Patent Description:
This disclosure describes systems and methods for fracturing a subsurface formation and, more particularly, fracturing a subsurface formation based on a breakdown pressure prediction from a numerical finite element model.

Hydraulic fracturing has been used to stimulate tight sandstone and shale gas reservoirs. Rock breakdown or fracture initiation is typically required for a successful hydraulic fracturing treatment. For hydraulic fracturing treatments, accurately estimating a breakdown pressure of a subsurface (or subterranean) formation may help determine correct selections of casing size, tubing size, and wellhead (for example, to correctly select their respective burst pressure limiting requirements), as well as a pump schedule design. Otherwise, the hydraulic fracturing operation may not properly inject a fracturing liquid to fracture the formation (for example, if the breakdown pressure was underestimated). Conventionally, hydraulic fracturing simulators may not accurately predict the breakdown pressure due to, for example, model simplifications.

<CIT> describes system and methods for optimizing multistage hydraulic fracturing design based on three-dimensional (3D) continuum damage mechanics are provided. A 3D global model of a field of hydrocarbon reservoirs is generated. A finite element analysis of the 3D global model is performed to calculate values of one or more mechanical variables for the field. A smaller-scale 3D sub-model of a selected portion of the field is generated from the 3D global model based on the calculated values of the mechanical variables.

<NPL>)) describe a finite element method to predict the fracture initiation pressure.

<NPL>) describe an analytical model for predicting fracture initiation pressure from a cased and perforated wellbore.

<NPL>) describe a fracture mechanics approach to breakdown pressure determination.

The systems and methods described in this disclosure are related to fracturing of a subsurface formation based on a breakdown pressure prediction from a numerical finite element model. A nonlinear three-dimensional (3D) finite element model includes a full 3D geometric model of a deviated, cased hole, and perforated wellbore, is able to predict fracture initiation and breakdown pressure (e.g., via simulation) of the perforated wellbore within the subsurface formation for a hydraulic fracturing treatment. For example, once an engineer has simulated the breakdown pressure for a wellbore with a particular geometry using the finite element model, the components of the hydraulic fracturing operation (e.g., pump, tubing, casing, etc.) can be properly sized to inject a fracturing liquid into well to fracture the formation.

The finite element model includes a full 3D finite element model of the casing of the wellbore, the cement of the wellbore, the perforation geometry of the wellbore, and the formation (e.g., rock) around the wellbore and around the perforations (e.g., 2D assumptions or symmetry assumptions are not necessary). The finite element model includes in-situ stresses, stress orientations, and both casing and cement geometries, mechanical (constitutive) properties, and contact properties (e.g., between the casing and the cement, and between the cement, and the formation). The finite element model includes the 3D geometry defining the perforation cluster and perforation phase angles of the perforated wellbore. Rock around the perforation tunnel is modeled using concrete damage plasticity with both compression and tensile damage such that rock damage progression as a function of wellbore pressure is simulated in the finite element model. "Breakdown pressure," in the context of the finite element model, refers to the internal casing pressure that leads to one of the perforation clusters developing the threshold amount of the tensile damage (e.g., <NUM> %, <NUM>%, etc.). In other words, when the threshold amount of damage is detected in at least one of the perforation clusters, the pressure applied to the internal casing pressure (e.g., representing the hydraulic fluid within the wellbore) that caused the threshold amount of damage is determined to be the breakdown pressure. The threshold amount of damage is typically predetermined based on model calibration procedures.

Methods of hydraulic fracturing a subsurface formation can include using a three-dimensional finite element model implemented on one or more processors to simulate a deviated well with a wellbore casing, a cement adjacent to the wellbore casing, and a perforation cluster with at least two perforations through a side wall of the wellbore casing, each of the at least two perforations extending through a rock of the subsurface formation via a respective perforation tunnel and having a different phase angle relative to an longitudinal axis of the casing; solving the three-dimensional finite element model implemented on one or more processors to determine a breakdown pressure of the deviated well based on an amount of tensile damage of the perforation cluster induced by an applied pressure representing injected hydraulic fluid; drilling and completing a deviated well with a wellbore casing size, tubing size, and wellhead selected at least in part based on the determined breakdown pressure; and injecting hydraulic fluid into the deviated well at an injection pressure determined at least in part on the determined breakdown pressure to cause hydraulic fracturing of the rock of the subsurface formation, wherein the three-dimensional finite element model includes a concrete damage plasticity model representing constitutive behavior of the rock and a contact relationship between an interface of the wellbore casing and the cement adjacent to the casing, the contact relationship modeling de-bonding of the interface of the casing and the cement.

Methods of hydraulic fracturing a subsurface formation can include implementing a three-dimensional finite element model on one or more processors to simulate a deviated well with a wellbore casing, a cement adjacent to the wellbore casing, and a perforation cluster with at least two perforations through a side wall of the wellbore casing, each of the at least two perforations extending through a rock of the subsurface formation via a respective perforation tunnel and having a different phase angle relative to an longitudinal axis of the casing; assigning a concrete damage plasticity model to define constitutive behavior of the rock of the subsurface formation, the concrete damage plasticity model including compression hardening, compression damage, tensile stiffening, and tensile damage behavior; assigning a contact relationship at an interface between the casing and the cement, the contact relationship modeling de-bonding of the interface between the casing and the cement; and solving the three-dimensional finite element model on the one or more processors to determine a breakdown pressure of the deviated well based on an amount of tensile damage of the rock induced by an applied pressure representing injected hydraulic fluid and the assigned concrete damage plasticity model and the assigned contact relationship.

Implementations of these systems and methods can include one or more of the following features.

In some implementations, the method includes extracting material properties of the rock from log data and/or lab test data, wherein the concrete damage plasticity model includes compression hardening, compression damage, tensile stiffening, and tensile damage behavior at least in part based on the extracted material properties.

In some implementations, the method includes injecting hydraulic fluid into the deviated well by pumping the hydraulic fluid into the deviated well using a pump schedule determined at least in part based on the determined breakdown pressure.

In some implementations, the method includes selecting a wellbore casing size, a tubing size, and wellhead at least in part based on the determined breakdown pressure.

In some implementations, the method includes determining a pump schedule of a pump for injecting hydraulic fluid into the wellbore at least in part based on the determined breakdown pressure.

In some implementations, each of the at least two perforations are axially spaced relative to each other and each of the at least two perforations are angularly spaced relative to each other around a circumference of the wellbore. In some cases, the at least two perforations are six perforations located within a <NUM> centimeter length of the wellbore casing and angularly spaced <NUM> degrees apart.

In some implementations, determining the breakdown pressure includes detecting when at least one finite element representing the rock has a tensile failure above a predetermined threshold. In some cases, the predetermined threshold is a scalar between <NUM> and <NUM>.

In some implementations, solving the three-dimensional finite element model comprises solving the three-dimensional finite element model using two quasi-static loading steps, a first step used for solving static equilibrium with a loading of in-situ stresses, gravity loading, overburden, and underburden, and a second step used for solving static equilibrium of an applied pressure to determine the breakdown pressure while including the loading of the first step.

In some implementations, the method includes rotating an orientation of the three-dimensional finite element model based on borehole image log data so that a well trajectory orientation and in-situ stresses are oriented in accordance with the borehole image log data.

In some implementations, an output of the solved three-dimensional finite element model is a spatially varying contour of the tensile damage of the rock.

In some implementations, a portion of the deviated well has a deviated angle of at least <NUM> degrees relative to a normal direction from a ground surface.

The systems and methods described in this specification provide various advantages.

The finite element model is applicable to deviated and cased holes with perforations, which is difficult to include in analytical due to model simplifications. Incorporating the effect of a deviation, the casing on the breakdown pressure prediction, and the inclusion of perforations with a particular geometry would make an analytical model intractable to solve. The finite element model accounts for these features and can be used to study model changes (e.g., such as relocating the perforations, changing the angle of the perforations, etc.) with ease.

The finite element model is not limited to vertical open-holes since the model geometry accounts for deviated and cased-holes. For example, the finite element model includes the high strength casing of the wellbore in the modeling which affects the distribution of stress throughout the wellbore and the perforation clusters which more accurately represents the physical conditions of hydraulic fracturing treatment. In scenarios where the casing is not included in the finite element model, the fluid pressure would act on the open-hole instead of the casing, which would incorrectly redistribute the stresses and change the breakdown pressure. Without a cased-hole, the finite element model would represent an open-hole fracking method which is different from a cased-hole fracking method.

The finite element model is not limited to initiating hydraulic fracture at the perforations and not limited to propagating the hydraulic fracture along a direction of maximum principal stress. For example, the finite element model is able to determine (e.g., via simulation) the location of fracture initiation which may be in a perforation cluster or in the rock around the perforation cluster. The finite element model includes perforation cluster geometry detail (e.g., a diameter of the perforation cluster is less than a diameter of a casing, etc.) and perforation geometry detail (e.g., the perforation cluster includes a first perforation with a phase angle of <NUM> degrees, a second perforation with a phase angle of <NUM> degrees, each with different lengths along the perforation cluster, etc.) so that the location of fracture initiation is determined among various perforation tunnels within a perforation cluster. In some cases, the finite element model includes a perforation cluster with six perforations and six perforations tunnels. Each perforation is angled <NUM> degrees relative to each other and axially spaced within a 2ft section along the wellbore. This configuration is used in hydraulic fracturing and increases the accuracy of the breakdown prediction of the finite element model. By modeling six perforations together, the model represents a "superposition effect" where multiple perforations are simulated together to determine breakdown pressure, which further increases accuracy of the predicted breakdown pressure.

The finite element model accounts for material non-linearity (e.g., the rock, the cement, etc.) and contact non-linearity (e.g., between the casing and the cement, the cement and the formation, etc.) via a robust numerical formulation which is difficult to account for in a boundary element and/or a finite volume formulation. In some examples, boundary element and/or a finite volume formulations require linear elastic constitutive models instead of constitutive models that account for non-linearity. In this scenario, a linear elastic rock constitutive model is unable to represent the physical fracture initiation mechanism for hydraulic fracturing treatment.

The finite element model includes a constitutive model for the rock that is a concrete damage plasticity model. The concrete damage plasticity model better represents the rock tensile damage progression induced by the pressure inside the casing and the perforation surfaces of the perforations. The finite element model is not limited to determining the breakdown pressure based on maximum tensile stress, which is often an assumption in analytical models. Instead, breakdown pressure is the pressure acting on the internal casing that develops tensile damage from one perforation of the perforation cluster. Maximum tensile stress criterion is also referred as the tension cut-off, which assumes the rock will be completely damaged or lose strength once the tension stress reaches the maximum tensile stress threshold. However during the damage process, stiffness gradually decreases. Before reaching maximum tensile stress, damage might have already began, the stiffness of rock has already degraded. Maximum tensile stress method cannot account for this process. The concrete damage plasticity model better represents the rock tensile damage progression.

The finite element model includes a denser mesh representing the perforation cluster of the wellbore and the surrounding rock and a less dense mesh representing the other regions of the model. This approach lowers the overall computational cost of simulating the finite element model. One measure of mesh density is an average characteristic length of the finite elements used. In this way, as mesh density increases, so does the number of degrees of freedom of the finite element model. Increasing the number of degrees of freedom increases the accuracy of the simulation, but also increases computational cost. The finite element model accounts for this computational cost vs. accuracy tradeoff by using a denser mesh portion in regions of the finite element model where failure is likely to occur based on engineering judgement and/or via previous simulations, while using a lower mesh density (e.g., the other regions of the finite element model), where failure is not likely to occur.

The finite element model accounts for casing-cement contact instead of being fully bonded. Incorporating the contact in the finite element model better represents the physical situation of the wellbore compared to assuming that the casing-cement is fully bonded (e.g., glued or tied).

The finite element model is easier to verify by aligning the minimum horizontal stress direction with the same direction of the horizontal wellbore orientation. In a subsurface, in-situ stress is in compression state. A minimum horizontal stress direction represents the lowest compression direction. During a drilling operation, it is much easier to split the rock in the direction perpendicular to this direction than a direction along this direction. For example, drilling a horizontal well along the minimum stress direction with perforation tunnels in a direction perpendicular to the minimum horizontal stress direction is preferable. In this scenario, injecting fluid with high pressures (above the breakdown pressure) pressurize the perforation tunnel and split the rock. However, the finite element model allows for in-situ stresses to be oriented at any direction with respect to the horizontal wellbore (e.g., aligned with the wellbore, perpendicular to the wellbore, <NUM> degrees relative to the axis of the wellbore, etc.).

By splitting the solution process of the finite element model into two load steps, the first load step is used to account for in-situ stresses, the gravity load, top surface overburden pressure as a base state. In other words, when the second load step begins and pressure is incrementally increased, the incremental solution already incorporates the full in-situ stresses, the gravity load, top surface overburden pressure. This "two-step process" represents the physical scenario of the wellbore in the formation.

The details of one or more embodiments of these systems and methods are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of these systems and methods will be apparent from the description and drawings, and from the claims.

<FIG> is a schematic diagram of an example implementation of a wellbore system <NUM> according to the present disclosure. In some aspects, the wellbore system <NUM> (all or part of it) may provide a wellbore system and computational framework (for example, embodied in control system <NUM>) for calculating a breakdown pressure of a subterranean formation. In some aspects, the wellbore system <NUM> (and the computational framework) provides such a determination for a deviated, cased-hole, and clustered perforation hydraulic fracturing treatment, while taking into account an impact of casing-cement mechanical properties on the breakdown pressure.

In some aspects, the computational framework (for example, executed on a control system <NUM>) of wellbore system accounts for one or more effects of casing-cement intermediate layers in a deviated wellbore to calculate the breakdown pressure of a subterranean formation. The computational framework of wellbore system <NUM> may also account for a potential effect of perforation quality on estimating the breakdown pressure of a subterranean formation. The computational framework of wellbore system <NUM> improves current available models on calculating breakdown pressure, for example, which are applicable to deviated, cased wellbores, and clustered perforation hydraulic fracturing treatments.

As illustrated, the wellbore system <NUM> includes a wellbore <NUM> formed (for example, drilled or otherwise) from a terranean surface <NUM> and to and into subterranean formation <NUM>. Although the terranean surface <NUM> is illustrated as a land surface, terranean surface <NUM> may be a sub-sea or other underwater surface, such as a lake or an ocean floor or other surface under a body of water. Thus, the present disclosure contemplates that the wellbore <NUM> may be formed under a body of water from a drilling location on or proximate the body of water.

The illustrated wellbore <NUM> is a directional wellbore in this example of wellbore system <NUM>. For instance, the wellbore <NUM> includes a substantially vertical portion <NUM> coupled to a radiussed or curved portion <NUM>, which in turn is coupled to a substantially horizontal portion <NUM>. As used in the present disclosure, "substantially" in the context of a wellbore orientation, refers to wellbores that may not be exactly vertical (for example, exactly perpendicular to the terranean surface <NUM>) or exactly horizontal (for example, exactly parallel to the terranean surface <NUM>). In other words, those of ordinary skill in the drill arts would recognize that vertical wellbores often undulate offset from a true vertical direction that they might be drilled at an angle that deviates from true vertical, and horizontal wellbores often undulate offset from a true horizontal direction. Further, the substantially horizontal portion <NUM>, in some aspects, may be a slant wellbore or other directional wellbore that is oriented between exactly vertical and exactly horizontal. The substantially horizontal portion <NUM>, in some aspects, may be oriented to follow a slant of the formation. As illustrated in this example, the three portions of the wellbore <NUM> - the vertical portion <NUM>, the radiussed portion <NUM>, and the horizontal portion <NUM> - form a continuous wellbore <NUM> that extends into the Earth. Thus, in this example implementation, at least a portion of the wellbore <NUM>, such as the radiussed portion <NUM> and the horizontal portion <NUM>, may be considered a deviated wellbore, in other words, a non-vertical wellbore.

The illustrated wellbore <NUM> has a surface casing <NUM> positioned and set around the wellbore <NUM> from the terranean surface <NUM> into a particular depth in the Earth. For example, the surface casing <NUM> may be a relatively large-diameter tubular member (or string of members) set (for example, cemented) around the wellbore <NUM> in a shallow formation. As used herein, "tubular" may refer to a member that has a circular cross-section, elliptical cross-section, or other shaped cross-section. As illustrated, a production casing <NUM> is positioned and set within the wellbore <NUM> downhole of the surface casing <NUM>. Although termed a "production" casing, in this example, the casing <NUM> may or may not have been subject to hydrocarbon production operations. Thus, the casing <NUM> refers to and includes any form of tubular member that is set (for example, cemented) in the wellbore <NUM> downhole of the surface casing <NUM>. In some examples of the wellbore system <NUM>, the production casing <NUM> may begin at an end of the radiussed portion <NUM> and extend throughout the substantially horizontal portion <NUM>. The casing <NUM> could also extend into the radiussed portion <NUM> and into the vertical portion <NUM>.

As shown, cement <NUM> is positioned (for example, pumped) around the casings <NUM> and <NUM> in an annulus between the casings <NUM> and <NUM> and the wellbore <NUM>. The cement <NUM>, for example, may secure the casings <NUM> and <NUM> (and any other casings or liners of the wellbore <NUM>) through the subterranean layers under the terranean surface <NUM>. In some aspects, the cement <NUM> may be installed along the entire length of the casings (for example, casings <NUM> and <NUM> and any other casings), or the cement <NUM> could be used along certain portions of the casings if adequate for the particular wellbore <NUM>. Other casings, such as conductor casings or intermediate casings, are also contemplated by the present disclosure for the wellbore system <NUM>.

As illustrated, the wellbore <NUM> extends through one or more subterranean layers (not specifically labeled) and lands in subterranean formation <NUM>. The subterranean formation <NUM>, in this example, may be chosen as the landing for the substantially horizontal portion <NUM>, for example, in order to initiate completion operations such as hydraulic fracturing operations and ultimately recover hydrocarbon fluids from the subterranean formation. In some examples, the subterranean formation <NUM> is composed of shale or tight sandstone. Shale, in some examples, may be source rocks that provide for hydrocarbon recovery from the subterranean formation <NUM>.

As shown in <FIG>, the wellbore system <NUM> includes one or more perforation tunnels <NUM> (also known as perforations <NUM>) that are formed from the wellbore <NUM>, through the casing <NUM> and the cement <NUM>, and extend into the subterranean formation <NUM>. Generally, the perforation tunnels <NUM> may be formed by, for example, shaped explosive charges, water jetting, laser, or other conventional perforating techniques. In some aspects, multiple perforation tunnels <NUM> may comprise a perforation stage <NUM>. Each perforation tunnel <NUM>, as well as each perforation cluster <NUM>, may provide a path (or paths) for a hydraulic fracturing liquid (with or without proppant) to enter the subterranean formation <NUM> from the wellbore <NUM> in order to initiate and propagate hydraulic fractures (extending from the perforation tunnels <NUM>) through the subterranean formation <NUM>.

As shown in <FIG>, the example implementation of the wellbore system <NUM> also includes a logging tool <NUM> that is communicably coupled to a downhole conveyance <NUM>, such as a wirelines, optical line, or other data communication cable. The downhole conveyance <NUM> provides data from the logging tool <NUM> to the control system <NUM>, for real time (for example, during logging operations) or later usage in determining a breakdown pressure of the subterranean formation <NUM>. In some aspects, the control system <NUM> comprises a microprocessor based control system that includes, for example, one or more hardware processors, one or more memory storage devices (for example, tangible, non-transitory computer-readable memory modules), one or more network interfaces, and one or more input/output devices, including, for example, a graphical user interface (GUI) to present one or more determinations or data from the computer framework for determining a breakdown pressure of a subterranean formation.

In-situ stresses act on a wellbore formed through a subsurface formation. For example, in some aspects, the logging tool <NUM> may derive or generate an image log of the subterranean formation <NUM>, from which the maximum horizontal principal stress angle of the subterranean formation <NUM> can be obtained. From the maximum horizontal principal stress angle and borehole image (breakout and breakdown), a maximum horizontal principal stress of the subterranean formation <NUM> can be estimated and calibrated together with the vertical stress and minimum horizontal principal stress and finally determined.

In some aspects, the in-situ stresses can be calculated according to a number of parameters. For example, such parameters may include the image log, which includes wellbore TVD, azimuth, and deviation. The parameters may also include density log, stress orientations. The parameters may also include the mechanical properties of, for example, the casing <NUM>, the cement <NUM>, and the subterranean formation <NUM> itself.

For example, an in-situ stress field of the subterranean formation <NUM> exists in the far field and takes the form as follows: <MAT>.

In Eq. <NUM>, σHmax and σHmin are the maximum and minimum horizontal stresses respectively, and σV is the principal vertical stress component. The dynamic Young's modulus and Poisson's ratio can be calculated using, for example, a sonic log from the logging tool <NUM>, then converted to a static modulus based on correlations. The vertical stress SV (total stress) or σV (effective stress) can be reasonably calculated based on, for example, a density log of the logging tool <NUM>, as: <MAT>.

Without considering tectonic stresses, the effective and total minimum horizontal stress can be approximately calculated by: <MAT>.

In Eq. <NUM>, α is the Biot's poroelastic parameter and P<NUM> is reservoir pressure. The maximum principal stress can be estimated based on, for example, the image log by calibrating the maximum horizontal stress magnitude against a drilling fluid ("mud") weight and observed breakout and breakdown zone exhibited in the image log data.

In a conventional analysis for a vertical open-hole, the maximum horizontal principal stress can be obtained based on the breakdown pressure from a leak off test during drilling. The above equation assumes the horizontal strain equal to zero. Under the tectonic regime with given horizontal strains εHmax and εHmin, the maximum and minimum horizontal stresses can be generally calculated by: <MAT> <MAT>.

Drilling the wellbore <NUM> in and through the subterranean formation <NUM> leads to a stress redistribution around the wellbore <NUM>. The wellbore <NUM> is generally supported by drilling fluid pressure acting on the wellbore wall. Accurately estimating the stresses around the wellbore <NUM> may be necessary for wellbore stability. Also, it may be helpful to determine the breakdown pressure for hydraulic fracturing design, which directly impacts the selection of casing size, treatment tubing size, wellhead, steel grade, pump schedule, and other equipment.

For example, <FIG> illustrates a schematic top view cross-section <NUM> of a cased, vertical wellbore with particular stresses. In cross-section <NUM>, Rw represents the wellbore radius, and R represents a radial distance from the concentric center of the casing <NUM> and cement <NUM>, along with the effective total minimum and maximum horizontal stresses, σHmax and σHmin.

For a conventional, vertical open-hole wellbore, it is a generally accepted convention that the three far field principal stresses and orientation are known for a conventional vertical, open-hole wellbore. The elastic solutions of the effective stresses around wellbore based on plane strain condition are given by: <MAT> <MAT>.

The above equations are also called by Kirsch Equation. <NUM> and <NUM>, σr is the radial stress acting outwards from the wellbore; σθ is the hoop stress around the wellbore; θ is the angle from the direction of σxx; Pw is the wellbore pressure; and P<NUM> is the reservoir pressure. For an open-hole wellbore, limiting this to the wellbore wall with R = Rw leads to: <MAT> <MAT> <MAT>.

For the hydraulic fracturing, tensile failure criteria is generally used to direct the fracture propagation trajectory; therefore a fracture propagates at the direction of maximum horizontal stress. The corresponding hoop stress at θ = <NUM> yields: <MAT>.

Breakdown pressure is determined based on tensile failure. If the hoop stress turns into tension at wellbore wall and exceeds the material's tensile strength T, the material (in other words, the rock) will fail in tensile mode: <MAT> <MAT>.

Eq. <NUM> is generally used to predict the required drilling mud weight in the conventional, vertical open-hole example, which can avoid wellbore breakdown issues during drillings. For horizontal wells drilled in the subterranean formations (such as deep and tight reservoirs), the horizontal parts are generally drilled in the minimum horizontal stress direction and thereafter is cased and cemented. After perforating the casing, fluid injection is executed to initiate hydraulic fractures from the perforation towards the maximum horizontal stress direction. In such situations, the computational framework of the present disclosure may calculate the hoop stress around the perforation tunnel for judging whether fracture can be initiated or not, even though the hoop stress with respect to the wellbore is not the main concern. For a cased and cemented wellbore with perforation clusters (such as wellbore <NUM>), the
breakdown pressure refers to the bottom-hole pressure inside the casing that leads to tensile failure within the area of the wellbore-perforation interface. Therefore, Eq. <NUM> cannot be directly used to estimate the breakdown pressure for deviated, cased wellbores with clustered perforations for a hydraulic fracturing treatment (such as wellbore <NUM>).

<FIG> is a perspective view of a 3D finite element model <NUM> for simulating (or predicting) breakdown pressure. The finite element model <NUM> includes the geometric configuration of a perforated wellbore <NUM> that includes a casing with cement around the casing and includes a formation <NUM> around the casing. In some cases, perforated wellbore <NUM> is the same as, or similar to, the wellbore <NUM> described with reference to <FIG>.

<FIG> shows an example 3D domain <NUM> of the finite element model <NUM> (and the mesh used to discretize the domain). The mesh includes mostly hexahedral finite elements. The domain <NUM> extends <NUM> meters in the x-direction of the coordinate system <NUM>, <NUM> meters in the y-direction of the coordinate system <NUM> (e.g., axial direction), and <NUM> meters in the z-direction of the coordinate system <NUM> (e.g., vertical direction). In other examples, the domain <NUM> extends less than or more than <NUM> meters in the x-direction (e.g., <NUM> meters, <NUM> meters, etc.), less than or more than <NUM> meters in the y-direction (e.g., <NUM> meters, <NUM> meters, etc.), and less than or more than <NUM> meters in the vertical direction (e.g., <NUM> meters, <NUM> meters, etc.). In some cases, symmetry and/or periodic boundary conditions are applied to reduce the computational effort.

<FIG> represents a finite element model <NUM> with the perforated wellbore <NUM> aligned in a horizontal direction (e.g., the y-direction). This scenario represents the wellbore <NUM> (shown in <FIG>) that includes the substantially vertical portion <NUM> coupled to the substantially horizontal portion <NUM> where the perforation cluster <NUM> is located. In other examples, the perforated wellbore <NUM> is modelled at an angle relative to the vertical portion <NUM> (e.g., <NUM> degrees, <NUM> degrees, <NUM> degrees, <NUM> degrees, <NUM> degrees, etc.) and the model is not limited to the horizontal direction as shown in <FIG>.

The finite element model <NUM> includes overburden 308A and underburden 308B which are assigned different properties to account for additional stresses exerted on the wellbore <NUM>. Properties for the overburden 308A include Young's modulus, Poisson's ratio, and density, and properties for the underburden 308B include Young's modulus, Poisson's ratio, density. In most cases, overburden and underburden are modeled using linear elasticity and assigned with in-situ stresses. The finite element model <NUM> includes a denser mesh partition <NUM> representing the axial position of the perforation cluster of the wellbore <NUM> and the rock formation radially surrounding the perforation cluster.

<FIG> is a perspective view of the finite element model <NUM> showing the wellbore <NUM> and the perforation cluster <NUM> of the wellbore <NUM> in detail. <FIG> shows the cased borehole <NUM>, the casing <NUM> around the borehole <NUM>, the cement <NUM> around the casing <NUM>, and a part of the formation <NUM> around the cement <NUM> (e.g., the remainder of the formation <NUM> is shown in <FIG> - see also <FIG> for additional geometric detail). The central bore of the wellbore <NUM> is not explicitly modeled because a pressure load is applied to the inner diameter of the casing in lieu of modeling the fluid explicitly.

In some cases, the perforation cluster <NUM> is the same as, or similar to, the perforation cluster <NUM> described with reference to <FIG>. While the finite element model <NUM> incudes one perforation cluster <NUM>, in some cases, the finite element model <NUM> (and/or wellbore <NUM>) includes more than one perforation cluster (e.g., two, three, four perforation clusters, etc.). The perforation cluster <NUM> includes perforations 402A, 402B (generally perforations <NUM>) that are modeled with a circular cross section. The perforations include various perforation angles around the circumference of the wellbore <NUM> (e.g., <NUM> degrees from the x-direction, <NUM> degrees from the x-direction, etc.) and various axial positions along the wellbore <NUM> (e.g., <NUM> centimeters apart from each other, etc.). In this example, the perforation cluster <NUM> includes six perforations with phase angle of <NUM> degrees (e.g., they are oriented with an angular spacing of <NUM> degrees apart from each other). Details of the angular spacing of the perforations is described with reference to <FIG>.

<FIG> is a cross section view of the perforated wellbore <NUM> that includes the cased borehole <NUM>, the casing <NUM>, the cement <NUM>, and part of the formation <NUM> as described with reference to <FIG>. In this example, the borehole <NUM> diameter is <NUM> millimeters (<NUM>-<NUM>/<NUM> inches) and the casing internal diameter is <NUM> millimeters (<NUM>-<NUM>/<NUM> inches). In other cases, the borehole <NUM> diameter is less than or greater than <NUM> millimeters (e.g., <NUM> millimeters, <NUM> millimeters, etc.) and the casing internal diameter is less than or greater than <NUM> millimeters (e.g., <NUM> millimeters, <NUM> millimeters, etc.). The casing-cement interface <NUM> is modeled using contact mechanics behavior. The cement-formation interface <NUM> is also modeled using contact mechanics behavior. In some cases, the contact mechanics behavior includes a tangential frictional coefficient of <NUM>. In other cases, a tangential frictional coefficient greater than or less than <NUM> is used (e.g., <NUM>, <NUM>, <NUM>, <NUM>, etc.).

<FIG> is a perspective view of the finite element model <NUM> showing a perforation tunnel 404B of the perforation 402B. The perforation tunnel for perforation 402A is obscured by the finite element model <NUM> in this view. A section cut through perforation tunnel 404B shows the detailed mesh strategy used to represent the perforated wellbore <NUM> which includes the casing <NUM>, the cement <NUM>, the formation <NUM>, and the perforation cluster <NUM>. The perforation cluster <NUM> includes multiple perforations 402A, 402B, and respective perforation tunnels 404B. In this example, the perforation cluster <NUM> includes six perforations <NUM> and each perforation <NUM> includes a perforation tunnel <NUM>. Each perforation tunnel <NUM> radially extends into the formation <NUM>. In other models, less than or greater than six perforations (e.g., <NUM>, <NUM>, <NUM>, <NUM> perforations, etc.) are included in the finite element model.

<FIG> is a perspective view of the finite element model <NUM> showing the casing <NUM> and the projected location for each of the six perforations <NUM> and perforation tunnels 404A-F. The perforations <NUM> have a phase angle of <NUM> degrees relative to each other and are axially spaced apart within a <NUM>,<NUM> (<NUM> ft) section along the wellbore <NUM>. In this example, the perforation cluster <NUM> is <NUM> millimeters (<NUM> inches) in the length and <NUM> millimeters (<NUM> inches) in diameter. In this example, the perforation diameter <NUM> has a diameter of <NUM> inches which is more than 10X smaller than a diameter of the borehole <NUM> (which is <NUM> inches).

The casing <NUM>, the cement <NUM>, and the formation are assigned material properties for a constitutive model. In some cases, the material properties used in the finite element model <NUM> are extracted or calculated from log data and/or lab test data, such as Young's modulus, Poisson's ratio, frictional angle, tensile strength, unconfined compressive strength, cohesion, etc. In some examples, the logging tool <NUM> as described with reference to <FIG> is used to determine the material properties.

In this example, the casing <NUM> is assumed to deform elastically so a linear elastic material model is used to represent the constitutive behavior of the casing. In most scenarios, failure of the casing <NUM> is unlikely since the stresses that develop within the casing <NUM> at the breakdown pressure are often within the elastic range.

The cement <NUM> and rock (or formation) <NUM> surrounding the perforation are defined with different concrete damage plasticity models. In this example, the cement <NUM> is considered plastic, but has different properties from rock <NUM>. Generally, is difficult to determine cement <NUM> properties accurately because of downhole variations in the properties of the cement <NUM> and difficultly obtaining the samples for testing to determine the material properties. In this scenario, the properties of the concrete damage plasticity models for the cement <NUM> are extracted from (or based on) lab testing and/or lab data. In most cases, nominal cement <NUM> properties are used that include an elastic modulus of <NUM>,<NUM> MPa and Poisson's ratio of <NUM>.

The material model for the rock <NUM> is oftentimes more important than the cement <NUM> because the rock <NUM> is where fracture is likely to occur. The material model for the rock <NUM> includes different yield strengths in tension and compression, softening behavior in tension as opposed to initial hardening followed by softening in compression, different degradation of the elastic stiffness in tension and compression, and stiffness recovery effects during loading and unloading. In this example, the rock's material model includes an elastic modulus of <NUM>,<NUM> MPa, a Poisson's ratio of <NUM>, and the frictional angle of rock is assumed to be <NUM> degrees. In some examples, the properties of the concrete damage plasticity models for the rock <NUM> are extracted from (or based on) lab testing and/or lab data.

Tables <NUM> and <NUM> represent the damage parameters used in the material model for the rock in the finite element model <NUM>.

Fracture initiation is tensile failure whenever the induced tensile stresses around perforations tunnels <NUM> exceeds the rock tensile strength. The damage parameter and corresponding plastic strain are calculated as a part of the model solution process (e.g., to satisfy equilibrium). Damage scalar ranges from zero to one such that no rock damage is present if the damage scalar equals to zero and the rock is completely damaged (e.g., broken-down) if the damage scalar equals one. The finite element model <NUM> is solved to determine breakdown pressure based on the locations around the perforations <NUM> and perforation tunnels <NUM> that develop tensile damage indicating that the rock is broken-down.

<FIG> is a perspective view of the finite element model <NUM> showing the loads considered in the finite element model <NUM>. The different types of loads include in-situ stresses, overburden stress, gravity, and internal pressure acting on the casing and perforation cluster. In this example, the in-situ stress is defined by two depth points: a top surface (e.g., negative z-direction as shown in <FIG>) vertical stress of <NUM> MPa and a bottom point (e.g., a positive z-direction) of vertical stress <NUM> MPa. The top-to-bottom distance is <NUM> meters corresponding to the size of the domain <NUM> in the z-direction. Gravity is typically defined as a volumetric load acting in the positive z-direction with a magnitude of <NUM>/s<NUM>. The pressure acting on the casing and the perforation cluster is defined using targeted maximum bottom-hole pressure.

In this example, the wellbore <NUM> is assumed to be drilled in a normal stress regime and the horizontal part is at the minimum horizontal stress direction for the purpose of good hydraulic fracturing stimulation. The maximum horizontal stress direction is aligned with the x-direction and the minimum horizontal stress direction is aligned with the y-direction. In this case, the two horizontal in-situ stresses are defined as <NUM>σV and <NUM>σV, in the x-direction and y-direction, respectively. The three principal stresses are related by σV > σHmax > σHmin, which initiates transverse fractures in the rock and propagates the fractures along the maximum horizontal stress direction.

The finite element model <NUM> includes two load steps to predict the breakdown pressure. The first load step defines the in-situ stresses, gravity load, top surface overburden pressure. The second load step defines the applied pressure on the internal casing surface and all the perforation surfaces.

<FIG> is a plot of the solution progression of the finite element solver when solving the finite element model <NUM>. <FIG> shows the applied pressure Pw acting on internal casing surface vs. pseudo load time of the second load step. "Pseudo load time" represents a fraction of the total load step (e.g., from zero to one) and does not represent physical time units. In this example, the applied pressure Pw is uniform and assumed to simultaneously act on the internal radial surface of the casing <NUM> and the internal radial surfaces of each perforation tunnel <NUM>. In other examples, the pressure acting on each of the perforation tunnels <NUM> is different from the pressure acting on the internal radial surface of the casing <NUM>.

During the solution process, the finite element solver incrementally increases the applied pressure to a predetermined pressure limit <NUM> while solving for pressures of all the elements within the finite element model <NUM> at each step. For example, in some cases, the solver uses <NUM> increments to reach the targeted maximum bottom-hole pressure <NUM>. At each increment, tensile damage for each of the elements within the rock is determined by solving for equilibrium (e.g., quasi-static equilibrium). Once one or more elements in the finite element model <NUM> reach a tensile damage threshold, the breakdown pressure is determined to the pressure of the particular increment. As shown in the example of <FIG>, if tensile damage is detected at a pseudo load time of <NUM> (e.g., via an engineer interpreting the results or a post-processing script executing on the computer), the breakdown pressure would be about <NUM> MPa (<NUM> psi). The relationship between the applied pressure and incremental solution time is linear and the pressure gradually ramps up from zero to the targeted maximum bottom-hole pressure <NUM>. In some cases, locations of rock <NUM> within the finite element model <NUM> with high tensile stress results represent regions of the rock <NUM> where fractures are likely to occur.

<FIG> are perspective views of the stress results of the finite element model <NUM> at an internal pressure level representing pressure breakdown. <FIG> is the σ<NUM> (S11) stress result, <FIG> is the σ<NUM> (S22) stress result, and <FIG> is the σ<NUM> (S33) stress result. The casing <NUM> is hidden from view. The rock <NUM> and the ends of the perforations tunnels <NUM> are shown. The stress contours represent the stress state in the rock <NUM>.

Referring to <FIG>, S11 represents the σ<NUM> component of the Cauchy stress tensor σ. The σ<NUM> component is aligned with the "<NUM>" direction, which in this example represents the x-direction of the finite element model <NUM> (e.g., as shown in <FIG>). In other words, the displayed contour results represent the stress component acting along a horizontal direction perpendicular to the longitudinal axis of the wellbore. The σ<NUM> components vary from a maximum compression of <NUM> MPa (negative means compression) around location <NUM> to a minimum compression of <NUM> MPa around location <NUM>. In this example, all finite elements of the rock <NUM> have a σ<NUM> component representing compression at this applied pressure level.

Referring to <FIG>, S22 represents the σ<NUM> component of the Cauchy stress tensor σ. The σ<NUM> component is aligned with the "<NUM>" direction, which in this example represents the y-direction of the finite element model <NUM> (e.g., as shown in <FIG>). In other words, the displayed contour results represent the stress component acting along the longitudinal axis of the wellbore. The σ<NUM> components vary from a maximum compression of <NUM> MPa (negative means compression) to a maximum tension of <NUM> MPa (positive means tension) around location <NUM> which is around the perforation tunnel 404A. In this example, a majority of finite elements of the rock <NUM> have a σ<NUM> component representing compression at this applied pressure level but some finite elements are in tension around the perforation tunnels <NUM>. In this example, the σ<NUM> components are aligned with the directions of minimum horizontal stress of the wellbore.

In some examples, the Kirsch equations (Eqs. <NUM>-<NUM>) are used to verify the finite element model <NUM>. In some cases, perforation 404B will initiate fracture first in accordance with the Kirsch equations based on the in-situ stress orientation. The finite element model <NUM> results indicate that perforation 404B enters into tensile damage first, which is consistent with Kirsch equations.

Referring to <FIG>, S33 represents the σ<NUM> component of the Cauchy stress tensor σ. The σ<NUM> component is aligned with the "<NUM>" direction, which in this example represents the z-direction of the finite element model <NUM> (e.g., as shown in <FIG>). In other words, the displayed contour results represent the stress component acting along the vertical axis (e.g., the direction that gravity acts). The σ<NUM> components vary from a maximum compression of <NUM> MPa to a minimum compression of <NUM> MPa In this example, all finite elements of the rock <NUM> have a σ<NUM> component representing compression at this applied pressure level.

<FIG> is a perspective view of plastic strain (PEEQ) results and <FIG> is a perspective view of tensile damage (DAMAGET) results of the finite element model at the internal pressure level. The positive tensile stress of σ<NUM> in location <NUM> (shown in <FIG>) is also present in location <NUM> which is also surrounding the perforations <NUM>. In the example shown, locations <NUM> exceed the in-situ stress σHmin and are very likely the tensile failure locations because the stress in the minimum horizontal stress direction changes from compression to tension. This change from compression to tension is an indication that the rock <NUM> is close to tensile failure. The tensile damage result of <FIG> confirms that location <NUM> has accumulated damage. This means that transverse fractures will be initiated from the perforations <NUM> based on the orientation of the narrow tensile stress band under the loading of internal pressure from fluid injection. <FIG> shows the equivalent plastic strain occurs around the perforation cluster and the remainder of the rock <NUM> is still in elastic (i.e., not plastic). <FIG> indicates the tensile damage scalar due to fluid injection occurs in the surrounding area close to perforations <NUM> while the remainder of the rock <NUM> has no tensile damage (e.g., DAMAGET is zero).

In order to determine the breakdown pressure, the tensile damage occurring along the perforation tunnel <NUM> is determined from the simulation results. In this example, each perforation tunnel <NUM> has a different 3D orientation with respect to the in-situ stress orientations. This difference in orientation causes a variance in stress results around each perforation tunnel <NUM> which means that not all perforation tunnels <NUM> will have rock <NUM> surrounding the tunnel that fails at the breakdown pressure. The perforation tunnels <NUM> that develop tensile stress around the perforation tunnel <NUM> sufficient to cause enough damage to first reach a threshold of damage (e.g., <NUM>, <NUM> as measured using simulation output variable DAMAGET) will define the breakdown pressure level. In cases with more than one perforation cluster <NUM>, one perforation cluster may control the breakdown pressure of the wellbore (e.g., by reaching failure first) while the other perforation clusters do not.

<FIG> are cross-section views of the σ<NUM> component acting through cross sections of each of four perforations tunnels with at the internal pressure level. While only four perforations are shown in <FIG>, this because some of the perforation tunnels have stress results that are the same, or similar to, other perforation tunnels. Each of the perforation tunnels shown in <FIG> are oriented at a different angular phase angle and represent four sequential perforation angles. In this example, <FIG> corresponds to perforation tunnel 404C, <FIG> corresponds to perforation tunnel 404B, <FIG> corresponds to perforation tunnel 404A, and <FIG> corresponds to perforation tunnel 404F. The results of perforation tunnel 404D and 404E are substantially similar to the results shown in <FIG> and <FIG>, respectively. A positive tensile stress (e.g., positive S22 contour results) appears around the perforation tunnels representing the locations where hydraulic fracture initiation is likely to occur.

<FIG> are cross-section views of the plastic strain (PEEQ) acting through the cross sections of the perforations tunnels <NUM> at the internal pressure level. The cross section views of <FIG> correspond to the respective cross section views of <FIG>. The results indicate that plastic deformation (e.g., PEEQ greater than zero) occurs in the areas surrounding the perforation tunnels <NUM>.

<FIG> are cross-section views of the damage (DAMAGET) acting through the cross sections of the perforations tunnels <NUM> at the internal pressure level. The cross section views of <FIG> correspond to the respective cross section views of <FIG>. The results indicate that damage (e.g., DAMAGET greater than zero) occurs in the areas surrounding the perforation tunnels <NUM>.

In this example, the damage associated with the perforation tunnel 404C (as shown in <FIG>) is <NUM> which is the largest of all the perforation tunnels <NUM>. In this example, all four of the perforation tunnels <NUM> shown have rock <NUM> that has failed (e.g., when DAMAGET is greater than zero (or greater than a threshold, e.g., <NUM>, <NUM>, <NUM>, etc.). In this example, if the threshold was set to <NUM>, and the results from the previous increment of the load step resulted in damage values less than <NUM> around each of the six perforation tunnels <NUM>, then the breakdown pressure would be determined based on the applied pressure for this increment since perforation tunnels 404C and 404B have exceeded the damage threshold. In this example, the breakdown pressure is obtained from <FIG>. Tensile rock failure is at a pseudo load time around <NUM> corresponding to a breakdown pressure of around <NUM> MPa (<NUM>,<NUM> psi).

<FIG> shows that, among the four perforations tunnels, the horizontal perforation tunnel has the most number of finite elements with damage for the pressure level. The perforation cluster of <FIG> is the first one to develop tensile damage for this in-situ stresses setting. This is expected based on the Kirsch equation and the perforation orientation. The overall breakdown pressure for this wellbore will be limited by this horizontal perforation tunnel 404B since it is the first one to initiate fracture.

In some cases, the finite element model <NUM> includes more than one perforation cluster, In this scenario, the breakdown pressure is determined when one perforation tunnel develops tensile damage above a threshold damage among all the perforation clusters.

<FIG> is a flowchart of a modeling method <NUM> for predicting breakdown pressure for hydraulic fracturing a subsurface formation. Input parameters s are determined <NUM> that include a well survey to locate wellbore in the model and sonic log data to calculate the mechanical properties, model parameters, and in-situ stresses and orientation.

The finite element model is built <NUM> by developing geometric models of the casing, the cement, and the formation. Corresponding material models are assigned. Based on the in-situ stresses orientation, well landing depth, and the domain used for the finite element modeling, the model including the casing, cement and one or more perforation clusters is assembled <NUM> and rotated to the correct orientation based on borehole image log processing (e.g., aligned with the orientations according to image log data). The in-situ stress direction is obtained for a particular formation from the image log (e.g., identifying breakout and breakdown in image log). The finite element model is rotated so that the well trajectory orientation and in-situ stress orientation are correctly aligned in the modeling. For example, when the perforations are aligned in the maximum horizontal in-situ stress direction, the rock <NUM> will initiate fracture at the lowest breakdown pressure compared to aligning the perforations at a different angle.

The load analysis steps are defined <NUM>. The corresponding boundary conditions and loads are defined <NUM> over the domain. The internal pressure is applied <NUM> to the internal casing surfaces and the perforation surfaces. The numerical simulation is solved <NUM> and post processed to determine <NUM> the breakdown pressure based on the tensile damage developing around each the perforation cluster.

<FIG> is a flowchart of an alternate modeling method <NUM> for predicting breakdown pressure for hydraulic fracturing a subsurface formation. The method <NUM> includes implementing <NUM> a three-dimensional finite element model on one or more processors to simulate a deviated well with a wellbore casing, a cement adjacent to the wellbore casing, and a perforation cluster with at least two perforations through a side wall of the wellbore casing, each of the at least two perforations extending through a rock of the subsurface formation via a respective perforation tunnel and having a different phase angle relative to an longitudinal axis of the casing.

The method <NUM> includes assigning <NUM> a concrete damage plasticity model to define constitutive behavior of the rock of the subsurface formation, the concrete damage plasticity model including compression hardening, compression damage, tensile stiffening, and tensile damage behavior. A contact relationship is assigned <NUM> at an interface between the casing and the cement, the contact relationship modeling de-bonding of the interface between the casing and the cement.

The method <NUM> includes solving <NUM> the three-dimensional finite element model on the one or more processors to determine a breakdown pressure of the deviated well based on an amount of tensile damage of the rock based on an applied pressure representing injected hydraulic fluid and the assigned concrete damage plasticity model and the assigned contact relationship.

In some implementations, the method <NUM> includes drilling and completing a deviated well with a wellbore casing size, tubing size, and wellhead selected at least in part based on the determined breakdown pressure. In some implementations, the method <NUM> includes selecting a wellbore casing size, a tubing size, and wellhead at least in part based on the determined breakdown pressure.

In some implementations, the method <NUM> includes injecting hydraulic fluid into the deviated well at an injection pressure that represents the required breakdown pressure to cause hydraulic fracturing of the rock of the subsurface formation. In this scenario, the injection pressure is determined based on the breakdown pressure determined from the finite element model. In some implementations, injecting hydraulic fluid into the deviated well includes pumping the hydraulic fluid into the deviated well using a pump schedule determined at least in part based on the determined breakdown pressure. In some cases, the method <NUM> includes determining the pump schedule of a pump for injecting the hydraulic fluid into the wellbore at least in part based on the determined breakdown pressure.

<FIG> is a flowchart of an alternate modeling method <NUM> for predicting breakdown pressure for hydraulic fracturing a subsurface formation. The method <NUM> includes using <NUM> a three-dimensional finite element model implemented on one or more processors to simulate a deviated well with a wellbore casing, a cement adjacent to the wellbore casing, and a perforation cluster with at least two perforations through a side wall of the wellbore casing, each of the at least two perforations extending through a rock of the subsurface formation via a respective perforation tunnel and having a different phase angle relative to an longitudinal axis of the casing.

The method <NUM> includes solving <NUM> the three-dimensional finite element model implemented on one or more processors to determine a breakdown pressure of the deviated well based on an amount of tensile damage of the perforation cluster induced by an applied pressure representing injected hydraulic fluid.

The method <NUM> includes drilling and completing <NUM> a deviated well with a wellbore casing size, tubing size, and wellhead selected at least in part based on the determined breakdown pressure.

The method <NUM> includes injecting <NUM> hydraulic fluid into the deviated well at an injection pressure determined at least in part on the determined breakdown pressure to cause hydraulic fracturing of the rock of the subsurface formation. In some implementations, injecting hydraulic fluid into the deviated well comprises pumping the hydraulic fluid into the deviated well using a pump schedule determined at least in part based on the determined breakdown pressure.

In some implementations, methods <NUM>, <NUM> include extracting material properties of the rock from log data and/or lab test data. In some cases, the compression hardening, compression damage, tensile stiffening, and tensile damage behavior of the concrete damage plasticity model of the rock are at least in part based on the extracted material properties from lab test data.

In some implementations, methods <NUM>, <NUM> include rotating an orientation of the finite element model based on borehole image log data so that a well trajectory orientation and in-situ stresses are oriented over a domain of the finite element model in accordance with the borehole image log data. In some cases, the in-situ stress are defined based on a maximum horizontal stress direction of the borehole image log data.

In some implementations, each of the at least two perforations are axially spaced relative to each other. In some cases, each of the at least two perforations are angularly spaced relative to each other around a circumference of the wellbore. In some cases, the at least two perforations are six perforations equally spaced and located within a <NUM> centimeter length of the wellbore casing and angularly spaced <NUM> degrees apart.

In some implementations, an output of the solved three-dimensional finite element model is a spatially varying contour of the tensile damage of the rock. In some implementations, determining the breakdown pressure includes detecting when at least one finite element representing the rock has a tensile failure above a predetermined threshold. In some cases, the predetermined threshold is a scalar between <NUM> and <NUM>, between <NUM> and <NUM>, between <NUM> and <NUM>, between <NUM> and <NUM>, or between <NUM> and <NUM>.

In some implementations, solving the three-dimensional finite element model includes solving the three-dimensional finite element model using two quasi-static loading steps, a first step used for solving static equilibrium with a loading of in-situ stresses, gravity loading, overburden, and underburden, and a second step used for solving static equilibrium of an applied pressure to determine the breakdown pressure while including the loading of the first step.

In some implementations, the three-dimensional finite element model includes a concrete damage plasticity model representing constitutive behavior of the rock and a contact relationship between an interface of the wellbore casing and the cement adjacent to the casing, the contact relationship modeling de-bonding of the interface of the casing and the cement.

<FIG> is a schematic illustration of an example controller <NUM> (or control system) for determining a subterranean formation breakdown pressure according to the present disclosure. For example, the controller <NUM> may include or be part of the control system <NUM> shown in <FIG>. The controller <NUM> is intended to include various forms of digital computers, such as printed circuit boards (PCB), processors, digital circuitry, or otherwise parts of a system for determining a subterranean formation breakdown pressure. Additionally the system can include portable storage media, such as, Universal Serial Bus (USB) flash drives. For example, the USB flash drives may store operating systems and other applications. The USB flash drives can include input/output components, such as a wireless transmitter or USB connector that may be inserted into a USB port of another computing device.

The controller <NUM> includes a processor <NUM>, a memory <NUM>, a storage device <NUM>, and an input/output device <NUM> (for displays, input devices, example, sensors, valves, pumps). Each of the components <NUM>, <NUM>, <NUM>, and <NUM> are interconnected using a system bus <NUM>. The processor <NUM> is capable of processing instructions for execution within the controller <NUM>. The processor may be designed using any of a number of architectures. For example, the processor <NUM> may be a CISC (Complex Instruction Set Computers) processor, a RISC (Reduced Instruction Set Computer) processor, or a MISC (Minimal Instruction Set Computer) processor.

In one implementation, the processor <NUM> is a single-threaded processor. In another implementation, the processor <NUM> is a multi-threaded processor. The processor <NUM> is capable of processing instructions stored in the memory <NUM> or on the storage device <NUM> to display graphical information for a user interface on the input/output device <NUM>.

The memory <NUM> stores information within the controller <NUM>.

The storage device <NUM> is capable of providing mass storage for the controller <NUM>.

The input/output device <NUM> provides input/output operations for the controller <NUM>.

The features described can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. The apparatus can be implemented in a computer program product tangibly embodied in an information carrier, for example, in a machine-readable storage device for execution by a programmable processor; and method steps can be performed by a programmable processor executing a program of instructions to perform functions of the described implementations by operating on input data and generating output. The described features can be implemented advantageously in one or more computer programs that are executable on a programmable system including at least one programmable processor coupled to receive data and instructions from, and to transmit data and instructions to, a data storage system, at least one input device, and at least one output device. A computer program is a set of instructions that can be used, directly or indirectly, in a computer to perform a certain activity or bring about a certain result.

Additionally, such activities can be implemented via touchscreen flat-panel displays and other appropriate mechanisms.

Claim 1:
A method of hydraulic fracturing a subsurface formation, the method comprising:
using (<NUM>) a three-dimensional finite element model implemented on one or more processors to simulate a deviated well with a wellbore casing (<NUM>, <NUM>), a cement (<NUM>, <NUM>) adjacent to the wellbore casing (<NUM>, <NUM>), and a perforation cluster (<NUM>, <NUM>) with at least two perforations (<NUM>, <NUM>) through a side wall of the wellbore casing (<NUM>, <NUM>), each of the at least two perforations (<NUM>, <NUM>) extending through a rock (<NUM>) of the subsurface formation via a respective perforation tunnel (<NUM>) and having a different phase angle relative to a longitudinal axis of the wellbore casing (<NUM>, <NUM>);
solving (<NUM>) the three-dimensional finite element model implemented on one or more processors to determine a breakdown pressure of the deviated well based on an amount of tensile damage of the perforation cluster (<NUM>, <NUM>) induced by an applied pressure representing injected hydraulic fluid;
drilling and completing (<NUM>) a deviated well with a wellbore casing size, tubing size, and wellhead selected at least in part based on the determined breakdown pressure; and
injecting (<NUM>) hydraulic fluid into the deviated well at an injection pressure determined at least in part on the determined breakdown pressure to cause hydraulic fracturing of the rock (<NUM>) of the subsurface formation,
wherein the three-dimensional finite element model includes a concrete damage plasticity model representing constitutive behavior of the rock (<NUM>) and a contact relationship between an interface of the wellbore casing (<NUM>, <NUM>) and the cement (<NUM>, <NUM>) adjacent to the wellbore casing (<NUM>, <NUM>), the contact relationship modeling de-bonding of the interface of the wellbore casing (<NUM>, <NUM>)and the cement (<NUM>, <NUM>).