A harmonic pulse testing method for leakage detection in ... · RESEARCH ARTICLE 10.1002/2014WR016567 A harmonic pulse testing method for leakage detection in deep subsurface storage

RESEARCH ARTICLE10.1002/2014WR016567

A harmonic pulse testing method for leakage detection in deepsubsurface storage formationsAlexander Y. Sun1, Jiemin Lu1, and Susan Hovorka1

1Bureau of Economic Geology, Jackson School of Geosciences, University of Texas at Austin, Austin, Texas, USA

Abstract Detection of leakage in deep geologic storage formations (e.g., carbon sequestration sites) is achallenging problem. This study investigates an easy-to-implement frequency domain leakage detectiontechnology based on harmonic pulse testing (HPT). Unlike conventional constant-rate pressure interferencetests, HPT stimulates a reservoir using periodic injection rates. The fundamental principle underlying HPT-based leakage detection is that leakage modifies a storage system’s frequency response function, thus pro-viding clues of system malfunction. During operations, routine HPTs can be conducted at multiple pulsingfrequencies to obtain experimental frequency response functions, using which the possible time-lapsechanges are examined. In this work, a set of analytical frequency response solutions is derived for predictingsystem responses with and without leaks for single-phase flow systems. Sensitivity studies show that HPTcan effectively reveal the presence of leaks. A search procedure is then prescribed for locating the actualleaks using amplitude and phase information obtained from HPT, and the resulting optimization problem issolved using the genetic algorithm. For multiphase flows, the applicability of HPT-based leakage detectionprocedure is exemplified numerically using a carbon sequestration problem. Results show that the detec-tion procedure is applicable if the average reservoir conditions in the testing zone stay relatively constantduring the tests, which is a working assumption under many other interpretation methods for pressureinterference tests. HPT is a cost-effective tool that only requires periodic modification of the nominal injec-tion rate. Thus it can be incorporated into existing monitoring plans with little additional investment.

1. Introduction

Injection of waste fluids into deep saline aquifers has long been used by the mining and energy industriesas an option for disposing of process byproducts. In recent years, the storage of captured carbon dioxide insaline aquifers is also being actively pursued as a geoengineering measure for significantly reducing globalgreenhouse gas emissions. A primary environmental concern of these practices is the potential migration ofinjected fluids through connected pathways (e.g., abandoned wells or geologic faults) into undergroundsources of drinking water. A number of previous works have focused on forward and inverse modeling ofleakage from deep subsurface storage formations [e.g., Avci, 1994; Cihan et al., 2011; Javandel et al., 1988;Nordbotten et al., 2004; Sun and Nicot, 2012]. Most of these analyses assume passive monitoring, in whichpressure gauges are simply used to ‘‘listen’’ for possible leak signals. In this work, we propose an active mon-itoring method that involves an injector and one or more observation wells. The method is based on har-monic pulse testing (HPT), in which a sinusoidal injection rate is superposed on the nominal injection rateto probe potential leaks in the reservoir volume surrounding the injector (in this paper, we use ‘‘reservoir’’ torefer to storage formations and ‘‘aquifer’’ to nonstorage formations). During monitoring, HPT is repeatedroutinely; after each test, pressure responses obtained from observations well(s) are analyzed in the fre-quency domain for deviations from a previous test that is conducted at the same frequency. By simply vary-ing the pulsing frequency, the HPT-based leakage detection method can significantly enhance the signal-to-noise ratio and generate more useful information than does the passive monitoring method. In the fol-lowing, the existing literature pertaining to pressure-based leakage detection and HPT is briefly reviewed.

1.1. Pressure-Based Leakage DetectionLeakage detection is a special type of inverse problems and has long been studied for pipeline leakage andaquifer contaminant source identification [Mahar and Datta, 2001; Mandal et al., 2012; Michalak and Kitani-dis, 2004; Puust et al., 2010; Sun et al., 2006a, 2006b]. Existing leakage detection methods based on pressuresignals can be classified broadly into time domain and frequency domain diagnoses.

Key Points:� Harmonic pulsing testing (HPT) is

proposed for leakage detection� Analytical and numerical models are

developed to prove the concept� HPT is a cost-effective tool for

monitoring deep subsurfacerepositories

Supporting Information:� Supporting Information S1

Correspondence to:A. Y. Sun,[email protected]

Citation:Sun, A. Y., J. Lu, and S. Hovorka (2015),A harmonic pulse testing method forleakage detection in deep subsurfacestorage formations, Water Resour. Res.,51, 4263–4281, doi:10.1002/2014WR016567.

Received 17 OCT 2014

Accepted 28 MAY 2015

Accepted article online 30 MAY 2015

Published online 16 JUN 2015

VC 2015. The Authors.

This is an open access article under the

terms of the Creative Commons

Attribution-NonCommercial-NoDerivs

License, which permits use and

distribution in any medium, provided

the original work is properly cited, the

use is non-commercial and no

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Time domain diagnosis inspects time series of observed pressures for potential anomalies. Methods that fallinto this category can be further divided into model driven and data driven. In model-driven methods, thesource release history is first parameterized, and the unknown parameters are sought by minimizing the dif-ference between model predictions and pressure observations. Sun and Nicot [2012] demonstrated the per-formance of several inverse algorithms for identifying leakage in geologic carbon storage formations. Theirapproach for identifying leakage rates is based on impulse-response analysis, through which observed pres-sure data are deconvoluted to infer the leakage history of sources; the complicated problem of joint inver-sion of source locations and leakage rates is solved iteratively by embedding the deconvolution algorithmin a global optimization solver. Jung et al. [2013] solved an optimization problem directly to identify bothuncertain permeability values and leak locations. Potential caveats of the model-driven methods are that (a)they generally require a high-fidelity forward model that is suitable for leakage inversion—in other words,the model must be able to accurately simulate pressure responses caused by any hypothetical leak loca-tions during the solution process, and (b) the resulting inverse problem is ill posed and its solution can beeasily interfered by both model uncertainties and measurement noise.

In data-driven approaches, a black-box model (e.g., artificial neural networks) is first trained offline using his-torical observations and then deployed online for anomaly detection. The black-box model typically con-sists of a forecast module that predicts the nominal system behaviors at the operational conditions, and afault detection module that can recognize system anomalies in real time and trigger warnings. The efficacyof such data-driven methods depends critically on past records of leakage. Because of data incompletenessand information vagueness, most methods adopt either a probabilistic or fuzzy-theoretic approach. Mandalet al. [2012] used the support vector machine in conjunction with an artificial bee colony algorithm to pre-dict leaks in oil pipelines. Arsene et al. [2012] developed a decision support system for detecting leaks inwater distribution systems on the basis of a three-layer fuzzy min-max neural network and graph theory.Zhang et al. [2010] applied fuzzy-logic theory to estimate the probability of leakage from faults, in whichthe leakage probability is related to domain size, CO2 plume size, and parameters describing conduit lengthdistribution and uncertainty. For deep subsurface disposal projects, however, data-driven models can be dif-ficult to train because of the sparsity of known leakage incidents. A possibility is to use a risk-based frame-work to combine model-driven and data-driven methods such that the former is used to simulate syntheticleakage events for the latter model to ‘‘learn’’ [Yang et al., 2011].

Like their time domain counterparts, frequency domain diagnosis methods may also be classified as model anddata driven, only that the pressure response data are transformed and analyzed in frequency domain. For lineartime-invariant systems, the frequency response function, defined as the ratio of the Fourier spectra of outputand input pressure signals, characterizes a physical system in terms of its amplitude attenuation and phase shift(see section 2.2). The amplitude attenuation measures how much signal strength is reduced when the sourcedsignal reaches an observation point, whereas phase shift measures how much the signal has been delayed aftertraveling to the observation point. Frequency domain diagnosis has been used to detect water pipeline leaks byanalyzing pressure responses to steady-oscillatory flows, during which process oscillatory flows are induced intoa problematic pipeline section by opening/closing an upstream valve periodically. The power spectra of suchfluid transients are affected by pipeline anomalies (e.g., leaks and blockages), thus giving clues that can be usedfor locating such anomalies [Ferrante and Brunone, 2003; Lee et al., 2005; Mpesha et al., 2001]. Unlike the timedomain diagnosis, frequency domain diagnosis is less prone to interference by noise and has long been used asan alternative analysis tool in reservoir characterization (see section 1.2). To the best of our knowledge, fre-quency domain diagnosis has not been applied to leakage identification in geologic repositories; thus, its effec-tiveness requires a quantitative assessment, which is the main focus of this study.

1.2. Harmonic Pulse Testing (HPT)Pulse testing presents an alternative to conventional constant-rate well testing and has been used for reser-voir characterization since the 1960s [Johnson et al., 1966]. Pulse testing commonly imposes either squareor rectangular duty cycles (i.e., step-like injection/shut-in periods) at the pulser well, whereas HPT appliessinusoidal injection rates until a pseudosteady regime is reached. Thus, HPT can be considered a generaliza-tion of the conventional pulse testing. In fact, any square wave can be decomposed as a sum of harmonicsof different frequencies using Fourier expansion. The main advantages of HPT are (a) the harmonic rate issuperimposed on the nominal rate, thus creating little interruption to regular production activities; (b) theinterpretation is less affected by noise and wellbore effects; and (c) HPT causes less plume disturbance than

Water Resources Research 10.1002/2014WR016567


does a constant-rate pumping test when used for characterization of contaminated aquifers [Ahn andHorne, 2010; Cardiff et al., 2013; Fokker and Verga, 2011; Hollaender et al., 2002]. The main potential draw-back of HPT is that a test can take longer than the time required by conventional well testing [Gringarten,2008]. Nevertheless, this drawback is partly offset by HPT’s zero net production/extraction volume [Fokkerand Verga, 2011] and by the advent of automated well controls.

As in conventional well testing, significant interest exists in developing analytical or semianalytical solutions forinterpreting HPT data. Black and Kipp [1981] developed an analytical, steady periodic solution for an infiniteaquifer subject to point-source or line-source oscillatory stimulation. Rasmussen et al. [2003] derived semianalyticsolutions for the transient response associated with oscillatory pumping for both confined and phreatic aquifers.Ahn and Horne [2010] considered using HPT to characterize multicomposite reservoirs and derived semianalyti-cal, steady periodic solutions. Cardiff et al. [2013] validated the Black and Kipp [1981] solutions numerically andthen provided an inversion algorithm for estimating hydraulic conductivity in heterogeneous aquifers using HPTdata, in which parameter sensitivities are obtained by solving an adjoint-state equation. Fokker and Verga [2011]identified dynamic effective reservoir properties in an oil reservoir that is subjected to water flooding. Recently,Cardiff and Barrash [2015] provided design guidelines for conducting HPT in hydrogeological applications.

This work investigates the feasibility of using HPT as a leakage detection tool. Although the main interest hereis the time-lapse frequency response of a geologic storage reservoir, HPT can be used both for formationparameter estimation during site characterization and for leakage detection later during operations. The paperis organized as follows. Section 2 introduces the general methodology underlying the HPT-based leakagedetection. For single-phase flows, a set of analytical frequency response functions is derived for predicting fre-quency responses of a multilayer system, whereas for multiphase flows a fully numerical approach is taken.Section 3 demonstrates the results numerically, and, finally, section 4 summarizes the findings.

2. Methodology

2.1. Frequency Domain Leak Detection Using HPTLet us assume that an HPT has been conducted at an injector (Figure 1). Let ~PinjðxÞ and ~PobsðxÞ be the Fou-rier transform of pressure perturbations obtained at the injector and an arbitrary observation well, respec-tively, where x52p=Tp is the fundamental frequency corresponding to the pulsing period Tp of the HPT.The frequency response function,HðxÞ, is defined as the ratio of ~PobsðxÞ to ~PinjðxÞ

HðxÞ5~PobsðxÞ~PinjðxÞ

: (1)

HðxÞ provides a characterization of reservoir properties that, in turn, dictate how source strength attenu-ates with distance in the system. Specifically, the signal strength attenuation can be quantified via theamplitude and phase ofHðxÞ

K5jHðxÞj; and U5/HðxÞ: (2)

Here K and U actually measure the amplitude attenuation and phase shift relative to the source. By defini-tion, K � 1. To normalize U, we take its modulo with respect to 2p and then divide the remainder by 2p

[Ahn and Horne, 2010]. An advantageof equation (1) is that it allows the sys-tem responses to transient stimulationsto be evaluated efficiently in terms ofthe periodic steady state responses tosinusoidal stimulations. Equation (2) isgeneral and can be applied to any pairof observation well and injector.

The HPT-based, frequency domainleakage detection is built on the pre-mise that if one or more leaks occurwithin the detectable range of an HPTsource, the associated HðxÞ will show

Figure 1. Illustration of a well pair used in HPT, where x52p=Tp is fundamentalpulsing frequency and Tp is pulsing period.



a deviation from the baseline. Thus, the mon-itoring task becomes characterizing HðxÞroutinely during the performance period of astorage reservoir for anomalies. Figure 2 pro-vides a general workflow diagram for usingHPT as such a leakage diagnosis tool.

The workflow consists of a three-step processthat can be repeatedly applied during moni-toring. In the first step, multiple HPTs areconducted at different pulsing periods, theresulting pressure response data (after trendremoval) are transformed into frequencydomain responses by using fast Fourier trans-form (FFT), and an experimental HðxÞ is con-structed using values at the fundamentalfrequencies of the tests. As we will show insection 3.1, both K and U vary smoothly and,for most practical pulsing frequency ranges,also monotonically as a function of x; thus,only a small number of tests are needed. Inaddition, we remark that multiple testsmainly serve for the purpose of cross valida-tion of test results. It is possible to optimizethe HPT design such that only a single test isneeded for leakage detection.

In the second step, the newly obtained experi-mentalHðxÞ is compared with an existing onefor possible deviations. Initially, the firstHðxÞ isused as the baseline (Method 1). Or, if a forwardmodel is available, it can be used to generate atheoretical baseline HðxÞ (Method 2). Themain advantage of Method 1 is that it does notrequire knowledge of reservoir parameters, nor

a model. However, it assumes that the reservoir volume surrounding a test area is free of leaks before injectionstarts; otherwise, only new leaks can be detected. Method 2 can be considered a model-driven approach, usingwhich the theoreticalHðxÞ curves corresponding to no-leak and leak cases can be constructed a priori withoutconducting actual HPT experiments; however, Method 2 is limited by model and parameter uncertainties. Inpractice, the two methods mentioned herein are complementary to each other: during a scoping analysis,Method 2 is used to give estimates of expected deviations and to help design the HPT parameters; during opera-tions, Method 1 is routinely applied to detect deviations in a time-lapse sense. Additional information gained dur-ing operations may also be used to reduce uncertainty in Method 2 so that it can be used for predictive analyses.

In the third step, if a leak is confirmed, the frequency domain data can be used to search for leak locations by usingan optimization algorithm. A working procedure involving the genetic algorithm (GA) is proposed for this task(see section 2.3). In the next subsection, we derive a set of analytical expressionsHðxÞ to demonstrate the feasibil-ity of HPT-based leakage diagnosis for single-phase flows. The analytical nature of solutions makes it possible toexamine the entire frequency range. For more complex problem settings and multiphase flows, however, numeri-cal models are necessary. We emphasize that constructing an experimentalHðxÞ does not require models per se.They are only needed when either a theoretical baselineHðxÞ or location search is necessary.

2.2. Analytical Solution for Single-Phase Flow in a Multilayer SystemWe consider a three-layer system consisting of a storage reservoir, an aquitard (confining layer), and anoverlying aquifer (Figure 3). The properties of each layer are assumed to be homogeneous. An injector islocated in the center of the domain, and an observation well is located at a distance rO away. For the leak

Figure 2. Workflow of the HPT-based leakage diagnosis process.



scenario, a leaky well is presentat a distance rI from the injec-tor, and the distance betweenthe leaky well and observationwell is denoted by rL . Such aproblem setting has beenwidely used in analytical andnumerical studies for leakagemodeling [e.g., Avci, 1994; Bir-kholzer et al., 2011; Cihan et al.,2011; Nordbotten et al., 2004;Sun and Nicot, 2012; Sun et al.,2013b; Zeidouni, 2014]. Most ofthe previous analytical studiesassume constant injectionrates. Thus, the first task in ourderivation is to expand the

solution for use with harmonic injection patterns. In the following derivations, subscript 1 denotes reservoirand subscript 2 denotes the upper aquifer.

In the absence of leaks, the governing equation for pressure response in the storage reservoir correspond-ing to a pulser well located at the origin (fully perforated in reservoir) is

@2p@r2

11r@p@r

5/lct;1

k1

@p@t; (3)

subject to initial and boundary conditions

pðr; t 5 0Þ5pinit

limr!1

pðr; tÞ5pinit

2pk1b1

lr@p@r

��r5rw

52QðtÞ

where p5pðr; tÞ is pressure; pinit is the initial pressure; the injection rate QðtÞ5QbðtÞ1Qoejxt is the sum of anonsinusoidal nominal injection term, Qb, and a complex HPT source term, Qoejxt , with Qo denoting themagnitude, j the imaginary unit, and x the fundamental frequency; rw is the radius of the injector; and /, l,k, b, and ct denote porosity, fluid viscosity, permeability, layer thickness, and total compressibility, respec-tively. The pressure response caused by Qb is a trend term and needs to be filtered out from the time seriesbefore frequency domain analysis. In the following, we shall disregard Qb unless otherwise noted and sim-ply focus on pressure responses resulting from the sinusoidal source term Qoejxt . The Laplace domain solu-tion to the diffusivity equation (3) is (see Appendix A)

P̂ðrÞ5 Qol

2pk1b1rwðs 2 jxÞffiffiffiffiffiffiffiffiffis=g1

p K0ðrffiffiffiffiffiffiffiffiffis=g1

pÞ

K1ðrw

ffiffiffiffiffiffiffiffiffis=g1

pÞ; (4)

in which r is the distance between injector and an arbitrary observation location; s is the Laplace transformvariable; K0 and K1 are the zero and first-order modified Bessel function of the second kind, respectively; g5

k=/lct is diffusivity; and P̂ is the Laplace transform of the pressure change, p2pinit .

When a leak is present, the Laplace-transformed pressure change in the reservoir is

P̂1;lðrlÞ5q̂l

2pk1b1ra


p K0 rl


p� �K1 ra


p� � ; (5)

in which rl is the distance between the leak well and the observation location, ra is the radius of the leakwell, and q̂ is the Laplace transform of the leakage rate. Thus, the Laplace transform of total pressure

Figure 3. Problem setting for analytical solution derivation, where a three-layer system con-sisting of a storage reservoir, an overlying aquifer, and an aquitard is involved.



change in the reservoir, pT1 , is obtained by summing equations(4) and (5)

P̂T15P̂1P̂1;l: (6)

Similarly, the Laplace-transformed total pressure change in the upper aquifer caused by the leak is

P̂T2ðrlÞ52

q̂l

2pk2b2ra


p K0 rl


p� �K1 ra


p� � : (7)

We can then express the Laplace transform of the unknown leakage rate using Darcy’s law and P̂T1 and P̂

T2 ,

q̂52kapr2

a

lP̂

T22P̂

T1

Dl

!; (8)

where Dl is the vertical length of the leakage pathway (approximated as a permeable feature) from the res-ervoir to the upper aquifer, ka is the permeability of the pathway, leakage flux is positive outward the reser-voir, and the system is assumed to be at hydrostatic equilibrium initially. Solving for q̂ from equations (4)–(8) gives

q̂5

2Qo K0ðrI

ffiffiffiffiffiffiffis=g1

pÞ

2pk1b1rw ðs2jxÞffiffiffiffiffiffiffis=g1

pK1ðrw


pÞ

X1K0ðra


pÞ

2pk1 b1ra


pK1ðra


pÞ1

K0ðra


pÞ

2pk2b2ra


pK1ðra


pÞ

; (9)

where both P̂T1 and P̂

T2 are evaluated at the boundary of the leaky well (i.e., rl5ra), and the flow resistance

term is X5Dl=ðpr2a kaÞ.

From equation (9), it is straightforward to obtain the Laplace-transformed total pressure change in the reser-voir by replacing r with rO, and rl with rL

P̂T15

q̂l

2pk1b1ra


p K0ðrL


pÞ

K1ðra


pÞ1

Qol

2pk1b1rwðs2jxÞffiffiffiffiffiffiffiffiffis=g1

p K0ðrO


pÞ

K1ðrw


pÞ: (10)

The Laplace-transformed total pressure change in the upper aquifer can be obtained similarly.

Let us introduce the following dimensionless groups,

sD5sr2

I

g1; qD5q=Qo; pD;i5

2pk1b1pTi

Qol; ði51; 2Þ;

XD52pk1b1X; rD;x5rx=rI ðx5a;w; L;OÞ; tD5g1

r2I

t; xD5xr2

I

g1:

Then q̂, P̂T1 , and P̂

T2 become, respectively

q̂D5

2K0ðffiffiffiffisDp Þ

ðsD2jxDÞrD;wffiffiffiffisDp

K1ðrD;wffiffiffiffisDp Þ

XD1K0ðrD;a

ffiffiffiffisDp Þ

rD;affiffiffiffisDp

K1ðrD;affiffiffiffisDp Þ1

K0ðrD;affiffiffiffiffiasDp Þ

brD;affiffiffiffiffiasDp


; (11)

P̂TD;15

q̂D

rD;affiffiffiffiffisDp K0ðrD;L

ffiffiffiffiffisDp Þ

K1ðrD;affiffiffiffiffisDp Þ1

1rD;wðsD2jxDÞ

ffiffiffiffiffisDp K0ðrD;O

ffiffiffiffiffisDp Þ

K1ðrD;wffiffiffiffiffisDp Þ ; (12)

P̂TD;252

q̂D

brD;affiffiffiffiffiffiffiasDp

K0 rD;LffiffiffiffiffiffiffiasDp� �

K1 rD;affiffiffiffiffiffiffiasDp� � ; (13)

where a5g1=g2 is the diffusivity ratio, and b5k2b2=ðk1b1Þ is the transmissivity ratio between the upperaquifer and reservoir. A solution for leak rate that includes the effect of Qb is



q̂D5

1ðsD2jxDÞ1

Qb=Qo

sD

� �K0ðffiffiffiffisDp Þ

rD;wffiffiffiffisDp

K1ðrD;wffiffiffiffisDp Þ

XD1K0ðrD;a

ffiffiffiffisDp Þ

rD;affiffiffiffisDp

K1ðrD;affiffiffiffisDp Þ1


brD;affiffiffiffiffiasDp


: (14)

Note that equation (14) is only used during the solution validation process (section 3.1.1).

In general, time domain solutions corresponding to equations (11)–(13) need to be obtained by performingthe inverse Laplace transform numerically. For the current case, however, we can make use of the close con-nection between the Laplace transform and Fourier transform to obtain the steady periodic solutionsdirectly from the Laplace solutions by replacing s with jxD and making necessary adjustments [Can and€Unal, 1988]. The results are

~qD5

2K0ðffiffiffiffiffiffijxD

pÞ

rD;w

ffiffiffiffiffiffijxD

pK1ðrD;w


pÞ

XD1K0ðrD;a


pÞ

rD;a


pK1ðrD;a


pÞ1

K0ðrD;a

ffiffiffiffiffiffiffiajxD

pÞ

brD;a


pK1ðrD;a


pÞ

; (15)

~PTD;15

qD

rD;affiffiffiffiffiffiffiffijxDp K0ðrD;L

ffiffiffiffiffiffiffiffijxDp

ÞK1ðrD;a


Þ11

rD;wffiffiffiffiffiffiffiffijxDp K0ðrD;O


ÞK1ðrD;w


Þ ; (16)

~PTD;252

qD

brD;affiffiffiffiffiffiffiffiffiffiajxDp K0 rD;L

ffiffiffiffiffiffiffiffiffiffiajxDp� �

K1 rD;affiffiffiffiffiffiffiffiffiffiajxDp� � : (17)

Using equations (15)–(17) and the definition in equation (1), the frequency response function for the leakcase, HleakðxDÞ, can be calculated directly for both the reservoir and upper aquifer. Note that the frequencyresponse at the injector, which is needed for calculating the denominator of HleakðxDÞ, is obtained byreplacing rD;O with the well radius rD;w in equation (16).

In the absence of leaks, the steady periodic solution in the reservoir is given by

~PD;15K0ð

ffiffiffiffiffiffiffiffijx0Dp

Þr0D;w


K1ðr0D;wffiffiffiffiffiffiffiffijx0Dp

Þ ; (18)

where slightly different dimensionless quantities are used, x0D5xr2O=g1 and r0D;w5rw=rO. Further, when

rw ! 0, equation (18) simplifies to

~PD;15K0ðffiffiffiffiffiffiffiffijx0D

pÞ; (19)

which is equivalent to the solution shown in Rasmussen et al. [2003, equation (6)] that assumes a pointsource. The reservoir layer frequency response function for the no-leak case, Hnoleakðx0DÞ, can be readily cal-culated in terms of the solution given in equation (18)

Hnoleakðx0DÞ5K0ð


ÞK0ðr0D;w


Þ : (20)

All analytical frequency response functions are validated numerically and the results are given in section 3.

2.3. The Location-Search ProblemAs discussed in section 2.1, the time-lapse leakage detection can indicate the presence of leakage, but it doesnot tell where the actual leakage is. To address the latter more challenging question, a forward model is nec-essary so that frequency responses to different trial leak locations can be predicted. In this case, the experi-mental HðxÞdata becomes observation information. We define the following objective function to minimizethe differences between observed and simulated amplitude attenuation ðKÞ and phase shift ðUÞ of

minh

OðhÞ5 minh

EKðhÞ1kEUðhÞf g; (21)

where the terms Ej (j5K, U) are given as a distance measure (e.g., root-mean-square error) between meas-ured and observed quantities, h is the unknown vector, and k is a weighting parameter used to scale thetwo terms because of the magnitude differences. In polar coordinates, the unknowns h may include thelength and angle of the vector pointing from the injector to the leak well location. We assume thatthe formation properties are available. If formation properties are unknown, they may be included in h. In



essence, an inverse problem is solved here usingamplitude and phase data in frequency domain as‘‘observations.’’ Like any other inverse problems, con-ceptual uncertainty and numerical errors in the forwardmodel can affect the inversion. For each test, only asubdomain of the entire reservoir needs to be dealtwith, thus the number of degrees of freedom is consid-erably smaller. Nevertheless, to solve equation (21),global optimization solvers are preferred to avoid trap-ping at local minima during the search of leak locations[Sun and Nicot, 2012]. This work uses GA, which is awidely used global optimization algorithm. A detaileddescription of GA can be found in Goldberg [1989].

3. Numerical Demonstration and Discussion

In this section, the HPT-based leakage diagnosis procedure is demonstrated numerically, using both single-phase analytical solutions and a multiphase numerical model that simulates injection of CO2 into a salineaquifer.

3.1. Single-Phase Flows3.1.1. A Benchmark ProblemWe first validate the analytical frequency responses derived in section 2.2 in three different ways. The firstmethod uses a published solution in time domain [Avci, 1994], the second method compares the analyticalsolutions to discrete HðxÞ values obtained using inverse Laplace transform and FFT, and the third methoduses a finite element model. Parameters used for this benchmark problem are listed in Table 1.

Figure 4a compares the dimensionless reservoir pressure buildup obtained under constant injection(Qb51000 m3/d) with that obtained under both Qb and HPT (Q05100 m3/d). The dimensionless pulsingperiod of sinusoidal injection is Tp;D566. The pressure response to Qb (solid line in Figure 4a) matches Avci’ssolution for a three-layer system with (a) a single leaky well, (b) zero flow resistance along the leaky well(i.e., XD50 or very large ka), (c) identical hydraulic conductivity, thickness, and specific storage values inlayers 1 and 2 (i.e., a51, b51), and (d) negligible injector radius [see Avci, 1994, Figure 5]. The HPT simply

Table 1. Benchmark Problem Parameters

Parameter (Unit) Value

Formation permeability (k1, k2) (md) (100, 100)Total compressibility (ct;1) (Pa21) 5.1 3 10211

Porosity 0.15Thickness (b1, b2, ba) (m) (10, 10, 5)Injection rate (Qo , Qb) (m3/d) (100, 1000)Period (Tp) (s) 600Viscosity (l) (Pa�s) 8.9 3 1024

Radius (rw , ra)a (m) (0.001, 0.01)Dist. between leak and injector (rI ) (m) 10Dist. between leak and observer (rL) (m) 10Dist. between injector and observer (rO) (m) 10

aUnless otherwise noted, rw and ra are set to 0.1 m forall examples after section 3.1.1.

Figure 4. Comparison of dimensionless (a) pressure buildup and (b) leakage rate for constant injection only (solid line) and for an HPT superposed on constant injection (dotted line).



introduces periodic oscillations around the pressure response to Qb. Figure 4b compares the correspondingnormalized leakage rate under constant injection (solid line) and under HPT, respectively. Again, the solidline matches Avci’s solution well [Avci, 1994, Figure 3]. All time domain solutions are obtained by perform-ing the inverse Laplace transform numerically using the Laplace solutions given in section 2.2.

Using the benchmark problem parameters and time domain solutions, we repeat HPT for six different puls-ing periods, ranging from Tp;D519:8 to 119, with a uniform increment of 19.8. The pressure responses are

Figure 5. (a) Amplitude attenuation and (b) phase shift of frequency response functions corresponding to leak and no-leak scenarios in the benchmark problem. Dimensionless well radiiare set to rD;w 51024 and rD;a51023.

Figure 6. Numerical validation of analytical frequency response functions: (a) amplitude attenuation and (b) phase shift.



then transformed to frequency domain using FFT. The results give six values (crosses and open circles) ofthe experimental frequency response function, which are plotted against the analytical functions (solidlines) in Figures 5a and 5b. A good match between the two sets of results is found. Also, Figure 5 suggeststhat the presence of a leak clearly causes both K and U to deviate from their no-leak baselines.

The previous two tests all utilize the Laplace solutions obtained in section 2.2. In the last validation test, wetake a full numerical approach by developing a three-dimensional model using the finite element software

Figure 7. Amplitude attenuation as a function of transmissivity ratio (b) and dimensionless frequency for flow resistance of (a) XD50 and (b) XD510. The ratio b5k2b2=k1b1 is labeledon the plots.

Figure 8. Amplitude attenuation in upper aquifer as a function of frequency for flow resistance of (a) XD50 and (b) XD510 and for b51 and 10.



COMSOLVR . The reservoir and aquifer layers are modeled using cylinders of 500 m radii, which are separatedby an aquitard (see SI 1 for screenshots). The lateral boundaries are set to constant head and are monitoredduring simulation to ensure they do not cause boundary effect. The injector and leak well are modeled aspermeable thin vertical cylinders, and their radii are set to 0.2 m in both analytical and numerical models.All other parameters are the same as those listed in Table 1. HPT simulations are conducted at three differ-ent pulse periods. Figures 6a and 6b show a good match between the analytical frequency response func-tions (solids lines) and numerical solutions (open circles). Thus, all three validation tests give satisfactoryresults.

3.1.2. Sensitivity StudiesThe purpose of this subsection is to investigate the sensitivity of K and U to different system parameters.First, we fix the distances between wells (as specified in Table 1) and study the sensitivity to pulsing fre-quencies, layer permeability contrasts, and vertical flow resistance. Figure 7a shows that K decreasesmonotonically with the fundamental pulsing frequency. Shorter pulsing periods (i.e., higher frequencies)

Figure 9. Deviation of (a and c) amplitude attenuation and (b and d) phase shift from the no-leak case for different pulsing periods: (a and b) rO510 m and (c and d) rO550 m. Pulsingperiods are in minute.



tend to attenuate the source energy much faster than the longer pulsing periods do. At the high-frequencyend, it is no longer possible to distinguish the K of leak cases from that of the no-leak case, all approachingzero. On the other hand, at the low-frequency end the deviations between the no-leak case and leak casesreach the maximum, and the shape of K becomes essentially flat. In practice, the intermediate frequencyranges are likely to be more relevant for finishing each HPT within a reasonable amount of testing time. Fig-ure 7a also shows that the amount of deviation (i.e., the difference between no-leak and leak cases) for afixed pulsing frequency is a function of b, which is equivalent to the transmissivity ratio between the upperaquifer and reservoir. Where all other parameters are fixed, the higher the value of b is, the greater the leak-age flux into the upper aquifer and, thus, the larger the deviation from the no-leak case. Figures 7a and 7bcontrast the effect of leakage resistance XD, which is related to the aquitard thickness and leaky well perme-ability. The value XD510 is equivalent to ka53310210 m2 (300 darcies). Greater flow resistance tends todampen the leakage flux and, thus, weaken the observable deviation from that of the no-leak case. As aresult, the amplitude deviation is reduced for the same set of b values used in Figure 7a. In this base case,the phase shifts (not shown) are relatively small because of the close distance between the leak andinjector.

To assess the detectability of a leak in the upper aquifer (also referred to as the above-zone monitoringinterval or AZMI in the literature), we use the ratio between the spectra of the upper aquifer and the injectorpressure responses instead. In this case, the purpose is to examine under what conditions upper-aquifermonitoring is favorable. Figures 8a and 8b show the amplitude attenuation for flow resistance XD50 andXD510, respectively, and for different upper-layer observation distances from the leak location (normalizedby rI). When b51, clear signals can be seen in both subplots. However, the signal is significantly weakenedwhen b increases to 10. Thus, if permeability values of the upper aquifer and the reservoir are about thesame, it is better to monitor the reservoir when the upper aquifer is thicker, and, conversely, it is better tomonitor the upper aquifer when it is thinner than the reservoir.

One of the main concerns during detectability assessment is how large the area of coverage an HPT canprovide. Figure 9 shows the sensitivity to leak distances for two different observation distances, rO510 and50 m, respectively. The angle between leak vector and observation vector, both originates from the injector,is fixed at 608. For each case, the deviation from the no-leak case (i.e., Kno leak2K and Uno leak2U) is plotted

Figure 10. Effect of white and colored noise on (a) K and (b) U.



as a function of leak well distance rI

and for different pulsing periods (inmin). An immediate observation is thatthe deviations corresponding to differ-ent rO (e.g., Figure 9a versus 9c) havedifferent decline rates with rI . The puls-ing periods used in Figure 9 rangefrom 10 min to 2 days. The longer thepulsing periods are, the farther HPT sig-nals penetrate in the lateral directions.For instance, a leak located more than150 m away (i.e., rI > 150 m) can onlybe detected by using pulsing periodslonger than or equal to 1 day amongall the pulsing periods considered.Unlike the monotonic decreasing pat-tern shown by K, the phase shift devia-tions first reach a peak beforedecreasing with rI (Figures 9b and 9d).The location of this maximum phaseshift deviation depends on observationlocations. The farther away an observa-tion point is, the greater the phase shiftis. This sensitivity of K and U on obser-vation locations is encouraging, sug-gesting that we can use a searchprocedure like the one described insection 2.3 to locate leak locations.

In the final sensitivity study, we dem-onstrate that the HPT time-lapse analy-sis is robust to measurement noise.

First, HPTs are conducted at pulsing periods 3, 6, 12, 15, and 20 min. The pressure responses are then ‘‘cor-rupted’’ by adding multiplicative random measurement noise of 5% and 10%, respectively. Figure 10 showsthat the impact of random noise is relatively insignificant on frequency responses. We also simulate ‘‘col-ored noise’’ by generating a sinusoidal wave with a 0.25 day period and 10% of the maximal magnitude ofthe HPT pressure signals and then adding it to the pressure observations. Figure 10 shows that the presenceof ‘‘colored noise’’ only causes the leak-case curves to shift slightly from the theoretical curves because, inthis case, all pulsing periods are significantly different from the 0.25 day period of the colored noise.

3.1.3. Leak Location IdentificationWe now demonstrate the leak location identification problem using amplitude and phase data. Figure 11shows the plan view of problem setup, where a leak well is located 25.5 m from the injector, and its angle is158 from the horizontal axis. Two observation wells are installed at (10 m, 608) and (40 m, 2308), respec-tively. All other parameters are given in Table 1. In this example, the analytical frequency response functionis used as a forward model to simulate the amplitude and phase for each hypothetical leak location. Theoptimization problem (21) is solved using the GA function from Matlab’s global optimization toolbox by set-ting the population size and the number of generations to 50 and 40, respectively. A large enough popula-tion size enables the trial solutions to cover the parameter space sufficiently, whereas a suitable generationsize ensures that the population converges to a near-optimal global solution. The crossover fractionbetween generations is set to 0.8 and the mutation rate is set to 1%. The lower and upper bounds of leakvector magnitude and orientation are [10, 60 m] and [58, 808], respectively (i.e., shaded area in Figure 11),and the weighting factor k is set to 100.0. Figure 12 shows the GA convergence history (best objective func-tion value in each generation) for three different sets of HPT experiments, with each consisting of only twopulsing periods (shown in the legend). In all three experiments, the actual leak location is successfully

Figure 11. Problem settings for GA-based leak location search, where the shadedarea shows the parameter space for searching the unknown distance and orienta-tion of the leak (measured from the origin). Locations of the two observationwells are fixed.



located. Figure 12 suggests thatlonger pulsing periods tend toenhance deviation signals andhelp find the final solutionsfaster. Although only a simpleexample is presented here, thefrequency domain optimizationproblem bears many similaritiesto the time domain contaminantsource identification and moni-toring network design problems[Sun and Nicot, 2012; Sun et al.,2013a], except that it is lessprone to noise interference.Therefore, optimization meth-ods that are developed for thetime domain can be used tohandle additional complexitiesin the frequency domain, suchas observation network design,spatial heterogeneity, andparameter uncertainty.

3.2. Leak Detection in Carbon Sequestration ReservoirsSo far we have focused on single-phase flow problems. When more than one phase is present, the problemat hand becomes more complicated. The governing flow equation generally becomes nonlinear and thesystem’s effective hydraulic diffusivity is modified because of changes in mobility and total compressibility.In pressure interference tests, it is commonly assumed that injected fluid displaces the ambient fluid in apiston-like fashion such that the system consists of two radially stationary zones (or fluid banks): the satura-tion gradient within each fluid bank is negligible, and the mobility and total compressibility are relativelyconstant [Abbaszadeh and Kamal, 1989]. The application of HPT under two-phase flow conditions has been

studied by Fokker and Verga[2011], who demonstrated thatHPT can be used to monitor theevolution of effective parame-ters (i.e., effective mobility andtotal compressibility) under theassumptions of mild nonlinear-ity and slightly compressible flu-ids. This section examines theuse of time-lapse HPT for detect-ing leakage in carbon sequestra-tion formations, by adoptingand expanding a numericalmodel recently presented in[Sun et al., 2014]. Similar to thebenchmark problem consideredin previous subsections, the sys-tem consists of a carbon storagereservoir, a confining layer, andan upper aquifer. Both the stor-age reservoir and the upperaquifer are 30 m thick, while theaquitard is 5 m thick. The lateral

Figure 12. GA convergence traces for three different HPT sets (Tp unit is min).

Figure 13. Plan view (not to scale) of the CO2 sequestration test problem. Multiple obser-vation wells are located along a 458 line from the leaky well.



dimensions are both equal to 4280.3 m. A three-dimensional numerical model is developed usingthe commercial multiphase flow simulator, CMG-IMEX. The system is discretized using a 51 3 51 3

11 nonuniform numerical grid in x, y, and z direc-tions. A leaky well, which is approximated as a thinvertical column, is located at the center of thedomain. Its horizontal cross section is square with a0.26 m side width. The grid resolution is refinedaround the leaky well and gradually coarsenedtoward the outer boundaries (see SI 2 for screen-shots). A CO2 injector is located 14.1 m from theleak and is fully perforated in the reservoir layer(Figure 13). Local grid refinement is applied for gridblocks adjacent to the injector by using 5 3 5 3 2local refinement on each of those blocks (SI 2). Thesystem is at hydrostatic equilibrium initially. The lat-eral boundaries are infinite-acting boundaries. The

Brooks-Corey relative permeability-saturation model is used and its fitting parameter (k) is set to 1.5. Capil-lary pressure is not considered. Table 2 lists all other model parameters used.

In the first numerical example, a constant injection rate of 1000 m3/d is applied at the reservoir conditions dur-ing the initial 10 days, after which HPT starts. Sun et al. [2014] assumed square pulsing patterns. In this work,we use sinusoidal injection patterns that are approximated by using 12 subperiods for each pulsing period.Pulses with a variation range of 6200 m3/d are superposed on the nominal injection rate. Four pairs (one withleak and one without) of HPTs are performed using pulsing periods of 0.25, 0.5, 1, and 2 days, respectively. Thesame set of experiments is then repeated by setting the upper aquifer permeability to 500 mD.

Table 2. Parameters for the Carbon Sequestration Problem

Parameter (Unit) Value

CO2 density (kg/m3) 479CO2 viscosity (Pa s) 3.95 3 1025

CO2 compressibility (Pa21) 2.18 3 10210

Aquifer permeability (mD) 800Reservoir permeability (mD) 100Aquifer thickness (m) 30Porosity 0.15Residual brine saturation 0.1Nominal injection rate (m3/d) 1000Brine density (kg/m3) 1045Brine viscosity (Pa s) 2.54 3 1024

Brine compressibility (Pa21) 6.53 3 10211

Aquifer thickness (m) 30Leaky well permeability (vertical, lateral) (mD) 1 3 106, 100Reservoir thickness (m) 30Aquitard thickness (m) 5Residual CO2 saturation 0.0Pulse rate (m3/d) 200

Figure 14. Sensitivity of amplitude attenuation to upper aquifer permeability: (a) 100 mD and (b) 500 mD. Observation well distances are labeled on the plots.



Figures 14a and 14b show the result-ing amplitude attenuation curves. Ingeneral, we observe similar mono-tonic trends in K versus x plots, aswe have seen in the single-phasecases: the amount of deviationbetween no-leak and leak casesdecreases with observation distan-ces; an increase in the upper-aquiferpermeability tends to amplify leak-

age flux and, thus, increase the magnitude of deviation. The change in phase shift (not shown) is minor. Inthis example, the time-lapse HPT works because the nonlinearity is minor and because each HPT starts fromthe same initial conditions. Although it is not realistic to expect identical initial conditions for each test, weexpect that the time-lapse approach is applicable as long as effective reservoir parameters in the testingzone do not change significantly between two back-to-back HPTs. To further illustrate this point, in the sec-ond example, we simulate a sequential HPT set by using the following well schedule: injecting at a constantrate of 1000 m3/d for the first 50 days, which is followed by the first HPT; a leak occurs right after the firstHPT, but the second HPT only starts 1 week after the first HPT. Constant injection is used between the tests.The same pulsing periods are used and all other parameters are the same as before.

Table 3 compares CO2 saturations at three observation locations and for three different times, namely, afterthe constant injection, and after the first and second HPT, respectively. Among all three locations consid-ered, the farthest location (i.e., 83 m) experiences the largest change in saturations (>10%). Figures 15a and15b show the resulting K and U curves. Compared to a similar case shown in Figure 14a, we see that devia-tion stays about the same at shorter distances (14 and 43 m), but becomes virtually unobservable at 82 m.This example implies that reservoir conditions around observation wells should not change significantlybetween two back-to-back HPTs for the time-lapse concept to apply.

In general, longer pulsing periods and testing time are necessary because CO2 is more compressible thanbrine. Thus, observation wells located farther away require longer time to respond. Nevertheless, longerpulsing periods are operationally easier to work with, especially when rate change needs to be controlled

Table 3. CO2 Saturations for Two Pulsing Periods

Pulse Period(day)

Obs Location(m)

CO2 Saturation

At 50 days After First HPT After Second HPT

0.25 14 0.61 0.61 0.6343 0.60 0.60 0.6282 0.53 0.55 0.58

2 14 0.61 0.64 0.6743 0.60 0.63 0.6682 0.53 0.61 0.65

Figure 15. (a) Attenuation and (b) phase shift for a pair of HPT performed sequentially.



manually at the wellhead. Because of its low cost, Hollaender et al. [2002] even suggested that HPT be main-tained indefinitely to allow continuous monitoring of reservoir conditions.

4. Conclusions

This study investigates the feasibility of using HPT as a cost-effective tool for monitoring the integrity ofdeep subsurface storage formations. Our results indicate that:

1. Leakage modifies a geologic storage system’s frequency response function and may be detected by per-forming routine HPTs.

2. The amplitude and phase of the frequency response function provide independent information that canbe used not only for detecting the presence of leakage, but also the locations of leaks.

3. Amplitude of the frequency response function decreases monotonically with pulsing frequencies, thuslonger HPT pulsing periods (lower frequencies) increase the coverage area and reduce false negativerates.

4. Smaller reservoir permeability or, equivalently, higher upper aquifer permeability favors detection of leak-age, if all other parameters are kept the same.

5. HPT is sensitive to factors that can dampen pressure signals (e.g., fluid compressibility); thus, in multiphaseenvironments it may require longer pulsing periods; nevertheless, with high-resolution, high-accuracydownhole gauges that are widely available today even weak pulses may be sensed a long distance away.

In this work, a workflow of deploying HPT as a long-term monitoring strategy is suggested and demon-strated using a stylized multilayer repository system. In practice, the efficacy of HPT may be affected by (a)inaccurate knowledge on the nominal injection schedule and nearby production activities, which can inter-fere with the trend removal; (b) infeasibility to impose long pulsing periods, which can affect an HPT’s cov-erage area; (c) leakage from more than one wells, which could impose challenge on the simple location-search algorithm proposed here; (d) significant reservoir state changes (not related to leakage) betweentwo HPTs; or (e) inappropriate use of HPTs by terminating a test prematurely. Thus, in case of suspectedanomalies, it is recommended that information from codeployed sensors be used to provide additional veri-fication. Currently, the team is performing further work to validate the HPT concept experimentally.

Appendix A

The Laplace transform of a function f ðtÞ is defined as

L f ðtÞ½ �5ð1

0e2st f ðtÞdt; (A1)

where s is the Laplace transform variable. Applying equations (A1)–(A3) and its initial and boundary condi-tions gives

@2P̂@r2

11r@P̂@r

2/lct;1

k1sP̂50; (A2)

P̂ðr !1; sÞ50; (A3)

2pk1b1

lr@P̂@rjr5rw

52Qo

s2jx: (A4)

Equation (A2) is the zero-order modified Bessel differential equation, and a general solution is in the form[Bruggeman, 1999]

P̂ðrÞ5AI0ðrffiffiffiffiffiffiffiffiffis=g1

pÞ1BK0ðr


pÞ; (A5)

where I0 and K0 are the zero-order modified Bessel function of the first and second kind, respectively, andg15k1=/lct;1. Because I0 ! 0 as r !1, the coefficient A needs to be zero. Substituting (A5) into (A4) andsolving for B gives



B5Qol


pK1ðrw


pÞ: (A6)

Thus, for an infinite reservoir without any leaks, the Laplace-transformed pressure change at an arbitraryobservation location r is

P̂T15

QolK0ðrffiffiffiffiffiffiffiffiffis=g1

pÞ


pK1ðrw


pÞ:

Notations

Symbol Explanation.j Imaginary unit.HðxÞ Frequency response function.k1,k2 Permeability of the layers [mD].ka Well permeability [mD].K0,K1 Zeroth and first-order modified Bessel function of the second kind.p Pressure [Pa].P̂ Laplace-transformed pressure.~P Frequency domain pressure response.Q Injection rate [m3/d].q Leakage rate [m3/d].ra Radius of abandoned well [m].rI Distance between injector and leak [m].rL Distance between leak and observation well [m].rO Distance between injector and observation well [m].rw Radius of injector [m].Tp Pulsing period [day].

Greek symbols

a Diffusivity ratio.b Transmissivity ratio.g Diffusivity./ Porosity.x Angular frequency [rad/s].K Amplitude attenuation.U Phase shift.X Flow resistance [m23].

Subscripts

D Dimensionless quantities

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AcknowledgmentsFunding for this research is providedby the U.S. Department of Energyunder award DE-FE0012231. Theauthors are grateful to the AssociateEditor and three anonymous reviewersfor their constructive comments. Theteam would like to thank the DOEProject Manager Brian Dressel for hissupport, and to Hamid R. Lashgari atthe Petroleum EngineeringDepartment at UT Austin for histechnical help on CMG software. Alldata used in this paper are syntheticand have been reported clearly in thetables and support information.



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http://dx.doi.org/10.1111/gwat.12308

http://dx.doi.org/10.1002/wrcr.20519

http://dx.doi.org/10.1029/2011WR010721

http://dx.doi.org/10.1029/2004WR003214

http://dx.doi.org/10.1029/2003WR002997

http://dx.doi.org/10.1029/2005WR004312

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