We may prefer to think different modules in coupled models are like matching pieces of a jigsaw puzzle and, by coupling, we are just fitting them to figure out the big picture. In reality, this is not probably the most accurate image of the problem as, usually, different modules in these models are for different physics with different field equations, different discretization methods, and even sometimes different time steps, a fact that modelers ought to remember to acquire precise results from their models. In practice, these unfitting pieces need to be stitched to each other by interfaces, i.e., the mediums with the role of transferring and translating data between these modules. Like any language translation, if translated loosely or if some data get lost in translation, misinterpretations and errors will dominate the solution. In this post, I will discuss some of the common communication issues that can threaten the accuracy of coupled modeling. Some others will be discussed in another post of this series to be published soon on geometry and gridding.
Cell-To-Node Projection, A Big Deal!
This is a less noticed (or better to say ‘less considered’) issue that has the potential of leading to significant errors especially in locations close to the boundaries of the permeable and low permeability rocks (for instance, a reservoir and its caprock). Most likely, for many justifiable reasons, your fluid flow/heat transfer module uses a version of finite volume method (FVM) for solving its field equations while your geomechanics model uses finite element method (FEM). These two methods are significantly different in the ways they discretize geometry and in how they treat the field equations, set aside the different physics involved. While in FVM the equilibrium equations are written for a volume (cell), in FEM, they are written for nodes of an element. As a result, in FVM, properties (e.g., pore pressure, temperature, or stresses) are assigned to the volume of discretized cells but, in FEM, they are assigned to the nodes of elements. As its name makes it evident, the cell-to-node projection becomes a concern when the common properties between these modules are transferred or projected between the cells of one module to the nodes of another. To get an idea of this process, see the simple example below.
Assume a basic fluid flow – geomechanics coupled model with very simple geometry as shown in Figure 3a for a reservoir and its caprock. For simplicity, both modules have the same mesh.
1. Imagine, as a result of production, pore pressure change of dp is given by the fluid flow module within the reservoir and, as expected, no pressure change is predicted by this module in the caprock (see Figure 3b).
2. The given cellular pressure needs to be translated into nodal properties of the geomechanics module. Using a simple technique, pore pressure at each node is calculated by averaging pore pressure in the cells around it (Figure 3c).
3. Now, let’s see how these nodal pressure changes are understood by the geomechanics module. This module simply calculates pore pressure change in each of its element by averaging the nodal properties and gives what we see in Figure 3d.
You see how a pore pressure change of dp/4 has been artificially introduced into the bottom layer of the caprock by just data transfer from one module to another. Also. pore pressure change within the top layer of the reservoir is 3dp/4 in the FEM model lower than the fluid flow module prediction . It works similarly for heat transfer.
To get a sense of the importance of this issue, imagine the model given in the example above has been developed for a caprock integrity assessment of an in-situ thermal project (e.g., SAGD). Having artificially created pore pressure and temperature in the riskiest location of the caprock easily leads to significant increase in the predicted chance of caprock failure and predicts lower maximum allowable pressure (MOP), something a producer does not like to hear. Goumiri and Prevost (2010) tried to quantify the error rising from cell-to-node projection.
Dealing with the cell-to-node projection is not probably the easiest task in modeling. One way to solve the problem is using specific numerical techniques such as using piecewise shape functions and low order integration in the FEM module. This is not the modeler’s job and must had been formulated in the software by the developer. This, of course, will take the load off your shoulders, so, if you are using a full thermo-poro-mechanical software suit, always check your software’s technical manuals to see how its interface treats the issue of data transfer between different modules. However, several modeling softwares have not been designed with this consideration, or in some cases, the modelers have to write their own interfaces to facilitate data transfer between different modules. In these cases, a solution is using very fine gridding at the vulnerable boundaries so you will be able to ignore the wrongly behaving elements with more confidence and, instead, track your model behaviour at the adjacent observation cells that are not significantly affected by the issue (see Figure 4, for an example). Splitting your thick caprock in 4 equal layers and tracking the response in the bottom cells, for instance, very likely will give you inaccurate results.
Strength of Coupling
Different coupling approaches may be used in modeling such as partly coupled, one-way, two-way, and iterative two-way. Serious problems might occur when coupling degree between different modules is not strong enough and it neglects or underestimates the effect of different physics on each other. For instance, for a mechanically-sensitive reservoir, implementing a one-way or even a simple two-way coupling approach might lead to significant errors in the results. Examples are unconsolidated rocks such as shallower reservoirs or oil sands or shale gas reserves and fractured reservoirs. I fully acknowledge this fact that we still have a long way to understand the detailed physics of fluid flow/heat transfer and geomechanics interaction for different reasons such as the science being young and research being costly and expensive but there is no excuse for ignoring what we already know.
Naturally, for a modeler coming from one discipline, it is a common tendency to underestimate the influences or complexities of other disciplines and ‘take it easy’ when it comes to ‘them’. Flipping through the literature, many examples may be found, e.g., reservoir engineers who see geomechanics as one of the ‘add ons’ of their reservoir modeling softwares and also geomechanics specialist with similar mentality about fluid flow modeling. We may even encounter experts from other disciplines (like mathematicians) who, without a solid understanding of any of these physics, still insist on doing all different parts of the job themselves assuming the story is just about solving some equations or using a software package. Occasionally, a similar attitude may even be observed in some software developing companies. Of course, in this diverse world of science, nobody can or expected to master every branch of science and coupled geomechanical modeling, as an inter-disciplinary practice, is not an exception and it, definitely, needs the collective knowledge of a diverse team of experts or let’s call it ‘expertise coupling’ in addition to coupling of different physics for its success.
To be continued.
In less than two weeks, at URTec 2015 in San Antonio, I will be presenting two papers on geomechanics of the Duvernay Formation based on the Canadian Discovery’s study on this play. The papers provide comprehensive information on different geomechanical components, workflows and results of the study.
The first paper (to be presented on Monday, July 20) discusses some of our interesting observations based on detailed reviews of drilling experience for several wells that confirm the existence of different regional drilling patterns in the play. The paper demonstrates how using in-situ stresses, wellbore stability analysis, and core descriptions can help with explaining the reasons behind these differences, a major one being the potential influence of existing carbonate reefs on the geomechanical response of the play.
The second paper (to be presented on Tuesday, July 21) reviews the integrated workflows developed and implemented to combine core description and geomechanical, geochemical, and petrophysical data to identify and characterize different fracture fabrics. The paper also shows how integrating these data can provide a reliable methodology for characterizing rock fraccability and brittleness. Such a methodology may be a strong base for using log and seismic data for fracture characterization through quantitative interpretation.
In case you are also attending the conference, it would be great to have you there for the presentations and hear your questions, comments, and suggestions. I may also be found at the CDL booth in the exhibit area for most of the conference time. The abstracts for the papers are given below.
A Regional Review of Geomechanical Drilling Experience and Problems in the Duvernay Formation in Alberta
Mehrdad Soltanzadeh, Amy Fox, Sarah Hawkes, David Hume
The Duvernay Formation has been an attractive unconventional play for several producers during the last few years, and the number of drilled wells in this play has been increasing rapidly. Nevertheless, drilling in this formation is usually considered a challenging practice due to the extremely high stresses and pore pressures that can be encountered. Drilling in the Duvernay and its overlying formation, the Ireton, has shown a wide variety of drilling incidents such as sloughing, tight spots, bridges and lost circulation. This paper summarizes the results of a comprehensive regional review of drilling experience for 43 Duvernay wells. In this review, the details of drilling experience for these wells were documented using a graphical approach that captures important information on the details of drilling incidents including depths and dates, along with information on mud weights and well trajectory. As an initial part of a broader regional geomechanical study of the Duvernay, these data were statistically analyzed to identify the stratigraphic and spatial variation of drilling patterns throughout the study area. The results revealed significant differences between drilling patterns in the three major active areas of the play including south (Willesden Green and Edson), central (Kaybob), and the northwest regions. In general, wells in the central region can be drilled with lower mud weights than other regions, and experience fewer drilling problems. Because pore pressure in the Duvernay Formation in this area is as high as in the other two regions, the difference in drilling experience was attributed to considerable differences in in situ stress, which appears to be related to the presence of Leduc reefs. These stress differences were confirmed by modelling and distribution of fractures in the study area.
Application of Mechanical and Mineralogical Rock Properties to Identify Fracture Fabrics in the Devonian Duvernay Formation in Alberta
Mehrdad Soltanzadeh, Graham Davies, Amy Fox, David Hume, Nasir Rahim
Mechanical rock properties, along with in situ stresses and pore pressure, play critical roles in forming fractures in reservoir rocks. As part of a regional geomechanical analysis of the Duvernay resource play, several Duvernay cores were analyzed in detail, including the identification of different types of natural and induced fractures. The observed natural fractures include uncategorized natural fractures and polished slip faces (PSF) with rare presence of cleavage. Coring-induced fractures included petal and petal-centreline fractures and bed parallel parting (BPP). Comparison of the presence of the different fracture fabrics with mechanical and mineralogical properties of the rock revealed strong correlations between rock properties and fracture types. Such correlations may be efficiently implemented for characterization of fracture fabrics in the rock using wireline logs or seismic surveys. The observed natural and induced fractures in the cores have also been utilized to revisit and verify the concept of rock brittleness. The analyses show that, as a result of high clay content and overpressuring, the conventional mineralogical and mechanical brittleness indices do not adequately describe the variability of the Duvernay Formation stratigraphic units. Alternative indices developed for this study (i.e., plane-strain Young’s modulus and clay-based brittleness index) seem to be able to represent the mechanical behaviour of rock much more precisely. This study suggests that using natural and induced fracture fabrics observed on image logs and in cores, along with mineralogical and mechanical rock properties, is a more practical approach to assist with identifying sweet spots in unconventional plays.
Reservoir-scale models usually integrate different physics like rock mechanics, multi-phase fluid flow, heat transfer, fracture mechanics and sometimes geochemistry and geophysics. Considering all these physics at the same time and solving their filed equations simultaneously (fully-coupled modeling), if possible, seems to be an ideal case but usually it is less practical for different reasons such as being computationally intensive, costly, complex, unstable, and difficult to learn, run, and troubleshoot. Therefore, instead, models developed based on coupling of separate modules are much more popular in industry. In this modeling scheme, each module handles only one or two physics at a time and feeds its results to the other modules in every step of modeling. Figure 1 shows a generic illustration of geomechanical models that couple rock mechanics, fluid flow, heat transfer, and fracture effect on fluid flow. This type of coupling is very attractive to the users as it allows them to find the most practical, advanced, available, and affordable software packages in each discipline and tie them to each other. Although attractive, coupling of different modules has the risk of leading to erroneous results mainly due to the lack of an integrated perspective of how these modules interact with each other. This series of posts briefly discusses some of the issues that may show up and need to be looked after in the process of coupled modeling.
The Stressful Stage of Stress Initialization
Probably one of the most challenging tasks of modeling of complex geologies is applying the right initial stresses to the model. Ideally, in case you run a model with the proper initial in-situ stresses in a stationary or steady state (no external loading or deformation applied), it is not suppose to show any further deformation or stress changes. In other words, it must be and stay in the equilibrium state since the beginning. Unluckily, in the real world of geomechanical modeling of complex structures, this is not an easy condition to hold as the in-situ stresses are usually determined by the user separately and without really caring about the static equilibrium of the 3D or 2D models. These data usually come from different sources such as field tests, frictional equilibrium, simplistic poroelastic models, or wellbore stability analysis and they come in the form of single data points or well profiles that need to be populated in model’s volume. So, when the model is running, the first thing it tries to do is taking the stresses to an equilibrium state that is usually accompanied by inducing new deformations and stress changes in the model. Eventually, the initially introduced stresses may change to a different stress state that can be different from the initially introduced stress state to the model and note that all of these happen without applying any loading, drilling, fracturing or injection/production.
There are different ways to tackle this problem. One simple way is ignoring the effect of initial stresses in numerical modeling, run the model without any initial stresses (zero-stress model) and simply superimpose the induced stress changes by field operations to the initial stress state. As much as off beam this method sounds, in fact, it can be working fine if i) you do not care about initial stress state of your model to be in equilibrium and ii) if your rock lives in a linear elastic life where its behaviour is not dependent on its past or current stress/strain state (see Figure 3 for an example). This might be the case for some consolidated sandstones or carbonates or even less consolidated rocks that undergo limited changes in loading, pressure and temperature.
Some modellers may prefer to apply their acquired initial stress data and run the initial model (without any external loading) to an equilibrium stress state where no further changes observed and, then, consider the newly developed stresses as their initial stress state for modeling. This is only correct if the changes in stress state from initial to equilibrium are not significant and can be ignored. Another, probably more reasonable, approach is applying the tectonic strains (deformations) derived from back analysis of stress measurements to the model and let the stresses develop (Figure 4). Minimizing the difference between the developed stresses and initial stress state based on user’s data can help with finding the best tectonic strain approximation.
No matter what approach is used for stress initialization, there are always some uncertainties that come along at this stage of the job. We need to remember that it is always important to verify the equilibrium state of the model before applying further changes in terms of drilling, fracturing, or pressure or temperature. We need to let our model run in its initial condition for a while and make sure the changes are trivial. If not, we need to take action and come up with a solution. When dealing with a complex geometry with several different formations which are not simple flat layers, the problem becomes more cumbersome. Presence of faults, makes the issue even more challenging.
A more complex issue shows up when the initial disequilibrium of the in-situ stresses in thermo-poro-mechanical models is not just the result of imbalanced stress initialization but it is also caused by the unsteady fluid flow or heat transfer in the initial model. For instance, having a non-steady pore pressure state in the model may lead to fluid flow from one zone to another leading to pore pressure change and, consequently, perturbation of initially introduced stresses.
In case of having no data or a small amount of data to compare with our model’s results, or when we are simplifying our geometry, or in cases when we are ignoring the effects of pressure and heat imbalance, we might get the impression that stress initialization is not a big deal as the model is doing the initialization job for us automatically but, in reality, it is the ignorance of the model that makes it look easier to handle and this is nothing to be excited about. Sometimes we cannot or decide not to do anything about the issue; this is fine as long as we acknowledge the shortcomings and potential errors that may arise from our decisions and lack of certainty.
It would be a great feeling to hold a borehole in the palm of your hand, turn it around, look at the propagated fractures around it, and ask yourself: “ok, why don’t these two fracture wings propagate in the same plane?” This was my experience when I was playing with the glass blocks with deviated boreholes drilled in them at Geoconvention this year. The fractured blocks were the results of photoelasticity tests performed at University of Alberta trying to simulate fracture initiation and propagation induced by drilling. The results show how different stress states can lead to formation of different types of fractures such as bi-wing, en echelon, bottomhole, and petal (see Jia et al., 2015 for more details). As can be observed in the figures here, the method proves to be an excellent mean for teaching geomechanics, a discipline that severely suffers from the lack of tangible educational resources.
Integration of Data and Disciplines for Unconventional Plays: Geomechanics, A Good Spot to Build A Junction
The emergence of unconventional plays has been so rapid during the last decade that, despite tremendous progresses, the science of petroleum engineering, which has been initially developed to treat conventional plays, has not had enough time to catch up and provide the most optimal solutions for these plays yet. The truth is the complicated nature of these plays, along with the complexity of the technologies applied for their exploitation, seem to require much more integration of data and disciplines in comparison to conventional reservoirs. We may have been able to luckily survive with the least amount of interdisciplinary work for many conventional plays due to their high recovery rates and inexpensive production costs but the story has been quite different in the case of unconventional plays especially in the current state of oil and gas prices that is endangering the economic profitability of these reserves.
Fortunately, seeing it happening all around, it seems that almost the entire industry has realized the value and significance of integration of data and disciplines for development of these plays. In supposedly properly-designed integrated studies, different expertise need to come together, carefully listen to each other, re-orient and adjust their methodologies, tools and products based on the mutual needs, and handle the problem in a holistic approach. Although some successful cases of integration can be counted, unfortunately, some of the so-called ‘interdisciplinary works’ in the industry are mainly focused on one discipline while trying to outreach others by implementing a simplistic understanding of them and, not surprisingly, mostly through broken knowledge of these disciplines.
As Richard Borden briefs us (see the quote below), human mind naturally tends to treat the science in an analytical approach and, therefore, integration has always been a major challenge for it. Ideally, for unconventional plays, we need to build integration teams (and I do not mean managing or lead positions in their current forms) who are specialists in doing this job but it seems that we are still far from this goal mainly because not all laws and methodologies of integration in unconventional plays are really clear and understood at the moment.
In the absence of professional integration teams, to design roadmaps for efficient development of unconventional plays, we need some meet-up places or ‘junctions’ where all the relevant data and expertise encounter each other, giving the experts an opportunity to exchange ideas, redefine their paths and continue on their ways. It is very important to locate these junctions properly to ensure the integration is performed efficiently. Definitely, more than one junction may be required for such a roadmap with so many unparalleled pathways and geomechanics (the actual science and not as has been narrow-mindedly butchered by some in industry) can be one of the excellent spots to build one of these junctions for different reasons and not just because of the geomechanically sensitive behaviour of the unconventional rocks but also due to the mechanical nature of operations such as drilling, completion and fracturin, the role of geomechanics in production from these plays, and the fact that, geomechanics has to deal with several different types of data from different disciplines to accomplish its goals successfully.
Last week, we published a Geomechanics Insight article in Discovery Digest on different methods of stress orientation characterization. The article reviews the cons and pros of different methods and gives examples for the most practical ones. These examples include using image logs, sonic anisotropy, caliper logs, core analysis, and microseismic data. All the examples are from the Monteny play in Alberta. One of my favourites in this article is the compilation of different natural and induced fractures and their relation to stress orientation with a great example from a Montney core.
Geoconvention 2015 will be in town soon and we will be proudly presenting some of the workflows, methodologies and conclusions from our integrated geomechanical study of the Duvernay Formation, a fast-growing and attractive unconventional play in Alberta. This study is a part of The Duvernay Project conducted by Canadian Discovery Ltd., Graham Davies Geological Consultants Ltd., and Trican Geological Solutions Ltd.
The extended abstracts for these presentations are already available online and, while required to be under four pages, we tried to keep them as inclusive and comprehensive as possible hoping they will be beneficial to our readers. One of these presentations/articles provides a general review of the study and its key findings and conclusions on different issues such as geology, hydrogeology, drilling experience, rock properties, stress characterization, and fractures. The other one pays specific attention to rock properties and the integrated approaches used for their characterization in this study and it includes relations between mineralogical/geochemical composition and mechanical properties, a new perspective of rock brittleness, and application of these findings for identifying fraccable rocks and their fracture types and other issues such as the effects of pore pressure on mechanical properties.
Here are the full titles:
A Regional Geomechanical Study of the Duvernay Formation in Alberta, Canada
Amy D. Fox and Mehrdad Soltanzadeh
Application of an Integrated Approach for the Characterization of Mechanical Rock Properties in the Duvernay Formation Mehrdad Soltanzadeh, Amy D. Fox and Nasir Rahim
Hope to see you for the talks and hear your comments/questions/suggestions.
Find the Complete PDF version of this series here: The Delicate Matter of Rock Brittleness
There’s something brittle in me that will break before it bends. (Mark Lawrence, Prince of Thorns)
Considering the definition of rock brittleness discussed in the previous post of this series, it is clear that the dynamic elastic properties measured by sonic or seismic waves and even the static ones measured in laboratories are not the most appropriate parameters for determining this property of rock as these parameters only describe the rock deformation before it reaches near failure or fracturing. This is truer for dynamic elastic properties as they only agitate the rock with very small deformations (see my other post for more on this). In fact, rock’s response during cracking or failure needs to be defined by fracturing or failure criteria. However, there must be some justifiable reasons for using dynamic elastic properties so frequently for identifying brittle rocks. Of course, the main reason is the abundant availability of these data from wireline logs and seismic surveys but, in addition to data availability, there are some technical evidence to justify using elastic properties as proxies for brittleness.
In a general sense, more brittle rocks (e.g., granite or over-consolidated cemented sandstones) are usually expected to show less deformation within their elastic limits and before yielding (or failure) in comparison to more ductile rocks. Although this general pattern may work for some rocks, in reality, brittleness/ductility is not equal to stiffness/softness. I will talk about the inconsistencies later in this post but let’s see how the idea of ‘less deformability=more brittleness’ is exploited to determine brittleness index using elastic properties. In elasticity, less axial deformation means higher Young’s modulus (E) and less lateral deformation means less Poisson’s ratio (v). Therefore, according to the discussed hypothesis, these two parameters seem to be good candidates for estimation of rock brittleness. This is the basis for the frequently used form of elastic brittleness index in petroleum industry as proposed by Rickman et al. (2009) who gave the same weight to the effects of Young’s modulus and Poisson’s ratio by simple arithmetic averaging:
max and min subscripts in this equation denote the maximum and minimum values of the elastic parameters for the formation(s) of interest. Figure 1 shows an example of using this equation where Young’s modulus and Poisson’s ratio are acquired from sonic and density logs. Figure 2 shows the cross-section of a shale formation showing the brittleness index variation. The dynamic elastic parameters in this case were derived from both sonic and seismic data.
Figure 1. An example of brittleness calculated from sonic and density log data. The direction of arrow shows increasing in brittleness index and the data points are color-coded by this parameter. (Modified after Varga et al., 2013).
Figure 2. An example of brittleness index profile derived from seismic and sonic data. Yellower colours stand for higher brittleness and bluer colours indicate less brittle rock (Source: Varga et al., 2013).
Reasons for Doubt
The idea of identifying brittle rocks using abundantly available log- or seismic-based dynamic elastic properties sounds very appealing but there are reasons that make it hard to accept the credibility of this method inclusively. I discussed one of these reasons before but there are more. One obvious reason is the hesitation in accepting the relation between rock brittleness and elastic properties. As it will be discussed later in this series, brittleness and fraccability are functions of more than just elastic properties even in materials that follow the rules of linear elastic fracture mechanics (LEFM). In LEFM, other independent parameters such as fracture toughness or fracture energy are the major factors that govern fracture propagation or blunting. This becomes even more complex in inelastic or plastic materials such as rocks.
On the other hand, during shear failure experiments, there are occasions that rocks with less elastic deformibality (i.e., higher Young’s modulus and lower Poisson’s ratio) show less brittle behaviour compared to the ones with more elastic deformability. Another contradictory observation is the change in rock behaviour by increase in confining stress during triaxial tests as discussed in the previous post of this series. This behaviour might not be the case for all the rocks but there are several types of rocks that follow this rule. The example given in Figure 3 shows that with increase in confining pressure on samples in triaxial test, the elastic deformability of rock is not significantly affected but it becomes more ductile in a shear failure mode. In other words, by increasing the confining stress, rocks become more ductile and less brittle while their Young’s moduli do not change; something that is not aligned with the assumptions of elastic brittleness index given above.
Figure 3. This figure shows how the mechanical behaviour of a limestone in a shear mode changes with increase in confining stress in triaxial test. Apparently, Young’s modulus does not change with confining pressure while rock’s behaviour becomes more ductile (source: www.higgs-palmer.com).
Different contradictory examples discussed here show that the idea of ‘lower v and higher E = higher brittleness’ may not work as expected all the time. In addition, there are others reasons that undermine credibility of this hypothesis. One is the effect of high pore pressure in hydrocarbon plays on dynamic elastic properties. High pore pressure is known to make the rock less consolidated and more deformable but the question is whether ‘dynamic Young’s modulus is always lower and dynamic Poisson’s ratio is always higher’ for rocks with higher pore pressure or not. The answer seems to be YES for Young’s modulus and NO for Poisson’s ratio as depicted in Figure 4. This figure shows Poisson’s ratio is only higher for higher pressures if the occupying fluid is incompressible enough (e.g., brine) but for more compressible fluids such as light oil or gas (as it is the case for many unconventional plays), increase in pore pressure leads to decrease in Poisson’s ratio. In the case of more compressible fluids, higher pore pressure still leads to less stiff and more deformable rocks but this does not result in higher dynamic Poisson’s ratio as is the case for brine-filled rocks.
One other thing about the brittleness index defined in the given equation is the fact that it is hard to agree with Young’s modulus and Poisson’s ratio having exactly the same share of influence on rock brittleness as imposed by arithmetic averaging in the equation. Also, we must remember that there are cases where a rock might have a very low elastic deformability but, at the same time, it can also have a very high resistance against fracturing due to its higher yield and failure strength. In these cases, for a certain amount of stress or fluid pressure, this rock does not fracture while rocks with higher elastic deformability (less brittleness index) and less resistance may fracture. Everything said here shows that the presented equation for brittleness might not be always trusted as a measure of rock brittleness but we cannot completely deny that there are cases where this index can work as a proxy for this property of rock.
Figure 4. These graphs show how pore pressure increase affects elastic properties such as (a) compressional wave velocity and (b) compressional wave impedance and Poisson’s ratio for different types of filling fluids including brine, oil, and gas (Source: Dvorkin).
Elastic Parameters and Hydraulic Fractures’ Characteristics
Now, let’s take a close look at some theories that relate elastic properties to fracture characteristics. It is important to remember that what we will discuss in the following is not directly related to brittleness or criteria for fracturing and it merely explains the effect of elastic properties on the fractures without considering their resistance against propagation. In other words, these fractures are considered to propagate as ‘knife cuts through butter’[i].
In practice, it is assumed that the volume of injected fracturing fluid is either lost to the formation through leakoff or helped in creating the volume of fracture. This forms a volume balance equation that is completed by knowing the fracture geometry. The necessary equations for finding fracture geometry are usually derived using the principals of LEFM that assumes elastic behaviour for the rock. Therefore, elastic parameters such as Young’s modulus and Poisson’s ratio have a direct role in determining the characteristics of hydraulic fractures. Depending on the complexity of modeling, different solutions have been suggested and used in industry. In here, without getting into detail, we will look at one of the simplest (although still practical) solutions called Perkins-Kern (PKN) that is based on assuming plane-strain geometry for the fracture. This model assumes an ellipsoidal shape for the fracture with a half-length of xf , maximum width of wf at the borehole wall, and constant depth of hf (Figure 5). If we inject a Newtonian fracturing fluid with an injection rate of q and viscosity of m for a time period of t, it is possible to find fracture geometry parameters using the following equations with assumption of no fluid loss to the formation:
The net pressure (i.e., the difference between fracturing fluid pressure and in-situ stress required for fracture propagation) at the initiating point of the fracture at the wellbore wall can be found using the following equation:
In these equations, Eps is an elastic parameter called plane-strain Young’s modulus that can be written as a function of basic elastic properties:
G in this equation is rock’s shear modulus. Now, let’s examine and see how changes in E and v will affect the geometry and net pressure of the fracture for a given injection rate for a certain period of time. Figure 6a to 6c show the effects of these changes on xf, wf, and pn. According to these figures, increase in both E and v leads to longer and narrower fractures that require higher net pressure for propagation. Nevertheless, unlike the discussed brittleness index, significance of Poisson’s ratio in determining fracture characteristics is much less than Young’s modulus. These results show that the dependency of fracture characteristics to elastic properties is more complex than ‘high is good, low is bad’.
Depending on hydraulic fracturing design priorities, different values of elastic parameters may be preferred but usually wider and longer fractures that can propagate with less pressure of fracturing fluid are favoured.
Figure 5. Geometry of Perkin-Kern (PKN) model for hydraulic fractures (source: modified after Economides and Valkó, 1996)
Figure 6. Variation of different fracture characteristics in the PKN model with Young’s modulus and Poisson’s ratio: (a) fracture’s half-length (xf,), (b) fracture’s maximum width (wf), and (c) fracture’s net pressure (pn).
Elastic Parameters and In-situ Stresses
Hydraulic fracturing engineers, geophysicists, and petrophysicists frequently use elastic properties to calculate horizontal in-situ stresses in rocks (See Figure 7 for an example). These calculations are usually performed by assuming poroelastic and uniaxial vertical deformation during sedimentary deposition of rocks. Using this approach, horizontal stresses are calculated simply by using vertical stress (Sv), pore pressure (Pp), and elastic properties. Considering tectonic strains (eHmax and eHmin) or stresses in these calculations, it is possible to account for the effect of tectonics and stress anisotropy in the rock. For instance, minimum and maximum horizontal stresses (Shmin and Shmax, respectively) for a homogeneous isotropic rock are calculated using the following equations:
a in this equation is Biot’s coefficient. In relaxed basins with low tectonic effects (i.e., eHmax and eHmin~0), Shmin and Shmax become equal. We must remember that these are not the favorite equations of many geomechanics experts as they believe the assumptions are over-simplistic and imprecise.
A hydraulic fracture propagates only if the pressure of the injected fracturing fluid exceeds both Shmin and fracture resistance. Therefore, higher Shmin means less potential for fracture propagation with a certain injection pressure. Now, let’s see according to the uniaxial poroelastic theory, how this critical parameter is affected by variation in elastic properties. Figure 8 shows an example of how Shmin in the given equation changes with variation in Young’s modulus (E) and Poisson’s ratio (v). According to this figure, increase in both E and v leads to higher Shmin or, in other words, fractures will have a harder time to propagate. This might indirectly contradict Rickman’s assumption that higher elastic Young’s modulus leads to more fraccability.
It seems that the simple assumption of using high or low elastic properties (or any similar assumption) for brittleness evaluation does not seem to have extensive theoretical and experimental support. Elastic properties are not satisfactory enough to characterize rock’s brittleness and more parameters are required for this purpose. On the other hand, considering their abundant availability from seismic surveys and sonic logs, it is not clever to completely rule them out as with proper and precise treatment they might be able to act as proxies for brittleness. If we want to use these parameters properly, probably the right option is using them besides other means such as laboratory measurements of brittleness and correlating them with observed fractures in the field from cores, image logs, and sonic scanners, or microseismic surveys. We can also compare them with other types of brittleness such as mineralogical brittleness to ensure their repressiveness (as will be discussed in the next post of this series). We also need to remember that instead of classifying the rocks based on high or low values of elastic parameters, we need to find specific ranges of them that suit rock fraccability.
Figure 7. This example shows minimum horizontal in-situ stress (Shmin) calculated using elastic equations for the some selected formations (source: Gray et al., 2012)
Figure 8. An example graph showing how minimum horizontal stress (Shmin) changes with variation in Young’s modulus and Poisson’s ratio according to the given equation in the text. In this example, vertical stress is 20 MPa, pore pressure is 10 MPa, Biot’s coefficient is 1.0, maximum tectonic strain is 0.0001, and minimum tectonic strain is assumed to be zero.
Canadian Discovery Ltd. has recently made one of our Duvernay geomechanics posters publicly available through its Discovery Digest Journal website . This poster is an example of the CDL’s integrated geomechanical characterization workflow for unconventional reservoirs applied to a specific well in the Kaybob Area, Alberta. It includes the results of mechanical properties and brittleness calculations, petrophysical analysis, fracture fabric characterization, drilling experience review, maximum horizontal in-situ stress (SHmax) determination, wellbore integrity modeling/history-matching, and critically-stressed fractures analysis. My favorite part is the fracture fabric characterization that tries to tie the depth of observed fractures (that are mostly coring-induced) on the core and image log with rock mechanical and mineralogical data from sonic logs and petrophysical analysis . The results are interesting as:
i) There exists a strong correlation between mechanical and mineralogical brittleness indices. Nevertheless, both of these parameters are defined differently from the regular brittleness indices used in the industry.
ii) Mechanical properties and brittleness indices can be efficiently used for fracture fabric characterization. In my view, if we want to find the characteristics of fraccable rock, the best approach is to find out where, in reality, the rock has been fractured (either naturally or induced).
iii) There are quantifiable ranges of brittleness indices or mechanical properties that determine the brittle rock that favors fracturing. Such ranges come very handy when quantitative interpretation (QI) method is implemented for characterization of brittle rock using seismic data.
One other thing on this poster is the CDL’s graphical representation of drilling experience that I personally like a lot due to its inclusive and abstract nature. This particular well is probably not the funnest one as it is a simply vertical well that does not show so many drilling problems but several of the 43 wells that we reviewed in this project showed sever drilling problems from all different types.
Find the Complete PDF version of this series here: The Delicate Matter of Rock Brittleness
Her reputation is no less brittle than it is beautiful. (Jane Austen, Pride and Prejudice)
It should not surprise us that oil producers would appreciate quick measures such as ‘brittleness index’ to tell them where to drill, complete, and fracture to achieve better hydrocarbon production results in low-permeability formations such as unconventional plays . Recently, we have seen different forms of this index introduced to the industry though the definition of ‘brittleness’, as a mechanical property of rock, has still remained a controversial matter among the experts from different disciplines and with different perspectives towards rock mechanics. In its typical application for characterization of unconventional plays, brittleness index is expected to define the mechanical behaviour of rock during fracture propagation. A brittle rock is usually expected to allow a pressurized fracture to propagate the same way ‘a knife cuts through butter’. The opposite case is a ‘ductile’ rock that tends to blunt a propagating fracture. In addition, a brittle rock is expected to be a good host for the proppants filling the fractures without swallowing them as may happen during propping a ductile rock. This will let fractures to remain functionally open and perform their role as fluid conduits. Some experts would rather using the term ‘fracability’ instead to define this behaviour of rock probably because ‘brittleness’, as a technical term, has been reserved by mechanical engineers to define a more general behaviour of materials as will be discussed later in this post since a long time ago. Another main reason for using ‘fracability’ instead of ‘brittleness’ is to emphasize that the process of fracture propagation is not just a function of mechanical rock properties but it also depends on other parameters such as in-situ stresses, fracturing fluid ‘s type, rate, and pressure, existence of natural fractures, etc. This article intends to review the concept of rock brittleness from different perspectives such as rock mechanics, fracture mechanics, geophysics, and petrophysics.
Old Definitions, New Applications
Mechanical engineers, in general, have been traditionally using the two terms of ‘brittle’ and ‘ductile’ to define the behaviour of rock before its final failure. In response to excessive loading, a brittle material (e.g., a steel knife blade) breaks abruptly without a significant deformation while a ductile material (e.g., a copper wire) deforms significantly before it breaks apart. Figure 1 shows examples of stress/strain behaviours of ductile and brittle metallic materials during standard tensile strength tests. In reality, fracture propagation in rocks during hydraulic fracturing is a tensile failure (i.e., Mode I failure) process but, unfortunately, direct tensile tests similar to what shown in Figure 1 are not usually possible to be conducted for sedimentary rocks due to very small tensile strength of these rocks [i]. Therefore, in rock mechanics, brittle/ductile behaviour of rocks is usually studied under compressive loading instead of tensile loading. In this case, most likely, the sample will fail in a shear mode (i.e., Mode 2 failure).
Figure 1. Stress-strain diagram obtained from the standard tensile tests (a) ductile material (b) brittle material (Modified after Budynas and Nisbett, 2008).
During compressive loading (e.g., triaxial tests), rocks usually fail on shear failure surfaces (Figure 2). Rock behaviour in response to this type of loading is very similar to what the earth crust experiences under tectonic movements and, therefore, the results of such tests have been traditionally used to study rock brittleness/ductility in geological structures (e.g., geological folds are usually formed in ductile rock and faults are formed in brittle rocks, see Figure 3, for instance), earthquakes (brittle rocks are likely to induce larger earthquakes/seismic events than ductile rocks), and reservoir sealing integrity (comparing to ductile sealing, the brittle ones are more likely to act as conduits after being deformed by reservoir depletion).
Figure 2. Brittle-ductile behaviour of rocks under compressive loading (Source: Evans et al., 1990)
Figure 3. Effect of rock brittleness/ductility in forming geological structures (a) folds formed in ductile rocks (b) fractures developed in brittle rocks. (Sources: a: www.esci.umn.edu; b: www.geoscience.wisc.edu).
While the compressive tests are designed to study the rock’s behaviour when it fails under shear, some rock mechanics engineers believe that these results can also act as proxies for the behaviour of rock in response to fracture propagation during hydraulic fracturing. We know that failure criteria of rocks during compressive loading are distantly different from that of tensile loading. Therefore, if we really want to measure the actual rock behaviour during hydraulic fracturing, we need to develop more precise and comprehensive testing procedures to quantify rock’s response during tensile loading [iii]. Nevertheless, the compressive failure tests using triaxial apparatus are commonly used to verify rock brittleness and ductility in the laboratories. Based on the results of these tests, the magnitude of brittleness index may be quantified in different ways by using strains, rock strengths, or the work/energy during these tests. Figure 4 shows a list of different equations. Drilling, geotechnical, and mining engineers have also shown interest in the concept of rock brittleness as they believe it is one of the parameters that governs the penetration rate during drilling wells, mine shafts, and tunnels (this character is called drillability). Some of the relationships used in these fields calculate the rock brittleness index as a function of rock compressive strength (Sc) and tensile strength (St). These parameters have been usually used in different combinations to calculate rock brittleness (e.g., BI=Sc/St; BI=Sc x St/2; BI=(Sc – St)/(Sc + St)). Even some standard tests are used in these disciplines to measure rock brittleness.
Figure 4. Different types of brittleness indices defined based on compressive loading tests (modified after: Yang et al., 2013)
We need to remember that compressive loading (e.g., triaxial) tests are strongly influenced by testing condition such as in-situ stresses and ambient pressure and temperature. For instance, experiments show that these tests are strongly dependent on the magnitude of confining stress applied during testing. Rocks usually (though there are exceptions) show more brittle responses at lower confining stresses and, with increasing in this parameter, their responses become more ductile (Figure 5). The combined effect of confining pressure (i.e., in-situ stresses in here) and temperature on rock brittleness has been the main subject of investigation for ‘brittle-ductile transition zone‘ studies performed in plate tectonics, seismology, and earthquake engineering.
Triaxial measurements are costly and time-consuming and the scattered data found by them may not be enough for characterizing the rock behaviour in geological scales. This might be the motivating reason for some experts in the petroleum industry to use the abundantly available wireline log and seismic data for assessment of rock fracability and brittleness in geological formations as will be discussed in the other parts of this article.
Figure 5. Dependency of stress-strain behaviour to confining pressure. The numbers on the curves in the top figure are confining pressures and, in the lower figure, confining pressure is increased from a to d. (source: Paterson and Wong, 2006).
[i] Rock tensile strength is usually measured indirectly using a loading pattern that is not tensile by definition. The most common method is called Brazilian test that imposes a lateral edge loading on a cylindrical sample of rock until it breaks under stress. Point loading is another approach that applies a concentrated load on a planar piece of rock. 3-point bending is another approach that is less used in rock mechanics.
[ii] The terms ’brittle’ and ‘ductile’ in this sentence have been used in their most general forms.
[iii] Defining fracture toughness for quantifying the resistance against fracture propagation has been one of these efforts though the methodology implemented for its measurement can be argued for different reasons.