to the welfare of the environment and society. Geomechanics needs to come up with creative methods to explain itself to the general audience with minimum technical knowledge.

Some of the addressed points may already be in place and practiced to some extent but it is still hard to overlook the urgent need for their development and improvement. Fortunately, several other disciplines (for instance, ‘oil and gas transportation’) have been wrestling with similar issues for their entire life and their experiences may be effectively used to ensure the practice of geomechanics is aligned with ethics and professional integrity and welfare of the society and environment.

Induced Seismicity Canada

Figure 2. Major seismic events felt close to a hydraulic fracturing operation site in British Columbia, Canada. Event Locations, event sequence and drilling pad locations shown within 10 km radius shaded circle (source:


Geomechanics of Compressibility – Part III-Different Compressibility Coefficients and Their Applications

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Read the first part of this series here.

Read the second part of this series here.

As discussed, the common concept of compressibility in geomechanics has been developed to study the changes in either of bulk volume (Vb) or pore volume (Vp) of rocks in response to variation in confining pressure (σc) or pore pressure (p). Also, I explained the concept of coupling between pore pressure and confining pressure and the fact that in drained conditions, effects of these two parameters on volume changes are uncoupled from each other though this assumption is not valid for most of problems in reservoir geomechanics.

However, assuming an uncoupled condition, it is possible to define four different types of compressibility coefficients to relate the two mentioned pressures to the two named volumes (Zimmerman, 1991) as listed below. In these definitions, one of pore pressure or confining pressure is assumed to remain unchanged while the other varies.

1- Bulk volume compressibility Coefficient (Cbc):  

Tectonic Model
Figure 1. A plate tectonic model that uses compressibility coefficient to calculate variation in density induced by change in confining pressure  (Nikolaeva et al., 2011)

Bulk volume compressibility coefficient (Cbc) equals to the change in the bulk volume of rock (Vb) with respect to the variation in the confining pressure (σc) while the pore pressure (p) is held unchanged:

Cbc= (-1/Vb) (∂Vb/∂σc )                                              

where   p=constant.

Cbc is usually used in large-scale tectonic modeling and also in wave propagation analysis. In tectonic modeling, this parameter is implemented to account for the dependency of rock compressibility (usually in high temperatures) to tectonic forces. In the case of wave propagation problems, wave velocities are closely dependent on the rock’s matrix compressibility (though it is usually stated in terms of other elastic parameters such as bulk modulus).

Probably a major importance of Cbc is the fact that it is analogous to the compressibility of non-porous media and so it can be compared to the compressibility of different solids and fluids.

2- Pseudo-bulk compressibility Coefficient (Cbp):

Well with subsidence
Figure 2. Evidence of ground surface subsidence around a well in Baytown in the Harris-Galveston District, Texas (source: )

This type of bulk volume compressibility coefficient (Cbp), also called ‘pseudo-bulk compressibility coefficient’ quantifies the change in bulk volume of the rock (Vb) with respect to variation in the pore pressure (p) while the confining pressure (σc) is held unchanged:

Cbp=(-1/Vb) (∂Vb/∂p)             

where   σc=constant.

Cbp is useful for heave/subsidence calculations induced by pore pressure change during production or injection. Several cases of such deformations have been documented in the histories of underground water extraction and hydrocarbon production. Some of the famous examples are San Joaquin Valley in California with 9m of subsidence between 1935 and 1977, Wilmington oil field in Long Beach ,California with 8.8m of subsidence between 1932 and 1965, Ekofisk oil field in North Sea with 8.5m of subsidence between mid 1970s and 2004, Wairakei geothermal field in the News Zealand with 14m of subsidence between 1950 and 1997, and Maracaibo Lake in Venezuela with 7m of subsidence between 1926 and 2004.

3- Formation compaction Coefficient (Cpc)

Pisa Tower
Figure 3. When it comes to settlement of buildings, there is nothing more famous than the leaning tower of Pisa with its uneven settlement. (source: Wikipedia)

This pore volume compressibility coefficient (Cpc) which is also called ‘formation compaction coefficient’ equals to the change in pore volume of the rock (Vp) with respect to the variation in the confining pressure (σc) while pore pressure (p) is held unchanged:

Cpc=(-1/Vp) (∂Vp/∂σc)  

where p=constant.

Cpc is used in subsidence (settlement) calculations induced by external loadings such as construction at the ground surface. Foundation settlement is an inevitable consequence of construction that needs to be controlled by geotechnical engineers. Almost all of us are familiar with the consequences of large and especially uneven settlement of foundations that can lead to ranges of effects from trivial to devastating on buildings and infrastructures. Probably the most  famous case of foundation settlement is the leaning tower of Pisa that has made it an attraction for the tourists but there are several other famous examples around the world.



4- Effective pore compressibility Coefficient (Cpp)

Effect of Compressibility on Flow
Figure 4. Results of fluid flow simulations using stream line method comparing cumulative oil production with and without compressibility effects (Source: Osako and Datta-Gupta, 2007).

Pore volume compressibility (Cpp), also called ‘effective pore compressibility’, equals to the variation in pore volume of the rock (Vp) with respect to the change in the pore pressure (p) while the confining pressure (σc) is unchanged:

Cpp=(-1/Vp) (∂Vp/∂p)  

where σc=constant

Cpp is frequently used in modeling of fluid flow in reservoirs and aquifers. Almost all the fluid flow simulations can take this effect in consideration. This parameter can become a critical parameter in less consolidated rocks where compaction acts as an important drive mechanism for hydrocarbon production.


5. Uniaxial Compressibility Coefficient (Cbu)

In petroleum geomechanics, it is common to assume reservoir’s deformations during production and injection to be uniaxial and in vertical direction. This assumption is not far from reality in many deep reservoirs that have a relatively small thickness compared to their lateral extension. In these cases, total vertical stress (which equals to the weight of overburden) does not change significantly as a result of pressure change.

In a uniaxial compressibility test, uniaxial pore volume compressibility (Cbu) is defined for a condition that the sample is not allowed to have lateral deformations during the test.

Cbu=(-1/H) ∂H/∂p

where lateral strain=0 and H in this equation is the sample’s height .

When Rock Behaves Elastically …

It is no secret that assuming elastic behaviour for rocks is not totally credible specially for large pressure changes and also in unconsolidated rocks. Nevertheless, it has been very common in the industry to assume an elastic behaviour for rocks due to its simplicity and also availability of elastic data from different sources (field tests, logs, seismic). None of these reasons, however, could give a green light to use such simplifications in rock behaviour without enough due diligence.

In cases where the rocks behave elastically, bulk compressibility of rocks (Cbc) is simply the inverse of its elastic bulk modulus (K). For an isotropic rock, this can be written as:

Cbc =1/K=E/[3(1-2v)]

where E and v are Young’s modulus and Poisson’s ratio of the rock, respectively.

Similarly, uniaxial bulk compressibility of rocks (Cbu) is the inverse of constrained elastic modulus (also called P-wave modulus) of rocks (M):

Cbu =1/M=E(1-v)/[(1+v)(1-2v)]

Relations Between Different Compressibility Coefficients

The following equations have been simply (and wrongly) suggested based on the relation between different volumetric components of rocks:

Cbp =φCpp+(1-φ)Cm 
Cbc =φCpc+(1-φ)Cm 

where φ is rock porosity and Cm is the compressibility of rock matrix (or grains in granular rock). As Zimmerman (1991) discussed, these equations have no theoretical basis and physical support.

Zimmerman (1991) showed that, assuming the validity of elastic behaviour, the following equations are also valid between compressibility coefficients measured at different applied pressure conditions:

Cbp = Cbc-Cm 

Cpp = Cpc-Cm 

Cpc =Cbp / φ=(Cbc-Cm)/φ = [Cbc-(1+φ)Cm]/φ

Once more, note that the oversimplification of mechanical rock behaviour and its compressibility using elastic parameters may lead to conclusions that are far from reality.

Geomechanics of Compressibility – Part II: Drained versus Undrained

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Read the first part of this series here.

Some Basics

The Four Main Parameters: The common concept of compressibility in geomechanics has been developed to study the variations in either of bulk volume or pore volume of rocks in response to variation in confining pressure or pore pressure. Let’s talk a bit about these four elements before going forward.

The Two Volumes: The two types of volumes used in different compressibility definitions are bulk volume (Vb) and pore volume (Vp). We usually like to know variations in Vb as it shows how our operations can affect ground deformation and we are interested in variation of Vp as it shows how porosity, a crucial parameter for fluid flow analysis, changes with underground operations. Apparently, these two volumes are related and by knowing rock’s porosity, they can be easily translated to each other.

The Two Pressures: Rock’s volume tends to change if either of confining pressure (σc) or pore pressure (p) varies. Traditionally, different types of compressibility coefficients used in the industry assume that pore pressure and confining pressure can change independently (this type of rock response is called drained). At first glance, it seems that rock’s deformation is only caused by pore pressure variation during injection or production but, in reality, in-situ stresses (which in essence are the pressures confining the rock) almost always change along with pore pressure variation (see Figure 1). So, the volume change is commonly the result of changes in both pore pressure and confining pressure simultaneously (this type of rock’s response is called undrained). Due to its importance, for more clarification, let’s pause here and talk more about the difference between the drained and undrained behaviours.

stress change - depletion
Figure 1. A simple schematic showing how total stresses within and around a reservoir change by pore pressure decrease (i.e., production). Each pair of arrows pointing towards each other demonstrates a compressive state of stress change that squeezes the element in the direction of arrows while the ones pointing away from each other denote an extensile state of stress change that is trying to pull the element apart in the direction of arrows.


Drained versus Undrained Behaviour

A Geotechnical Classic: Let’s start with a classic example from the field of geotechnical engineering (Figure 2) that models the ground behaviour by using a cylinder and piston system. The fluid under the piston stands for the pore fluid and the spring mimics the soil/rock’s matrix behaviour. The fluid flow through valve represents the hydraulic conductivity (which increases with permeability of the soil/rock). While the building in Figrue 2 is constructed at the ground surface, if the draining valve is closed and there is no pathway for the fluid to escape, the applied weight of the building will be taken in parts by both of the rock’s skeleton (as effective stress) and the pore fluid (as pressure), simultaneously. This is called undrained behaviour. Now, if the fluid can escape because the draining valve is open, the excessive pore pressure disappears almost instantly and we can assume pore pressure change is negligible and, therefore, all the load is taken by the skeleton. In his case, the effective load taken by the skeleton equals to the total load applied to the rock. This is called drained behaviour. Of course, it is possible to have other conditions between these two extremes if the valve is half open.

Figure 2. This classic and great schematic tries to show the difference between drained and undrained response of soil/rock to the construction load and also the concept of effective stress as introduced by Terzaghi.  In this simple model, the saturated soil/rock system is represented by a container filled with fluid and it is capped by a piston that resembles the ground surface. The load of piston can be carried by either or both of the fluid and the spring (that resembles the soil/rock skeleton). In undrained condition,  the valve is closed so there is no escape path for the fluid and, so, it has to bear the load along with the skeleton. As soon as the fluid can escape through the valve, it will not take any of the construction load and all the load has to be taken by the skeleton.

A More General Definition: A more inclusive definition f0r the drained and undrained behaviour can be explained based on the concept of dependency or coupling of confining pressure (total stresses) and pore pressure. Based on this definition, if pore pressure and confining pressure are not coupled and can change independently, the rock’s behaviour is called drained while if these two are coupled and can affect each other, the rock’s behaviour is called undrained.

In Reservoir Geomechanics: As in the case of production and injection, pressure variation almost always leads to changes in total stresses as shown in Figure 1, according to the given definition, we will have an undrained behaviour.

Drained/Undrained Conditions and Compressibility Tests: In practice, most of compressibility tests used to measure rocks’ compressibility coefficients are performed by increasing confining hydrostatic pressure (i.e., an external omindirectional stress) on dry/unsaturated samples. This is a form of drained behaviour as pore pressure does not play a role in these tests. Some of the more developed drained tests try to mimic what happens in the field by letting the dry/unsaturated sample deform only in the vertical direction (i.e., uniaxial deformation).

In a more realistic version of compressibility testing, estimated in-situ stresses are initially applied to the rock sample and, while keeping the deformation uniaxial to represent common reservoirs’ behaviour, pore pressure is gradually changed to simulate injection or production. This type of test can be called an undrained compressibility test as stresses change with changing pore pressure.

Read the third part of this series here.

Geomechanics of Compressibility – Part I: Why Is It Important?

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After encountering frequent misunderstandings of rock compressibility and its applications, I decided to write a series of posts on ‘geomechanics of compressibility’ to explain its rock mechanical definition, its different types, methods of compressibility measurement and their differences and the parameters affecting this property especially stress path and hysteresis.  As an introductory post, let’s start with explaining why compressibility is important to us. But wait a moment! If you feel the equations are eyesores just ignore them, I don’t think it will really matter.

Compressibility Graph
Figure 1. An old but popular graph based on laboratory tests used for estimation of rock compressibility for fluid flow analysis in reservoirs. (source: Hall, 1953).

Fluid Flow Analysis

Pore volume compressibility (Cp) has long been recognized as an important factor in fluid flow simulation for aquifers and reservoirs as shown by the following fundamental equation of fluid flow in porous media:

(Cp+Cl)ρφ ∂p/∂t+∇.(-k/μ(∇p+ρg∇z)=q 

[In this equation, Cl is fluid compressibility, ρ is pore fluid density, φ is rock porosity, g is gravity acceleration, q is the source term, k is permeability, t is time and z is the elevation measured in the vertical direction oriented downward.]

Generally, in either of fluid flow simulations or material-balance calculations, the role of pore volume compressibility coefficient (Cp) becomes increasingly important as the fluid compressibility decreases. The importance of pore volume compressibility is even more crucial for closed systems where, in absence of fluid flux, fluid flow and pressure changes are controlled mainly by pore volume changes.

Hydrocarbon Reserve Estimation and Storage Capacity Evaluation

Assessment of Carbon Storage in US
Figure 2. Assessment of potential CO2 storage in United States (Source:


Pore volume compressibility is also important in volumetric estimation of hydrocarbon reserves (find more in here) and evaluation of fluid storage capacity and efficiency in aquifers (for instance, waste fluid disposal or CO2 sequestration). As a simple example, in estimation of storage capacity of closed-system aquifers, the efficiency factor (Ei) for storage is introduced by Zhou et al. (2008) as:

Ei=(Cp+Cw )Δp

[where Cw is the compressibility of water and Δp is the average pressure increase within the aquifer induced by injection.]

Porosity Change - Gilwood
Figure 3. Variation of porosity with two consecutive cycles of pore pressure change (injection/production) in a uniaxial pore volume compressibility test (UPVT) for a sample from a sandstone formation in Alberta, Canada  (Source: Soltanzadeh, 2016).

Approximation of Porosity Variation

Another application of pore volume compressibility is for the estimation of porosity change induced by pore pressure variation within a reservoir. The following equation is widely used for porosity approximation of consolidated and cemented reservoir rocks (e.g., Satter et al., 2008):

φ21 exp(Cp(p2-p1))

[where φ1 and φ2 are the values of porosity, at reservoir pressures of p1 and p2, respectively.]

Such relations are the most simplistic way of involving geomechanics in fluid flow simulation. However, as it can bee seen in Figure 3, such relations must be used with the most caution as I will discuss it in detail later in this series.

San Joaquin Valley Sunsidence
Figure 4. Historic 1977 photo depicting the location of maximum land subsidence in the U.S., near Mendota, CA in the San Joaquin Valley. Joseph Poland (pictured), USGS, scientific subsidence studies pioneer, placed the date signs to indicate previous elevations (Source:

Estimation of Ground Deformation

Bulk volume compressibility coefficient, when measured using a uniaxial pore volume test, can be directly used for calculation of reservoir or aquifer contraction or expansion induced by production or injection.

In general, the expansion of ΔH induced by the average pore pressure increase of Δp in a reservoir or an aquifer with an average height (thickness) of H may be calculated from the following equation:

ΔH= CbuΔpH

[where Cbu is uniaxial compressibility.]

Some rocks, such as consolidated sandstones, behave elastically when stresses are less than critical yield stresses. Rocks show more elastic responses when pore pressure is increased e.g., in the case of waste fluid disposal or CO2 sequestration (Fjær et al., 2008). When rock behaviour is isotropic and elastic, the following relation exists between uniaxial bulk compressibility and rock elastic parameters:


[where υ is Poisson’s ratio and E is Young’s modulus of the rock.]

Hence, in absence of other reliable data, bulk volume compressibility can be used as an auxiliary parameter for estimating elastic properties of the rock.

Figure 5. Maurice Anthony Biot (1905-1985) in his apartment at 300 Central Park West, New York, September 1964 (Source:

Calculation of Biot’s Coefficient

Bulk volume compressibility coefficient (Cb) may also be implemented in the following equation to estimate Biot’s coefficient (α) as a key parameter required for any geomechanical analysis:


where Cm is the matrix (or grain) compressibility and can be measured using an unjacketed hydrostatic test, or it can be estimated from the mineralogical composition of the rock (Zimmerman,1991).

Read the second part of this series here.


Fjaer, E., Holt, R.M., Horsrud, P., Raaen, A.M., and Risnes, R., 2008. Petroleum related rock mechanics. 2nd Edition, Elsevier, Amsterdam.

Hall, H.N., 1953. Compressibility of Reservoir Rocks, Journal of Petroleum Engineering, 5(1).

Satter, A., Iqbal, G.M., and Buchwalter, J.L. 2008, Practical Enhanced Reservoir Engineering: Assisted with Simulation Software. Pennwell Corporation, Oklahoma, 688 p.

Zimmerman, R.W. 1991. Compressibility of Sandstones, Elsevier, Amsterdam, 173 p.

Zhou, Q., Birkholzer, J., Tsang, C.-F., Rutqvist, J., 2008. A method for quick assessment of CO2 storage capacity in closed and semi-closed saline formations. Journal of Greenhouse Gas Control 2, 626–2,639.

A Primer on the Geomechanics behind Fracturing Pressure Curves

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PDF version of this article: A Primer on the Geomechanics behind Fracturing Pressure

In this post, I will break an idealized pressure curve (like what is recorded during hydraulic fracturing or pressurize fracturing tests) in different segments and explain the geomechanics behind each. I will try to answer questions such as:

  • Why does pressure increase, decrease or remain constant in each segment?
  • At what pressure a fracture initiates? At what stage it can be considered a mature fracture? what pressure is required for a fracture to grow?
  • What are the important pressure values on this curve and how they are representing the mechanical state of the rock? How well-known pressure values such as leak-offinitiation, breakdown, propagation, shut-in, and closure pressures are defined and what is their geomechanical significance?

I will deliberately avoid getting deep in explaining mechanical models behind fracture initiation and propagation for the sake of simplicity.

Pressure Curve - Hydraulic Fracturing
Figure 1. A pressure curve idealizing what is usually measured during hydraulic fracturing or pressurized fracturing tests. Remember that the graph is schematic and not-to-scale.


To avoid complexities that are out of the scope of this primer article, I will use an ideal case of hydraulic fracturing similar to pressurized fracturing tests such as mini-frac, extended leak-off, or DFIT tests (Figure 1). The curve in Figure 1 simply shows how fracturing fluid pressure (on the vertical axis) varies with time (on the horizontal axis). The rate of injection is assumed to remain constant before the pumps are turned off. Note that this is an ideal curve and similar to many other idealization in engineering, the real curves may not look as smooth as this one. Also, the graph has not been drawn to scale on any of its axes to ensure all the major details and variations could be demonstrated. In addition, it was assumed that no natural or induced fractures exist in the zone of interest prior to fracturing.


The operation is performed on an isolated zone of a wellbore that can be either cased or open. In a simple pressurized fracturing test, the fracturing fluid is injected at a specific and constant rate for a period of time (that is to be known by the response of the rock to injection during the test) and then pumping is stopped although the pressure measurement is continued. In massive hydraulic fracturing, the injection rate varies by time and varying volumes of proppants are also injected along with the fracturing fluid.

Fluid pressure is measured throughout the entire test most likely at the wellhead and occasionally downhole. If pressure is measured at the wellhead, it needs to be converted to downhole pressure by accounting for the hydrostatic column of the fluid and all the dynamic pressure losses caused by friction and other effects during injection. This conversion becomes more cumbersome in massive fracturing jobs performed with high injection rates, special fluids (viscose, energized, foam, nitrogen, etc.) and proppants.

Ascending Straight Up (A-B)

After injection is started, the low-permeability target interval is usually intact with no fractures to let the injected fluid escape. At this condition, by continuing injection in the isolated volume of the borehole, the fluid will be compressed and, as a result, pressure has to increase. The rate of pressure increase (e.i., the slope of the line A-B) depends on different parameters mainly the compressibility of fracturing fluid (e.g., you can inject a larger volume of a less compressible fluid with less increase in pressure) and the rigidity of your container (the well). The rigidity of the container varies based on whether the well is cased or not and, also, dependent on how packers used for zone isolation and other tools will deform in response to pressurization. This straight line might be affected by high permeability of the formation, pre-existing fractures, or fluid pathways related to the cement job.

A Little Bit Extra!

If injection is stopped at a desired pressure along this period , the test is called Formation Integrity Test (or FIT). This test is used to ensure the target formation is competent enough to stand the maximum pressure needed for drilling or enhanced recovery. Nowadays, however, conducting full-cycle tests is more favored as it provides much more useful information.

When It Bends (B) – Leakoff Pressure

The discussed straight line does not continue forever and there comes a so called ‘leak-off’ point where this line bends. This is the time when induced fractures are starting to form. Initiating fractures means there will be more room for the injected fluid to occupy. Having this extra room, fluid will not get as much pressurized as before and the slope of the line is reduced and it will appear as a bending point.

Although fractures are already initiated at this bending point, they should not be considered as maturely extended fractures. These initiated fractures are small in both length and width and they are not likely to propagate far without being exposed to greater pressures. Note that Leakoff pressure is usually greater than minimum in-situ stress and the reason is speculated to be the stress concentration around the borehole.

The Uphill (B-C)

By keeping on injection, the initiated fractures will open wider and extend farther from the well and, as a result, more room will be created for the injected fluid. This extra room means less pressure increase and more bending (the curve slope will reduce) in response to more fluid injection. This segment of the curve might be quite short for the highly brittle rocks. Fluid injection type, rate and viscosity along with the complexity of the fracture also play roles in forming this uphill segment.

An important point to remember here is that, at this stage, the fracture is ‘stable’ in contrast to what we will see soon in the next segment of the curve. A stable fracture needs higher pressure to overcome the rock’s resistance against propagation and if the pressure does not increase, the fracture will not grow anymore. At this stage, more injection and pressure is required to extend the fractures meaning that the operator is in full control of the fracture’s destiny. However, as soon as the climax of the curve is passed, we are going to lose control as will be discussed in the following.

The Climax (C) – Breakdown Pressure

This is a climax necessary for creation of a trustworthy fracture. For a long time, definition of breakdown pressure and its difference with fracture initiation pressure has been a source of debate mainly due to the complex physics behind the problem. There are some less popular theories that speculate that the time of breakdown is when a fracture actually initiates (e.g., Boone and Ingraffea, 1989). However, the commonly accepted theories in fracture mechanics believe in existence of fracture prior to this time. These theories, however, differentiate the status of the fracture before and after breakdown. According to these theories, breakdown is a point where the fracture moves from a ‘stable’ to an ‘unstable’ condition (Guo et al., 1993, is a great read on this if you are interested). They also sensibly argue that even the fluid entrance into the fracture and pressure distribution within the fracture are different in these two distinct states.

Breakdown pressure has been observed to be dependent on fracturing fluid type and viscosity, injection rate and borehole size.  Efforts to simply calculate breakdown pressure from elastic models (commonly used in borehole stability and drilling models) have not been very successful. Also, there have been some efforts in the industry to use the recorded breakdown pressure in these models to estimate magnitudes of in-situ stresses using elastic models, mostly showing less success.

A Little Bit Extra!

The similarity between fluid pressure-time graphs recorded during fracturing (Figure 1) and stress-strain curves measured during compressive failure of rock (Figure 2) is interesting. It might have been the reason that, in earlier times, some experts (e.g., Morgenstern, 1962theorized that breakdown of the rock might be the result of shear failure. Some experiments have also shown that geometry complexities of the curved or parallel fractures have major influences on the magnitude of breakdown pressure (see Figure 3 for an example). 

Compressive Triaxial Test
Figure 2. A schematic of stress-strain curve as recorded during compressive triaxial test. In this test, the rock is believed to mostly fail in shear. Although the general appearance of this curve looks similar to the fluid pressure-time curve recorded during hydraulic fracturing, the modes of fracturing in these two cases are known to be very different.

Losing Control (C-D) – Relief-In Pressure

At the breakdown point, the energy provided by pressurization helps the fracture to become mature enough and grow unstably. This unstable fracture is not in control anymore and employs the previously stored energy along with the currently injected one to grow wider and farther. As a result of this extensive fracture propagation, the fracturing fluid has a lot of room to occupy and so, it relaxes some of its high pressure and the fluid pressure drops substantially. There is another reason for pressure drop: the unevenly distributed pressure in the previous immature fracture is now re-distributed much more uniformly in the current wide and long fracture.

As we will see in the next section, like any instability with a limited amount of energy, this one has to come to a stable state if given enough time.

A Little Bit Extra!

Based on several lab simulations of hydraulic fracturing for wells with different orientations , Abass et al. (1996) showed that the pressure loss during this unstable period (so called ‘relief-in pressure’) is related to fracture geometry complexities such as curving.

Breakdown and Relief-in Pressure vs, Well Orientation
Figure 3. Results of experimental hydraulic fracturing tests performed by Abass et al. (1996) showing variation of breakdown pressure and relief-in pressure versus change in horizontal wellbore orientation with respect to the in-situ horizontal stresses.

The Flat Ride (D-E) – Propagation Pressure

Ultimately, with no change in the rate of injection, fluid, fracture and rock will all come to a stable and balanced condition where, first, the existing pressure at the tip of the fracture is exactly what is required to extend the fracture and, second, the volume of the injected fluid is exactly in balance with the fracture volume generated by fracture extension.

Having everything in balance, pressure does not need to vary significantly if injection rate does not change. This equilibrium pressure is called fracture propagation pressure or fracture extension pressure or simply fracturing pressure. Fracturing pressure is higher than minimum in-situ stress and it is usually used to determine the allowable upperbound pressure during drilling or injection to avoid fluid loss or leakage.

Enough Pumping! (E) 

So far, many things have been revealed throughout a course of injection of a fluid in an isolated interval of a well. Things such as how the wellbore as a container reacts to injection, how much pressure is required to initiate the first fractures in the rock, at what pressure we can create a ‘mature’ fracture, and finally, the balanced pressure at which the fracture keeps propagating. In case of hydraulic fracturing jobs, the operator has a desired fracture geometry in mind so s/he keeps injecting until s/he is convinced that the desired fracture geometry is achieved based on the designs (I leave it to him/her to tell us how much s/he trusts the results). In the case of formation tests, keeping on injecting for long is not going to reveal much more. In contrast, there is still so much valuable knowledge to be learned by stopping injection and simply observing the pressure response of the system.

Free Fall (E-F) – Instantaneous Shut-In Pressure (ISIP)

As soon as the pumps are off, a sudden drop will happen in the pressure curve and pressure will fall to a value called Instantaneous Shut-in Pressure (ISIP). This drop happens because the pressure caused by flow turbulence and friction during injection instantly disappears after pumping is stopped. With no influences from the dynamic flow, the mechanical characters of rock and fracture are probably less masked in ISIP in comparison to the previously recorded parameters. This is the reason that ISIP has gained so much popularity in the industry.

Some may argue that ISIP is the ‘real’ fracture propagation pressure as it does not include the dynamic effects of the flow. This reasoning might not be very convincing as fracture propagation pressure cannot be really considered valid if fracture does not propagate. In other words, existence of flow and its characteristics can hardly be separated from fracture propagation.

Curtains Closing (F and Beyond) – Closure Pressure

After shut-in, the fracture will stop propagating and instead, in absence of the required pressure for its propagation, it will start closing. Fracture closure is the consequence of pressure drop in the fracture as fluid flows back into the well and penetrates into the rock, simultaneously. This period is probably the most favorite part of the operation for geomechanics as it provides a great opportunity to find closure pressure, which is a great proxy for minimum in-situ stress. Let me emphasize here that closure pressure is not exactly minimum in-situ stress (i.e., a parameter that we might never be able to measure it exactly) but it can be very close to this stress component. One other thing to keep in mind is that the exact location of closure pressure on the curve is not always easily identifiable and industry has come up with several different approaches to estimate it.

After closure, pressure will still decline due to the permeable behaviour of the fracture and rock but contribution of geomechanics to the process becomes trivial. The rest of the curve is highly favored by the engineers who want to find out more about fluid efficiency, formation leakoff capacity, permeability, and reservoir pressure.


Abass, H.H., Hedayati, S., Meadows, D.L. 1996. Nonplanar Fracture Propagation From a Horizontal Wellbore: Experimental Study. SPE 24823.

Boone T.J. and lngraffea A.R. 1989. Simulation and visualization of hydraulic fracture propagation in poroelastic rock. The Report to NSF Grant 8351914.

Guo, F., Morgenstern, N.R., Scott, J.D. 1993. Interpretation of Hydraulic Fracturing Breakdown Pressure Int. J. Rock Mech. Min. Sci. & Geomeeh. Abstr. 30, 6, pp. 617-626.

Morgenstern N.R. 1962. A relation between hydraulic fracture pressure and tectonic stresses. Geofis. Pura Applic. 52, 104.

Curving and Re-orientation of Hydraulic Fractures

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What do we know about the geometries of hydraulic fractures? Are they really as simple as the straight planes we usually assume in our reservoir simulations and fracture designs? What factors influence their geometries? What are the geomechanical mechanisms behind geometry complexities? How do our completion designs affect fractures geometries and what completion designs can reduce the adverse effects of geometry complications? What is the influence of fracture geometry on production?

Our recently published article in CDL’s Discovery Digest titled ‘Fracture Re-Orientation: The Impact on Completion and Production’ is trying to investigate these questions and present a general overview of the current knowledge of fracture re-orientation and curving in a simple language. The article briefly reviews several mechanisms responsible for fracture re-orientation and curving and their effects on the efficiency of hydraulic fracturing.

Below are some of the topics discussed in the article:

  • Near-wellbore tortuosity and curving;
  • Fracture bending, branching, merging and link-up;
  • Curving potential in offset and infill wells;
  • Effects of natural fractures and faults on hydraulic fracture geometry;
  • Behaviour of fractures at lithological interfaces;
  • Influence of production on fracturing and re-fracturing.

Fracture Curving Article - Page One