Fracturing_Modeling.pdf

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FRACTURING ENGINEERING MANUAL

Fracture Modeling

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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............................................................................................................... 7Fig. 2. Fracture divided into elements.......................................................................................... 9Fig. 3. Representation of a planar fracture. ............................................................................... 10Fig. 4. KGD geometry. ............................................................................................................... 11Fig. 5. PKN geometry................................................................................................................. 12Fig. 6. 2D and radial Sneddon cracks. ....................................................................................... 13Fig. 7. Elliptical profile (P-3D)..................................................................................................... 17Fig. 8. Example grid (PL-3D model)........................................................................................... 18Fig. 9. Fracture profile (PL-3D model). ...................................................................................... 19Fig. 10. Permeability, thickness and stress profile. .................................................................... 20Fig. 11. Computed values for Young's modulus and Poisson's ratio. ........................................ 21Fig. 12. Profile of bottomhole, casing and tubing pressures. ..................................................... 24Fig. 13. Pressure match for bottomhole and casing pressure. .................................................. 24Fig. 14. Fracture profile.............................................................................................................. 25Fig. 15. Fracture width profile. ................................................................................................... 25Fig. 16. Match of net pressure for calibration fracture and main fracture. ................................. 26Fig. 17. Fracture profile.............................................................................................................. 26Fig. 18. Reservoir model for final history match......................................................................... 28

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Fig. 19. TRIFRAC length and width profile. ................................................................................32Fig. 20. STIMPLAN length and width. ........................................................................................32Fig. 21. FRACPRO length and width profile. ..............................................................................33Fig. 22. GOHFER length and width profile. ................................................................................33Fig. 23. TERRAFRAC length profile. ..........................................................................................34Fig. 24. STIMPLAN length and width profile...............................................................................34Fig. 25. MEYER length and width profile. ...................................................................................35Fig. 26. Ohio state length profile.................................................................................................35

TABLES

Table 1. Comparison Of Stress ..................................................................................................22Table 2. Permeability and Fluid Loss .........................................................................................22Table 3. Design Information .......................................................................................................23Table 4. Fracture Model Comparison Runs................................................................................30Table 5. Fracture Model Comparison Runs................................................................................31

1 Introductory Summary

The prediction of fracture geometry has been a central issue in engineering designand evaluation of hydraulic fractures, and many models have been developed overthe years. These models determine fracture geometry by attempting to relate manyvariables such as rock properties, fluid properties, fluid volume pumped and stressdata. Some models use a fixed fracture height and others continuously calculate theheight during the simulation. Each change aimed at more closely matching the realconditions requires more sophistication in modeling the fluid flow in the entirefracture, effect of proppant and elasticity of the entire system. To be practical,however, the calculations must be made at reasonable increments along the fractureand computational time must not be excessive. The degree of sophistication of amodel is therefore somewhat controlled by the practical application. The models arealso data limited.

The comparison of different models can be difficult and confusing because of theway 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. Decisionson how to handle elasticity, fluid flow, type of grid or cross section, vertical stressdifferences and toughness, for example, can have a large impact on the calculatedfracture geometry. There is still much work to be done in obtaining meaningful datafor input into the more sophisticated models.

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Basically, there are four types of fracture models either being used or beingdeveloped 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 thatshould be clarified. Actually, the term planar means that the fracture occurs in aplane. This condition is true for all fracture models except the fully 3D. Planar wassimply used to name the model that is more advanced than the pseudo 3D, but notquite as sophisticated as the fully 3D. A fully 3D model would have the capability ofbeing nonplanar (fracture could curve or change planes) if the correct stress dataand other information were available for input.

This section on fracture models will be limited to a brief discussion of some of theconcepts that must be considered to build a model, as well as a brief discussion ofeach model. It is beyond the scope of this section to cover each model in detailbecause of the number of models available, and not having the documentation orcode to examine each model. Also, models change as more data become availablefrom evaluation, in-situ testing and calibration of logs, and from special industryprojects to calibrate the various models based on the best available information. Asuccessful model is one that has the ability to match the pressure from the treatmentby using realistic variables based on in-situ data, and to calculate a fracture heightconsistent 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 oflaws 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 describethe behavior of a system under a certain number of conditions.

Coupling these two sets with the appropriate “boundary conditions” produces somevery complicated mathematical formulations. To solve the coupled problem requiresdiscretization 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. Theconservation of mass for modeling a fracture treatment is used to give the overallmaterial balance. This material balance is illustrated by Eq. 1.

V V Vi fp LP= + (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) thebody forces such as gravity that act on the whole volume, and (2) the surfaceforces, such as pressure forces and fluid friction, that act only along the boundaryof the domain.

The conservation of momentum principle relates the time change in the totalmomentum of a body to the applied forces (both on the volume and on thesurface). Many fracture treatments are modeled as quasistatic. This implies thatthe rates of the change of velocities are negligible, and therefore the summationof surface and volume forces is zero.

• Conservation of Energy: This pertains primarily to the two fundamental laws ofthermodynamics for a system; (1) the change in total energy of a system is equalto 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 thatdepend on time and position.

• mass

• temperature

• velocity (three components)

• stresses (six components).

These variables make a minimum of 11 unknowns while the number of componentsin the three conservation laws is only five. Therefore, constitutive equations to solvethe system need to be defined. Some equations used in the less complicatedmodels can be simplified to the point where they no longer apply, thereby reducingthe 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 (incompressiblefluid) causes the equation of mass conservation to simplify and become zero.

• Stress/Strain: The section on rock mechanics showed the relationship betweenstress (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 theshape of the fracture rather than the tip positions at any pressure. Analyse todetermine the effect of modulus contrast are extremely time consuming becausethey require large finite-element or similar solutions. Formulas for the averagewidth are used in many cases to scale the width profile.

For the general case of a fracture growing into the surrounding layers of greaterstress and/or fracture toughness than the pay zone, the pressure, height andshape of the fracture cross section depend on the stress, toughness, thicknessand modulus of the individual layers. The more simple models would assume aconstant modulus in all layers, plane strain elasticity and no flow-inducedpressure drop in the vertical direction.

When considering fluid flow, the stress tensor includes pressure and a viscoustensor (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 severalof the fracture models is assumed to be in one direction only (x direction) and thevelocity field is therefore unidirectional. This means that both the shear stress(τxy) and rate of deformation: �γ xy are related as:

τ µγxy xy= � (3)

• Coupling conservation relations with the constitutive equation leads to a verycomplicated fluid equation. The more sophisticated PL-3D models may attemptto handle this equation while the 2D and many P-3D models must makesimplifying assumptions, particularly on velocity, before numerically solving thesystem. Handling fluid flow in the fracture to determine the pressure distributionis very important and necessary to determine the fracture displacement, as wellas using a method to determine how a particular model will handle all of thevariables associated with adding the proppant to the system. An accuratecalculation of the pressure distribution within the fracture is critical for any realisticsimulation.

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• Flux Laws: These include the relationships of pressure drop in a porous mediumto velocity (Darcy's law), of heat flux to temperature (Fourier's law) or the rate ofreaction 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 theformation will modify the stress distribution and pressure within the formation,creating a fracture in which the injected fluid flows. This injected fluid exchangesheat with the formation and leaks off into the formation. The poroelastic effect canbe interpreted as a time-dependent back-stress; whenever a fracture profile isobtained and a pressure is calculated, an additional term must be added to thepressure. The fluid is also considered to be a multiphase type because of theproppant being carried. The proppant also alters the fluid viscosity, which affects thefluid-flow model. A good model needs rigorous calculation of the fluid fronts as wellas handling the settling of the proppant as in the simpler 2D models up to convectionhindered settling and other more complicated accounting of the proppant in some ofthe PL-3D models.

Simulating the propagation of a hydraulic fracture requires consideration of the linearelastic fracture mechanics (LEFM) formulation, fracture fluid flow and continuityequation. These three sets need to be coupled to simulate the propagation; themathematical problem is complex because of the different types of equations and thepresence of a moving boundary (fracture edge).

The strains produced in the formation caused by the deformation from inducing ahydraulic fracture are actually quite small. The small value of strain allows theassumption that the formation deforms in a linear elastic manner. Knowing the stateof 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 ellipticalsurface with semiaxes a and b. This condition forces the fracture to assume anelliptical 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-shapedcrack. This will be covered in more detail in the discussion of 2D models. Theclassical techniques in elasticity do have severe limitations because of the stresssingularity near the crack tip, and special methods have to be used to improve theaccuracy in computing the stress intensity factors. Boundary integral techniqueshave become popular because the problem is solved only at the boundary, reducingthe dimension of the problem by one, and provides a simple method to determinestress intensity factors. However, implementation can be difficult fornonhomogeneous media with varying elastic moduli.

Remember from the section on rock mechanics that the stress intensity factor (KI) isfor the opening mode of the fracture and describes the magnitude of the stressesnear the crack tip — it depends on the crack geometry, surrounding material andapplied loads. Fig. 1 shows the three fundamental modes of loading for a fracture.

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Fig. 1. Modes of loading.

The opening mode corresponds to normal tension, in front of the crack. The slidingmode is associated with transverse shear while the tearing mode is a longitudinalshear. Usually, only the opening mode applies for a plane fracture. The othermodes may be applicable in complex fracturing, out of plane fractures or when nearnatural fissures.

For a given system of stresses, it is necessary to determine whether a fracture willinitiate and then propagate, and in which direction, and from which point in thecreated fracture. Griffith (1921) addressed the problem by defining the surfaceenergy of a fracture. The Griffith criterion stated that there is an equilibrium if thespecific surface energy γ is related to the change in internal energy (U) by:

2γ∂∂

= − UA

(4)

where A is the fracture area. For a uniformly loaded fracture, the critical load (σc)and the surface energy are related, such as

σ γπ νc

f

E

x=

−

2

1 2

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 criticalstress-intensity factor (KIc), also known as fracture toughness. The condition for afracture to be in equilibrium requires that the stress intensity factor (KI) associatedwith the load be equal to a critical value (KIc). A simple relation can be derivedbetween the surface energy and KIc for a uniformly pressurized fracture —

γ ν= −( )1 2 2KE

Ic (6)

This expression indicates (for linear elastic behavior) that the surface energycriterion and the critical intensity factor are related and form a unified criterion forpropagation.

The calculation of the pressure distribution in the fracture, due to fluid flow, isnecessary to determine the fracture displacements. As mentioned earlier, the fluid-flow problems to include non-Newtonian as well as Newtonian fluids present someproblems and require a large amount of computational time. Because of thisproblem, the fracture may be discretized into a series of parallel lines (surfaces) andthe flow considered as quasistatic. The PL-3D models do handle the flow problem alittle more rigorously and, consequently, require more time for computation.

The continuity equation is the last relation and simply describes the conservation ofmass previously discussed. The continuity equation may be written for each fractureelement (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 facesof the resulting crack. This pressure causes the crack to open. The width at anypoint is therefore dependent on the pressure applied to the faces. The condition ofthe plane strain and rectangular elements is used to model the fracture for the 2Dcases. 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 constantover each element. For problems solved by this method, the width distribution istypically obtained for a specified distribution of pressure over the elements.Alternatively, the pressure distribution can easily be found from a specified widthprofile. Some P-3D models may use a little more sophisticated grid system, but fluidflow is again only 1D and the vertical height growth is handled by shape factors andother methods although elasticity is considered to be 2D. The PL-3D models use amore complicated grid system (small squares or triangles). The junction of the gridelements becomes the node for the solution of the problem. This allows the appliedpressure 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 amodel 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 complicatedand various models handle this flow differently. Usually flow along the fracturelength is modeled in detail, while horizontal flow across the width of the fracture andvertical flow are neglected (2D, P-3D). Horizontal flow may be calculated in anaverage sense by assuming that an average velocity adequately represents the flow.An overall fluid-flow model could therefore be obtained by coupling 1D massconservation and momentum conservation equations. Volume conservation wouldreplace 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 andwidth can be calculated from the volume and flow characteristics governed by thecode 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 (KGDmodel), (2) Perkins and Kern, and later Nordgren (PKN model), and (3) the radial orpenny-shaped model.

A common simplifying assumption is that the lateral effects of a fracture are smallcompared to the vertical effects and can be neglected. This condition is termedplane strain and implies that each cross section acts independently of any othersection, 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. Twocases of plane strain can be distinguished for the 2D formulations.

• Horizontal Plane-Strain Geometry: The fracture zone will deform independentlyof the upper and lower layers. This will be possible for free slippage on theselayers and represent a fracture with horizontal penetration that is much smallerthan the vertical, and where the fracture shape does not depend on the verticalposition. This describes the KGD fracture geometry as shown in Fig. 4. Thisgeometry has a constant height with a cross section that is rectangular in shape.

Fig. 4. KGD geometry.

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• Vertical Plane-Strain Geometry: This condition exists for a large confinementwhere the fracture is limited to a given zone. Each vertical cross section deformsindependently of the others. The fracture widths in the vertical direction arecoupled through the continuity and fluid-flow equations; however, because thereis no vertical extension (in each of the vertical sections) during simulation, thepressure is uniform and the cross-sectional shape of the fracture height iselliptical. 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 elasticassumption) was mentioned earlier during the discussion of concepts. Fig. 6 showsthe 2D and radial Sneddon cracks. These solutions are for a 2D crack having onedimension of infinite extent, and the other dimension of finite extent (d in Fig. 6). Theradial or penny-shaped crack is defined by the radius (R). The resulting width iselliptical in shape for both types of cracks, and is proportional to one of thecharacteristic 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 proportionalto the plane-strain modulus given as

′ = −E E / ( )1 2ν (7)

Young's modulus and Poisson's ratio are at in-situ conditions (E is defined by thetangent Young's modulus Et).

Sneddon's method has been used in different ways to model 2D fractures. Thecharacteristic 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 theinfinite dimension has to correspond to the fracture height. The other assumption isthat the characteristic dimension, d, is the fracture height. This is the condition forthe PKN model; since d is the fracture height, then the infinite dimension has to bethe fracture length.

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The KGD and PKN models consider the propagation of a vertical fracture of givenheight. However, in some cases, the vertical stress is lower than the horizontalstress and the fracture will propagate in a horizontal or inclined direction. BothPerkins and Kern as well as Geertsma and de Klerk gave simplified expressions forthe propagation of a radial horizontal fracture. Similarly, vertical fractures maypropagate radially in thick formations where there are no barriers to height growth.This situation leads to the same equations when the injection interval is smallcompared to the fracture extension. The KGD model is valid when h>>xf. The PKNmodel is valid when xf >>h. The radial model is most appropriate when the totallength (2xf or 2R) is approximately equal to the height. Again, the three sets ofequations to be coupled are the elasticity, continuity and fluid flow.

Some of the important characteristics and differences pertaining to the 2D modelsare

• KGD Fracture Model

− A fixed fracture height is assumed, and fluid flow is horizontal only (in thedirection of the propagation).

− Crack opening is solved in the horizontal plane. As a result, the fracturewidth does not vary with the fracture height, except by the boundarycondition set at the wellbore that specifies a constant total injection rate.

− Width is constant in the vertical direction because of the plane-straincondition and individual horizontal planes.

− The model gives wider fracture widths and shorter fracture lengths whencompared to the PKN model.

− The flow resistance in the narrow rectangular vertical width is whatdetermines the fluid pressure gradient in the propagating direction.

− The excess pressure (net pressure) decreases with time, and in log-logcoordinates has a slope equal to -1/3.

− The model is most appropriate when the fracture length is smaller than thefracture height.

• PKN Fracture Model

− A fixed fracture height is assumed and fluid flow is horizontal only (in thedirection of the propagation).

− Crack opening is solved in the vertical plane. As a result, the fracturingfluid pressure is constant in vertical cross sections perpendicular to thedirection of the fracture propagation.

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− Each vertical cross section deforms individually and obtains an ellipticshape with the maximum width in the center. The only coupling betweenthe different vertical cross sections is due to fluid flow in the fracture. Thereis no elastic coupling between the planes.

− This model is used to describe the behavior of planar fractures that have alength-to-height ratio greater than three.

− The model gives narrower widths and longer fracture lengths compared tothe KGD model.

− The excess pressure increases with times; in log-log coordinates, the slopeis positive.

− The model is most appropriate when the fracture length is much larger thanthe fracture height.

The 2D fracture models require less input and perform the computations veryquickly. This advantage may be used when the fracture is known to be contained inheight, and the value of the height is known. However, the fracture shape is stillrectangular because no difference is calculated in the height at the tip of the fracturecompared to the height at the wellbore. Because of these conditions as well as the1D fluid flow and noncoupling of the length and width, the 2D model fracture length isusually longer, the net pressure is higher and the width is wider than actual or whencompared 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 elasticproperties of the rock layers, fluid flow and fracture initiation. These assumptionsreduce 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 flowis one-dimensional in the direction of the propagation. Many P-3D simulatorsassume all layers have the same elastic, reservoir and fluid-loss properties. SomeP-3D simulators allow the user to input a description for each layer and then usesome method for averaging, while others use calculated shape factors to arrive atthe effect the layers will have on the geometry. For example, a different Young'smodulus could be input for 15 layers. The simulator, depending on the choice of theauthor, may then use averaging or other factors to determine the actual impact overthe entire interval. The values of 15 different layers are now reduced to one. Thismethod can have significant implications in sections with thick layers of coal, shale orother lithology. Most P-3D simulators have problems running when a large contrastin elastic properties exists. The assumptions and use of averaging and factors allowthe P-3D simulators to generate a solution in a very short time, even on the smallpersonal computers.

The P-3D simulators divide the fracture into a fixed number of vertical elements inthe grid. Each vertical element extends from the top of the fracture to the bottom.

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All of the values describing the fracture geometry, pressure in the fracture and fluidloss are the same throughout that one single element. This is one reason why mostP-3D simulators estimate more vertical height growth when compared to the PL-3Dsimulators. Also, there is no pressure drop within the element from the perforationsto the vertical limits of the fracture. The pressure at the vertical tip of a P-3D fracturesimulation is also higher than that of a PL-3D simulation. The P-3D simulators mayproduce solutions with excessive vertical height growth; however, the simplerdiscretization scheme does significantly reduce the complexity and computationaltime required to run a simulation.

Most P-3D simulators handle the fluid flow only in the horizontal direction within thefracture. This fluid flow in the horizontal direction (the direction of the propagation) isbecause of the single grid element in the vertical direction. This 1D flow also limitsthe ability to properly describe the proppant transport within the fracture. Tocompensate for the problem of proppant transport, some P-3D models use acorrelation that creates an elliptical proppant front.

P-3D simulations usually produce an elliptical fracture profile (side view) because ofthe previously discussed methods for handling vertical growth, and because the P-3D models are usually lumped models. Lumped models are those where the griddata are integrated from the tip to the wellbore and averaged. This method typicallyproduces 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 inboth simple and very heterogeneous lithologic sections. These models require all ofthe parameters describing the reservoir, elastic and fluid-loss properties for eachlayer. Simulation can take 15 min or several hours depending on the complexity ofthe problem and/or the computer being used.

PL-3D models divide the fracture into a mesh of vertical and horizontal gridelements. The elements may take different shapes (usually small squares ortriangles). Fig. 8 shows an example of a mesh composed of quadrilateral elementswith 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 todevelop as a plane (elasticity equations relate the pressure in the elements on thecrack faces to the width) and to calculate fracture-height growth. The differentvalues of pressure, fluid loss and other important parameters are calculated bothvertically and horizontally. The PL-3D model has 3D elasticity and 2D fluid flow.Fracture (crack) propagation is controlled by the criterion of linear elastic fracturemechanics. 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 crackextension at each grid node. The theory of linear elasticity also allows problems tobe solved by the superposition of solutions to subproblems. Superposition is amethod whereby a PL-3D model can use part of Sneddon's solution to calculate thevertical width profile.

Most PL-3D simulators allow the user to control the refinement of the grid. A coarsegrid with fewer elements will require less computing time, but there may be moreerror in the solution. A very fine mesh grid will provide a solution with a very lowmaterial balance error. The more complex grid system also allows the description ofthe proppant transport in a more detailed manner.

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The 2D grid can produce some problems with complex stress profiles. Some PL-3Dsimulators have difficulty handling bounding layers with a value of stress lower thanthe 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 thefracture grows into the lower stress bounding layer. This continues until thecomputational errors are extremely large and the simulator terminates the run.

The advantage of any PL-3D simulator is the ability to model the pressure droplaterally 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 thewellbore. This allows the simulator to model a fracture with a greater vertical heightat the wellbore than at the fracture tip. The profile of this type of fracture isdetermined by the grid and is not a lumped solution. Fig. 9 is a profile from a PL-3Dsimulation.

Fig. 9. Fracture profile (PL-3D model).

3.4 Fully Three-Dimensional (3D)

Techniques for general 3D fracture propagation (including out-of-plane) have beenpresented by some authors. The numerous problems and the system of equationsneeded 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 specialconditions that cannot be handled by other models. However, at the present time,no working model exists in the industry — except in research.

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4 Examples

The information used to compile the examples for the different models comes from acase study where several calibration tests were performed. Extensive stress testing,core analysis, reservoir evaluation and fluid analysis were also performed. Thereservoir parameters, stress values and other critical parameters are considered tobe the best available because of the many techniques used to obtain andcorroborate the data. Routine treatment designs usually never have such a largevolume of reliable data available with which to work.

Trying to examine all of the input and output data in detail for the examples, and foreach model, is simply too voluminous to cover in this section. Tables and profileswill be used to show data and comparisons (case history). The actual modelcomparisons 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 modelcomparison. 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 4.2that follows this case history.

The reservoir stress and elastic properties that are used in the examples are given inFig. 10 and Fig. 11.

Fig. 10. Permeability, thickness and stress profile.

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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 layerinput. The modified stresses were obtained by modifying the log stresses to moreclosely 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 7300 7300 0

9070 7800 8200 400

9115 7150 7350 200

9155 6600 6600 0

9170 6050 6050 0

9200 5600 5800 200

9250 5250 5250 0

9310 5850 6050 200

9340 6550 6550 0

9360 7300 7300 0

9380 5800 6200 400

9435 6400 6700 300

9455 7550 7950 400

9475 8400 8400 0

9575 7850 7850 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 usedduring the treatment. The average injection rate was 50 bbl/min and a total of1,168,910 lbm of sand was placed in the fracture. The average treating pressurewas 3000 psi.

Table 2. Permeability and Fluid Loss

Formation Permeability (md) 0.0065

Initial Total Leakoff Coefficient ( min)ft 0.0010

Reservoir Fluid to Filtrate Permeability Ratio 10.0

Reservoir to Filter-Cake Permeability Ratio 100.0

Leakoff Interval Entire Fracture

The treatment was pumped down the casing/tubing annulus, and the bottomholepressure was measured by a pressure gauge run inside the tubing. Fig. 12 showsthe pressure profile.

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

Table 3. Design InformationFluid Volume, BBL Proppant Volume,

lbmStage Design Actual Fluid Type Proppant

Conc.(lbm/gal)

Design Actual ProppantType

1 1500 1571 40-lbm linear 0 0 0 None2 0 0 Shut-in 0 0 0 None3 1000 576 Slickwater 0 0 0 None4 3000 2908 50-lbm x-link 0 0 0 None5 300 293 50-lbm x-link 1 12,600 11,110 100-Mesh

Sand6 400 477 40-lbm x-link 0 0 0 None7 400 337 40-lbm x-link 1 16,800 14,150 20/40

Ottawa sand8 500 500 40-lbm x-link 2 42,000 41,120 20/40







Ottawa sand15 300 240 Slickwater 0 0 0 None

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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 byResource 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 lowstress zone (see Stress Log Interval 9380 to 9455). Fig. 17 is the fracture profiledata 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 toattempt to determine the fracture height after the treatment. The method that wasaccepted as being the most accurate in this case was the Continuous MicroseismicRadiation (CMR) log. This microseismic height log was run four months after thetreatment, and therefore will more accurately indicate the propped fracture heightrather than the created fracture height.

The CMR log indicated the height was from 9125 ft to 9375 ft, allowing ± 25 ft at bothtop and bottom. This makes the propped height range from 250 ft to 300 ft. Thecreated fracture-height differences for the various models are difficult to evaluatebecause created height can be significantly different from the propped height. Thisdifference can be caused by the way each model handles the information on thevarious layers, and the actual width profile that was calculated. Anotherconsideration is how the model treats the proppant movement, settling or the manyother complicated aspects present when the proppant is added to the system.

Several postfracture reservoir evaluation techniques were used to analyze theresults of the treatment. The analysis testing was started after 89 days of productionfrom the well. A reservoir model was used to obtain the final history match of thereservoir variables. The reservoir model used data obtained from buildup analysisas well as the production match for both gas and water (two-phase flow). The resultof this analysis is illustrated in Fig. 18.

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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 betruly 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, andcontinued to be used through the completion of the project. Comparison with othermodels (to be shown later) is more difficult because many times the final geometryinterpretation may be influenced by special data or variables input to obtain thematch. 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 piecesobtained 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 inputused for FRACPRO.

4.2 Model Comparisons

This portion will show comparisons for a similar set of input parameters. The criticaloutput parameters obtained from simulating a common treatment (using the samedata as in Table 2 and Table 3) will be given, as well as profiles or graphs illustratingthe 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, butthese serve to show again the differences between the model code. Differentcompanies running the same model also obtain different results. This difference maybe caused by how each company handled the stress layers, and the fact that insome cases the various companies were asked to model the calibration fracture anduse fluid-loss coefficients derived from that analysis. This left room for some to use awall-building coefficient initially, while others may have started with a total fluid-losscoefficient. Hopes are that sometime in the future, one inclusive set of data will begiven to everybody and the models run again for comparison.

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The comparisons show results for KDG and PKN modes as well as for the P-3D andPL-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 sameheight 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. Thesevalues give a viscosity of approximately 450 cp at 37.5 sec -1. Case 3 and Case 4show the difference between fluid-loss values, which is also evident by the calculatedefficiency. For the sake of space, example profiles for these KGD and PKNgeometries will not be shown.

The list of output data from P-3D and PL-3D simulations is given in Table 5. Thesesimulations use the same data previously shown in Table 2 and Table 3. The sameinformation concerning the viscosities applies here as it did from the previousdiscussion.

The additional data and comparisons for these P-3D and PL-3D simulations involvethe number of layers used in the particular simulation. The example profiles andgraphs 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 toexamine.

Table 5 should be self-explanatory for comparing the information in Case 8 with thefigures. However, to eliminate confusion each simulation will carry the figure numbercorresponding 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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Table 4. Fracture Model Comparison Runs

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Table 5. Fracture Model Comparison Runs

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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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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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