sensitivity analysis for the appraisal of hydrofractures...

$: Sensitivity analysis for the appraisal of hydrofractures ...faculty.engr.utexas.edu/sites/default/files/torresverdin/files/10_2013_journal_pub...Sensitivity analysis for the appraisal$
Sensitivity analysis for the appraisal of hydrofractures in horizontalwells with borehole resistivity measurements

David Pardo1 and Carlos Torres-Verdín2

ABSTRACT

We numerically evaluate the possibility of using boreholeelectromagnetic measurements to diagnose and quantify hy-draulic fractures that have been artificially generated in ahorizontal well. Hydrofractures are modeled as thin disksperpendicular to the well and filled with either sand-based or electrically conductive proppant. The study focuseson the effect of thickness and length (radius) of hydrofrac-tures to assess their effects on specific configurations ofborehole-resistivity instruments. Numerical results indicatethat several measurements (e.g., those obtained with low-and high-frequency solenoids) could be used to assess thethickness of a fracture. However, only low-frequency mea-surements performed with electrodes and large-spacingbetween transmitter and receivers (18 m) exhibit the neces-sary sensitivity to reliably and accurately estimate thelength of long hydrofractures (up to 150 m) in open-holewells. In the case of steel-cased wells, the casing acts asa long electrode, whereby conventional low-frequencyshort-spaced, through-casing measurements are suitablefor the accurate diagnosis of long hydrofractures (up to150 m in length).

INTRODUCTION

Generation of one or several artificial hydrofractures within awell is a common technique to enhance hydrocarbon recovery intight (low permeability) rock formations. After creating an artificialhydrofracture, it is often necessary to appraise its main geometricalproperties, including shape, length, thickness, and principal direc-tions of propagation. The appraisal of a hydrofracture enables the

quantification of its benefit in terms of hydrocarbon recovery; it alsoallows the determination of optimal locations to initiating additionalhydrofractures in multistage hydrofracturing processes.The most common technique used for the characterization of hy-

drofractures is based on microseismic monitoring (see, for example,Nolen-Hoeksema and Ru, 2001; Warpinski, 2009; Song et al.,2010). However, this technique is used only during the stimulationphase of the fracture; occasionally, a posterior assessment may beconvenient to appraise the effectiveness of the hydrofracture pro-cess. Other acoustic-based methods for hydrofracture diagnosishave been recently proposed in Tang and Patterson (2009) usingshear-wave propagation, and in Matuszyk et al. (2011) by examin-ing resonant frequencies. Alternative less common fracture charac-terization techniques include pressure transient analysis (seeWagenhofer [1996] and references therein).The main goal of this paper is to study the possibility of accu-

rately estimating thickness and length of long hydrofractures (up to100 m) using borehole electromagnetic (EM) measurements. Hy-drofractures are modeled as disks that are perpendicular and axialsymmetric with respect to a horizontal borehole; they may exhibitvariable thickness and electrical resistivity. Due to the great depthsat which most of these fractures take place and the lossy electricalnature of the earth’s subsurface, we focus only on borehole mea-surements for detection and appraisal, avoiding the use of bore-hole-to-surface resistivity measurements such as those describedin Wang et al. (1991). We also avoid the use of crosswell measure-ments because a second nearby well may not be available in allreservoirs.Early results on the diagnosis of open-hole hydrofracture proper-

ties using borehole resistivity measurements include the work ofSibbit and Faivre (1985), where large resistive fractures were char-acterized using dual-laterolog measurements. More recent resultsinvoking different induction logging instruments can be foundin Rabinovich et al. (2004), Wang et al. (2005), Tang et al.(2006), Xue et al. (2008), and Hu et al. (2010). Because these

Manuscript received by the Editor 12 January 2013; published online 20 June 2013.1University of the Basque Country (UPV/EHU), Department of Applied Mathematics, Statistics, and Operational Research, Leioa, Spain; IKERBASQUE

(Basque Foundation for Sciences), Bilbao, Spain. E-mail: [email protected] University of Texas at Austin, Austin, Texas, USA. E-mail: [email protected].

© 2013 Society of Exploration Geophysicists. All rights reserved.

D209

GEOPHYSICS, VOL. 78, NO. 4 (JULY-AUGUST 2013); P. D209–D222, 12 FIGS.10.1190/GEO2013-0014.1

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studies assume acquisition frequencies above 10 kHz, the sensitiv-ity of the corresponding resistivity measurements to the length oflong hydrofractures is limited.In contrast to previous studies, this paper considers the case of low-

frequency (below 1 kHz) and possibly long-spaced antennas. In ad-dition, we consider proppants that exhibit high electrical conductivityas well as conventional proppant made with granular sand. The use ofelectrically conductive proppants in field operations is not new; theyhave been used in the past for enhanced hydrocarbon recovery (seeMurdoch and Chen, 1997). However, their possible use for fracturecharacterization has not been exploited yet. In this paper, we assumethat electrically conductive proppant is used to enhance the resistivitycontrast between the surrounding rock formation and the hydrofrac-ture itself to increase the sensitivity of borehole resistivity measure-ments to the hydrofracture’s geometrical features.More importantly, this paper documents for the first time a sensi-

tivity analysis of through-casing resistivity measurements (Kauf-mann, 1990) to the presence, thickness, and length of longhydrofractures. Appendix A is also included where we describethe main advantages and limitations of using magnetic proppant(see Schmidt and Tour [2009] for a description on magnetic prop-pants) for hydrofracture characterization.Numerical simulations are performed with a 2D self-adaptive,

goal-oriented hp-finite element method, where h indicates theelement size, and p the polynomial order of approximation, bothvarying locally throughout the computational grid (for details, seeDemkowicz, 2006; Pardo et al., 2007a). This method enables accu-rate and reliable simulations of resistivity measurements in differentborehole environments, including steel-cased wells and highcontrasts in material properties (see Pardo et al., 2006a, 2006b,2007b, 2008b; Nam et al., 2008).

MODEL PROBLEM

We consider the open-hole model problem described in Figure 1(horizontal location is higher for the receivers than for the transmit-ter). When indicated, the well may also possibly incorporate a 1 cm-thick casing with resistivity equal to 2.3 · 10−7 Ω · m. Although thegeometry of a fracture could be rather complex (e.g., Gudmundssonand Brenner, 2001); Flekkøy et al. (2002), in this paper we assumethat its geometry can be described with a perfect disk perpendicularto the borehole. Therefore, in the study we are not concerned withthe main direction of propagation of the hydrofracture, which is as-sumed to be known. The objective of this simplified model is toquantify the limits of spatial resolution of borehole resistivity mea-

surements to basic hydrofracture models. If borehole resistivitymeasurements are not sensitive to such simple hydrofracture modelsthen it becomes moot to appraise the limits of detectability of themeasurements to more complicated models (e.g., multiple adjoiningfractures) that may also give rise to electromagnetic clutter.We consider the following four types of proppant filling the frac-

ture: (1) a sand-based proppant such that the resultant hydrofrac-ture resistivity is equal to 0.1Ω · m, (2) an electrically conductivecoke-breeze-based proppant with a resultant fracture resistivityequal to 3 × 10−4 Ω · m (see Graphite, 2012), (3) an electricallyconductive proppant such that the resultant hydrofracture resistivityis equal to 10−6 Ω · m, and (4) a ferrite-based proppant such thatthe resultant hydrofracture relative magnetic permeability is equalto three (this type of proppant is analyzed in Appendix A). Theseresistivities assume a clean fracture fully saturated with proppant.In addition, the highly conductive proppant cannot be currentlymanufactured in a cost-effective manner. Several companies areworking on the design of more electrically conductive proppants;we nevertheless include results corresponding to such specific casein this paper because they may serve as additional motivation forthe fabrication of more conductive proppants. In the assumed sim-ple hydrofracture model associated with a horizontal well, fracturethickness is equal to 0.005 m (unless otherwise indicated). Thelength of the hydrofracture varies from zero (no fracture) to152 m. As subsequently verified in this paper, resistivity measure-ments are only sensitive to the conductance (multiplication ofthickness times electrical conductivity) of the hydrofracture. Thus,results shown here also apply to different fracture thicknesses and/or proppant resistivities, as long as the electrical conductance ofproppant remains constant.

Sources and receivers

Using cylindrical coordinates ðρ;ϕ; zÞ, we consider two types ofaxisymmetric sources. The first one uses solenoidal coils modeledby prescribing an impressed volume electric current densityJimp ¼ δðzÞδðρ − aÞIJϕ, where IJ ¼ 1 and a ¼ 0.07m is the radiusof the coil, where the resulting electric field is such thatEρ ¼ Ez ¼ 0, i.e.; this source excites only the so-called transverseelectric (TE) mode. The second one employs electrodes that gen-erate only aHϕ component of the magnetic field, i.e.,Hρ ¼ Hz ¼ 0

is the transverse magnetic (TM) mode. See Liu et al. (1994) fordetails. We model this source by prescribing an impressed volumemagnetic current density over a coil of radius 0.07 m whose far-fieldsolution in a homogeneous space is equivalent to that produced by avertical electric dipole (VED), as described in Lovell (1993) and

Pardo et al. (2006b).

SIMULATION METHOD

Our simulation method is based on 2D hp-finite-element (FE) discretizations of EM prob-lems. Here, h stands for element size whereasp denotes the polynomial element order (degree)of approximation, both varying locally throughoutthe grid. Our method also incorporates a goal-or-iented hp-grid-refinement algorithm that automa-tically delivers a sequence of optimal hp-grids foreach problem. Starting from a given coarse grid,the self-adaptive procedure performs optimal and

a) b)

f

Figure 1. Model problem.

D210 Pardo and Torres-Verdín

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local hp mesh refinements by minimizing the error of a prescribedquantity of interest (solution at the receivers) with respect to theadded number of unknowns. Notice that in resistivity logging mea-surements it is critical to minimize the error in the quantity of interestrather than in a global norm because the solution at the receivers istypically several orders of magnitude smaller than that near thesource, whereby a relatively small (global) absolute error does notnecessarily guarantee a small relative error at the receivers.

The main advantage of this goal-oriented grid-refinement method isgiven by the following result: The sequence of grids delivered by thehp goal-oriented adaptive procedure converges exponentially fast interms of the error in the quantity of interest (solution at the receivers)versus the problem size (number of unknowns). Thus, it provideshighly accurate solutions using limited computational resources.The outstanding performance of the hp-FE method for simulating

diverse resistivity logging measurements as well as its verification has

4 4.1 4.2 4.3 4.4

× 10−12

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Hor

izon

tal p

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m)

Absolute value of Eφ (V/m)

0-m-long (no fracture)0.6-m-long fracture3-m-long fracture15-m-long fracture61-m-long fracture

Solenoids, 1.2 – 0.3-m spacing

0 1 2 3

× 10−4

Hor

izon

tal p

ositi

on (

m)

Absolute value of Eaxial

(V/m)


Electrodes, 1.2– 0.3-m spacing

5.46 5.48 5.5 5.52 5.54 5.56 5.58× 10

−14

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)


0-m-long (no fracture)0.6-m-long fracture3-m-long fracture15-m-long fracture61-m-long fracture152-m-long fracture

Solenoids, 18–1.2-m spacing

0 2 4 6 8× 10

−9

Hor

izon

tal p

ositi

on (

m)


(V/m)


Electrodes, 18 –1.2-m spacing

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

−6

−4

−2

0

2

4

6

a) b)

c) d)

Figure 2. Sensitivity of different logging instruments to the length of a hydrofracture filled with sand proppant. Frequency of operation:100 Hz. Background (shale) resistivity: 3 Ω · m.

EM appraisal of hydrofractures D211

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been documented in Pardo et al. (2006a, 2006b, 2007b). A detaileddescription of the hp-FE method and its exponential convergenceproperties can be found in Demkowicz (2006). We refer to Pardo et al.(2007a) for technical details on the goal-oriented adaptive algorithmapplied to electrodynamic simulation problems.For truncation of the computational domain, we impose a zero Di-

richlet boundary condition at 200 m (for open-hole) or 1000 m (for

steel-cased wells) from the source. This truncation method is custom-ary in most resistivity borehole simulations, thereby making unneces-sary the use of more sophisticated methods such as perfectly matchedlayers Michler et al. (2007) and Pardo et al. (2008a). The use of a zeroDirichlet boundary condition is justified by the rapid spatial decay ofthe EM fields as the observation points recedes away from the sourcedue to the lossy electrical nature of the earth’s subsurface.

0.2 0.4 0.6 0.8 1 1.2

× 10−10Absolute value of Eφ (V/m)


Solenoids, 1.2 –0.3-m spacing

0 0.5 1 1.5 2

× 10−3Absolute value of E

axial (V/m)



0 0.5 1 1.5 2 2.5

× 10−13Absolute value of Eφ (V/m)


Solenoids, 18 –1.2-m spacing

0 1 2 3 4 5

× 10−7Absolute value of E

axial (V/m)


Electrodes, 18–1.2-m spacing

0

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

−6

−4

−2

0

2

4

6

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

−6

−4

−2

0

2

4

6

Hor

izon

tal p

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on (

m)

Hor

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tal p

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on (

m)

Hor

izon

tal p

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on (

m)

Hor

izon

tal p

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on (

m)

a) b)

c) d)

Figure 3. Sensitivity of different logging instruments to the length of a hydrofracture filled with coke-breeze-based conductive proppant(3 × 10−4 Ω · m). Frequency of operation: 100 Hz. Background (shale) resistivity: 3 Ω · m.


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NUMERICAL RESULTS

Open hole wells

Figure 2a describes the sensitivity of borehole resistivity mea-surements to the presence and length of an artificially generatedfracture filled with sand-based proppant. The measurement systemis based on solenoidal coils operating at 100 Hz in an open-hole

horizontal well. Fracture location can be readily identified fromthe simulated measurements. However, its length cannot be esti-mated above 15 m. When employing electrode sources (Figure 2b),the sensitivity to the fracture’s length decreases even further. As thespacing between transmitters and receivers increases (see Figure 2cand 2d), measurements become more sensitive to the length of thehydrofracture. Regardless of such a behavior, hydrofractures with

0 0.5 1 1.5 2

× 10−8

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Hor

izon

tal p

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on (

m)



Solenoids, 1.2– 0.3-m spacing

0 0.5 1 1.5 2

× 10−3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Hor

izon

tal p

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on (

m)


(V/m)



0 2 4 6 8

× 10−12

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)



Solenoids, 18 –1.2-m spacing

0 2 4 6 8

× 10−7

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)


(V/m)


Electrodes, 18–1.2-m spacing

a) b)

c) d)

Figure 4. Sensitivity of different logging instruments to the length of a hydrofracture filled with a highly electrically conductive proppant(10−6 Ω · m). Frequency of operation: 100 Hz. Background (shale) resistivity: 3 Ω · m.


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lateral extent of 61 m (200 ft) and 152 m (500 ft) give rise to(almost) identical simulation results for all cases. Therefore, weconclude that the length of long hydrofractures filled withsand-based proppant cannot be accurately estimated using boreholeresistivity measurements. Additionally, notice that measurementsperformed with solenoids exhibit smaller differences than those

obtained with electrodes (compare scales of Figure 2a and 2c versusFigure 2b and 2d).When considering a more electrically conductive proppant filling

the hydrofracture (e.g., coke breeze-based proppant), solenoids(Figure 3a and 3c) provide similar (although slightly inferior) re-sults than electrodes (Figure 3b and 3d). Electrode-based systems

−1 0 1 2 3 4 5

× 10−7

−6

−4

−2

0

2

4

6

a) b)

c) d)

Hor

izon

tal p

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on (

m)

Real part of Eaxial

(V/m)


Coke-breeze proppant (3 × 10−4 Ω· m)

Conductive proppant (10−6 Ω· m) Conductive proppant (10−6 Ω· m)

Coke-breeze proppant (3 × 10−4 Ω· m)

−15 −10 −5 0 5

× 10−9

−6

−4

−2

0

2

4

6

Hor

izon

tal p

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m)

Imaginary part of Eaxial

(V/m)


−2 0 2 4 6 8

× 10−7

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)

Real part of Eaxial

(V/m)


−3 −2 −1 0 1

× 10−8

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)


(V/m)


Figure 5. Sensitivity of different logging instruments to the length of a hydrofracture. Frequency of operation: 100 Hz. Background (shale)resistivity: 3 Ω · m. Spacing between transmitter and first receiver: 18 m. Spacing between first and second receiver: 1.2 m.


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provide the best results in terms of sensitivity with respect tofracture length, and exhibit the necessary sensitivity to discern be-tween a 15 m and a 61 m long fracture in the case of long-spacedelectrodes (Figure 3d). These differences increase as one increasesthe electrical conductivity of the proppant filling the hydrofracture,as described in 4d. Therefore, we conclude that a measurement

system based on long-spaced electrodes can be employed to accu-rately estimate the length of a hydrofracture filled with electricallyconductive proppant.In the case of an electrically highly conductive fracture, real and

imaginary parts of the EM fields are sensitive to the presence of ahydrofracture. Furthermore, for long hydrofractures filled with

−15 −10 −5 0 5

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)


−15 −10 −5 0 5

× 10−8Imaginary part of E

axial (V/m)


−20 −15 −10 −5 0 5

× 10−9

Hor

izon

tal p

ositi

on (

m)


(V/m)


−4 −3 −2 −1 0 1 2

× 10−9Imaginary part of E

axial (V/m)


−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)

Hor

izon

tal p

ositi

on (

m)

Electrodes, 10 Hz, background resistivity: 3 Ω· m Electrodes, 1000 Hz, background resistivity: 3 Ω· m

Electrodes, 100 Hz, background resistivity: 10 Ω· m Electrodes, 100 Hz, background resistivity: 0.5 Ω· m

a) b)

c) d)

Figure 6. Sensitivity of different logging instruments to the length of a hydrofracture filled with a coke breeze electrically conductive proppant(3 × 10−4 Ω · m). Spacing between transmitter and first receiver: 18 m. Spacing between first and second receiver: 1.2 m.


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highly conductive proppants, the imaginary component of theelectric field becomes dominant, leading to a shielding effect.Consequently, we observe that the separation of curves correspond-ing to long hydrofractures decreases in the case of highly conduc-tive proppant (Figure 4c) compared to the case of coke breeze-basedproppant (Figure 3c).

Figure 5 shows that the imaginary part has a higher sensitivity tothe hydrofracture’s length than the real part for the considered cases.In the remainder of the paper, we display only those components ofthe solution with maximum sensitivity to hydrofracture length.Notice that such components may exhibit a small amplitude com-pared to the absolute value, whereby the effect of noise should be

5.45 5.5 5.55 5.6 5.65

× 10−14

−6

−4

−2

0

2

4

6

a) b)

c) d)

Hor

izon

tal p

ositi

on (

m)


0-m long (no frac)152-m frac.152-m frac. (half thickness)152-m frac. (double cond.)152-m frac. (double cond., half thickness)

Solenoids, sand proppant

0 0.5 1 1.5 2

× 10−8

Hor

izon

tal p

ositi

on (

m)


(V/m)


Electrodes, sand proppant

0 1 2 3 4 5

× 10−13

−6

−4

−2

0

2

4

6

Hor

izon

tal p

ositi

on (

m)



Solenoids, coke-breeze proppant (3 × 10−4 Ω· m) Electrodes, coke-breeze proppant (3 × 10−4 Ω· m)

0 1 2 3 4 5 6

× 10−7

Hor

izon

tal p

ositi

on (

m)


(V/m)


−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Figure 7. Sensitivity of different logging instruments to proppant thickness and resistivity. Frequency of operation: 100 Hz. Background(shale) resistivity: 3 Ω · m. Spacing between transmitter and first receiver: 18 m. Spacing between first and second receiver: 1.2 m.


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accounted for in actual field operations, especially at low fre-quencies.As emphasized earlier, the sensitivity of measurements to hydro-

fracture length depends primarily of the electrical conductivity ofproppant. There also exist other factors that increase or diminish thissensitivity. Specifically, it is possible to quantify larger hydrofracturelengths by decreasing the frequency of operation (Figure 6a), whilean increase in frequency considerably decreases the depth of inves-

tigation (Figure 6b) of the measurements. Similarly, an increase ofbackground resistivity (Figure 6c) increases the depth of investigationand increases the conductivity contrast between the formation andbackground; the opposite effect takes place when decreasing thebackground conductivity (Figure 6d).Figure 7 describes the sensitivity of measurements to hydrofrac-

ture thicknesses and electrical resistivity for a 152-m-long fracture.These results show that measurements are insensitive to variations

4 5 6 7 8

× 10−13

Hor

izon

tal p

ositi

on (

m)

Amplitude of first difference of Eaxial

(V/m)


Sand proppant, 1.2–0.3-m spacing

0 1 2 3 4

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)


Conductive proppant (10−6 Ω· m), 1.2−0.3-m spacing

1 1.05 1.1 1.15 1.2

× 10−12

Hor

izon

tal p

ositi

on (

m)


(V/m)


Sand proppant, 18−1.2-m spacing

0 0.5 1 1.5 2 2.5 3

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)


Conductive proppant (10−6 Ω· m), 18−1.2-m spacing

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

a) b)

c) d)

−1.5

−1

−0.5

0

0.5

1

1.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1.5

−1

−0.5

0

0.5

1

1.5

Figure 8. Steel-cased well. Sensitivity of different logging instruments to hydrofracture length. Frequency of operation: 10 Hz. Background(shale) resistivity: 3 Ω · m.


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in hydrofracture thickness and resistivity when the hydrofractureelectrical conductance (multiplication of conductivity times thick-ness) remains constant.

Steel-cased wells

In this subsection, we consider numerical simulations of 1 cm-thick casing of electrical resistivity equal to 2.3 · 10−7 Ω · m.

Figure 8 describes the sensitivity of through-casing resistivity mea-surements to the presence and length of a hydrofracture filled withsand-based proppant. Results indicate that while the fracture is un-iquely identified, its length cannot be accurately estimated when itis greater than 1 m. An increase in proppant electrical conductivity(see Figure 8b), causes the measurements to become more sensitiveto hydrofracture length.

0 1 2 3 4 5 6

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)


1 Hz, background resistivity: 3 Ω· m

0 0.2 0.4 0.6 0.8 1 1.2

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)



0 0.5 1 1.5 2 2.5 3

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)



0 2 4 6 8

× 10−10

Hor

izon

tal p

ositi

on (

m)


(V/m)


10 Hz, background resistivity: 0.5 Ω· m

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

a) b)

c) d)

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 9. Steel-cased well. Sensitivity of different logging instruments to the length of a hydrofracture filled with electrically conductiveproppant (10−6 Ω · m). Spacing between transmitter and first receiver: 1.2 m. Spacing between first and second receiver: 0.3 m.


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Contrary to the case of open-hole measurements, increasing thespacing between transmitter and receivers does not significantly in-crease the depth of investigation (see Figure 8c and 8d) of the mea-surements for the case of steel-cased wells. Because casing acts as along electrode, the depth of investigation of the measurementssignificantly increases in the presence of steel-cased wells, whereby

even conventional (short-spacing) through-casing resistivity measure-ments become suitable for estimation of long hydrofracture lengths.Figure 9a indicates that sensitivity of through-casing resistivity

measurements to different hydrofracture lengths increases with adecrease in frequency, while an increase in frequency considerablydecreases the depth of investigation of the measurements (see

1 2 3 4 5 6

× 10−11

Hor

izon

tal p

ositi

on (

m)


(V/m)


1 Hz, background resistivity: 3 Ω·m

0 1 2 3 4 5 6

× 10−11

Hor

izon

tal p

ositi

on (

m)


(V/m)



0 1 2 3 4

× 10−11

Hor

izon

tal p

ositi

on (

m)


(V/m)



0 2 4 6 8

× 10−11

Hor

izon

tal p

ositi

on (

m)


(V/m)


10 Hz, background resistivity: 0.5 Ω· m

0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

a) b)

c) d)

Figure 10. Steel-cased well. Sensitivity of different logging instruments to the length of a hydrofracture filled with coke breeze-based proppant(3 × 10−4 Ω · m). Spacing between transmitter and first receiver: 1.2 m. Spacing between first and second receiver: 0.3 m.


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Figure 9b). Similarly, an increase of background resistivity(Figure 9c) causes the conductivity contrast and the depth of inves-tigation of the measurements to increase; the opposite effect isachieved when decreasing the background resistivity (Figure 9d).Similar effects are observed with a coke breeze-based proppant

(see Figure 10), as well as in the case of open-hole resistivitymeasurements.

Figure 11 quantifies the relative differences (in percentage) of theeffect of hydrofracture properties on the simulated resistivity mea-surements (i.e., the relative difference of measurements acquiredwith and without the presence of the fracture in the formation).These results indicate that measurements are sensitive to the pre-sence of a hydrofracture when the latter is filled with an electricallyconductive proppant.

0 200 400 600 800 1000

Hor

izon

tal p

ositi

on (

m)

Relative difference of imaginary part of Eaxial

(%)


Coke-breeze proppant (3 × 10−4 Ω.m),18–1.2-mspacing

Coke-breeze proppant (3 × 10−4 Ω.m),1.2–0.3-mspacing

Coke-breeze proppant (3 × 10−4 Ω.m),1.2 – 0.3-mspacing

Highly conductive proppant (10−6 Ω.m), 18–1.2-mspacing

0 500 1000 1500 2000

Hor

izon

tal p

ositi

on (

m)

Relative difference of imaginary part of Eaxial

(%)


0 2000 4000 6000 8000 10000

Hor

izon

tal p

ositi

on (

m)

Relative difference of first difference of Eaxial

(%)


0 1000 2000 3000 4000

Hor

izon

tal p

ositi

on (

m)

Relative difference of first difference of Eaxial

(%)


−6

−4

−2

0

2

4

6

a) b)

c) d)

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−6

−4

−2

0

2

4

6

−1.5

−1

−0.5

0

0.5

1

1.5

Figure 11. (a and b) Open-hole well, 100 Hz. (c and d) Steel-cased well, 10 Hz. Sensitivity of different logging instruments to hydrofracturelength. Background (shale) resistivity: 3 Ω · m.


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Moreover, it is possible to quantify large hydrofracture lengths byemploying low-frequency electrodes with an electrically conductiveproppant. This conclusion is valid for open-hole and steel-casedwells.

CONCLUSIONS

Numerical simulations indicate that is possible to detect andquantify the thickness and length of single hydraulic fractureswith borehole resistivity measurements. The sensitivity of themeasurements to hydrofracture properties increases considerablywhen proppant is electrically conductive (in this case, rock con-ductivity is enforced through proppant rather than through thefluid filling the space between proppant particles). For the caseof open-hole measurements, numerical simulations indicate thatlong-spaced, low-frequency electrodes are often superior to sole-noidal-based systems to quantify hydrofracture properties in hor-izontal wells. For the case of steel-cased measurements,conventional (possibly short-spaced) low-frequency through-cas-ing measurements exhibit sufficient sensitivity to diagnose hydro-fracture properties. Because casing acts as a long electrode, itnaturally increases the depth of investigation of the measurements,whereby the use of long-spaced electrodes be-comes unnecessary. Such a measurement con-figuration can be used to accurately assesshydrofracture lengths up to 150 m provided thatbackground resistivity and fracture conductanceare large enough. However, we note that thedepth of investigation of the measurements de-creases with an increase of background conduc-tivity and/or frequency of operation.The above conclusions are valid for the case of

simple hydrofracture models generated in a hor-izontal well. Future extensions of this work in-clude the characterization of preferentialhydrofracture orientation via 3D borehole resis-tivity measurements.

ACKNOWLEDGMENTS

The work reported in this paper was funded byThe University of Texas at Austin’s ResearchConsortium on Formation Evaluation, jointlysponsored by Anadarko, Apache, Aramco, Ba-ker-Hughes, BG, BHP Billiton, BP, Chevron,ConocoPhillips, ENI, ExxonMobil, Halliburton,Hess, Maersk, Marathon Oil Corporation, Mex-ican Institute for Petroleum, Nexen, ONGC, Pet-robras, Repsol, RWE, Schlumberger, Shell,Statoil, Total, and Weatherford. The first authorwas also partially funded under projectMTM2010-16511 of the Spanish Ministry ofSciences and Innovation, the Basque Govern-ment Consolidated Research Group GrantIT649-13, and the CYTED 2011 projectP711RT0278. A note of special gratitude goesto Mukul M. Sharma (University of Texas atAustin) for insightful discussions about hydro-fracture properties. We are indebted to Sofia Da-vydycheva, David Alumbaugh, an anonymous

reviewer, and the associate editors for their constructive technicaland editorial comments that improved the first version of the manu-script.

APPENDIX A

NUMERICAL SIMULATIONS WITH MAGNETICPROPPANT

This appendix considers the case of magnetic proppant, as pro-posed in Schmidt and Tour (2009). Specifically, we consider twotypes of proppant. First, a sand- and ferrite-based proppant suchthat the resultant hydrofracture material properties are: (1) relativemagnetic permeability equal to μr ¼ 3, and (2) electrical resistivityequal to 0.1 Ω · m. Second, a coke breeze-based proppant with fer-rite such that the resultant hydrofracture material properties are: (1)relative magnetic permeability equal to μr ¼ 3, and (2) electricalresistivity equal to 3 × 10−4 Ω · m.Long-spaced open-hole measurements for the case of sand- and

ferrite-based proppant are described in Figure A-1a and A-1b. Re-sults indicate that using magnetic proppant does not increase the

4.5 5 5.5 6 6.5

× 10−14

Hor

izon

tal p

ositi

on (

m)

Real part of Eφ (V/m)


Solenoids, real parta) b)

c) d)

−2 −1 0 1 2 3

× 10−13

Hor

izon

tal p

ositi

on (

m)

Imaginary part of Eφ (V/m)


Solenoids, imaginary part

−1 0 1 2 3 4 5

× 10−7

Hor

izon

tal p

ositi

on (

m)

Real part of Eaxial

(V/m)


−15 −10 −5 0 5× 10

−9

Hor

izon

tal p

ositi

on (

m)


(V/m)


Electrodes, real part Electrodes, imaginary part

−8

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

−8

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Figure A-1. Open-hole well. Sensitivity of different logging instruments to the hydro-fracture length; (a and b) sand- and ferrite-based proppant; (c and d) coke breeze-basedproppant with ferrite. Frequency of operation: 100 Hz. Background (shale) resistivity:3 Ω · m.


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depth of investigation of the measurements. Therefore, hydrofrac-ture length can only be estimated up to at most 10 m when using thistype of proppant.In the case of a more conductive coke breeze-based proppant,

the use of ferrite to modify the proppant’s magnetic permeabilityhas no significant effect on the simulated measurements (compareFigure A-1d [with ferrite] to Figure 3d [no ferrite]).Even for the case of a proppant with relative magnetic permeability

as large as 50, further results show that their sensitivity is insufficientto estimate the length of long hydrofractures. Those results have beenomitted here for the sake of brevity and because such a high relativemagnetic permeability is unavailable in commercial proppants.The above results indicate that the use of magnetic proppant does

not substantially increase the sensitivity of borehole resistivitymeasurements to estimate hydrofracture length.This behavior arises because the contrast of magnetic permeabil-

ity between proppant and background formation is much lower thanthat attained with electrical conductivity.

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sensitivity analysis for the appraisal of hydrofractures...

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