Fracturing_Modeling.pdf

FRACTURING ENGINEERING MANUAL Schlumberger

Dowell

Fracture Modeling

Section 300 May 1998 Page 1 of 35

FRACTURE MODELING 1 Introductory Summary............................................................................................................. 2 2 Concepts................................................................................................................................... 3 2.1 Fundamental Laws............................................................................................................... 4 2.2 Constitutive Laws ................................................................................................................. 4 2.3 Fracture Propagation ........................................................................................................... 6 3 Hydraulic Fracturing Models .................................................................................................. 9 3.1 Two-Dimensional (2D) ....................................................................................................... 11 3.2 Pseudo Three-Dimensional (P-3D) .................................................................................... 15 3.3 Planar Three-Dimensional (PL-3D).................................................................................... 18 3.4 Fully Three-Dimensional (3D) ............................................................................................ 19 4 Examples ................................................................................................................................ 20 4.1 Case History ...................................................................................................................... 20 4.2 Model Comparisons ........................................................................................................... 28 FIGURES Fig. 1. Modes of loading............................................................................................................... 7 Fig. 2. Fracture divided into elements.......................................................................................... 9 Fig. 3. Representation of a planar fracture. ............................................................................... 10 Fig. 4. KGD geometry. ............................................................................................................... 11 Fig. 5. PKN geometry................................................................................................................. 12 Fig. 6. 2D and radial Sneddon cracks. ....................................................................................... 13 Fig. 7. Elliptical profile (P-3D)..................................................................................................... 17 Fig. 8. Example grid (PL-3D model)........................................................................................... 18 Fig. 9. Fracture profile (PL-3D model). ...................................................................................... 19 Fig. 10. Permeability, thickness and stress profile. .................................................................... 20 Fig. 11. Computed values for Young's modulus and Poisson's ratio. ........................................ 21 Fig. 12. Profile of bottomhole, casing and tubing pressures. ..................................................... 24 Fig. 13. Pressure match for bottomhole and casing pressure. .................................................. 24 Fig. 14. Fracture profile.............................................................................................................. 25 Fig. 15. Fracture width profile. ................................................................................................... 25 Fig. 16. Match of net pressure for calibration fracture and main fracture. ................................. 26 Fig. 17. Fracture profile.............................................................................................................. 26 Fig. 18. Reservoir model for final history match......................................................................... 28 DOWELL CONFIDENTIAL

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TRIFRAC length and width profile. ................................................................................32 STIMPLAN length and width. ........................................................................................32 FRACPRO length and width profile. ..............................................................................33 GOHFER length and width profile. ................................................................................33 TERRAFRAC length profile. ..........................................................................................34 STIMPLAN length and width profile...............................................................................34 MEYER length and width profile. ...................................................................................35 Ohio state length profile.................................................................................................35 TABLES

Table 1. Table 2. Table 3. Table 4. Table 5.

Comparison Of Stress ..................................................................................................22 Permeability and Fluid Loss .........................................................................................22 Design Information .......................................................................................................23 Fracture Model Comparison Runs................................................................................30 Fracture Model Comparison Runs................................................................................31

1 Introductory Summary The prediction of fracture geometry has been a central issue in engineering design and evaluation of hydraulic fractures, and many models have been developed over the years. These models determine fracture geometry by attempting to relate many variables such as rock properties, fluid properties, fluid volume pumped and stress data. Some models use a fixed fracture height and others continuously calculate the height during the simulation. Each change aimed at more closely matching the real conditions requires more sophistication in modeling the fluid flow in the entire fracture, effect of proppant and elasticity of the entire system. To be practical, however, the calculations must be made at reasonable increments along the fracture and computational time must not be excessive. The degree of sophistication of a model is therefore somewhat controlled by the practical application. The models are also data limited. The comparison of different models can be difficult and confusing because of the way the various authors handle the variety of conditions, what they feel is important, what assumptions they make and how portions of the model are coupled. Decisions on how to handle elasticity, fluid flow, type of grid or cross section, vertical stress differences and toughness, for example, can have a large impact on the calculated fracture geometry. There is still much work to be done in obtaining meaningful data for input into the more sophisticated models.

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Basically, there are four types of fracture models either being used or being developed in the industry today. • Two-Dimensional (2D) •

Pseudo Three-Dimensional (P-3D)

•

Planar Three-Dimensional (PL-3D)

•

Fully Three-Dimensional (3D).

The designation of the third listed model as “Planar” is an area of confusion that should be clarified. Actually, the term planar means that the fracture occurs in a plane. This condition is true for all fracture models except the fully 3D. Planar was simply used to name the model that is more advanced than the pseudo 3D, but not quite as sophisticated as the fully 3D. A fully 3D model would have the capability of being nonplanar (fracture could curve or change planes) if the correct stress data and other information were available for input. This section on fracture models will be limited to a brief discussion of some of the concepts that must be considered to build a model, as well as a brief discussion of each model. It is beyond the scope of this section to cover each model in detail because of the number of models available, and not having the documentation or code to examine each model. Also, models change as more data become available from evaluation, in-situ testing and calibration of logs, and from special industry projects to calibrate the various models based on the best available information. A successful model is one that has the ability to match the pressure from the treatment by using realistic variables based on in-situ data, and to calculate a fracture height consistent with other methods used on the actual treatment.

2 Concepts Modeling fracturing treatments requires a blending of many different components, such as rock mechanics, fluid mechanics, rheology and heat transfer. Two sets of laws are required for this process. • The Fundamental Laws dealing with mass, momentum and energy conservation. These relate to the physical principles. •

The Constitutive Laws include rock elasticity and fluid rheology. These describe the behavior of a system under a certain number of conditions.

Coupling these two sets with the appropriate “boundary conditions” produces some very complicated mathematical formulations. To solve the coupled problem requires discretization of the system (break into small geometric components such as a grid), and then writing equations in a form that can be solved with digital computing.

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2.1 Fundamental Laws •

Conservation of Mass: The mass of a system does not change with time. The conservation of mass for modeling a fracture treatment is used to give the overall material balance. This material balance is illustrated by Eq. 1. Vi = V fp + VLP

(1)

Where: Vi = volume injected Vfp = fracture volume •

•

VLp = injected volume lost to the formation. Conservation of Momentum: Two types of forces can be distinguished; (1) the body forces such as gravity that act on the whole volume, and (2) the surface forces, such as pressure forces and fluid friction, that act only along the boundary of the domain. The conservation of momentum principle relates the time change in the total momentum of a body to the applied forces (both on the volume and on the surface). Many fracture treatments are modeled as quasistatic. This implies that the rates of the change of velocities are negligible, and therefore the summation of surface and volume forces is zero. Conservation of Energy: This pertains primarily to the two fundamental laws of thermodynamics for a system; (1) the change in total energy of a system is equal to the work of the forces applied on the system plus the rate of heat transfer, and (2) the internal energy of a system is a function only of its entropy.

2.2 Constitutive Laws A mechanical system is completely defined by a certain number of variables that depend on time and position. • mass •

temperature

•

velocity (three components)

•

stresses (six components).

These variables make a minimum of 11 unknowns while the number of components in the three conservation laws is only five. Therefore, constitutive equations to solve the system need to be defined. Some equations used in the less complicated models can be simplified to the point where they no longer apply, thereby reducing the number of computations needed to complete a simulation. Three types of constitutive relations are considered for the system.

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1. Fluid incompressibility is assumed. 2. A relation exists between stress and strain (or rates of strain). 3. Scalar (magnitude only) quantities are related to fluxes. • Incompressibility: For many fluids, the density does not change with time, pressure or temperature. This assumption of constant density (incompressible fluid) causes the equation of mass conservation to simplify and become zero. •

Stress/Strain: The section on rock mechanics showed the relationship between stress (σ) and strain (∈) for an elastic solid, giving the coefficient of proportionality (σ = E∈) where E is Young's modulus. Modulus contrasts act mainly to alter the shape of the fracture rather than the tip positions at any pressure. Analyse to determine the effect of modulus contrast are extremely time consuming because they require large finite-element or similar solutions. Formulas for the average width are used in many cases to scale the width profile. For the general case of a fracture growing into the surrounding layers of greater stress and/or fracture toughness than the pay zone, the pressure, height and shape of the fracture cross section depend on the stress, toughness, thickness and modulus of the individual layers. The more simple models would assume a constant modulus in all layers, plane strain elasticity and no flow-induced pressure drop in the vertical direction. When considering fluid flow, the stress tensor includes pressure and a viscous tensor (τ). Newton's law for viscous incompressible fluids is simplified to τ = µγ

(2)

where µ is the viscosity of the fluid and γ is the strain rate tensor. Flow in several of the fracture models is assumed to be in one direction only (x direction) and the velocity field is therefore unidirectional. This means that both the shear stress (τxy) and rate of deformation: γxy are related as: τ xy = µγxy

•

(3)

Coupling conservation relations with the constitutive equation leads to a very complicated fluid equation. The more sophisticated PL-3D models may attempt to handle this equation while the 2D and many P-3D models must make simplifying assumptions, particularly on velocity, before numerically solving the system. Handling fluid flow in the fracture to determine the pressure distribution is very important and necessary to determine the fracture displacement, as well as using a method to determine how a particular model will handle all of the variables associated with adding the proppant to the system. An accurate calculation of the pressure distribution within the fracture is critical for any realistic simulation.

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Flux Laws: These include the relationships of pressure drop in a porous medium to velocity (Darcy's law), of heat flux to temperature (Fourier's law) or the rate of reaction to the change in concentration (Fick's law) for acid fracturing.

2.3 Fracture Propagation Modeling the fracturing process is a very complicated task. Injecting fluid into the formation will modify the stress distribution and pressure within the formation, creating a fracture in which the injected fluid flows. This injected fluid exchanges heat with the formation and leaks off into the formation. The poroelastic effect can be interpreted as a time-dependent back-stress; whenever a fracture profile is obtained and a pressure is calculated, an additional term must be added to the pressure. The fluid is also considered to be a multiphase type because of the proppant being carried. The proppant also alters the fluid viscosity, which affects the fluid-flow model. A good model needs rigorous calculation of the fluid fronts as well as handling the settling of the proppant as in the simpler 2D models up to convection hindered settling and other more complicated accounting of the proppant in some of the PL-3D models. Simulating the propagation of a hydraulic fracture requires consideration of the linear elastic fracture mechanics (LEFM) formulation, fracture fluid flow and continuity equation. These three sets need to be coupled to simulate the propagation; the mathematical problem is complex because of the different types of equations and the presence of a moving boundary (fracture edge). The strains produced in the formation caused by the deformation from inducing a hydraulic fracture are actually quite small. The small value of strain allows the assumption that the formation deforms in a linear elastic manner. Knowing the state of stress (or pressure) induced by the fluid in the fracture and the confining stresses (boundary conditions) allows the calculation of the fracture width (displacement). One simple method is to consider the crack as a uniformly pressurized elliptical surface with semiaxes a and b. This condition forces the fracture to assume an elliptical shape (Sneddon). Different cases may be derived from Sneddon's solution. These cases can relate to the Griffith crack as well as the radial or penny-shaped crack. This will be covered in more detail in the discussion of 2D models. The classical techniques in elasticity do have severe limitations because of the stress singularity near the crack tip, and special methods have to be used to improve the accuracy in computing the stress intensity factors. Boundary integral techniques have become popular because the problem is solved only at the boundary, reducing the dimension of the problem by one, and provides a simple method to determine stress intensity factors. However, implementation can be difficult for nonhomogeneous media with varying elastic moduli. Remember from the section on rock mechanics that the stress intensity factor (KI) is for the opening mode of the fracture and describes the magnitude of the stresses near the crack tip — it depends on the crack geometry, surrounding material and applied loads. Fig. 1 shows the three fundamental modes of loading for a fracture. DOWELL CONFIDENTIAL


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Fig. 1. Modes of loading. The opening mode corresponds to normal tension, in front of the crack. The sliding mode is associated with transverse shear while the tearing mode is a longitudinal shear. Usually, only the opening mode applies for a plane fracture. The other modes may be applicable in complex fracturing, out of plane fractures or when near natural fissures. For a given system of stresses, it is necessary to determine whether a fracture will initiate and then propagate, and in which direction, and from which point in the created fracture. Griffith (1921) addressed the problem by defining the surface energy of a fracture. The Griffith criterion stated that there is an equilibrium if the specific surface energy γ is related to the change in internal energy (U) by: 2γ = −

∂U ∂A

(4)

where A is the fracture area. For a uniformly loaded fracture, the critical load (σc) and the surface energy are related, such as   2 Eγ  σc =   πx ( 1 − ν 2 )   f 

1/ 2

(5)

where xf is the fracture half-length and E and ν are the elastic constants.

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Another important concept related to the surface energy criterion is that of the critical stress-intensity factor (KIc), also known as fracture toughness. The condition for a fracture to be in equilibrium requires that the stress intensity factor (KI) associated with the load be equal to a critical value (KIc). A simple relation can be derived between the surface energy and KIc for a uniformly pressurized fracture — γ=

(1 − ν 2 )K 2 Ic E

(6)

This expression indicates (for linear elastic behavior) that the surface energy criterion and the critical intensity factor are related and form a unified criterion for propagation. The calculation of the pressure distribution in the fracture, due to fluid flow, is necessary to determine the fracture displacements. As mentioned earlier, the fluidflow problems to include non-Newtonian as well as Newtonian fluids present some problems and require a large amount of computational time. Because of this problem, the fracture may be discretized into a series of parallel lines (surfaces) and the flow considered as quasistatic. The PL-3D models do handle the flow problem a little more rigorously and, consequently, require more time for computation. The continuity equation is the last relation and simply describes the conservation of mass previously discussed. The continuity equation may be written for each fracture element (in the grid) — Flow Rate In = Flow Rate Out + Accumulation.

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3 Hydraulic Fracturing Models When a fracture is created in a rock mass, the fluid applies pressure to the two faces of the resulting crack. This pressure causes the crack to open. The width at any point is therefore dependent on the pressure applied to the faces. The condition of the plane strain and rectangular elements is used to model the fracture for the 2D cases. Fig. 2 illustrates a boundary element technique applied to a planar fracture.

Fig. 2. Fracture divided into elements. The simplest representation is when the width is assumed to be piecewise constant over each element. For problems solved by this method, the width distribution is typically obtained for a specified distribution of pressure over the elements. Alternatively, the pressure distribution can easily be found from a specified width profile. Some P-3D models may use a little more sophisticated grid system, but fluid flow is again only 1D and the vertical height growth is handled by shape factors and other methods although elasticity is considered to be 2D. The PL-3D models use a more complicated grid system (small squares or triangles). The junction of the grid elements becomes the node for the solution of the problem. This allows the applied pressure and displacements everywhere on the fracture faces to be related. Elasticity is also 3D in these PL-3D models.

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The planar fracture was briefly introduced in the introductory summary to clarify a model description. Fractures are planar features as shown in Fig. 3.

Fig. 3. Representation of a planar fracture. The fracture lies in the xz plane, and the width is represented by the y direction. Fluid may flow in all directions in this fracture, but 3D fluid flow is very complicated and various models handle this flow differently. Usually flow along the fracture length is modeled in detail, while horizontal flow across the width of the fracture and vertical flow are neglected (2D, P-3D). Horizontal flow may be calculated in an average sense by assuming that an average velocity adequately represents the flow. An overall fluid-flow model could therefore be obtained by coupling 1D mass conservation and momentum conservation equations. Volume conservation would replace mass conservation if only incompressible fluids are considered.

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3.1 Two-Dimensional (2D) Most 2D models require that a value for fracture height be input so the length and width can be calculated from the volume and flow characteristics governed by the code for the particular model. The models that will be discussed here are the (1) Khristianovic and Zheltov, with later contributions by Geertsma and de Klerk (KGD model), (2) Perkins and Kern, and later Nordgren (PKN model), and (3) the radial or penny-shaped model. A common simplifying assumption is that the lateral effects of a fracture are small compared to the vertical effects and can be neglected. This condition is termed plane strain and implies that each cross section acts independently of any other section, so that the mechanical analysis need only be performed in two dimensions. This plane-strain assumption is an integral part of the 2D fracture models. Two cases of plane strain can be distinguished for the 2D formulations. • Horizontal Plane-Strain Geometry: The fracture zone will deform independently of the upper and lower layers. This will be possible for free slippage on these layers and represent a fracture with horizontal penetration that is much smaller than the vertical, and where the fracture shape does not depend on the vertical position. This describes the KGD fracture geometry as shown in Fig. 4. This geometry has a constant height with a cross section that is rectangular in shape.

Fig. 4. KGD geometry. DOWELL CONFIDENTIAL

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Vertical Plane-Strain Geometry: This condition exists for a large confinement where the fracture is limited to a given zone. Each vertical cross section deforms independently of the others. The fracture widths in the vertical direction are coupled through the continuity and fluid-flow equations; however, because there is no vertical extension (in each of the vertical sections) during simulation, the pressure is uniform and the cross-sectional shape of the fracture height is elliptical. This describes the PKN geometry as illustrated in Fig. 5.

Fig. 5. PKN geometry.

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Sneddon's solution for modeling the behavior of a fracture (for linear elastic assumption) was mentioned earlier during the discussion of concepts. Fig. 6 shows the 2D and radial Sneddon cracks. These solutions are for a 2D crack having one dimension of infinite extent, and the other dimension of finite extent (d in Fig. 6). The radial or penny-shaped crack is defined by the radius (R). The resulting width is elliptical in shape for both types of cracks, and is proportional to one of the characteristic dimensions (either d or R).

Fig. 6. 2D and radial Sneddon cracks. The width is also proportional to the net pressure (pf - σmin) and inversely proportional to the plane-strain modulus given as E ′ = E / (1 − ν 2 )

(7)

Young's modulus and Poisson's ratio are at in-situ conditions (E is defined by the tangent Young's modulus Et). Sneddon's method has been used in different ways to model 2D fractures. The characteristic dimension, d, is assumed to be the total tip-to-tip fracture length (2xf) for the KGD model. Since d is assumed to be the total fracture length, then the infinite dimension has to correspond to the fracture height. The other assumption is that the characteristic dimension, d, is the fracture height. This is the condition for the PKN model; since d is the fracture height, then the infinite dimension has to be the fracture length.

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The KGD and PKN models consider the propagation of a vertical fracture of given height. However, in some cases, the vertical stress is lower than the horizontal stress and the fracture will propagate in a horizontal or inclined direction. Both Perkins and Kern as well as Geertsma and de Klerk gave simplified expressions for the propagation of a radial horizontal fracture. Similarly, vertical fractures may propagate radially in thick formations where there are no barriers to height growth. This situation leads to the same equations when the injection interval is small compared to the fracture extension. The KGD model is valid when h>>xf. The PKN model is valid when xf >>h. The radial model is most appropriate when the total length (2xf or 2R) is approximately equal to the height. Again, the three sets of equations to be coupled are the elasticity, continuity and fluid flow. Some of the important characteristics and differences pertaining to the 2D models are • KGD Fracture Model − A fixed fracture height is assumed, and fluid flow is horizontal only (in the direction of the propagation). − Crack opening is solved in the horizontal plane. As a result, the fracture width does not vary with the fracture height, except by the boundary condition set at the wellbore that specifies a constant total injection rate. − Width is constant in the vertical direction because of the plane-strain condition and individual horizontal planes. − The model gives wider fracture widths and shorter fracture lengths when compared to the PKN model. − The flow resistance in the narrow rectangular vertical width is what determines the fluid pressure gradient in the propagating direction. − The excess pressure (net pressure) decreases with time, and in log-log coordinates has a slope equal to -1/3.

•

− The model is most appropriate when the fracture length is smaller than the fracture height. PKN Fracture Model − A fixed fracture height is assumed and fluid flow is horizontal only (in the direction of the propagation). − Crack opening is solved in the vertical plane. As a result, the fracturing fluid pressure is constant in vertical cross sections perpendicular to the direction of the fracture propagation.

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− Each vertical cross section deforms individually and obtains an elliptic shape with the maximum width in the center. The only coupling between the different vertical cross sections is due to fluid flow in the fracture. There is no elastic coupling between the planes. − This model is used to describe the behavior of planar fractures that have a length-to-height ratio greater than three. − The model gives narrower widths and longer fracture lengths compared to the KGD model. − The excess pressure increases with times; in log-log coordinates, the slope is positive. − The model is most appropriate when the fracture length is much larger than the fracture height. The 2D fracture models require less input and perform the computations very quickly. This advantage may be used when the fracture is known to be contained in height, and the value of the height is known. However, the fracture shape is still rectangular because no difference is calculated in the height at the tip of the fracture compared to the height at the wellbore. Because of these conditions as well as the 1D fluid flow and noncoupling of the length and width, the 2D model fracture length is usually longer, the net pressure is higher and the width is wider than actual or when compared to the various P-3D or PL-3D models. 3.2 Pseudo Three-Dimensional (P-3D) Most of the P-3D models incorporate basic assumptions regarding the elastic properties of the rock layers, fluid flow and fracture initiation. These assumptions reduce the complexity and number of calculations needed for a fracture simulation. The elasticity is considered in two dimensions (but not in cross section) and fluid flow is one-dimensional in the direction of the propagation. Many P-3D simulators assume all layers have the same elastic, reservoir and fluid-loss properties. Some P-3D simulators allow the user to input a description for each layer and then use some method for averaging, while others use calculated shape factors to arrive at the effect the layers will have on the geometry. For example, a different Young's modulus could be input for 15 layers. The simulator, depending on the choice of the author, may then use averaging or other factors to determine the actual impact over the entire interval. The values of 15 different layers are now reduced to one. This method can have significant implications in sections with thick layers of coal, shale or other lithology. Most P-3D simulators have problems running when a large contrast in elastic properties exists. The assumptions and use of averaging and factors allow the P-3D simulators to generate a solution in a very short time, even on the small personal computers. The P-3D simulators divide the fracture into a fixed number of vertical elements in the grid. Each vertical element extends from the top of the fracture to the bottom. DOWELL CONFIDENTIAL



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All of the values describing the fracture geometry, pressure in the fracture and fluid loss are the same throughout that one single element. This is one reason why most P-3D simulators estimate more vertical height growth when compared to the PL-3D simulators. Also, there is no pressure drop within the element from the perforations to the vertical limits of the fracture. The pressure at the vertical tip of a P-3D fracture simulation is also higher than that of a PL-3D simulation. The P-3D simulators may produce solutions with excessive vertical height growth; however, the simpler discretization scheme does significantly reduce the complexity and computational time required to run a simulation. Most P-3D simulators handle the fluid flow only in the horizontal direction within the fracture. This fluid flow in the horizontal direction (the direction of the propagation) is because of the single grid element in the vertical direction. This 1D flow also limits the ability to properly describe the proppant transport within the fracture. To compensate for the problem of proppant transport, some P-3D models use a correlation that creates an elliptical proppant front. P-3D simulations usually produce an elliptical fracture profile (side view) because of the previously discussed methods for handling vertical growth, and because the P3D models are usually lumped models. Lumped models are those where the grid data are integrated from the tip to the wellbore and averaged. This method typically produces the elliptical fracture profile as illustrated in Fig. 7.

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Fig. 7. Elliptical profile (P-3D).

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3.3 Planar Three-Dimensional (PL-3D) PL-3D models are able to provide a detailed description of a hydraulic fracture in both simple and very heterogeneous lithologic sections. These models require all of the parameters describing the reservoir, elastic and fluid-loss properties for each layer. Simulation can take 15 min or several hours depending on the complexity of the problem and/or the computer being used. PL-3D models divide the fracture into a mesh of vertical and horizontal grid elements. The elements may take different shapes (usually small squares or triangles). Fig. 8 shows an example of a mesh composed of quadrilateral elements with triangular subelements.

Fig. 8. Example grid (PL-3D model). A very fine mesh may be used around the edges and at the tip for greater accuracy, while a coarser grid is used in the center of the fracture. The fracture is assumed to develop as a plane (elasticity equations relate the pressure in the elements on the crack faces to the width) and to calculate fracture-height growth. The different values of pressure, fluid loss and other important parameters are calculated both vertically and horizontally. The PL-3D model has 3D elasticity and 2D fluid flow. Fracture (crack) propagation is controlled by the criterion of linear elastic fracture mechanics. The fracture advances by a method where the stress intensity factor (KI) is maintained nearly equal to the critical stress-intensity factor (KIc) during the crack extension at each grid node. The theory of linear elasticity also allows problems to be solved by the superposition of solutions to subproblems. Superposition is a method whereby a PL-3D model can use part of Sneddon's solution to calculate the vertical width profile. Most PL-3D simulators allow the user to control the refinement of the grid. A coarse grid with fewer elements will require less computing time, but there may be more error in the solution. A very fine mesh grid will provide a solution with a very low material balance error. The more complex grid system also allows the description of the proppant transport in a more detailed manner. DOWELL CONFIDENTIAL


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The 2D grid can produce some problems with complex stress profiles. Some PL-3D simulators have difficulty handling bounding layers with a value of stress lower than the pay, which are separated from the pay by a bounding layer with a higher stress. This situation causes the grid elements to become increasingly skewed as the fracture grows into the lower stress bounding layer. This continues until the computational errors are extremely large and the simulator terminates the run. The advantage of any PL-3D simulator is the ability to model the pressure drop laterally within the fracture. As the fracture grows with each step of the simulation, all of the parameters are recalculated in each grid element. Using the grid system, the pressure at the lateral tip of the fracture can be lower than the pressure at the wellbore. This allows the simulator to model a fracture with a greater vertical height at the wellbore than at the fracture tip. The profile of this type of fracture is determined by the grid and is not a lumped solution. Fig. 9 is a profile from a PL-3D simulation.

Fig. 9. Fracture profile (PL-3D model). 3.4 Fully Three-Dimensional (3D) Techniques for general 3D fracture propagation (including out-of-plane) have been presented by some authors. The numerous problems and the system of equations needed have not been developed to the point of providing a realistic, operational, fully 3D model. A model that is truly a fully 3D model is needed to simulate special conditions that cannot be handled by other models. However, at the present time, no working model exists in the industry — except in research. DOWELL CONFIDENTIAL



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4 Examples The information used to compile the examples for the different models comes from a case study where several calibration tests were performed. Extensive stress testing, core analysis, reservoir evaluation and fluid analysis were also performed. The reservoir parameters, stress values and other critical parameters are considered to be the best available because of the many techniques used to obtain and corroborate the data. Routine treatment designs usually never have such a large volume of reliable data available with which to work. Trying to examine all of the input and output data in detail for the examples, and for each model, is simply too voluminous to cover in this section. Tables and profiles will be used to show data and comparisons (case history). The actual model comparisons will show profiles and an output summary for each model simulation. 4.1 Case History This case history uses the same basic data that will be used later for the model comparison. However, this case history uses 15 to 22 layers in the simulation, whereas only 5 layers were used for the model comparisons given in Subsection 4.2 that follows this case history. The reservoir stress and elastic properties that are used in the examples are given in Fig. 10 and Fig. 11.

Fig. 10. Permeability, thickness and stress profile. DOWELL CONFIDENTIAL


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Fig. 11. Computed values for Young's modulus and Poisson's ratio.

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Table 1 shows the depth, log stress and modified stress that can be used for layer input. The modified stresses were obtained by modifying the log stresses to more closely match the measured net pressure response from the calibration fractures. These modified stress data were the values used for the model simulations.

Table 1. Comparison of Stress Top of Zone (ft)

Log Stress (psi)

Modified Stress (psi)

Delta Stress (psi)

9030 9070 9115 9155 9170 9200 9250 9310 9340 9360 9380 9435 9455 9475 9575

7300 7800 7150 6600 6050 5600 5250 5850 6550 7300 5800 6400 7550 8400 7850

7300 8200 7350 6600 6050 5800 5250 6050 6550 7300 6200 6700 7950 8400 7850

0 400 200 0 0 200 0 200 0 0 400 300 400 0 0

Table 2 gives the permeability and fluid-loss coefficient used in the initial simulations. Table 3 shows the design information for the treatment and the actual volumes used during the treatment. The average injection rate was 50 bbl/min and a total of 1,168,910 lbm of sand was placed in the fracture. The average treating pressure was 3000 psi.

Table 2. Permeability and Fluid Loss Formation Permeability (md) Initial Total Leakoff Coefficient ( ft min )

0.0065 0.0010

Reservoir Fluid to Filtrate Permeability Ratio Reservoir to Filter-Cake Permeability Ratio Leakoff Interval

10.0 100.0 Entire Fracture

The treatment was pumped down the casing/tubing annulus, and the bottomhole pressure was measured by a pressure gauge run inside the tubing. Fig. 12 shows the pressure profile. DOWELL CONFIDENTIAL

Section 300


May 1998

Fracture Modeling

Dowell

Page 23 of 35

Note that the second calibration test performed on the well is also included on this profile, and ends at a little over 100 min. A shut-in time is obtained and then the main fracturing treatment starts at approximately 250 min. A PL-3D model (GOHFER) from Marathon Oil Company was used to match the bottomhole pressure as well as the surface casing (annulus) pressure. The simulation used stress and elasticity data from the top of the log to the bottom. These data were used to give 22 layers for input. Fig. 13 shows the pressure match (dotted lines) from the start of the treatment to the point of shutdown. Fig. 14 and Fig. 15 show the fracture length and fracture width profiles from the simulation.

Table 3. Design Information Fluid Volume, BBL Stage

Design

Actual

Fluid Type

1 2 3 4 5

1500 0 1000 3000 300

1571 0 576 2908 293

40-lbm linear Shut-in Slickwater 50-lbm x-link 50-lbm x-link

Proppant Conc. (lbm/gal) 0 0 0 0 1

6 7

400 400

477 337

40-lbm x-link 40-lbm x-link

8

500

500

9

500

10

Proppant Volume, lbm Design Actual

0 0 0 0 12,600

0 0 0 0 11,110

0 1

0 16,800

0 14,150

40-lbm x-link

2

42,000

41,120

437

40-lbm x-link

3

63,000

54,910

600

623

40-lbm x-link

4

100,800

105,330

11

800

755

40-lbm x-link

5

168,000

133,630

12

900

807

40-lbm x-link

6

226,800

227,250

13

900

934

40-lbm x-link

7

264,600

300,100

14

900

935

40-lbm x-link

8

302,400

292,420

15

300

240

Slickwater

0

0

0

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Proppant Type None None None None 100-Mesh Sand None 20/40 Ottawa sand 20/40 Ottawa sand 20/40 Ottawa sand 20/40 Ottawa sand 20/40 Ottawa sand 20/40 Ottawa sand 20/40 Ottawa sand 20/40 Ottawa sand None



Fracture Modeling

Fig. 12. Profile of bottomhole, casing and tubing pressures.

Fig. 13. Pressure match for bottomhole and casing pressure.

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Fig. 14. Fracture profile.

Fig. 15. Fracture width profile.

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Net pressure was also matched using the P-3D Cleary model (FRACPRO) as run by Resource Engineering Systems (RES). The net pressure match is shown in Fig. 16.

Fig. 16. Match of net pressure for calibration fracture and main fracture. The drop in pressure at about 300 min, also shown on the bottomhole pressure plot, corresponds to the time that the fracture height during the simulation reached a low stress zone (see Stress Log Interval 9380 to 9455). Fig. 17 is the fracture profile data from the simulation for the net pressure match.

Fig. 17. Fracture profile.

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Calculated Parameters • Fracture Length (ft)


1819

•

Propped Length (ft)

1441

•

Fracture Height (ft)

358

•

Propped Height (ft)

284

•

Fracture Width (max.) (in.)

0.72

•

Proppant Concentration (lbm/ft2)

1.88

These are the dimensions at the end of shut-in. Several methods were used to attempt to determine the fracture height after the treatment. The method that was accepted as being the most accurate in this case was the Continuous Microseismic Radiation (CMR) log. This microseismic height log was run four months after the treatment, and therefore will more accurately indicate the propped fracture height rather than the created fracture height. The CMR log indicated the height was from 9125 ft to 9375 ft, allowing ± 25 ft at both top and bottom. This makes the propped height range from 250 ft to 300 ft. The created fracture-height differences for the various models are difficult to evaluate because created height can be significantly different from the propped height. This difference can be caused by the way each model handles the information on the various layers, and the actual width profile that was calculated. Another consideration is how the model treats the proppant movement, settling or the many other complicated aspects present when the proppant is added to the system. Several postfracture reservoir evaluation techniques were used to analyze the results of the treatment. The analysis testing was started after 89 days of production from the well. A reservoir model was used to obtain the final history match of the reservoir variables. The reservoir model used data obtained from buildup analysis as well as the production match for both gas and water (two-phase flow). The result of this analysis is illustrated in Fig. 18.

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Section 300


May 1998

Fracture Modeling

Page 28 of 35

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Fig. 18. Reservoir model for final history match. Note that conductivity was calculated to 1515 ft, but possibly only 1100 ft may be truly effective. Total fracture half-length is shown as 1845 ft. It must be pointed out that FRACPRO was the initial model used for the project, and continued to be used through the completion of the project. Comparison with other models (to be shown later) is more difficult because many times the final geometry interpretation may be influenced by special data or variables input to obtain the match. This is not wrong, and is the way that models should be calibrated. However, other models may not have the advantage of using the information pieces obtained throughout the time of the project. Also, in the case history just discussed, the data given to run the simulation with GOHFER differed some from the exact input used for FRACPRO. 4.2 Model Comparisons This portion will show comparisons for a similar set of input parameters. The critical output parameters obtained from simulating a common treatment (using the same data as in Table 2 and Table 3) will be given, as well as profiles or graphs illustrating the resulting fracture geometry. These comparisons were made by S.A. Holditch & Associates, Inc., using original data from a project for the Gas Research Institute. The reader will see some significant differences in some of the output values, but these serve to show again the differences between the model code. Different companies running the same model also obtain different results. This difference may be caused by how each company handled the stress layers, and the fact that in some cases the various companies were asked to model the calibration fracture and use fluid-loss coefficients derived from that analysis. This left room for some to use a wall-building coefficient initially, while others may have started with a total fluid-loss coefficient. Hopes are that sometime in the future, one inclusive set of data will be given to everybody and the models run again for comparison. DOWELL CONFIDENTIAL


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Fracture Modeling


The comparisons show results for KDG and PKN modes as well as for the P-3D and PL-3D runs. Table 4 gives the list of output data from the runs using both GDK (KGD)) and PKN geometry. Note that since these models use fixed-height assumptions, most have the same height listed because it is an input parameter. Comparisons are made for two cases — (1) fluid viscosity set at 200 cp, and (2) viscosity based on n' and k' values. These values give a viscosity of approximately 450 cp at 37.5 sec -1. Case 3 and Case 4 show the difference between fluid-loss values, which is also evident by the calculated efficiency. For the sake of space, example profiles for these KGD and PKN geometries will not be shown. The list of output data from P-3D and PL-3D simulations is given in Table 5. These simulations use the same data previously shown in Table 2 and Table 3. The same information concerning the viscosities applies here as it did from the previous discussion. The additional data and comparisons for these P-3D and PL-3D simulations involve the number of layers used in the particular simulation. The example profiles and graphs for these simulations will be taken from Case 8 of Table 5 (Variable Viscosity, 5-layer). The detailed differences will not be discussed, but left to the reader to examine. Table 5 should be self-explanatory for comparing the information in Case 8 with the figures. However, to eliminate confusion each simulation will carry the figure number corresponding to the following list — • Fig. 19. SAH (TRIFRAC) length and width profile. •

Fig. 20. NSI (STIMPLAN) length and width graphs (P-3D).

•

Fig. 21. RES (FRACPRO) length and width profile (P-3D).

•

Fig. 22. Marathon (GOHFER) length and width profile (PL-3D).

•

Fig. 23. ARCO (TERRAFRAC) length profile only (PL-3D).

•

Fig. 24. ARCO (STIMPLAN) length and width profile (P-3D).

•

Fig. 25. MEYER length and width profile P-3D).

•

Fig. 26. Ohio State length profile only (P-3D).

Copies of the simulation plots for length and width were not available for MEYER (Bells) or TEXACO (FRACPRO).

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Fracture Modeling

Table 4. Fracture Model Comparison Runs

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Table 5. Fracture Model Comparison Runs

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Fracture Modeling

Fig. 19. TRIFRAC length and width profile.

Fig. 20. STIMPLAN length and width.

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Fig. 21. FRACPRO length and width profile.

Fig. 22. GOHFER length and width profile.

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Fracture Modeling

Fig. 23. TERRAFRAC length profile.

Fig. 24. STIMPLAN length and width profile.

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Fig. 25. MEYER length and width profile.

Fig. 26. Ohio state length profile.

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Fracturing_Modeling.pdf

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