26th US Symposium on Rock Mechanics / Rapid City, SD / 26-28 June 1985

Energy analysis of hydraulic fracturing

J.SHLYAPOBERSKY Shell Development Company, Houston, Texas, USA

INTRODUCTION

Hydraulic fracturing is increasingly being used as a stimulation technique for oil and gas production from low permeability reservoirs. Most industrial fracture treatment designs are still based on two simple hydraulic fracture models: the Khristianovitch-Geertsma-de Klerk (KGK) model (Geertsma & de Klerk 1969) and Perkins-Kern-Nordgren (PKN) model (Perkins & Kern 1961, Nordgren 1972). These models have been compared and preferred conditions for their application to treatment design have been discussed (Geertsma & Haafkens 1979). Both models have two main shortcomings. They neglect rock strength and cannot predict the frac- ture height growth. The latter fact has, in recent years, stimulated intensive research to generate complex hydraulic fracture growth and containment models. Computer codes are being developed by Terra Tek (Clifton & Abou-Sayed 1981), MIT (Cleary 1980, Cleary et al. 1983), ORU (Palmer & Luiskutty 1984) and other research groups. Because of simpli- fying assumptions about hydraulic fracturing, these models predict different fracture geometries for identical reservoir and treatment conditions (Palmer & Luiskutty 1984).

The propped fracture length estimated using pressure build-up tests and/or production data is often much less than predicted by hydraulic fracture simulators. The propagating overpressure (the fracture treat- ment bottom hole pressure minus the fracture closure pressure) observed in the field is frequently much higher than that predicted by hydraulic fracture simulators (Warpinski 1984, Medlin & Fitch 1983). If a hydrau- lic fracture model is "tuned" on high propagating overpressure, the predicted fracture length will usually be shorter and more consistent with the well's production response.

Two mechanisms could cause these high treatment pressures. One is due to viscous flow through a much more complex multiple fracture system than the planar fracture assumed in hydraulic fracture models (Warpinski & Teufel 1984; Medlin & Fitch 1983). Another possible mechanism is due to a layer of relatively small cracks around a large hydraulic fracture that causes a significant increase of the apparent fracture energy with fracture growth and thus, increase of treatment pressures required for hydraulic fracture propagation.

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SCALE EFFECT IN HYDRAULIC FRACTURING

Scale effects are inherent in geomechanical problems. Traditionally, mathematical theories describing rock behavior have been formulated based on analysis of laboratory experiments with small rock specimens. Legitimate questions arise about the applicability of such theories to predict behavior of large rock bodies, especially in cases when rock characteristics required by the theory demonstrate a strong dependence on specimen size and test conditions.

Contemporary fracture mechanics are based on Griffith's idea that failure of brittle solids is due to microcracks always present in real materials. Griffith formulated a crack propagation criterion and intro- duced a new material constant, specific fracture surface energy , which characterizes material resistance to crack growth (fracture tough- ness). Using the Griffith energy balance, it was shown that the criti- cal pressure necessary to start propagating a penny-shaped crack of radius R is (Sack 1946)

p* = /(/2)''/R (1) where E' = E/(1-u 2) is plain strain Young's Modulus.

The energy per unit crack area ( is the energy required to rupture atomic bonds when the crack is formed. It is a true material constant only for ideal crystals. Thus, the Griffith theory can be, strictly speaking, applied only to crack propagation in ideal, defect free, single crystal solids. The applications of the theory to fracture propagation in rocks and other real solids are questionable. These solids contain different types of flaws which makes the fracturing process extremely complicated.

Microscopic observations of fracture propagation in rocks, polymers and metals show that the material in the region surrounding the main crack undergoes a microstructural transformation. In rocks, an exten- sive array of microcracks was observed (Friedman et al. 1971); in poly- mers microcrazes and recrystallized zones appeared (Bevis & Hull 1970); in metals, creation of microcracks (Chudnovsky & Bessendorf 1983) and the presence of plastic zones, associated with extensive dislocation motion, was noted.

The complicated structure of the damage in the material near an induced fracture is typical. Similar features of rock transformation in a zone near a propagating hydraulic fracture can be anticipated only on a large field scale. Indeed, recent observations in mineback experi- ments (Warpinski & Teufel 1984) have shown that there are considerable surface roughness and wavyhess, echelon fracturing, and significant offsets when natural fractures are intersected. Detection of hydraulic fractures by borehole geophones supports the presence of a relatively narrow zone adjacent to the fracture with high microseismic activity (Hart et al. 1984). The microseismic signals can be generated either by shear slippage along pre-existing joints or by cracks propagating in shear-tensile combined mode. In any case, these signals indicate energy dissipation due to irreversible rock transformation in a near-fracture active zone called crack layer.

To properly describe fracture propagation in real solids, the Griffith theory has to be modified to account for energy dissipated in creation of the crack layer. It was proposed (0rowan 1952; Irwin 1958) that this dissipated energy be added to the specific fracture surface energy and a new characteristic called the apparent fracture surface energy be deter- mined from laboratory tests on pre-cracked specimens using formulas

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similar to (1). This apparent fracture surface energy which is an empirical characteristic of fracture toughness, is the total energy required to create a unit area of the main fracture and the crack layer associated with this area. A difficulty in applying this Griffith- Orowan-Irwin theory is that the apparent fracture surface energy is not a material constant, but depends on the whole history of crack layer development during fracture propagation (Bakar et al. 1984). One may anticipate that as the crack layer grows with fracture propagation, the fracture toughness increases also. This effect of crack growth on the rock fracture toughness has been observed in fracture lab tests (Ouchteralony 1983). Microseismic activity during hydraulic fracture treatment, measured by borehole geophones and direct observations in mineback experiments also demonstrate that the size of the crack layer around hydraulic fracture can be much larger than seen in lab experi- ments. The corresponding rock toughness characterizing hydraulic frac- ture propagation should also be significantly higher than its laboratory measured value.

DETERMINATION OF FRACTURE TOUGHNESS FROM TREATING PRESSURE

This scale effect on rock toughness in hydraulic fracturing process manifests itself indirectly through very high fracture treating pres- sures observed during frac jobs. Accurate measurements of the fracture propagation overpressure and comparison with numerically simulated overpressures show that hydraulic fracture models, which either neglect rock toughness or use its laboratory measured values, predict much smaller overpressures than actually appear in field conditions (Warpinski 1984a,b). The rock toughness characteristics can be deter- mined from field pressure data by matching these and the overpressure simulated with a hydraulic fracture model, in which fracture toughness coefficient ( K1C) is considered as a free matching parameter. The proposed approach requires much effort in field performance and contains uncertainties in test procedures and data interpretation.

The fracture closure pressure (p.), assumed to be equal to the minimum in situ stress, has to be determkned first. An accurate estimate of this pressure is essential for the whole analysis. To measure Pc' minifrac and microfrac shut-in and flowback tests are used (Nolte 1984, McLennan & Roegiers 1982, Warpinski 1984a). In a shut-in test, the pressure decline curve has no evident features which would indicate fracture closure. A detailed reservoir analysis of the pressure fall- off is thus required to avoid a subjective and erroneous definition of the closure pressure. Different approaches in defining p_ from shut-in pressure data have been reviewed (McLennon & Roegiers 198). An alter- native technique to determine the fracture closure pressure is the constant rate flowback test. The flowback pressure decline curve usually has two distinctive points which may indicate fracture closure. The inflection point A on the pressure decline curve (Figure 1) has been successfully used to estimate the fracture closure pressure (Nolte 1984). However, this approach may sometimes yield unequal Pc values in two subsequent flowback tests performed with different flow- back rates (Warpinski 1984a). A mathematical analysis of the flow back tests and several field tests suggests that point C on the flowback pressure decline curve may be a better indication of the fracture clo- sure pressure because under certain conditions it represents a well reproducible rate independent pressure value.

Once Pc is known, the fracture overpressure near the well bore, APw Pw Pc (Figure 2) can be calculated from the bottom hole

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pressure (BHP) and theoretically estimated friction pressure drop (p_) through perforations: Ap = p - p = BHP - p - p .If the inJeg-

W W C C tion BHP data are used in the match procedure, an tificially high rock toughness can be expected because:

(1) Some fluid flow restrictions may exist in the perforated zone (Warpinski 1984b) which create higher than theoretically estimated perforation friction pressure drop at the fracture entrance and, respectively, lower pressure APw in the fracture. (2) The averaged overpressure p = ]Ap(x)dA/A and the critical stress intensity factor Ki[Ap] may be smaller for the real fracture than calculated by a hydraulic fracture model (Figure 3).

These two effects of fluid flow are difficult to quantify. Their significance in the pressure analysis can be reduced by using shut-in pressure data. During the early shut-in period the pressure gradient in the fracture is almost eliminated and p = p (small pressure drop in the fracture may still exist due to fluid overflow for some time after shut-in). If one notices that the average overpressure () for a given fracture geometry is determined by the fracture volume and the latter changes insignificantly after a short time of pressure equaliza- tion after shut-in, a surprizing result is that the ISIP is a measure of the average fracture propagating pressure, not the fracture closure pressure (the minimum in situ stress). Apparent rock toughness can be estimated by the Griffith-Sack type equation

KlC = p ief f (2) where the fracture shape dimensionless constant is of the order of one, the effective fracture radius Ref f is the radius of an uncontained fracture or the half height of a contained fracture. The effective fracture size has to be known to use Equation (2). In microfrac tests, the fracture shape is assumed to be circular and its radius can be calculated from pressure decline curve (modified Nolte analysis). In minifrac tests, the fracture is assumed to be contained in the pay zone wth half height considered as the effective fracture radius giving a conservative estimate of fracture toughness.

FIELD EXAMPLE

We now apply the proposed method to three minifrac tests conducted in the MWX Paludal Zone Phase I stimulation (Warpinsk 1984a). Prior to these tests, the minimum in situ stress of about 40.7 MPa has been esti- mated by different techniques (microfrac tests, step rate test, flow back test). The pressure decline curve for the first test shows Pc - 41.4 MPa and ISIP 44.2 MPa , yielding pp - 2.8 MPa .

The pay zone height of about 24m gives Reff= 12m, which is less than the temperature surveys show (fracture height H T - 30m in the first test and H - 45m in the last one). The use of equation (2) gives K - 9.7 MPa or in terms of the apparent fracture surface energy

1C 2 , Z , F = KI /E - 3 kJ/m (E = 31GPa). Results of the analysis for two othe tests summarized in Table 1 indicate that rock toughness appearing in hydraulic fracture treatments is much larger than one would expect to measure in laboratory fracture tests (Ouchterlony 1983); one and two orders of magnitude for K and F , respectively. These C results agree with the general tren ofhe apparent fracture surface energy rowth with c%ack size; -1 J/m for micro grain size crack

. o Z Z Z ( 10- m), 100 J/m for lab size cracks ( 10 m) and 10,O00 J/m for "small" hydraulic fractures (

likely to be caused by the crack layer accompanying the hydraulic fracture propagation and is clearly seen even for small changes of pumped volumes in Table 1. How strong this effect can be in large treatments is an important practical issue. Some limited data reported in the literature (Medlin & Fitch 1983) demonstrate tremendous increases of post treatment ISIP's of more than 16MPa with respect to their initial values which resulted in screenouts. It is suggested that pressure screenouts be studied not only as sandout phenomena, but also as a phenomenon of crack layer evolution during hydraulic fracturing treatments.

ENERGY ANALYSIS OF HYDRAULIC FRACTURING

The analysis of the fracture propagation process and hydraulic fracturing treatment data suggests that the rock toughness effect is very important and, under certain conditions, may even become dominant for fracture growth. Therefore, any hydraulic fracture theory has to account for at least four interacting processes--fluid flow, rock defor- mation, rock toughness, and fluid loss through fracture walls to ade- quately describe hydraulic fracture growth and to predict realistic fracture dimensions. An approximate theory that incorporates the rock toughness effect in conventional fracture models is presented below for a circular fracture.

A circular fracture of radius R at time t is considered during quasi- equilibrium growth. The following volum4e balance accounting for the fluid pumped (qt), leak-off v_ume (2Rv CT r) (Nolte 1984), and the current fracture volume (R) :

R2 = qt - 2R2VCTr (3) Here v is a constant (4/3 v /2)dependent on the total fluid loss coefficient CT, fluid visousity , and the apparent fracture energy . is th average fracture width determined from energy considertions.

For quasiequilibrium fracture propagation, the total energy dissipation rate is minimized. Three processes give the major contribu- tion to the total energy loss in the hydraulic fracture; (1) creation of the new fracture surface and the crack layer ( ), (2) fracture

.c

opening by deforming the surrounding reservoir ( d )' (3) viscous dissipation in fluid (f). Based on an elasticity solution for a penny-shaped crack and fluid flow between parallel plates, all energy rates can be calculated analytically and expressed in terms of average fracture width:

= rq/, d = (3/32)E'q/R, f = (12/)t21n/() 3 (4) c where 0 = R/R o, 2R o = the height of the perforated interval.

The condition of quasiequilibrium fracture propagation requires

which yields an expression for the average fracture width

([)2 = w 2 + /w + w} (6) c 2 (16/3a)m/E', w = (1281no/(3a))qR/E' where w c =

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Solving equations (4) & (6), the fracture radius R and the average fracture width at time t are determined. The average overpressure

Ap = (3/16)E'w/R and the pressure profile for flow between parallel plates

p(r) = Pw- (6q/)(w--)-31n(r/R) o can be calculated.

SCALING LAWS OF HYDRAULIC FRACTURING

The presented hydraulic fracture model allows introduction of dimensionless characteristics with clear physica maning. The .energy balance suggests two energy parameters k f = c /of and k d = f/,. The width Aequation suggests a dimensioness eometrica parameter k = Wc/W. Finally the material balance gives dimensionless leak-

of parameter k 1 = 2R CTV_/(q/) that is related to the fluid effi- ciency ef (the ratio of creted fracture volume to injected volume) as k 1 = 1 - ef (Nolte 1984). The three independent dimensionless para- meters k , and k 1 quantify three interactive mechanisms between four physicalCgrockesses involved in hydraulic fracturing. Thus, they repre- sent scaling laws for hydraulic fracturing which have to be obeyed in properly scaled lab hydraulic fracture experiments.

CONCLUS ION

The scale effect on fracture toughness is discussed in context of hydraulic fracturing. The crack layer causes this effect and results in significant increase of fracture toughness and treatment pressure with crack growth. The laboratory fracture toughness measurements cannot quantify rock fracture toughness of large hydraulic fractures. A method to estimate fracture toughness from treatment pressure data is proposed and a field example is discussed. A variational principle of the mini- mum energy dissipation rate is used to incorporate fracture toughness in classic hydraulic fracture models. Dimensionless characteristics of the hydraulic fracturing process are introduced which are the scaling laws of the process.

ACKNOWLEDGEMENTS

The author wishes to thank the management of Shell Oil Company for permission to publish the paper and Dr. W. F. J. Deeg for many valuable discussions and assistance in preparing the paper.

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Chudnovsky, A., & M. Bessendorf 1983. Crack Layer Morphology and Toughness Characterization in Steel. NASA Report 168154. Case Western Reserve Univ. Cleveland, OH.

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Clifton, R.$. & l.S. Abou-Sayed 1981. A Variational Approach to the Prediction of the Three-Dimensional Geometry of Hydraulic Fractures. SPE/DOE Paper 9879. Symposium on Low Permeability Reservoirs. Denver, CO. May.

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Table 1. Apparent Fracture Toughness From Mini Frac Tests in WX Paludal Zone Phase I Stimulation*

Volume ISIP Ap (HTm Ref f K1C (m 3) (MPa) (MPa) ) (m) MPa yr KJ/m 2 22 44.2 2.8 28 12 9.7 3 57 45.9 4.5 41 12 15.6 7.8

114 47.6 6.2 46 12 21.5 14.9

*Pc 41.4 MPa, E' = 31 GPa, HpA Y = 24m (Warpinski 1984a)

. I i s "1' F

ISl p i r

-p( : A B

qB'

Figure 1. Injection - shut-in - flowback pressure curve of typical micro/mini fracture test.

qB

'///////

INJECTION BHP pwJ5 PT HUT - )N

BHP ,,= DW ' PT FLOW - BACK

Figure 2. Bottom hole pressure measurements and friction drop during fracture test.

DISTANCE ALONG THE FRACTURE

IP-

DISTANCE ALONG THE FRACTURE

Figure 3. Hypothetical pressure profile during injection and shut-in for (A) idea1 well-fracture system, (B) fracture treatment.

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13. Poster session A