Analytical model of well leakage pressure perturbations in a closed aquifer system

Advances in Water Resources 69 (2014) 13–22

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Analytical model of well leakage pressure perturbations in a closed aquifer system Mehdi Zeidouni ⇑ Bureau of Economic Geology, The University of Texas at Austin, TX, USA

a r t i c l e

i n f o

Article history: Received 9 September 2013 Received in revised form 18 March 2014 Accepted 19 March 2014 Available online 29 March 2014 Keywords: Well leakage Closed aquifer system Analytical solutions Pressure monitoring CO2 storage

a b s t r a c t Deep saline aquifers are commonly used for disposal and storage of various surface fluids. The target injection zone must be hydraulically isolated from overlying zones in order to ensure containment of the injected fluids. Improperly plugged nonoperational abandoned wells that penetrate the injection zone are the main potential leakage pathways. Leakage through such wells may cause an observable pressure signal in a zone overlying the injection zone; such a signal can be used to detect the leakage. In this paper we develop an analytical model to evaluate the pressure change induced by leakage through a well in a multilayer system. Unlike previous analytical models on the topic, our model uses a closed system, which may significantly affect the strength and behavior of the pressure signal induced by leakage. The analytical model is first presented for a two-layer system centered at the leaky well location. We evaluate the leakage-induced pressure change using the Laplace transform of Duhamel’s superposition integral, yielding the solution in the Laplace domain. We then derive a late-time asymptotic solution using the final value theorem, which suggests that the leakage rate becomes constant after sufficient time. We then obtain the multilayer solution by extending the two-layer solution and presenting it in matrix form in the Laplace domain. We apply the solution to three examples. In the first example, we apply the analytical model to a two-layer system, investigating its behavior and comparing the results with a numerical solution. In order to demonstrate behavior and potential applications of the multilayer analytical model, we present two multilayer examples: one with identical layers and another, replicating a CO2 storage site, with dissimilar layers. The leakage-induced pressure change does not necessarily decrease as the distance increases from the injection zone toward the surface. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Deep geological formations are widely used for disposal of waste fluids produced by various industries, such as energy and chemical industries (see [1] and references therein). These formations are currently candidates for CO2 sequestration as a means of mitigating climate change by reducing atmospheric CO2 emissions [2,3]. However, such formations commonly lie in sedimentary basins that have been extensively drilled for exploration and extraction of hydrocarbon resources [4,5]. Many wells used for such drilling, although no longer in use, may not have been properly plugged and abandoned or may have degraded. In the United States, many old wells are reported to have been improperly abandoned, the location of which may not be precisely known [6,7]. When such wells penetrate the injection formation, they constitute major threats to the security of disposal/storage [8–10]. Leakage ⇑ Tel.: +1 512 471 4560; fax: +1 512 471 0140. E-mail address: [email protected] http://dx.doi.org/10.1016/j.advwatres.2014.03.004 0309-1708/Ó 2014 Elsevier Ltd. All rights reserved.

can occur through and along wellbores that penetrate or lie near the injection zone (IZ). Because of the fast propagation of the pressure, pressure monitoring has been investigated as a tool for detection and characterization of leakage from the IZ to overlying zones [11–17]. For a given injection site, the amount of leakage is likely to be small compared with the injected volume. Hepple and Benson [18] posit that, in order to warrant efficient CO2 storage, leakage rates must be lower than 0.01%/year. The pressure change induced by leakage in the IZ is therefore small compared with that caused by injection. However, pressure change in a permeable zone overlying the IZ (separated by a confining layer) may not be directly related to injection in the IZ and may be the result of leakage and/or geomechanical effects related to deformation [19,20]. Under controlled field conditions, the geomechanical effects may be minimized and negligible. The pressure changes in the overlying zone will then be mostly related to leakage. For this reason, efforts in leakage detection on the basis of pressure monitoring are focused on using pressure measurements from an above-zone monitoring interval (AZMI) (see [21] and references therein).

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M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

Nomenclature C ct H h g k L P Q q R d rw re rL r1 r2 r3 s T t

vector defined by Eq. (22); elements are dimensionless total compressibility, 1/Pa coefficient matrix defined by Eq. (22); elements are dimensionless thickness, m gravity = 9.8 m2/s permeability, m2 leak-monitor distance, m pressure, Pa vector defined by Eq. (22); elements are dimensionless volumetric flow rate, m3/s leak-injector distance, m center-injector distance, m well radius, m external (outer) radius, m leak radius, m center-monitor distance, m injector-monitor distance, m inverse-monitor distance, m Laplace transform dummy variable transmissivity = permeability  thickness, m3 time, s

Deployment of pressure gauges in all overlying zones is too expensive to be practical and may not even necessarily be useful. Pressure measurements are most useful if a strong pressure signal is obtained in response to a leak. Intuitively, one may expect that the distance from the IZ would determine the strength of the pressure signal in an overlying zone. It would follow that the zone immediately above the IZ would show the largest pressure signal in response to leakage. However, this relationship is generally incorrect; instead, the pressure signal is a function of the petrophysical properties of both the IZ (from which fluid is leaking) and the overlying zone (to which the leakage is occurring). Preliminary modeling and screening are required to determine which overlying zones provide the strongest pressure signals in response to a given leakage. One or more pressure gauges may then be deployed at favorable overlying permeable zones so that pressure measurements can be analyzed for leakage detection and characterization. Both numerical and analytical modeling tools can be used for this screening. However, because basin-scale simulations are required, the use of numerical simulation tools may be computationally very expensive and inefficient. Analytical modeling tools provide a better choice for the screening process they can run fast and require less input information. For short time periods, the IZ and potential AZMIs may be considered to be infinite acting. Consequently, previously developed analytical models [22–24] for calculation of leakage rate and corresponding pressure changes may be applied. The pressure will eventually reach the boundaries of the reservoirs, however, causing larger pressure changes compared with those derived under infinite-acting conditions. The time for the pressure to reach the boundary is independent of the injection rate and depends only on the size of the reservoir (distance to boundary) and the diffusivity coefficient. Once the boundary effect prevails, it causes additional pressure change in the IZ and AZMIs. Injection-induced pressure changes (in the IZ) considering the boundary effect were obtained for CO2 storage applications [25–28]. The reservoir boundaries are usually known from pressure transient tests analyzed using diagnostic plots (e.g. [29–31]), combined with geological and geophysical data. Analytical models considering closed boundary conditions are required to assess

a D

g l q /

w f

leakage coefficient = leak permeability  (leak radius)2/ (2  IZ permeability  IZ thickness  leak interval); dimensionless difference from the initial value diffusivity coefficient = permeability/(porosity  fluid viscosity  total compressibility), m2/s viscosity, Pa s brine density, kg/m3 porosity, fraction see Eq. (22); dimensionless see Eq. (22); dimensionless

Subscripts: D dimensionless i overlying zone number, i = 1 for the deepest overlying zone and i = N for the shallowest L leak c confining layer I injection zone (IZ) A above-zone monitoring interval (AZMI) u unit-rate pressure response 0 initial

long-term pressure monitoring in permeable intervals overlying an injection formation. In this study, we developed an analytical model considering closed boundary conditions in all the layers connected by a leaky well. The paper is organized as follows: Section 2 presents the problem, geometry, and assumptions. Section 3 provides the flow equations in each of the system’s components in a two-layer system, along with the Laplace-domain solution. Section 4 presents the real-time asymptotic behavior of the solution. The analytical solution is extended to a multilayer system in Section 5. In Section 6, we apply the analytical solutions to example problems, compare the results to those derived from a numerical solution, and investigate the solution behavior and potential application. 2. Problem description In developing the analytical model, we started with a simplified two-layer system, the geometry of which is shown in Fig. 1. We considered two zones, IZ and AZMI, separated by a confining layer. The injection fluid is assumed to be identical to the aquifer’s native fluid and is injected at a constant rate (q). The aquifers are connected by a leaky well, and both zones are circular homogeneous and isotropic media having uniform thickness. For convenience, the leaky well is considered to be at the center of both the IZ and the AZMI. The radii of the IZ and AZMI are reI and reA, respectively, and their thicknesses are, respectively, hI and hA (subscript I denotes the IZ, and A denotes the AZMI). The injection well fully penetrates the IZ and is located at distance R from the leaky well. The pressure is monitored at distance L from the leak in the AZMI. The injection well and leaky well radii are denoted by rw and rL, respectively. The leaky well permeability is kL, and the leakage interval is hL (=(hA + hI)/2 + hc) where hc is the thickness of the confining layer. The flow through the leaky well is considered Darcian. 3. Analytical model The leakage rate at a leaky well connecting the IZ to the AZMI is given based on integrating Darcy’s law over the leakage pathway:

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M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

Fig. 1. Schematic of problem, showing leaky well connecting IZ to AZMI.

qL ¼



pkL r2L DPI  DPA qghL  ðPI0  PA0 Þ þ hL l hL

 ð1Þ

where DPA = PA0  PA and DPI = PI0  PI are the pressure changes from initial pressure at AZMI, PA0, and IZ, PI0, respectively. q is brine density and g is gravity constant. l is viscosity of the fluid and is considered constant and, as mentioned, is identical for the injectate and the native fluids. The second term in the parenthesis in Eq. (1) is a measure of the initial hydrostatic-pressure difference between the aquifers. If the aquifers are initially at hydrostatic equilibrium, this term equals zero. If a leak existed over a long period of time, we can assume that the two aquifers have achieved hydraulic equilibrium, at least in the vicinity of the leak, and this term can be neglected. Writing Eq. (1) in dimensionless form considering initial hydrostatic equilibrium and defining qLD = qL/q provides:

qLD ¼ aðPID  PAD ÞjrLD

ð2Þ

where 2a is referred to as the leakage coefficient [24] given by a ¼ 2kkILhrILhL . The dimensionless pressures at the IZ and AZMI, respec2pkI hI ql

2pkI hI ql

tively, are given by PID ¼ DP I and PAD ¼ DPA . The dimensionless leaky well radius is rLD (rL/rw), and the injection well radius is rw. The pressure in the IZ at the location of the leak is the superposition of the pressure change caused by injection and that caused by leakage. The pressure response at the location of the leaky well (center of the IZ) to injection at a constant rate (q) through the offcenter well is required. Eq. (3) was derived by Temeng and Horne [32] for the dimensionless pressure at a monitoring well, based on which the pressure change in response to injection through an offcenter well varies linearly with time at every point of the system after the effect of the boundary prevails. Temeng and Horne [32] developed their solution by generalizing and applying the method of images (conventionally used for applications involving straight boundaries) for circular boundary. The image wells for circular boundary, however, change with time and become stationary after sufficiently long time. By applying Laplace transform to the governing diffusivity equation and its initial and boundary conditions, Temeng and Horne [32] arrive at Eq. (3), which is valid after the image wells become stationary. The monitoring well is located at dimensionless distance r1D (=r1/rw) from the center and dimensionless distance r2D (=r2/rw) from the injection well, and the injection well is located at distance dD (=d/rw) from the center.

PD ¼

  2 2t D 3 r21D þ dD r 3eID  þ þ ln r 2D r 3D dD r 2eID 4 2r2eID

ð3Þ

where r3D is the dimensionless distance from an inverse point located at r 2eID /dD to the monitoring point (see [32] for further details and definitions). For our problem described in Fig. 1, r1D = 1, r2D = dD = RD, and r3D = r2eID /dD, where RD(=R/rw), reID(=reI/rw) are the dimensionless injector-leak distance and the dimensionless IZ radius, respectively. Therefore, the dimensionless pressure at the location of the leaky well caused by injection is obtained by:

PD ¼

  2t D 3 R2D r eID  þ þ ln RD r 2eID 4 2r 2eID

ð4Þ

where tD is dimensionless time and is defined by tD = gt/r2w , and g is the diffusivity coefficient given by g = k/(/lct). Porosity and total compressibility are denoted by / and ct, respectively. The total compressibility is the sum of rock and fluid compressibilities (see e.g. [33]). The pressure in response to the time-dependent leakage rate qLD(tD) is given by Duhamel’s superposition integral:

PD ¼

Z

tD 0

qLD ðtD  sD Þ

dPDu ðsD ÞdsD dsD

ð5Þ

where PDu is the dimensionless pressure response corresponding to a constant leakage rate (qLD = 1). Because the leak is located at the center of the aquifer, PDu should be evaluated at the leaky wellbore (rLD) and is given by [34,35]:

PDu ¼

  2tD 3 r 2LD r eID  þ þ ln rLD r2eID 4 2r 2eID

ð6aÞ

Because rLD<
PDu ¼

2tD 3  þ lnðr eID Þ r2eID 4

ð6bÞ

By taking the Laplace transform, the Duhamel’s integral simplifies to the multiplication below:

D ¼ q  Du  PDu ðtD ¼ 0ÞÞ ¼ q  Du LD ðsP LD sP P

ð7Þ

where the bar sign on PD and qLD indicates Laplace domain variables and s is the Laplace transform dummy variable. Note that PDu vanishes as tD ? 0 although this is not visible from Eq. (6a) which is late-time solution. The full real-time solution (Eq. (XII) of Horner [34]) also includes a series term that makes PDu vanish as time goes to zero. Combining Eqs. (6) and (7) gives the following for the pressure change at the leak location caused by leakage:

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M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

D ¼ q LD P



2 3  þ lnðr eID Þ sr2eID 4

 ð8Þ

Taking Eq. (4) into the Laplace domain and combining it with Eq. (8) provide the following equation for the overall pressure change (due to both leakage and injection) at the leaky well in the IZ:

 !  1 2 3 R2D reID   PID r ¼  þ þ ln LD s sr2eID 4 2r 2eID RD   2 3 LD  þ lnðr eID Þ q sr2eID 4

ð9Þ

The pressure at the leaky well in the AZMI is only caused by leakage. Therefore, the pressure change obtained in the AZMI will be similar to that caused by leakage in the IZ, leading to:

     AD  ¼ qLD 2gD  3 þ lnðr eAD Þ P rLD T D sr 2eAD 4

ð10Þ

where TD and gD are the transmissivity ratio and diffusivity ratio, respectively, and are given by: gD ¼ ggA and T D ¼ kkAI hhAI . I Taking Eq. (2) into the Laplace domain, combining it with Eqs. (9) and (10), and solving for the leakage rate provides:

  R2  34 þ 2r2D þ ln rReIDD eID     34 þ lnðr eID Þ þ T1D sr2g2 D  34 þ lnðr eAD Þ 1 s

LD ¼ q 1

aþ



2 sr 2eID



ð11Þ

eAD

 ! LD 2gD 3 q L2D r eAD  PAD ¼  þ þ ln T D sr 2eAD 4 2r2eAD LD

ð12Þ

4. Asymptotic behavior In order to gain insight into the characteristics of leakage rates and corresponding pressure changes, we now investigate late-time asymptotic behavior of the model. According to the final value theorem, s is small for a late-time period in which tD is large, and:

LD lim qLD ¼ lim sq

ð13Þ

s!0

LD as s ? 0 provides: Rewriting Eq. (11) and taking the limit of sq

tD !1

1



limqLD ¼ lim s!0

1

a



1 2 sr2 eID

 34þ

R2 D þln 2r 2 eID



reID RD

 þ 

2 sr2 eID 2 sr2 eID



 34þlnðreID Þ

 34þ

R2 D þln 2r 2 eID



reID RD



 þ T D  1

2gD sr 2 eAD

2 sr2 eID

  34þlnðreAD Þ

 34þ

R2 D þln 2r 2 eID



reID RD



ð14Þ

The limits of the 2first, second, and third terms in the denominator r are 0, 1, and gT DD r2eID , respectively. Therefore, the late-time (steadyeAD state) leakage rate is given by:

qlD ¼

1 gD r2eID

1 þ TD

ð16Þ

Therefore, the AZMI’s pressure response to leakage will eventually vary linearly with time. Prior to that, the pressure change with time will be non-linear as a result of the time-dependence of the leakage rate. Note that the leakage will be initiated as soon as the pressure perturbations induced by injection are felt at the location of the leak. Therefore, the leakage is initially not affected by the boundary of the IZ and AZMI. As such, before the effect of the boundary dominates, infinite boundary condition should be considered to determine the leakage behavior [22–24,37]. The leakage-induced pressure changes in the AZMI will therefore follow three distinct flow periods: (1) at early time the pressure perturbations have not yet reached the boundary (in IZ or AZMI) and infinite boundary condition should be considered; (2) the flow in both IZ and AZMI is boundary dominated but the leakage rate has not yet reached steady-state conditions; and (3) the flow in both IZ and AZMI is boundary dominated and the leakage rate has reached steady-state conditions.

5. Extension to multilayer system

2 sr2eID

The leakage rate can then be determined by Laplace inversion (e.g. using the Stehfest algorithm [36]). The pressure at the AZMI at a dimensionless distance, LD, is then calculated by:

t D !1

PAD

 ! 1 2gD tD 3 L2D r eAD ¼  þ 2 þ ln 4 2reAD LD T D þ gD r 2eID =r 2eAD r 2eAD

ð15Þ

r 2eAD

Based on Zeidouni et al. [24], the late-time leakage rate equals 1/ (1 + 1/TD) when the IZ and AZMI are considered infinite, which indicates its independence from the transmissivity ratio. Therefore, unlike the leakage rate in an infinite system, the leakage rate in a closed system depends not only on the boundaries of the zones but also on the diffusivity ratio. After the constant leakage rate is reached, the pressure in the AZMI is obtained by taking the rate as constant:

So far we have assumed that the leaky well connects the IZ only to a single AZMI. However, multiple potential AZMIs that are otherwise separated by confining layers may be linked to the IZ via the leaky pathway. In this section we modify the twolayer model above to account for leakage to multiple overlying zones. Injection is performed in the lowermost layer. Again, all the layers are assumed to be centered at the leaky well. As fluid flows up the leaky well from the IZ, it first reaches the AZMI immediately above the IZ. The leakage constitutes a local source and a local pressure increase that drives fluid into the AZMI away from the leaky well. The remaining fluid flows up to the next AZMI and the same process is repeated. From this process (referred to as ‘‘elevator model’’ by Nordbotten et al. [22]) it can be deduced that the driving force for the total leakage is the pressure difference between the injection zone and the AZMI immediately above it at the location of the leak. Therefore, the pressure change at the location of the leaky well at the IZ will be obtained by modifying Eq. (9) to:

 !  1 2 3 R2D r eID    þ þ ln PID r ¼ LD s sr2eID 4 2r 2eID RD   N X 2 3 LD;i q  þ lnðr Þ  eID sr 2eID 4 i¼1

ð17Þ

where qLD,j is the leakage rate to the j’th overlying zone with j = 1 for the deepest overlying zone and j = N for the shallowest. Moreover, the pressure difference between two successive zones determines the sum of the leakage rate to overlying zones, i.e.: N X   AD;i1  P  AD;i Þ LD;j ¼ ai ðP q r j¼i

LD

i ¼ 1; 2; . . . ; N

ð18Þ

k r2

where PAD,0 = PID and ai ¼ 2kIL;ihI hLL;i . The pressure in each of the overlying zones at the leaky wellbore is given by:

   AD;i  ¼ qLD;i 2gD;i  3 þ lnðr eAD;i Þ P r LD T D;i sr2eAD;i 4 g

where gD;i ¼ gA;i and T D;i ¼ I Eq. (19) gives:

kA;i hA;i . kI hI

! ð19Þ

Combining Eqs. (17) and (18) with

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M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

!

!!

N X  LD;i 2gD;i 3 2gD;i1 3 q q LD;j ¼ ai LD;i1  þ lnðreAD;i1 Þ   þ lnðreAD;i Þ q T D;i1 sr 2eAD;i1 4 T D;i sr 2eAD;i 4 j¼i

for i ¼ 2;. . .; N

ð20Þ

 2   AD;i ¼ qLD;i 2gD;i  3 þ LD þ ln r eAD;i P T D;i sr2eAD;i 4 2r2eAD LD for i ¼ 1; 2; . . . ; N

!

ð24Þ

and N X

 ! 1 2 3 R2D r eID  þ þ ln RD s sr2eID 4 2r 2eID !!   N X LD;1 2gD;1 3 q 2 3 LD;i q  þ lnðr Þ  þ lnðr Þ   eID eAD;1 T D;1 sr2eAD;1 4 sr 2eID 4 i¼1

LD;i ¼ a1 q

i¼1

ð21Þ Eqs. (20) and (21) make a linear system of equations that can be written in matrix form as follows:

HQ ¼C

ð22Þ

where

LD;1 Q T ¼ ½q " CT ¼

LD;2 q

LD;2 q

LD;N1 q



LD;N ; q

 ! 2 3 R2D r eID 0 0  þ þ ln RD sr2eID 4 2r 2eID

a1 s

2

w1 6w 6 2 6 60 6 6 6 6 H¼6 6 6 6 6 6 6 6 40 0

f1

f1

f1

   f1

f1

f2

1

1

 1

1

w3

f3

1

 1

1

0

0

0

   wN1

fN1

0

0

0

 0

wN

f1

# 

0 0

3

1 7 7 7 1 7 7 7 7 7 7 7 7 7 7 7 7 7 1 5 fN

and

w1 ¼ 1 þ a1

f1 ¼ 1 þ a1

wi ¼ 

!  2 3 a1 2gD;1 3 þ  þ lnðr Þ  þ lnðr Þ eID eAD;1 T D;1 sr 2eAD;1 4 sr2eID 4





2 3  þ lnðreID Þ sr 2eID 4

T D;i1

fi ¼ 1 þ

T D;i

6.1. Example 1: two-layer system For this example problem, water is injected at a rate of 0.01 m3/ s into a 30-m-thick aquifer overlain by a 10-m-thick AZMI. The IZ and AZMI are separated by a 4-m-thick confining layer. The porosity and total compressibility of the IZ and AZMI are identical and equal 0.1 and 109/Pa, respectively. The leaky well is located 500 m from the injection well. Its radius is 2 m, and its permeability ranges from 5  1014 to 5  1012 m2. The pressure is monitored in a well perforated in the AZMI located 500 m from the leak and 1000 m from the injection well. The outer radii of the IZ and AZMI are identical and equal 1000 m. The IZ permeability is 5  1012 m2, and that of the AZMI is 5  1013 m2. This example problem is numerically simulated using KAPPA’s Ecrin Rubis simulation program [38]. Rubis is a 3D numerical modeling software based on automatic unstructured Voronoi grids, solved through finite volumes. Such gridding simplifies building the model, and provides the flexibility to simulate effects over various spatial and temporal scales. A 2-D plan view of the model at the AZMI is shown in Fig. 2. The numerical results are compared with the analytical solution in Fig. 3a. Good agreement between numerically and analytically calculated pressure changes is observed. The pressure increased by more than 6 MPa for the 5  1012 m2 leaky well compared with 0.7 MPa for the leak in the area having a permeability of 5  1014 m2. For further analysis, we fixed the leak permeability at 5  1012 m2, and varied AZMI thickness from 10 to 20 to 30 m. The resulting pressure change at the AZMI for various AZMI thicknesses is shown in Fig. 3b. Again, we observed good agreement between analytical and numerical solutions. The corresponding normalized leakage rates for these cases are also shown in Fig. 4. For the varying permeability values of the leaky well, the leakage rate is shown in Fig. 4a. For kl = 5  1012 m2



! 2gD;i1 3  þ lnðr eAD;i1 Þ for i ¼ 2; 3; . . . ; N sr2eAD;i1 4

ai

ai

6. Examples

! 2gD;i 3  þ lnðr eAD;i Þ for i ¼ 2; 3; . . . ; N sr 2eAD;i 4

Multiplying the inverse of matrix H into vector C provides the dimensionless leakage rate vector (Q). Similar asymptotic evaluations can be made to arrive at the following equation for the latetime constant leakage rate: 2 T D;i r eAD;i

gD;i r2eID

qlD;i ¼ 1þ

PN

2 T D;j r eAD;j j¼1 gD;j r 2 eID

for i ¼ 1; 2; . . . ; N

ð23Þ

According to Eq. (12), in order to obtain the pressure response at distance L from the leak in each of the overlying zones (potential AZMIs), the leakage rate vector should be used on the basis of the following:

Fig. 2. Plan view of gridded AZMI. Water is injected at Well #1 into IZ, and pressure is monitored at Well #2 at AZMI. Magnified red circle indicates leaky well gridding. Note that our analytical model does not require the wells to be aligned. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

Fig. 3. Comparison of pressure change at AZMI using analytical solution developed in this study and numerical solution for (a) various leakage pathway permeabilities and (b) various AZMI thicknesses.

the leakage rate increases very quickly and reaches a late-time constant value after <50 days. This rate can be determined on the basis of Eq. (15) and equals 0.25 (=1/(1 + 0.1/0.0333)). The leakage rate increases with time for the other leakage permeability values at a slower pace. However, the leakage rate eventually reaches the same value after a sufficiently long time because the ratio gD/TD is independent of leak permeability. Considering various AZMI thicknesses (Fig. 4b) the late-time leakage rate will no longer be the same because gD/TD will vary with varying AZMI thickness. The leakage rate is proportional to the difference between the IZ and the AZMI at the location of the leak (Eq. (1)). The larger thickness of the AZMI reduces the pressure rise caused by leakage (as shown in Fig. 3b) and, therefore, increases the pressure difference between the IZ and the AZMI, which determines the leakage rate. Therefore, the leakage rate will be larger for a thicker AZMI. The time to reach the constant late-time leakage rate will also be larger for a larger AZMI thickness. Note that, at the advent of the constant leakage rate, the pressure (Fig. 3a) becomes a linear function of time, as expected from Eq. (6) for pressure response to constant rate. The pressure change and leakage rate results are shown in Fig. 5 for various AZMI radii. As expected, the pressure change will be larger for a smaller AZMI radius. The larger pressure change reflects a smaller pressure difference between the IZ and the AZMI and, therefore, a smaller leakage rate. 6.2. Example 2: multilayer system Leakage in a multilayer closed system is examined in this example. Consider an IZ connected by a leaky well to nine overlying

Fig. 4. Normalized leakage rate (with respect to injection rate) based on analytical solution for (a) various leakage pathway permeabilities and (b) various AZMI thicknesses.

zones (potential AZMIs) that are otherwise separated by impermeable confining layers. The properties of the IZ as well as the leaky well and the location of the (leaky, monitoring, and injection) wells and injection rate are the same as those given in Example 1. Also, the properties of the nine overlying zones are identical and are the same as those of the AZMI in Example 1. The radii of the IZ and overlying zones are considered 5000 m, and the thickness of each separating confining layer is 10 m. It should be noted that identical properties are not necessary, and our analytical solution can handle distinct properties for all the modeling components. Identical properties are used here only for convenience. Normalized leakage rate (qLD) is plotted against time for 30 years of injection (Fig. 6b). Results are shown for three potential AZMIs: the bottom AZMI (immediately above the IZ, #1), the middle AZMI (#5), and the top AZMI (#9), which is the shallowest layer intersected by the leaky well. The leakage rate to the bottom AZMI reaches about 14% of the injection rate, whereas that to the top layer will be less than 0.4% after 2 years. However, the leakage rate to the bottom AZMI starts decreasing after reaching a maximum, whereas that to the middle and top layers continues to increase. Eventually, the leakage rate will reach a constant value for all the layers. Because the properties of the AZMIs are identical in this example, the leakage rate will be identical for all layers. Otherwise, the leakage rate would be distributed to the AZMIs such that2 the leakage rate to r T the i’th zone is proportional to the ratio gD;i reAD;i 2 . In this example, D;i eID the late-time constant leakage rate to each overlying zone, calculated using Eq. (23), is 8.3% of the injection rate. Total late-time leakage rate is 75% of the injection rate.

M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

Fig. 5. (a) Pressure rise in AZMI and (b) normalized leakage rate for various AZMI radii.

The reason for the decrease of the leakage rate to the bottom AZMI after that initial increase is a result of the reduced pressure difference between the IZ and the bottom AZMI. Because pressure difference is the driving force for leakage, a decreased pressure difference leads to a reduced leakage rate. This result has also been observed in analysis of leaky faults [39]. The pressure increase in the bottom, middle, and top AZMIs is shown in Fig. 6a. The pressure increase is greatest in the bottom layer and reduces toward the shallower layers. This result is, of course, due to less leakage to the top layer compared with that to the bottom layer. 6.3. Example 3: CO2 storage site case Injection of anthropogenic CO2 is considered in a mature and depleted oil field in the Texas Gulf Coast [40]. It is planned to inject 4 Mt of CO2 over the course of 3 years. The field is highly compartmentalized by faults. CO2 is to be injected in a block delimited by faults and covering an area of 4.64 km2 (equivalent reservoir radius is 1200 m). On the basis of the hydrocarbon accumulation and field history, the limiting faults may be sealing and the reservoir is considered closed [40,41], restricting the amount of CO2 injection because of geomechanical constraints [42]. A high overpressure may jeopardize mechanical stability of the caprock (e.g. [43]) and even induce seismic events [44]. The average reservoir pressure must not exceed the fracturing pressure, which can be estimated using Heller and Taber’s [45] correlation: 4

Pf ¼ ð22:62  9:24e4:3610

d

Þd

ð25Þ

19

Fig. 6. Results for bottom, middle, and top AZMIs of Example 2 in terms of (a) pressure rise and (b) normalized leakage rate.

where Pf is the fracturing pressure in Pa and d is depth in m. The reservoir depth is 1800 m, for which the fracturing pressure based on the above correlation is 33 MPa. Given the initial pressure of the reservoir (15 MPa), the maximum pressure rise is, therefore, DPmax = Pf  Pi = 18 MPa. The amount of CO2 that this reservoir block can accommodate can be approximated using [46]: VCO2 = DPmax Vr ct where Vr is the reservoir pore volume and ct is the total compressibility considering the effect of CO2 compressibility as defined by Ehlig-Economides and Economides [46]. For convenience, we ignore the effect of CO2 compressibility and we use the total compressibility as defined earlier (sum of the brine and rock compressibilities). The porosity and thickness of the reservoir are 0.3 and 40 m, respectively. Considering total compressibility ct = 109/ Pa, reservoir pore volume, Vr = 4.64  106  40  0.3 = 5.57  107 m3, and pressure rise DPmax = 18 MPa, the storage capacity is estimated to be VCO2 = 106 m3. Considering 3 years for the injection period, the equivalent injection rate will be 0.01 m3/s at reservoir condition. Note that the storage capacity calculated here is a conservative estimate, because a higher compressibility (accounting for CO2 compressibility) could have been used. Also, the reservoir is here assumed to be fully closed, which may not be realistic because the boundaries surrounding the reservoir may have small but nonzero permeabilities allowing flow and pressure communication [47]. The success of this CO2 storage project depends on the containment of the CO2 in the injection formation. Improperly plugged abandoned wells, especially the old ones that were plugged under unknown standards, are major potential pathways for release of CO2 from the storage reservoir. The pressure may be used to

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M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

monitor any leakage from the IZ. Brine leakage, if detected, will be very useful to the avoidance of any CO2 leakage from the reservoir because it will precede any CO2 leakage. On the basis of pressure data in overlying zones or potential AZMIs, the single-phase analytical model developed here can be used to study the monitorability of leakage before the CO2 leakage starts. Previous studies have shown that single-phase models may be sufficient for estimation of vertically averaged farfield pressure changes outside the CO2 plume if the injected CO2 is represented by an equivalent volume of brine [48–50]. Vilarrasa et al. [26] presented a methodology to calculate an average CO2 density to obtain the brine-equivalent of injected CO2 volume. Nevertheless, the single-phase solution may not match the two-phase solution for farfield pressure, e.g. if the compressibility contrast between CO2 and brine is relatively large – which may be the case at shallower depths [51]. The mobility contrast between the two fluids may also affect the solutions agreement. In evaluating the AZMI pressure response to fault leakage, Porse et al. [52] made some geophysical simplifications and presented an idealized configuration for the IZ and potential AZMIs that we use here. The permeability and thickness of the IZ and potential AZMIs, based on the configuration of Porse et al. [22], are given in Table 1. Note that AZMI #1 is the zone immediately above the IZ and AZMI #13 is the shallowest zone. The porosity and total compressibility of all AZMIs are taken to be identical and equal to 0.3 and 109/Pa, respectively. The thickness of each separating confining layer is considered to be 10 m. Injection at a rate of 0.01 m3/s stops after 3 years, but pressure monitoring continues for 7 more years. The radii of the overlying zones are considered the same as the radius of the IZ (1200 m). The leak-injector and leak-monitor distances are considered to be 600 m. The pressure change at the location of the monitoring well and the leakage rate are obtained for this case considering a leaky well having a radius of 0.5 m and permeability of 1011 m2 (Fig. 7). Considering a pressure change larger than 1 kPa as a sufficiently strong pressure signal for leakage detection (see [11] for details on pressure data quality), the leak will be detectable in AZMI #1 soon after the start of injection and in AZMIs #2, 3, and 4 after 1, 3.7, and 9 years, respectively, as shown in Fig. 7a. It should be noted that the pressure does not necessarily reduce toward the shallower zones and depends on their properties. For instance, the pressure change is larger in AZMI #7 than in AZMI #6. This is due mainly to high transmissivity contrast between these layers, and occurs in spite of the fact that the leakage rate to AZMI #7 is smaller than that to AZMI #6 (Fig. 7b). Also note that the leakage rate to AZMI #8 becomes larger than that to AZMIs #6 and 7 over time, while the pressure rise in this layer remains lower.

Fig. 7. Pressure change and leakage rate results for 13 AZMIs of CO2 storage site case.

Table 1 Permeability and thickness for the IZ and 13 potential AZMIs, as described by Porse et al. [52]. Layer IZ AZMI AZMI AZMI AZMI AZMI AZMI AZMI AZMI AZMI AZMI AZMI AZMI AZMI

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13

Permeability (m2)

Thickness (m)

4.53e13 4.9e13 4.3e13 1.5e13 1.8e13 1.2e12 2.1e15 6.5e13 5.6e13 1.3e12 9.9e15 8.9e13 7.1e13 4e10

40 45.7 300.5 62.8 133.8 89.9 40.8 2.7 46.3 56.7 48.8 35.1 182.9 304.8

Fig. 8. Pressure changes considering various radial extension of AZMI #4 of CO2 storage site case.

The AZMI layers may not cover identical areas. This difference in size may be related to discontinued bounding fault(s) beyond a certain depth, or may be due to lateral and vertical stratigraphic changes. The effect of the AZMI radius on the pressure signal is investigated by varying the radius of AZMI #4 while keeping other radii unchanged. The resulting pressure change is shown in Fig. 8. As expected, decreasing the size of the overlying zone increases the pressure change, leading to earlier detection of the leak. As a result of decreasing the leak size from 1200 to 800 to 400 m, the time for

M. Zeidouni / Advances in Water Resources 69 (2014) 13–22

the leak to become detectable reduces from 9.0 to 7.3 to 5.6 years, respectively. 7. Summary and conclusions We developed an analytical model to evaluate pressure response and leakage rate caused by a leaky well in a closed system. First, the solution was obtained for a two-layer system linked by a leaky well located at the center of the layers. Fluid was injected at a constant rate in the lower reservoir, and the pressure was monitored in an above-zone monitoring interval (AZMI). Superposition of the injection- and leakage-induced pressure changes gave the pressure change in the injection zone (IZ), whereas the pressure change in the AZMI was the result only of leakage. Owing to time dependence of the leakage rate, the pressure caused by leakage was obtained by applying Duhamel’s integral on the constant rate solution. To remove the integration operator, the Laplace transform was performed and the leakage-induced pressure change obtained in the IZ and AZMI. By writing the pressure changes at the location of the leak and applying Darcy’s law, we obtained the leakage rate. Next we examined the late-time asymptotic behavior of the solution and developed relationships for late-time leakage rate and pressure change. Next, we extended the solution for a multilayer system connected by a leaky well in which fluid was injected in the lowermost reservoir. In order to examine the behavior and potential application of the analytical model, three examples were presented. In the first example, the solution was validated against a numerical solution obtained using Ecrin Rubis reservoir modeling software. Very good agreement was observed between the analytical and numerical results considering various permeability and thickness values of the AZMI. Moreover, the validity of the asymptotic solution was tested. In Example 2, the solution was applied to a multilayer system having nine identical overlying zones. It was shown that unlike the pressure change, the leakage rates to the AZMIs vary nonmonotonically and are significantly different at the beginning. However, over time the difference is reduced until a constant leakage rate is reached that is proportional to the ratio

2 T D;i r eAD;i

gD;i r2eID

of each zone. Fi-

nally, in a third example, we applied the analytical model to a multilayer analog of a CO2 storage site on the Texas Gulf Coast. Over the next 3 years, CO2 will be injected in a reservoir block that can be considered closed due to its surrounding sealing faults. We showed that a leaky well having permeability more than 1011 m2 greatly perturbs the pressure in the overlying zone immediately above the IZ, and leakage may be detectable by pressure monitoring soon after the start of injection. Leakage can also be detected in another three zones successively above the first AZMI during the 10-year monitoring period. Acknowledgments This project is funded partly by EPA STAR Grant R834384. The author thanks J.-P. Nicot for useful discussion and constructive comments. Thank you to four anonymous referees for their detailed comments which have improved this paper. Thank you also to Susie Doenges and Chris Parker, who edited the manuscript. Publication authorized by the Director, Bureau of Economic Geology, The University of Texas at Austin. References [1] Tsang C-F, Birkholzer J, Rutqvist J. A comparative review of hydrologic issues involved in geologic storage of CO2 and injection disposal of liquid waste. Environ Geol 2008;54:1723–37. http://dx.doi.org/10.1007/s00254-007-0949-6. [2] IPCC. Intergovernmental panel on climate change, Special Report on Carbon Dioxide Capture and Storage; 2005.

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