Copyright 2001, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September–3 October 2001. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledg-ment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract Hydraulic fracture geometry (i.e., critical results of length and proppant placement) is driven by four major in situ parame-ters: Fracture Height (H), Modulus (E), Fluid Loss (C), and “Apparent” Fracture Toughness (KIc-app). In many (even most) cases, “Height” is the most important of these parameters – due to the need for some height confinement to achieve long fractures, or the need for height growth to insure good pay coverage. Due to this importance, industry research effort and most field measuring techniques concentrate on “Height.” In particular, the growing use of seismic imaging is offering a tool to measure height growth away from the wellbore. Results from such diagnostics have often shown, as one expects, that in situ stress variations control height. However, results have also shown situations where this is apparently not the case.

This paper examines another in situ parameter, “Layered Modulus,” which also affects height. In addition, by control-ling the “local” width of a fracture, layered modulus (i.e., lay-ered formations with different layers having significantly dif-ferent modulus) can have a critical effect on final proppant placement.

The importance of layered modulus in directly controlling fracture height is illustrated in this paper, and this is compared with published solutions. In general, it is found that, just as concluded in the past, modulus contrast is probably not an im-portant parameter in terms of direct control of fracture height. The greater importance lies in the effects on local fracture width. These local width changes can have a significant influ-ence on controlling proppant placement – and this can be criti-cal for low net pressure cases such as “water fracs” or fractur-

ing in “soft” formations. It is also noted that layered modulus significantly impacts the average width of a fracture, and thus impacts the critical material balance aspects of fracture mod-eling if not properly accounted for.

Finally, some of the theoretical solution problems created by “Layered Modulus” formations for fracture modeling are discussed and compared. This is done by comparing with 3-D Finite Element (static) solutions, and shows how some com-mon industry “approximations” for layered modulus give in-correct results. Based on this, examples with a fracture propa-gation model using a finite element-generated stiffness matrix are used to define types of cases where a simple “average” modulus is acceptable, versus cases where more complex cal-culations are needed.

Introduction Six major variables control hydraulic fracturing, fracture geometry, proppant placement, etc. Two of these are the “con-trollable” variables of fluid viscosity, µ, and pump rate, Q. The remaining four variables are “natural” variables and include: • Height. Fracture height (or more generally fracture geom-

etry) is possibly the most important unknown for fracture design and post-frac production success. Generally, it is recognized that in situ stress differences (the in situ stress profile) is the major controlling factor for this behavior. [1] At a minimum, in situ stress differences control the maxi-mum fracture height, i.e., if the net pressure is not avail-able to grow through high stress shale layers, then fracture height must be contained. The importance of fracture height/geometry is clear, and there are many research ef-forts and technical publications addressing this issue. [1-6]

• Fluid Loss. Fluid loss is typically characterized for hydrau-lic fracturing by a fluid loss coefficient, C, which charac-terizes linear flow fluid loss out of the fracture. This gives the familiar C/√(t-τ) form of fluid loss behavior. Again, this variable has been exhaustively discussed in the litera-ture including wall building characteristics of specific fluid systems, effects of natural fractures, behavior of fluid loss additives, etc. [7-16]

• Tip Effects. This controls the net pressure required, at the fracture perimeter or fracture tip, to propagate the fracture.

SPE 71654

Layered Modulus Effects on Fracture Propagation, Proppant Placement, and Fracture Modeling M. B. Smith, SPE, NSI Technologies; A. B. Bale, SPE, STATOIL; L. K. Britt, SPE, Amoco Production Co. (now with NSI Technologies); H. H. Klein, SPE, Jaycor, Inc.; E. Siebrits, SPE, Schlumberger; and X. Dang, Jaycor, Inc.

2 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

This is currently an area of some controversy, and again there is significant discussion of the variable, KIc-app (the apparent fracture toughness), in the literature. [17-24]

• Modulus. The “final” variable is the stiffness (sometimes informally referred to as the hardness) of the rock, the Young’s modulus, E. This variable has been a stepchild of fracturing, with generally much less attention, and this is the subject of this discussion.

General Effects of Modulus. In general terms, the effects of modulus are recognized. That is, for a “hard” (i.e., high modulus) rock, net treating pressure will generally be greater, and the fracture width will be somewhat narrower. This can be illustrated using basic fracturing equations for fracture net pressure (at the well) for confined height fractures:

4/1

4

24

4/1

2

4

4

4

+

∝

+

∝

−

−

EHK

ExQ

w

HK

ExQ

HEp

appIcf

appIcfnet

µ

µ

(1)

where the first term reflects the viscous pressure drop of a Newtonian viscous fluid flowing along a narrow fracture, and the second term reflects the “tip extension pressure” assuming the fracture has a “½ penny shaped” tip with a radius of H/2. (Note that this is not a “quantitative” equation, but rather a “parameterized” version of the rigorous analysis by Nolte [25].) In addition to the direct effect on width, the high treat-ing pressure caused by modulus can, of course, lead to addi-tional height growth. Also, as discussed by Miller [26], the effect of the narrow fracture width in a hard rock formation can cause gel degradation, and thus have dramatic effects on fluid loss.

In addition, recent work by Warpinski [27] has shown definitively that it is the “static” modulus, i.e., the value determined from laboratory stress-strain tests that would be used in the relations above. This is very important since “E” thus becomes the ONLY hydraulic fracturing variable directly measurable, in advance, via lab testing.

However, what about normal cases where modulus will not be a single value? Just as in situ stress and fluid loss usually vary from layer to layer due to normal geologic changes, modulus might also be expected vary! While theoretical and experimental work (as reviewed below) has considered layer modulus effects, the effect of such natural variations has been virtually ignored in hydraulic fracture modeling and design. One exception is the work by Van Eekelen [5] who discusses the effects of layered modulus on fracture height growth.

Literature Review. A large number of papers were published in the 1970s and 1980s on the effect of variations in elastic properties on fracture height containment. The consensus at the time seems to have been that Young’s modulus was a

second-order effect when compared with variations in confining stress in a layered reservoir (e.g., see Teufel and Clark [28]), and we agree with that conclusion. Indeed, the mineback experiments by Warpinski et al. [1] showed that material property changes had little effect on containment. However, Warpinski et al. state that “there were, however, significant width changes in regions of differing modulus, with larger widths found in low-modulus regions.” This finding has important repercussions in terms of accurate numerical modeling of hydraulic fractures in layered reservoirs, and can be particularly important to final proppant placement.

Even though modulus may not be a dominant player in terms of height containment, it nevertheless plays an important role in predicting final fracture height. The local fracture width and hence the net pressures in each layer are affected by modulus. A simulator which properly models the modulus changes will predict a more accurate width variation in the height direction, which will affect the fluid flow and net pres-sure distribution, and hence affect height growth.

There are references to the effect of interfacial strength on fracture height containment (e.g., Daneshy [29], Simonson et al. [6], Biot et al. [30]). While interfacial strength may be an important parameter in some formations, our investigations were limited to the effect of modulus only. All hydraulic fracturing simulators are designed so that interfaces are assumed to remain fully bonded, and we have thus restricted our investigations accordingly.

Other effects such as leakoff contrast across interfaces can also play a role in height containment. Blair et al. [31] per-formed tests on combined gypsum cement and sandstone blocks. They grew hydraulic fractures from the less permeable gypsum into a more permeable sandstone tablet or lens, embedded in the gypsum. Once the fracture reached the sand-stone tablet, the increased leak-off there caused pressure to drop in the fracture (i.e., fluid loss was temporarily greater than injection) and fracture height growth halted. However, the small sandstone lens in the laboratory experiment quickly saturated. Thereafter, the less permeable gypsum was again the controlling factor, allowing pressure to increase and the fracture to cross the interface and penetrate the sandstone tab-let.

A good review article on hydraulic fracturing in layered rocks was that of Mendelsohn [32]. He highlighted the com-plexities in modeling hydraulic fracture growth in a layered reservoir, and noted the complicating effects of the approach angle of the fracture to the interface, the potential for branch-ing, jogging, interfacial failure, mixed mode tip propagation conditions, etc.

From the above, we infer that modulus contrast as well as other factors, such as leakoff, toughness, interfacial strength, and confining stress, all act in combination to determine whether or not a hydraulic fracture will cross an interface.

SPE 71654 LAYERED MODULUS EFFECTS ON FRACTURE PROPAGATION, PROPPANT PLACEMENT, AND FRACTURE MODELING 3

Accuracy of Fracture Models Layered modulus represents a significant complication for fracture modeling, possibly one reason why this geologic real-ity has been ignored.

Any geometry model has to perform three distinct, seem-ingly simple, tasks. First, net pressure in the fracture must be calculated based on fluid flow and fracture propagation crite-ria. Second, the net pressure must be used to calculate the fracture width. Finally, this net pressure must be used to determine where/when the fracture next propagates. The major complication, of course, is that all three solutions depend upon one another, thus the three “equations” must be solved simul-taneously.

Modulus fits into the second of these solutions, i.e., in the relation between net pressure and fracture width. For this step, all current fracture models rely on the theory of elasticity for some analytical relation between fracture geometry (H or height), modulus, net pressure, and width. Probably the most common solution is Sneddon’s solution for a 2-D plane strain fracture under constant pressure loading [33]:

22 1NetP H Ew , E' ,E' v

π= =− (2)

where w is the “average” fracture width (actually fracture volume divided by H), PNet is the fluid pressure inside the fracture minus the fracture closure stress, E’ is the plane-strain modulus, E is Young’s modulus, and ν is Poisson’s ratio. Since ν is typically about 0.2, E’ is (for practical purposes) essentially the same as E. This solution has been widely used for 2-D and Pseudo 3-D (P3D) fracture geometry models. Note, however, that this assumes a single, uniform value for modulus! How can this be reconciled with geologic reality?

Another fracture geometry model (Barree [34]) utilizes the Boussinesq displacement (i.e., fracture half-width) solution for a pressure, p acting on an elastic, semi-infinite ½ space. This is given by Jaeger and Cook [35] as 22 )()(,),(

'1),( ηξηξηξ

π−+−== ∫∫ yxrdd

rp

Eyxu . (3)

This integral equation calculates the fracture half-width

(i.e., the displacement, u, at the surface of the ½ space) at a location (x,y) due to a distributed pressure loading p applied at any points (ξ,η) on the fracture area “A.” This assumes, of course that appropriate boundary conditions of “0” width are applied around the boundary of the fracture area A.

It has been suggested [34] that this equation can be gener-alized to nonhomogeneous materials by including the elastic material constant, “E” inside the integrals. However, this would be incorrect. The integral equation is only valid for a homogeneous, elastic material. The displacement at (x,y) is determined by integrating or summing the effects of all possi-ble source points (ξ,η) inside the region A. The contribution from each source point requires the same elastic constant, E’. In order to model fully bonded (or even partially bonded) lay-

ered materials using integral equations requires a more rigor-ous approach such as discussed by Crouch and Starfield. [36] This reference discusses a simple and rigorous, but inefficient, boundary integral approach for layered materials. More effi-cient approaches are possible as discussed by Peirce and Siebrits. [37, 38]

A slightly more rigorous solution used is that for a “dislocation” located in a bimaterial space (Wang and Clifton [39]). This solution extends the computations to two-layer systems. However, most geologic formations have multiple layers, so this is still not adequate for most reservoirs. Some approximations must thus be used when performing calcula-tions for actual field cases. These approximations for multiple layers can lead to large errors in the prediction of fracture width.

Methods of calculating the correct width profile in a lay-ered modulus formation include finite element and boundary element models. As mentioned above, an efficient numerical solution has been developed for modeling stacks of parallel layers that are bonded together (Peirce and Siebrits [37,38]) in a layered reservoir environment. The use of such rigorous approaches leads to more credible fracture widths as well as more credible fracture geometries, which in turn leads to more accurate proppant placement predictions and better treatment designs.

Average Modulus Approximation. Probably a common approach for simulating fracture geometry is to determine and use an “average” modulus. That is, calculations may not honor local width variations, but for a given pressure distribution, the correct average width will be calculated. This is a valid approach, and at least correctly preserves the material balance (and thus the fluid efficiency) of the fracture simulation. This then insures a valid pump schedule [40] (i.e., we will not try to put more proppant into the ground than the fracture can hold) – certainly a nontrivial issue. However, even determining “av-erage” modulus can be tricky! First, though, the term itself, “Average Modulus,” should be defined. For this discussion, the following definition is used: “The correct ‘Average Modulus’, EAvg, is the modulus

value, which when used in an elasticity relation based on uniform modulus, yields a fracture width profile giving exactly the same average width as the width profile pre-dicted from a rigorous, layered modulus solution.” As an example, consider the case illustrated in Figure 1.

What might the average modulus be? A simple height-aver-aged approximation would be 1.5 × 106 psi (i.e., [60 feet × 1 × 106 + 20 feet × 3 × 106]/80 feet); however that would be incorrect! The actual average modulus, i.e., the value of E that gives the correct average width is 1.86 × 106 psi. The figure also shows the predicted width profile based on an average width (i.e., with a “correct” average modulus of 1.86 × 106), and based on the actual layered modulus values using FEM (finite element) calculations.

Finally, the figure plots the width profiles based on a sim-ple “height weighted average,” i.e., E = 1.5 × 106 psi. Thus,

4 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

the use of a simple height weighted average modulus” leads to an automatic 20% error in calculating fracture width (and thus, depending on fluid loss, a significant error in predicting fluid efficiency for a treatment).

The behavior illustrated here is discussed in more detail in later sections. In brief, however, the effect of a “hard” or “soft” (i.e., high or low modulus) layer depends on its loca-tion, as well as its modulus value. That is, a hard layer located near the top of a fracture has a different effect than a similar layer located near the vertical middle of a fracture. In addition, modulus of the over- and under-lying formations affects the fracture width even before the fracture penetrates those layers.

Other Approximations. As will be seen in the examples below, clearly an “average” modulus is not the only approx-imation used in numerical simulators. The approach here was to run various simulators for real field cases, and then to compare the predicted width distribution with that calculated using finite element modeling (FEM). To do this, a snapshot of the fracture geometry and internal net pressure distribution (as predicted by the simulator) at a particular point in time was taken. This geometry/net pressure combination was then used to calculate the fracture width distribution using a commercial 3-D finite element package. The resulting width distributions were then compared. To keep the situation as simple as possible, the time right at the end of pumping pad was used for all calculations. The study included two Planar 3D models and one Planar “cell type” pseudo-3D model (per the model definitions by Warpinski [41]). The pseudo-3D model used an “Average Modulus” (as defined previously) approximation, with the average defined by numerical (i.e., finite element) relations, i.e., this simulation should produce the “correct” average width for layered modulus cases, but does not predict any local width variations.

Case 1 – High Permeability Oil Well. The first case con-sidered was an oil well as seen in Figure 2. For this case, the goal was for a propped fracture to penetrate upwards from perforations in the deeper Rannoch-1 formation. This complete case is described elsewhere [42], but the existence of “calcite layers” was of serious concern for initial treatments. For the Rannoch sandstone, modulus varies from 0.7 × 106 psi for the Rannoch-1 (the “lower,” 50 md, permeability, “lower” porosity zone) down to 0.3 × 106 for the very high perme-ability (1,000+ md) Rannoch-3. These values come from laboratory stress-strain tests on core samples. However, embedded within the Rannoch are 0.5 to 3 m thick, “0” poro-sity calcite zones, layers of pure, 100 % calcium carbonate, with a modulus of about 6 × 106 psi. It was feared that these layers would prove, just due to their high modulus, a barrier to fracture growth. Also, should the fracture penetrate through these layers, it was feared that the resulting propped width over these intervals would be so narrow as to effectively exclude production from the overlying Rannoch-3 zone. Because of this, effects of layered modulus were of critical importance.

This case was simulated using three different fracture geometry models, all using some different approximation for

layered modulus cases. The exact form of the approximation for 3D Model 1 and 3D Model 2 is not known. The critically important results for fracture height growth are included in Figure 3, with two of the calculations showing generally simi-lar geometry. One calculation, however, predicts the fracture does not fully penetrate into the Rannoch-3. (Note that post-frac production rates of 30,000+ bopd and well tests show conclusively that the fracture did provide good communication to that high permeability zone.)

The reason for this incorrect result (i.e., Model 2) is easily seen in Figure 4. This plots the vertical width profile (at the wellbore), and shows the 3-D fracture geometry model has calculated an extreme local width effect for the layered modu-lus. As shown, this 3-D model predicts a near “0” width over the first calcite layer at about 1,860 m which impedes vertical flow of the viscous fracturing fluid, and essentially halts ver-tical height growth.

To check this calculation, a snapshot of the fracture geometry and the predicted net pressure distribution at the end of pumping pad was input into a commercial 3-D finite ele-ment package. As seen in the figure, the correct width distri-bution for this case is considerably different. The width is somewhat narrower over the calcite layer, but there is not suf-ficient width restriction to drastically affect height growth. (Note that the term “correct” in the preceding sentences, and in subsequent discussions, is ONLY with reference to the geometry and internal net pressure distribution as predicted by the model. It is not meant to imply that the 3D FEM results are “correct” in some absolute sense. Just that given this geometry and pressure distribution, the approximation for the layered modulus effects is yielding an invalid width profile; this then alters subsequent fracture propagation calculations, and leads to an invalid solution.)

For this case, the layered modulus approximation has led to a critically important, erroneous conclusion. In addition, the average width from the layered modulus approximation is about 20 % less than the correct average width. Thus, fluid efficiency calculated by the model would be in error, and the final pump schedule would not be appropriate.

The results from another 3D model (i.e., Model 1), using a different layered modulus approximation, were quite different. The fracture geometry simulator predicted vertical width pro-file is compared to the correct width profile in Figure 5. As shown, the layered modulus approximation used for this cal-culation did not show the extreme local width reduction. However, the error in average width was even worse. This would lead to a significant error for the computed fluid effi-ciency, and thus, to an inappropriate treatment design.

This figure also displays results using a 2-D FEM solution. Due to the near radial fracture geometry, the 2-D FEM plane strain width calculations show, as expected, slightly too much width. However, this does confirm the general magnitude of the local width restrictions as calculated with the 3-D FEM model.

The final comparison with the pseudo 3D model is not pictured here. The average modulus approximation simply showed no local width changes related to the high modulus

SPE 71654 LAYERED MODULUS EFFECTS ON FRACTURE PROPAGATION, PROPPANT PLACEMENT, AND FRACTURE MODELING 5

layers, but did calculate the correct average width. As noted by Warpinski [40], pseudo-3-D models are not, strictly speaking, applicable for near radial fracture geometry such as found for this example. Thus, the model was actually applying two approximations: a) an average modulus calculation for the layered modulus, and b) an approximation due to the essen-tially radial fracture geometry. The fracture geometry and internal net pressure from this solution were again input in the 3-D FEM, and the good agreement for the predicted average width shows that both approximations were correct. However, due to the very nature of an “Average Modulus” approxima-tion, the geometry prediction showed no local width variations over the “hard” layers.

Case 2 – Confined Fracture Height. The second case studied was an idealized version of an actual moderate permeability gas well. In this case (compared to Case 1 above) the fracture was relatively well confined, with layers of mod-erate porosity/permeability mixed with layers of very low porosity, high modulus rock. Again, the modulus values for this case are taken from laboratory stress-strain testing. The overall net pressure behavior and fracture geometry as predicted by several simulations is included in Figure 6. This shows general agreement on the predicted net pressure behavior, and with the general prediction of height confine-ment. “Modulus Approximation 2,” i.e., “3D Model 2,” again predicts a drastic local width effect from the layered modulus, and this may (or may not) also be causing calculation of a sig-nificantly wider average width (and thus less length and height).

Again, the predicted fracture geometry (and the predicted internal pressure distribution) from each calculation was used as input into a commercial 3-D FEM package. The FEM pre-dicted widths were then compared with the fracture model widths. As seen in Figure 7, “Modulus Approximation 1” pre-dicts minimal local width variations, and this is supported by the FEM calculations. In this case, the average width from the approximate solution is also apparently close, BUT only at the expense of over-predicting downward height growth. That is, given the model predicted pressure distribution, the approxi-mate width solution shows positive width down to about 6,200 feet. However, the correct width solution (for this geometry and internal fluid pressure distribution) shows that this is incorrect. Thus, the downward height growth would not have occurred. In this case, the modulus approximation has signifi-cantly impacted predicted height. The apparent average width agreement is possibly also misleading. Without the downward height growth below 5,120 feet, and the resultant negative widths (due to negative net pressure) calculated there by the 3-D FEM package, the FEM solution would have shown a much wider width over the main interval, and thus shown an average width much greater than predicted using the layered modulus approximation.

Similar results for “Approximation 2” are included in Fig-ure 8. Similar to the results from Example 1 previously, this approximation predicted extreme effects on the local widths, with these predictions not borne out by the more rigorous FEM solution. The average width for Example 2 as predicted

by this approximation was about 20% greater than the “cor-rect” average width (as compared to the Example 1 results where this approximation produced an average width about 20% too low).

As seen in the preceding two comparisons, the FEM solu-tions for this case predict very little local width variation. Thus, the comparison with the average width solution was very close, with again the average width based solution pro-ducing the correct average, but again predicting “0” local width variation.

Approximations – Summary. The purpose above was not to say, “this is bad” or “this is good.” Rather, the purpose was to point out a simple fact. Any fracture width solution based on “normal” pressure/width relations must use some approximation in order to deal with geologic reality. In all the cases examined above, it is presumed that the selected approximation was appropriate for the conditions assumed to exist when that approximate solution was selected. However, it is also clear that under a different set of conditions, the approximation can lead to serious errors. Probably the only generally safe approximation is to use the correct “Average Modulus,” i.e., the modulus that maintains the correct average width. This then correctly models the material balance and fluid efficiency of the actual injection, and at least insures a reasonably correct final treatment design pump schedule. However, as discussed below, determining this correct “Average Modulus” may itself require a numerical solution of some form.

However, this “Average Modulus” approximation will still ignore “local” width restrictions due to hard (i.e., stiff, high modulus) layers, and these effects can be critically important to proppant placement (as seen in a later example). Thus, some limit must be placed on when such a compromise solu-tion may be useful, as examined in a later section. In addition, actually calculating the correct “Average Modulus” turns out not to be a straightforward procedure, and this is also exam-ined in a subsequent section of the paper.

Approximations – Average Modulus Calculations and Error. As discussed below, layered modulus can lead to two sources of error (as compared to some “average” modulus). • First, the use of an average modulus results in predicting a

simple elliptical width profile (ignoring possible width effects due to formation layers with higher/lower closure stress). Thus, this will clearly miss any local width changes due to “hard” or “soft” formation layers.

• Second, and possibly more critical, the use of an incorrect average modulus can lead to serious over- or under-pre-dictions of average fracture width (thus, errors in calculat-ing fracture volume, fluid efficiency, etc.), and these errors can approach 50+% for quite realistic geologic scenarios. As mentioned in the discussion of Figure 1, the correct

“Average Modulus” may be quite different from a simple “height weighted average over the fracture height.” Partially this difference is related to the location of the “hard” layers, since the effect of any single layer depends on where it is,

6 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

along with how stiff it is. In addition, over- and under-lying layers also affect the fracture pressure/width relation even without the fracture penetrating these layers. That is, as the fracture opens in the five middle layers in the Figure 1 exam-ple, it is also “straining” the higher modulus layers in the over- and under-lying formations. The high (5 × 106 psi) modulus of these layers then tends to reduce fracture width – i.e., to in-crease the “Average Modulus.”

The effect of these over/under-lying layers is quantified in Figure 9. This plots the correct “Average Modulus” (EAvg as defined above) as determined using a 2-D FEM program, for a simple case of a fracture contained in a formation with a uni-form modulus value, but over/under-lain by formations with differing modulus. As seen in the figure, the correct “Average Modulus” can be increased/decreased by up to 50% depending on modulus values for surrounding formations. The effect of thin individual layers on “Average Modulus” is discussed in more detail below.

To assess the effects of individual layers, the simple 5-layer formation illustrated in Figure 10 was used as a case stu-dy. A uniform pressure inside the fracture was assumed, and a 2-D plane strain FEM program was used to calculate the width profile for varying values of E2 /E1. For each width profile, an average width was calculated, and Equation 1 was used to cal-culate the correct “Average Modulus,” EAvg. This EAvg is then plotted in Figure 10. Finally, a simple “height weighted aver-age over the fracture height” was calculated as an example of a “simple” average modulus, and this is also plotted. The diff-erence between these then reflects a possible error in average width (and fracture volume/fluid efficiency) calculations. As an example, consider a case of E2 /E1=0.333. This might repre-sent a formation with a modulus of 6 × 106 psi, with a couple of high porosity streaks with a modulus of 2 × 106 psi – cer-tainly a possible geologic scenario. From Figure 10, the correct average modulus is about 0.88 E1, while the use of a simple “height weighted average” would suggest the use of a modulus of about 0.61 E1, an error in calculating the average width of 31%. This error would result in calculating too wide a fracture width, and possibly lead to unexpected proppant bridging in the fracture and total treatment failure.

As another example, consider a case with two relatively thin (10 foot) “hard” (i.e., high modulus) layers, with an E2 /E1 ratio of 5:1. From Figure 10, the “correct” average modulus is 1.36xE1, while a simple height weighted average is 2 E1 – an error in any subsequent fracture width calculations of nearly 50%. For this 5:1 case, Figure 11 plots the actual width profile (from the layered modulus FEM calculations) along with the theoretical elliptical width profiles for the two average modulus values.

The local width error between the actual (FEM) width pro-file, and the theoretical profile calculated from the correct EAvg is calculated on a point-by-point basis over the fracture height. The maximum width error (actually error in “½ width”) is then divided by the average ½ width to give a “percent error.” For this case (for a uniform internal pressure of 300 psi), the aver-age ½ width is about 0.15 inches, corresponding to a correct “Average Modulus” of 1.36 E1 psi. The maximum error in ½

width was bout 0.03 inches, giving a 20% error for this exam-ple. This was then repeated for a range of E2/E1 ratios, with the maximum local width error for each case included in Figure 12. Note that this is the percent error in the “local” width pro-file between the actual layered modulus width profile (calcu-lated using FEM methods), and the theoretical (elliptical) pro-file giving the same average width, i.e., using the “correct” EAvg.

For this 5-Layer case (2 “streaks” of different modulus comprising about 25% of the total zone thickness), the discus-sion above shows that for “hard” (i.e., high modulus) layers (i.e., E2 /E1 > 1), a simple “height weighted average” is possi-bly acceptable for 1 < E2 /E1 < 2. That is, the error in the final average width calculations (i.e., the difference between the “correct” average modulus and the simple height weighted average) is on the order of 10% or less, and the error in the local width profile is also on the order of 10%. For a modulus ratio (E2 /E1) greater than about 3, the overall width error from using a simple average exceeds 25%, and thus, is not accept-able. In addition, even when the correct EAvg is used, the local width error begins to approach 20%, and (depending on the magnitude of the average width) may create an unacceptable error. Finally, of course, for this case the over- and underlying formations are assumed to have a modulus of E1, thus if the modulus of these formations was different that would add additional complexity to the problem.

Interestingly, the error is somewhat worse for “soft” (i.e., low modulus) layers. For two relatively thin, “soft” layers, a ratio of E2 /E1 of 0.5 produces unacceptable errors on the order of 20%+. That is, an 80 foot formation with a modulus for most of the formation of say 4 × 106 psi, with two, 10 foot, “soft” layers with a modulus of 2 × 106 psi, would create ser-ous computation errors. This is clearly a potentially common geologic scenario, and for such a case widths calculated using a simple “height weighted average” modulus over predicts the average width (i.e., give too low a modulus value) by about 20%. That is, a “correct” EAvg = 0.91 E1, as compared to a sim-ple height weighted average modulus of 0.73 E1. The simple average would then over predict the fluid efficiency, and po-tentially lead to an inappropriate treatment design. In addition, for the ratio of E2 /E1 = 0.5, local width errors again begin to exceed 10% even if the correct EAvg is used.

Based on this, it appears that for layered modulus forma-tions, if the modulus ratio between any two layers falls outside the band 0.7 < E2 /E1 < 2, then, at a minimum, a rigorous, lay-ered modulus solution must be used to, at least, correctly define a correct “Average Modulus.” If the layered modulus ratio falls outside the range of 0.3 < E2 /E1 < 5, then even using a correct “Average Modulus” may not be acceptable, with local width errors (as a percentage of the average width) exceeding 20%. However, it must be remembered that this conclusion is based on a single layered geometry, and worse errors (and thus more restrictive limits) might apply for other cases. In fact, a subsequent section of the paper examines a field case where the local width variations due to a 2:1 shale/sand modulus ratio significantly impacts final proppant placement.

SPE 71654 LAYERED MODULUS EFFECTS ON FRACTURE PROPAGATION, PROPPANT PLACEMENT, AND FRACTURE MODELING 7

Effects of Layered Modulus The previous discussions have shown that layered modulus, i.e., geologic reality, presents a serious complication for frac-ture modeling. The question then becomes, is this a problem? Effect on Fracture Height Growth. Modulus can theore-tically affect fracture height growth in two major ways. First, fracture width at the top/bottom of a fracture growing up through stiff (i.e., high modulus) layers will be very narrow. This is illustrated in Figure 13 for an example case discussed below, and in this figure, the width profile was predicted by a Planar 3-D model using FEM calculations for the fracture pressure/width relation. This narrow width causes a significant pressure drop for any vertical fluid flow, thus reducing pres-sure near the top/bottom of the fracture, and thus reducing the rate of fracture height growth.

The second possible effect on height growth was discussed by Simonson et al. [6] with an analysis of fracture contain-ment as a 2-D problem. This study concluded, for massive hydraulic fracture treatments in the presence of modulus con-trasts, that hydraulic fractures in a pay zone sandwiched be-tween two stiffer barrier zones tend to be contained, and vice-versa. This behavior and the stress changes that occur as a fracture approaches an interface between dissimilar elastic materials have been extensively reported in the literature (e.g., Erdogan and Biricikoglu [43], Cook and Erdogan [19]). As a hydraulic (or for that matter a “dry”) fracture approaches an interface between two materials with different elastic moduli, the stress intensity factors at the fracture tip start to change. If the opposite material is “softer” (lower E) then the stress intensity factor increases as the tip approaches the interface, and vice-versa (e.g., see Figure 5 of Simonson et al. [6]). This implies that if the opposite material is stiffer, it tends to be-have as a “near perfect” barrier to fracture growth, even if the rock toughness is unchanged between the two materials. While this is a rigorous fracture mechanics solution, it depends on a “singularity” to give the perfect containment (as well as re-quiring a probably geologically rare “sharp” boundary be-tween layers), and in fact, further analysis shows that for low tensile strength materials (such as rocks) a fracture may be able to bypass this behavior.

This possibility arises since the tensile stresses directly ahead of the fracture tip will increase in the stiffer material, providing a fracture with the opportunity to jump ahead into the stiffer material. This is depicted in Figure 14, which shows the tensile stress ahead of the tip for a value of (R/a) of 0.07 (where R is the distance from the fracture tip to the interface and a is the fracture ½ height). The location of the interface relative to the fracture is illustrated in the figure.

For the example in Figure 14, the high (theoretically infi-nite) tensile stress just ahead of the fracture tip is evident. This tensile stress then decays very quickly away from the fracture tip. As the tip approaches ever closer to the interface, fracture propagation becomes more difficult. However, the tensile stress actually increases across the interface in the “hard” lay-er. The figure gives a “stress parameter” value of about 7, and for R/a of 0.07, with an internal net pressure of 600 psi, the

tensile stress just across the interface is about 2,000 psi. This is probably sufficient to initiate a new tensile fracture in many rock types, thus allowing a fracture to “skip” across the inter-face.

In a practical sense, this phenomenon of a “hard” layer act-ing as a perfect barrier would probably be quite rare since the necessary geologic conditions of a “sharp” interface between hard and soft layers is probably quite rare. The only field evi-dence known to these authors is the “Example 1” North Sea case discussed above. In this case, the geologic interface be-tween the “soft” sandstone layers and the “hard” calcite layers was very sharp. Despite this, and despite a modulus contrast greater than 10:1, the fracture very clearly penetrated through the high modulus layers (and generated sufficient propped width in these layers to produce from the high permeability Rannoch-3 zone).

The other effect on height growth, i.e., the first effect men-tioned above, is somewhat more “straightforward.” As dis-cussed by Van Eekelen [5], and illustrated in Figure 13, the fracture width where the fracture has penetrated high modulus layers will be narrow. This then reduces the flow of viscous fluid up/down to the bottom of the fracture, and thus, reduces the rate of fracture height growth. Van Eekelen discussed the following relation as a measure of this effect

2/1

1

2

1

22

log19241

2193

+==hH

EE

hx

orEE

LHh

dHdx ff (4)

based on an ideal, “lumped” fracture geometry located in a three layer system as illustrated in Figure 15. Note that this simply proposes a relation between the rates of length (dxf) growth versus the rate of height growth (dH), thus the actual ratio of length to height (2xf /H) varies with time as xf in-creases. The maximum length/height ratio is plotted versus the modulus ratio in this figure as well.

Van Eekelen’s results predicted very little height confine-ment due to modulus contrasts, and this seems borne out by numerical modeling. The predicted results are compared to a numerical model, with the pressure/width relations based on a rigorous finite element solution, in Figure 15. This shows even less confinement than predicted from the analytical (approxi-mate) relations above.

It should be noted than the Van Eekelen results suggest that the height confinement ratio, 2 xf /H, should first increase as xf grows, until a maximum is reached for some length. For continued growth, the confinement, i.e., the ratio of 2 xf /H, actually begins to decrease. The ratios plotted in the figure are the maximum possible values for various modulus ratios. The numerical model did show a very slow increase in 2 xf /H with continued growth, but the increase was very slow. No effort was made to find if a maximum might eventually be reached. Rather, the ratio included in the figure for the numerical model results simply represents the ratio 2 xf /H for xf = 200 feet.

In addition, the Van Eekelen relation implicitly assumes that the fluid viscosity dominates both height, and length growth. As seen in the figure, for a low viscosity case fracture

8 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

tip effects dominate propagation, and the high modulus layers have no effect on fracture growth. For viscosity dominated cases (as seen for the 500 cp calculations in the figure), the actual value of fluid viscosity has no effect on the final ratio of length to height, and calculations using a 1500 cp fluid gave identical results. However, the higher viscosity does produce higher net pressure (of course), and thus a wider fracture, and thus a greater volume injection was required to achieve a penetration (½ length or xf). However, once that ½ length was reached, the height was identical to the 500 cp case. This inde-pendence of 2 xf /H from viscosity is predicted by the Van Eekelen relations.

Summarizing, the conclusions are obvious, and similar to conclusions presented from past work. Modulus contrasts play little role “directly” in controlling fracture height. However, its indirect influence on height growth can still be significant through the influence on net pressure. In addition, the effect of modulus on width can have a significant impact on proppant movement and final proppant placement as seen below. While not of great practical importance, it is interesting to note that for the case of modulus contrasts, height growth is minimized by pumping a very viscous fluid at high pump rate!

Local Width Effect on Proppant Placement. Even if layered modulus has minimal direct effect on fracture height growth, it can still have a dominant effect on final proppant placement. A particular case might be where fracture width is very narrow anyway, with one example being low viscosity treatments such as “water fracs.” A previous discussion (based on a 5-layer geology) concluded that for a maximum modulus ratio, E2 /E1, between layers, a correctly computed average modulus, EAvg, should give reasonable fracture geometry predictions. However, as seen for this field case, local width variations due to a 2:1 shale/sand modulus ratio significantly impacted final proppant placement for a “water frac” treatment.

After being introduced by UPRC [44], low viscosity “water frac” treatments became a primary completion tech-nique for many operators in the East Texas Cotton Valley tight gas formation. There is significant controversy as to whether such treatments are the “best” answer for this formation, but clearly such treatments are producing long, conductive frac-tures. How can this be? Why does the proppant not settle to the bottom of the fracture? The answer may partially lie in the nature of the formation and the 2:1 shale/sand modulus ratio. The particular water frac treatment discussed here is described more fully elsewhere [45], along with details of the fracture parameters (i.e., stress profile, modulus, fluid loss, etc.). In short, the stress profile used for the modeling is based on cali-brated dipole sonic log data and the modulus values are de-rived from correlations based on more than 70 laboratory stress-strain tests.

As discussed in [45], the geometry model simulations pro-vided a perfect match to the measured bottomhole pressure behavior, and to the fracture height as measured by micro-seismic imaging. The micro-seismic imaging did suggest some asymmetry (though not enough to have any measurable affect on post-frac production), but twice the geometry model pre-

dicted ½ length was in essentially perfect agreement with the tip-to-tip length from the image data. The fracture geometry predicted for this case is included in Figure 16 for two cases. First, the geometry is modeled using a correct average modulus, EAvg as defined previously. This shows excellent proppant coverage over the lower sand (9,290 – 9,320 feet), with about 1,000 feet of good propped ½ length. However, for the upper sand (9,180 – 9,205 feet), sand is only out about 150 to 200 feet from the wellbore, and in fact very little of the sand is covered by proppant at all. That is, due to proppant settling the treatment would probably not have provided much stimu-lation for the upper sand.

However, modeling this treatment using FEM calculations to provide a rigorous layered modulus width solution gives a more positive view (as seen in the lower part of Figure 16). This shows good proppant coverage over both sands out to about 600 to 750 feet for the upper sand and out to about 1,000 feet for the lower sand. The difference is the additional width restriction between the two sands caused by “hard” shale layers. This creates a very narrow fracture width, and proppant settling is impeded if not halted entirely. In addition, the overall narrow width caused by the high closure stress and high modulus of the “middle shale” (between 9,205 and 9,260 feet), along with the 50 bpm pump rate, leads to very high fluid velocity over the sandstone layers (on the order of 150 to 200 feet/minute), and this is sufficient velocity to transport sand 100’s of feet into the fracture. Effect on Pressure Decline Behavior. Though beyond the scope of this paper, it is also expected that layered modulus can significantly affect shut-in pressure decline behavior. Conclusions • Just as concluded with earlier examinations, it appears that

modulus contrasts play little role in directly controlling fracture height growth. It still, of course plays an important role through the control of modulus on net treating pres-sure. That is, a “hard” (i.e., high modulus) formation gen-erally causes high net pressure, and this can lead to height growth.

• For fracture geometry models based on a “normal” pres-sure-width relation, probably the only safe approximation is the use of the correct “average” modulus, since any approximation other than this can lead to serious errors in predicting fracture width and overall fracture geometry. However, even determining a correct “Average Modulus” probably requires some form of rigorous layered modulus solution. For example, geologically common layered modulus cases with relatively minor modulus contrasts can create overall width calculation errors of 50% or more when comparing a “correct” average with a simple “height weighted average.” Such errors will also cause significant error in the overall material balance, fracture volume, and fluid efficiency calculations. These errors will then lead to an inappropriate treatment design.

SPE 71654 LAYERED MODULUS EFFECTS ON FRACTURE PROPAGATION, PROPPANT PLACEMENT, AND FRACTURE MODELING 9

• For the cases studied here, it appears that when the maxi-mum modulus ratio between various formation layers is in the range of 0.7 < E2 /E1 < 2.0, a “correct” Average Modulus solution should be sufficient. For geologic cases where the maximum modulus ratio between various for-mation layers falls outside the range of 0.3 < E2 /E1 < 5.0, then some rigorous layered modulus solution, such as ob-tained from a finite element of a true multi-layer boundary element scheme, for the relation between pressure and fracture width must be used. Note, however, that these conclusions for ranges are somewhat tenuous, and come from the examination of a single 5-layer case. For another case with a modulus ration, E2 /E1 equal 2, local width variations due to the layered modulus significantly altered on final proppant placement.

• In general, the ONLY fracture design variable that can be directly measured in advance via laboratory testing is stress-strain data for modulus, and probably this type of data should always be available before use of a sophistica-ted 3-D fracture geometry model is justified. However, if the geometry model is based on an approximation for lay-ered modulus effects, then the stress-strain data itself may not be useful since any approximation not based on a rig-orous layered modulus solution can easily lead to serious fracture width calculation errors.

Nomenclature a = Fracture ½ height (L) A = Fracture area (L2), feet2 C = Fluid loss coefficient L/t½ , ft/√min E = Young’s Modulus, m/Lt2, psi H = Fracture height, L, feet KIc-app = Apparent fracture toughness, mL½/Lt2, psi√inch p = Distributed pressure loading, m/Lt2, psi Q = pump rate, L3/t, bpm R = Distance from the fracture tip to the interface, L, feet u = Displacement, L, inches w = Fracture width, L, inches xf = Fracture ½ Length, L, feet µ = Fluid viscosity, m/Lt, cp ν = Poisson’s ratio Acknowledgements The authors would like to think Amoco Production Company and Schlumberger for supporting this project, and for allowing publication of this material. References 1. Warpinski, N.R., Schmidt, R.A. and Northrop, D.A., “In-

situ stresses: the predominant influence on hydraulic fracture containment,” JPT, March 1982, 653-664.

2. Warpinski, N.R., Clark, J.A., Schmidt, R., and Huddle, C.W., ”Laboratory Investigation of the Effect of In-Situ Stresses on Hydraulic Fracture Containment,” SPEJ, June 1982, 333-340.

3. Fung, R.L., Vijayakumar, S. and Cormack, D.E., ”The Calculation of Vertical Fracture Containment in Layered Formations,” SPE Formation Evaluation, December, 1987, 518-522.

4. Usman, A., ”Fracture Height Prediction,” JPT, July, 1988, 813-815.

5. Van Eekelen, H.A., “Hydraulic fracture geometry: frac-ture containment in layered formations,” paper SPE 9261, presented at 55th Annual Fall Technical Conference and Exhibition of the SPE of AIME, Dallas, Sep 21-24, 1980; or SPEJ, Jun 1982, 341-349.

6. Simonson, E.R., Abou-Sayed, A.S. and Clifton, R.J., “Containment of massive hydraulic fractures,” paper SPE 6089, presented at 51st Annual Fall Technical Conference and Exhibition of the SPE of AIME, New Orleans, Oct 3-6, 1976.

7. Howard, G.C. and Fast, C.R., ”Optimum Fluid Character-istics for Fracture Extension,” (Appendix by R.D. Carter) presented at the Mid Continent Sist. Spring Meeting, Tulsa, OK, 1957.

8. Williams, B.B., ”Fluid Loss from Hydraulically Induced Fractures,” Trans. AIME (1970) Vol. 249, 882-888.

9. Barree, R.D. and Mukherjee, H., ”Determination of Pres-sure Dependant Leakoff and its Effect on Fracture Geometry,” paper SPE 36424 presented at the 1996 SPE Annual Technical Conference and Exhibition held in Denver, CO, Oct. 6-9.

10. Harris, P.C., ”Dynamic Fluid Loss Characteristics of Foam Fracturing Fluids,” paper SPE 11065 presented at the 1982 SPE Annual Technical Conference and Exhibi-tion, New Orleans, LA, Sept. 26-29.

11. King, G.E., ”Factors Affecting Dynamic Fluid Leakoff with Foam Fracturing Fluids, paper SPE 6817 presented at the 1977 SPE Annual Conference and Exhibition, Denver, CO, Oct., 9-12.

12. Mcgowen, J.M., and Vitthal, S., ”Fracturing Fluid Leakoff Under Dynamic Conditions Part 1: Development of a Realistic Laboratory Testing Procedure,” paper SPE 36492 presented at the 1996 SPE Annual Technical Con-ference and Exhibition held in Denver, CO, Oct. 6-9.

13. Vitthal, S. and Mcgowen, J.M., ”Fracturing Fluid Leakoff Under Dynamic Conditions Part 2: Effect of Shear Rate, Permeability, and Pressure,” paper SPE 36493 presented at the 1996 SPE Annual Technical Conference and Exhibition held in Denver, CO, Oct. 6-9.

14. Penny, G.S., ”Nondamaging Fluid Loss additives for Use in Hydraulic Fracturing of Gas Wells,” paper SPE 10659 presented at the 1982 SPE Formation Damage Control Symposium, Lafayette, March 24-25.

15. Mcdaniel, R.R., Deysarkar, A.K., and Callanan, M.J., ”An Improved Method for Measuring Fluid Loss at Simulated Fracture Conditions,” paper SPE 10259 presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, TX, Oct. 4-7.

16. Hall, C. D. Jr. and Dollarhide, F.E., ”Fracturing Fluid-Loss Agent Performance Under Dynamic Conditions,” J. Pet. Tech. (July, 1968) 763-768.

10 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

17. Irwin, G., ”Analysis of Stresses and Strains Near the End of a Crack Transversing a Plate,” J. Applied Mech. (1957) Vol. 24, 361.

18. Erdogan, F., ”On the Stress Distribution of Plates with Collinear Cuts Under Arbitrary Loads,” Proc., Fourth U.S. Nat. Cong. Of Applied Mech. (1962) 547.

19. Cook, T.S. and Erdogan, F., “Stresses in bonded materials with a crack perpendicular to the interface,” Int. J. Engrg Sci., 10, 1972, 677-697.

20. Rice, J.R., ”Mathematical Analysis in the Mechanics of Fracture,” Treatise on Fracture, Academic Press Inc., New York (1962) Ch. 3, Vol. 2, 191.

21. Jeffrey, R.G., ”The Combined Effect of Fluid Lag and Fracture Toughness on Hydraulic Fracture Propagation,” paper SPE 18957 presented at the 1988 Joint Rocky Mountain Regional/Low Permeability Reservoir Sympo-sium, Denver, CO, March 6-8.

22. Gardner, D.C., ”High Fracturing Pressures for Shales and Which Tip Effects May be Responsible,” paper SPE 24852 presented at the 1992 SPE Annual Conference and Exhibition, Washington, DC, Oct.4-7.

23. Thiercelin, M., Jeffrey, R.G., and Ben Naceur, K., ”Influ-ence of Fracture Toughness on the Geometry of Hydraulic Fractures,” SPEPE (Nov. 1988) 435-442.

24. Yew, C.H. and Liu, G., ”The Fracture Tip and KIC of a Hydraulically Induced Fracture,” paper SPE 22875 pre-sented at the 1991 SPE Annual Conference and Exhibi-tion, Dallas, TX, Oct. 6-9.

25. Nolte, K. G., “Fracturing Pressure Analysis for Non-Ideal Behavior,” SPE 20704, J. Pet. Tech., Feb. 1991.

26. Miller II, W. K., Roberts, G. A., Carnell, S. J., “Fracturing Fluid Loss and Treatment Design Under High Shear Conditions in a Partially Depleted, Moderate Permeability Gas Reservoir”, SPE 37012, presented at SPE Asia Pacific Oil & Gas Conference held in Adelaide, Australia, 28-31 October 1996.

27. Warpinski, N.R., Peterson, R.E., Branagan, P.T., Engler, B.P., Wolhart, S.L., “In-Situ Stress and Moduli: Com-parison of Values Derived from Multiple Techniques,” SPE 49190 presented at the 1998 Society of Petroleum Engineers Annual Technical Conference and Exhibition held in New Orleans, LA, September 27-30.

28. Teufel, L.W. and Clark, J.A., “Hydraulic fracture propagation in layered rock: experimental studies of fracture containment,” SPEJ 9878, Feb 1984, 19-32.

29. Daneshy, A.A., “Hydraulic fracture propagation in lay-ered formations,” paper SPE 6088, presented at the 51st Annual Fall Technical Conference and Exhibition, New Orleans, Oct 3-6, 1976.

30. Biot, M.A., Medlin, W.L. and Masse, L., “Fracture pene-tration through an interface,” SPEJ, Dec, 1983, 857-869.

31. Blair, S.C., Thorpe, R.K. and Heuze, F.E. “Propagation if fluid-driven fractures in jointed rock. Part 2--Physical tests on blocks with an interface or lens,” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 27(4), 1990.

32. Mendelsohn, D.A., “A review of hydraulic fracture mod-eling-II: 3D modeling and vertical growth in layered

rock,” J. of Energy Resources Tech., Trans. of the ASME, 106, 543-553.

33. Sneddon, I.N., “The distribution of stress in the neighbor-hood of a crack in an elastic solid,” Proc. Royal Society, A. 187, 229-260, 1946.

34. Barree, R.D., “A practical numerical simulator for three-dimensional fracture propagation in heterogeneous media,” paper SPE 12273, presented at Reservoir Simu-lation Symposium, San Francisco, Nov 15-18, 1983.

35. Jaeger, J.C. and Cook, N.G.W., "Fundamentals of Rock Mechanics", 3rd edn, Chapman and Hall, London, p. 295, 1979.

36. Crouch, S. L. and Starfield, A. M., “Boundary Element Methods in Solid Mechanics,” Allen & Unwin, Boston, 2nd edition, 1974, 171-184.

37. Peirce, A.P. and Siebrits, E., “Uniform asymptotic solu-tions for accurate modeling of cracks in layered media,” Int. J. of Fracture, accepted, 2001.

38. Peirce, A.P. and Siebrits, E. “The scaled flexibility matrix method for the efficient solution of boundary value problems in 2D and 3D layered elastic media,” Computer Meth. in Appl. Mech. and Engrg, accepted, 2001.

39. Wang, J.-J. and Clifton, R.J., “Numerical modeling of hydraulic fracturing in layered formations with multiple elastic moduli,” in: Rock Mechanics Contributions and Challenges (eds: Hustrulid & Johnson), Balkema, Rot-terdam, 303-310.

40. Nolte, K. G., "Determination of Proppant and Fluid Schedules From Fracturing Pressure Decline," SPE 13278, presented at 59th Annual Meeting of SPE, Houston, Texas, September 16-19, 1984.

41. Warpinski, N. R., Moschovidis, Z. A., Parker, C. D., and Abou-Sayed, I. S., “Comparison Study of Hydraulic Frac-turing Models – Test Case: GRI Staged Field Experiment No. 3,” SPE Production & Facilities, February 1994.

42. Bale, Arthur, Owren, Kjell, Smith, M.B., “Propped Frac-turing as a Tool for Sand Control and Reservoir Manage-ment” SPE 24992, presented at European Petroleum Con-ference held in Cannes, France 16-18, November 1992.

43. Erdogan, F. and Biricikoglu, V., “Two bonded half planes with a crack going through the interface,” Int. J. Engrg. Sci., 11, 1973, 745-766.

44. Meehan, D. N., Mayerhofer, M.J., Richardson, M.F., Walker Jr., R.N., Oehler, M.W., and Browning Jr., R.R., “Proppants? We Don't Need No Proppants,” SPE 38611, presented at SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October 1997.

45. Britt, L. K., Smith, M. B., Cunningham, L. E., Hellman, T. J., Zinno, R. J., and Urbancic, T. I., “Fracture Optimi-zation and Design Via Integration of Hydraulic Fracture Imaging and Fracture Modeling: East Texas Cotton Val-ley,” SPE 67205, presented at SPE Production and Ope-rations Symposium, Oklahoma City, 24–27 March 2001.

SPE 71654 LAYERED MODULUS EFFECTS ON FRACTURE PROPAGATION, PROPPANT PLACEMENT, AND FRACTURE MODELING 11

Figure 1 – “Average” Modulus.

Figure 2 – Example 1, Layered Modulus Effects.

Figure 3 – Example 1, “H” Versus Time.

Figure 4 – Example 1, Modulus Approximation 2.

0.05 0.10 0.15

10

20

30

40

50

60

70

1/2 Width (inches)

Dep

th (f

eet) Actual

Theoreticalwith E=1.86

Theoreticalwith

E=1.5

E=1

E=1

E=5x10 psi

20 feet

20 feet

20 feetE=1

6

E=5x10 psi6

10 feet

10 feet

E=3

E=3

ν = 0.3, all layers

5 10 15 20 25

1,820

1,840

1,860

1,880

1,900

1,920

TIME (min)

TVD

(m)

Model 1 Model 2 Average E

Higher Stress

w (in)(at wellbore)

TVD

(m)

10.0 20.0 30.0 40.0

Xf (m)4500.0 5000.00.2 0.4 0.6 0.8 1.0 1.2 1.4

1,820

1,840

1,860

1,880

Pcl (psi)

Model 2 3-D FEM (50x70x15)

E=6

E=6E=6

E=.3

E=.3

E=.5

E=6E=.7

E=.7

12 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

Figure 5 – Example 1, Modulus Approximation 1

Figure 6 – Example 2, Layered Modulus Effects

Figure 7 – Example 2, Modulus Approximation 1

Figure 8 – Example 2, Modulus Approximation 2

Figure 9 – Effect of Over-/Under-Lying Layers on EAvg

w (in)(at wellbore)

TVD

(m)

10.0 20.0 30.0 40.0

Xf (m)4500.0 5000.00.2 0.4 0.6 0.8 1.0 1.2 1.4

1,820

1,840

1,860

1,880

Pcl (psi)

Model 1 3-D FEM (50x70x15) Abaqus 2-D FEM

E=6

E=6E=6

E=.3

E=.3

E=.5

E=6E=.7

E=.7

w (in)(at wellbore)

TVD

(ft)

100.0 200.0 300.0

Xf (ft)3500.0 4000.0 4500.00.2 0.4 0.6 0.8

4,800

4,900

5,000

5,100

5,200

5,300

Pcl (psi)

Model 1a Model 1b Avg. E Model 2

20 40 60 80 100

200

400

600

800

TIME (min)

Pnet

(psi

) Model 1a Model 1b Avg E Model 2

w (in)(at wellbore)

TVD

(ft)

100.0 200.0 300.0

Xf (ft)3500.0 4000.0 4500.00.2 0.4 0.6 0.8 1.0

4,800

4,900

5,000

5,100

5,200

5,300

Pcl (psi)

Modulus Approximation 2 3-D FEM (100x100x15 mesh)

1 2 3 4 5 6 7 8 9

0.5

1.0

1.5

2.0

2.5

3.0

3.5

E / E

E

/ E

2 1

1Av

g

E1

E2

E2

w (in)(at wellbore)

TVD

(ft)

100.0 200.0 300.0

Xf (ft)3500.0 4000.0 4500.0-0.2 0.0 0.2 0.4

4,800

4,900

5,000

5,100

5,200

5,300

Pcl (psi)

Model 1 3-D FEM (100x100x15 mesh)

SPE 71654 LAYERED MODULUS EFFECTS ON FRACTURE PROPAGATION, PROPPANT PLACEMENT, AND FRACTURE MODELING 13

Figure 10 – Layered Modulus Effects, Average Modulus for 5-Layer Case

Figure 11 – Example 5-Layer Case.

Figure 12 – Error in “Local Width” for 5-Layer Case.

Figure 13 – Fracture Width Profile Due to Fracture Penetrating High Modulus Layers

E

20 feet

20 feet

20 feet

1

10 feet

10 feet

ν = 0.3, all layers

E1

E2

E2

E1

E1

E1

0.2 0.3 0.5 1.0 2.0 3.0 5.0

0.5

1.0

1.5

2.0

2.5

3.0

E / E

E

/ E

2 1

1Av

g

CorrectAverage E

SimpleHeight Weighted

Average E

1/2 Width (inches)

Dep

th (f

eet)

0.10 0.20

10

20

30

40

50

60

70

Actual 5:1 Width Profile

Width Profile Based onCorrect E

E = 1.36 E Avg

Avg 1

Width Profile Based onSimple Height Weighted

Average E = 2 E 1

Max Local Width Error= 0.03 inches

0.03/0.15 = 20%

0.2 0.3 0.5 1.0 2.0 3.0 5.0

0.5

1.0

1.5

2.0

2.5

3.0

10

20

30

40

50

60

E / E

E

/ E

2 1

1Av

g

Loca

l Wid

th E

rror

(%)

CorrectAverage E

LocalWidthError

6900

7000

7100

7200

-0.2 0.0 0.2Fracture Width (in)

Dept

h (ft

)

2 1E / E = 5µ = 500 cpσ Grad = 0.7 psi/ftσ Differences = 0

14 M. B. SMITH, A. B. BALE, L. K. BRITT, H. H. KLEIN, E. SIEBRITS, X. DANG SPE 71654

Figure 14 – Tensile Stress In Front of Fracture Tip as it Approaches an Interface with a Stiffer Layer

Figure 15 – Theoretical Effect of Modulus on Height (after Van Eekelen)

Figure 16 – Effect of Local Width Variations Due To A 2:1 Shale/Sand E2 /E1 Ratio on Prop Settling

DimensionlessTensile Stress

Dim

ensi

onle

ss D

ista

nce

from

Inte

rface

(Dis

tanc

e/a)

σp √ π R/a

0.0

0.2

0.2

0.4

0.6

0.8

5 10 15R/a = 0.07

E / E = 102 1

E1

E2

hH2X f

E2

5 10 20 30 40

1

2

3

E / E

Max (2 Xf / H)Van Eekelen

2 1

2 Xf

/ H

Numerical ModelQ = 25 bpm, µ = 1 cp

Numerical ModelQ = 25 bpm, µ = 500 cp