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Transp Porous Med (2010) 81:219240DOI 10.1007/s11242-009-9397-y

Compressed Air Flow within Aquifer Reservoirs

of CAES Plants

R. Kushnir A. Ullmann A. Dayan

Received: 18 February 2009 / Accepted: 9 April 2009 / Published online: 20 May 2009 Springer Science+Business Media B.V. 2009

Abstract A model on the air flow within aquifer reservoirs of Compressed Air Energy

Storage (CAES) plants was developed. The design of such CAES plants requires knowledge

of the reservoir air pressure distribution during both the charging and discharging phases.

Also, it must assure air/water interface stability to prevent water suction during discharge. An

approximate analytical solution for the pressure variations within the anisotropic reservoir

porous space was developed, subject to the Darcy equation and for conditions of partially

penetrating wells. Sensitivity analyses were conducted to identify the dominant parametersaffecting the well pressure and the critical flow rate (water suction threshold). It is dem-

onstrated that water coning is a factor that could severely limit the discharge air flow rate.

A significant diminishment of that limitation and reduction of the pressure fluctuation can

be achieved by enlargement of the air layer height and discharge period. Likewise, aquifers

with larger horizontal permeability impose less restrictive critical flows. A conclusion on the

preferred screen length could not be merely drawn from technological considerations, but

should also involve important economic aspects.

Keywords Compressed air energy storage (CAES) Porous reservoirs

Partially penetrating well Water coning Critical flow rates

List of Symbols

C Constant, C= e = 1.781072

CD Charging discharging time ratio

f Porosity

F(t) Dimensionless air mass flow-rate at the well

g Gravitational acceleration

g Dimensionless hydrostatic pressure,w g H/P0h Well screen length

R. Kushnir A. Ullmann (B) A. Dayan

Department of Fluid Mechanics and Heat Transfer, School of Mechanical Engineering,

Tel Aviv University, Tel Aviv 69978, Israel

e-mail: [email protected]

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220 R. Kushnir et al.

h Dimensionless well screen length,h/H

H Gas layer height

k Permeability

mc Air mass flow rate through the compressor

mc Dimensionless air mass flow-rate through the compressor,

mc ZRTH krP

20

p Pressure

P0 Initial air pressure in the reservoir

p Dimensionless pressure, p/P0r Radial coordinate

r Dimensionless radial coordinate,

kzkr

rH

rw Radius of the well

rw Dimensionless well radius,kzkr

rwH

R Specific gas constants Laplace transform parameter

t Time

tp Time period of the cycle

ti i = 1, 2, 3,process duration times, see Fig.2

t Dimensionless time,t/tpt Dimensionless elapsed time from the beginning of each stage

T Temperature

v Superficial velocity of the gas

y Relative interface rise, HHh

z Vertical coordinate

z Dimensionless vertical coordinate,z /H

Z Gas compressibility factor

Greek Symbol

Pneumatic diffusivity,kP0/(f)

Modified pressure, defined in Eq. 21

Dimensionless modified pressure,/P0 Interface location

Dimensionless interface location,/H

Viscosity of the gas

n Lag angle

Density of the gas

w Density of the water

z Dimensionless cycle time period,tp z /H2

Subscriptcr Critical

f First cycle

r Radial

s Steady periodic cycle

z Vertical

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Compressed Air Flow of CAES Plants 221

1 Introduction

Typical diurnal electric power consumption exhibits significant variations. It reaches its

peak during daylight and drops to its trough at nighttime. Storage of excess power capac-

ity during off-peak hours is desirable both economically and environmentally. In principle,off-peak excess electrical energy is stored for subsequent use during hours of peak demand.

In this respect, incorporation of a Compressed Air Energy Storage (CAES) facility is highly

attractive. In a CAES plant, air is compressed into an underground reservoir through the con-

sumption of inexpensive excess electrical power. Later, during peak hours, the compressed

air is heated (fired) and then driven to expansion in a gas turbine for electrical power genera-

tion. Three geologically different types of underground reservoirs are feasible: salt caverns,

hard rock caverns, and porous reservoirs (such as aquifers or depleted reservoirs). For the

design of a CAES plant, prediction of the underground reservoir pressure oscillations during

plant operation is required, and specifically the oscillations extreme values. The maximum

pressure constitutes a constraint that the compressor train must meet. The minimum pressure

essentially determines the turbine inlet pressure.

This study addresses the air storage in aquifer reservoirs. To date, there is no commercial

CAES plant that is linked to such reservoirs. However, experimental testing in a single-well

and two-well environments was conducted andconfirmed that an aquifer reservoir is indeed a

suitable medium forcompressed airenergy storage (Allen etal. 1984, ANRStorageCompany

1986). The experiments also showed that incorporation of a compressor after-cooler reduces

air temperature variations within the reservoir (the gas flow becomes essentially isothermal).

Based on Darcys law, an isothermal transient gas flow in porous media is described by

a single, nonlinear, partial differential equation (Muskat 1937). An approximate analyticalsteady periodic solution of that equation was derived for one-dimensional radial flow by

Kushnir et al.(2008). The solution was developed for typical operating conditions, namely,

two periods of constant well mass flow-rate for the charging and discharging phases, and no

flow in between.

In general, the radial flow solution is applicable to studies of dry reservoirs, in which the

well can fully penetrate the air zone. In wet aquifer reservoirs, partially penetrating wells are

used to prevent undesired water suction during the discharge phase. Consequently, the air

flow near the well is two dimensional and axis symmetrical. Furthermore, instead of a fixed

impervious bottom boundary condition, a fluctuating airwater interface must be addressed.

During the discharge stage, a local pressure drop causes the interface to rise toward the well,forming a cone shape.Muskat and Wyckoff(1935), who coined the term water coning,

indicated the existence of a critical cone height and a corresponding critical flow rate. Beyond

that height, the interface becomes unstable and water would flow into the well. Accordingly,

it is necessary to avoid the condition of interface instability.

The air injection and withdrawal induces a two-phase (airwater) flow process. Numerical

solutions of the two-phase flow equations for a partially penetrating single well subject to

a weekly cycle were presented byWiles and McCann(1981). The work made part of an

extensive CAES research effort conducted at the Pacific Northwest Laboratory, U.S.A. (e.g.,

Wiles 1979). The reservoir pressure fluctuations were found to be highly sensitive to the wellscreen length. Also, it was shown that for certain operating conditions and reservoir char-

acteristics, water can enter the well.Meiri and Karadi(1982) developed a numerical model

for similar conditions (two phase, two dimensional, single well), but subject to a daily cycle.

They examined reservoir permeability effects and found that the airwater displacement is

strongly influenced by it. The breadth of the transition zone (the zone where both air and

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water are present in the void space at various degrees of saturation) narrows with increasing

permeability.

If the transition zone is considerably narrower than the air flow domain, it is justified to

assumethat a sharp airwater interface would be representative, where a single-phase air flow

exists in the upper domain. Based on that assumption,Braester and Bear(1984) developeda two-dimensional numerical model for a partially penetrating single well subject to a daily

periodic cycling. The model accounts for variations of the airwater interface location. In the

range of the parameters studied, the airwater interface fluctuations were found to be small.

Calculations indicated that the well pressure fluctuations were pronounced in cases of short

well penetrations (smaller than 20% of the gas layer height).

Clearly, application of each of the aforementioned numerical solutions is restricted to the

specific range of parameters for which it was developed. As such, those investigations cannot

reveal the general sensitivity of both the well pressure and water coning to the reservoir char-

acteristicsandoperatingconditions.Furthermore, no quantitative data on thecriticalflow rate

have been reported. Additionally, excluding the study ofMeiri and Karadi(1982), all other

investigations addressed only the first cycle solution, to minimize computation cost. There-

fore, there exists a need for explicit formulae that cover a wide range of parameters. In order

to address that need, this study was undertaken with the purpose of providing an analytical

tool to calculate the reservoir pressure distribution, and to determine the interface stability

during a plant operation. An approximate analytical solution for the reservoir pressure distri-

bution was developed for typical operating conditions, namely, two periods of constant well

mass flow-rate for the charging and discharging phases and no flow in between. Based on

the solution, the critical flow rate and interface height were determined in accordance with

the concepts ofMuskat and Wyckoff(1935).

2 Formulation of the Problem

Consider a partially penetrating well located in an aquifer reservoir as shown in Fig. 1. Ini-

tially, the porous reservoir contains a stationary air layer of height H, bounded by a water

aquifer below and an impervious caprock above. During a CAES plant operation, air flows

into and out of the reservoir cyclically. Owing to that flow, the airwater interface moves

downward in the charging stage and upward during the withdrawal stage. Two-phase flow

phenomena are expected to periodically affect a relatively small volume of the reservoir,located near the airwater interface. Such effects are, therefore, likely to be of limited signif-

icance. Hence, it is assumed that the airwater interface is sharp, meaning that the air-bubble

domain does not contain movable water (i.e., only water at irreducible saturation). For such

conditions, the continuity and momentum equations (neglecting gravitational effects) for the

air region, subject to the generalized gas state equation, are as follows:

( f)

t+

1

r

(rvr)

r+

(vz )

z= 0 (1)

vr =

kr

p

r , vz =

kz

p

z (2)

=p

ZRT(3)

where vrandvzare the gas radial and vertical superficial velocities, fis the porosity, and

krandkzdenote the medium permeability in the radial (horizontal) and vertical directions,

respectively. The remaining symbols are consistent with common notations. The model is

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H

h

rw

rz

Undisturbed interface

p(r,z,t)

CompressedAir

Discharge stage interface

Chargingstage interface

Water aquifer

(r,t)

Fig. 1 Schematic of a partially penetrating well in a CAES aquifer reservoir

based on the assumption that the reservoir can adequately be represented as a homogeneous

porous space with both constant effective porosity and permeability.

The air flow is essentially isothermal owing to both air cooling following the compression

stage and the immense thermal inertia of the porous medium (as compared to that of the gas).

Indeed, as aforementioned, the field test data (Allen et al. 1984) revealed that air temperature

variations in the reservoir are minor when a compressor after cooler is present. Substitution

of Eqs. 2and3,for a constant temperature, into Eq.1yields the following nonlinear partial

differential equation:

p

t=

kr

2f

2p2

r2 +

1

r

p2

r

+

kz

2f

2p2

z2 . (4)

The initial reservoir gas pressure, P0, is uniform and equal to the local hydrostatic head

(the reservoir gas static head is of negligible influence). The cyclical air injection and with-

drawal produce pressure fluctuations in the stored air. Given that the pressure fluctuations

are smaller than P0and subject to the isothermal-flow assumption, it is reasonable to con-

sider the fluid viscosity and compressibility factor as constants and, respectively, equal to

= (T, P0),Z = Z(T, P0).

The pertinent initial and boundary conditions for the indicated coordinate system are

t = 0, p = P0 (5)

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Fig. 2 The dimensionless air

mass flow-rate at the well during

a CAES plant cycle

Discharge-CD

1StorageStorage

Charging

tp1t t2t 3t

F(t)

time to the discharging time). The mass flow boundary condition is applied atr 0 rather

than at the well radius r = rw. This is justified since for typical operating conditions and

reservoir characteristics, seconds within the operation initiation of each stage, asymptotic

conditions are reached with a reservoir pressure distribution that is independent of the wells

radius (Kushnir et al. 2008). Consequently, the well is represented as a line source instead

of a cylindrical source. The boundary condition (9)addresses a large reservoir whose effec-

tive radius exceeds the fluctuating gas penetration radii (the effective region for gas storage

around the well). However, as will be discussed later, the solution can be extended to account

for finite reservoir boundaries and cases of multiple wells.

In order to complete the model, the boundary condition at the airwater interface must

be formulated. As indicated in Fig. 1,the interface coordinates are specified by (r, t). The

kinematic boundary condition requires that

D( z)

Dt=

t+

vr

f

r

vz

f= 0 on z = . (10)

Neglecting the capillary pressure, at equilibrium condition, the pressures on both sides of the

interface are identical. As seen later, excluding short transition periods at the beginning of

each stage, the pressure variation rate near the well is moderate. Consequently, the interface

movement rate is slow. It is, therefore, reasonable to assume that the water layer beneath the

air flow zone is quasi-static. Hence, the position of the interface is described by

p = P0 wg (H z) on z = . (11)

The left-hand side of Eq. 11represents the air pressure at the interface, and the right-hand

side (RHS) stands for the water pressure at the interface under hydrostatic conditions, withwbeing the water density. In order to obtain the kinematic boundary condition in terms of

the air pressure, the derivatives of Eq. 10(with respect to) are derived from Eq. 11.

Choosing P0andtpas the pressure and time scales, respectively, and H as the length scale

in the vertical direction, the dimensionless form of Eqs.411becomes

p

t =

z

2

2p2

r2 +

1

rp2

r +

2p2

z2

(12)

t = 0, p = 1 (13)

z

= 0, 0< r

< ,

p

z = 0 (14)

0 z h, r 0, r

h0

p2

r d z = mc F(t

) (15)

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0 z 1, r , p 1 (17)

z = ,p

t z

p

r

2+

p

z

2+ gz

p

z = 0, (18)

where the location of the interface is

z = , p = 1 g1 z

(19)

and

p p

P0, t

t

tp, r

kz

kr

r

H,z

z

H, h

h

H,

H

z tpkzP0

H2f

tp z

H2 , mc

mcZRT

H krP20

, g wgH

P0

(20)

Equations1219 could be reduced to a simpler formthrough the introductionof a modified

pressure , according to

p2 = P20 + P0, i.e., p2 = 1 + . (21)

The substitution of Eq. 21into Eqs. 1219yields

t = pz

2

r2 +

1

r

r +

2

z2

(22)

t = 0, = 0 (23)

z = 0, 0< r < ,

z = 0 (24)

0 z h, r 0, hr

r = mc F(t

) (25)

h < z 1, r = 0,

r = 0 (26)

0 z 1, r , 0 (27)

z = ,

t

z

2p

r 2

+

z 2

+ gz

z = 0 (28)

and the location of the interface is

z = ,1 +

1/2= 1 g

1 z

. (29)

The integrated well flux (Eq. 15)is represented by a constant average flux (Eq. 25). This

representation entails a negligible error for homogenous reservoirs (Muskat 1937;Ruud and

Kabala1997), and is clearly of advantageous simplicity.

In terms of the modified pressure, the nonlinearity of the differential equation is repre-

sentedsolelyby the coefficientpz . For efficient operation, namely, one that is characterized

by small pressure fluctuations relative to P0, Eq. 22can be linearized by substituting p

=1.Indeed, for typical operating conditions and reservoir characteristics, the nonlinearity effects

on Eq. 22are small (subsequently discussed). Hence, for practical conditions, the solution

of the linearized equation is sufficiently accurate to represent the full nonlinear equation

solution.

Nonlinearity exists in the airwater interface boundary condition (Eq. 28), which further-

more refers to the unknown surface z = . Nonetheless, for small pressure fluctuations,

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the airwater interface level would exhibit corresponding small variations. As a first approx-

imation, one may neglect the influence of the interface shape on the pressure distribution by

assuming a fixed airwater interface (represented as an impervious boundary to the air flow

domain). With that simplification, Eq. 28is replaced by

z = 1, 0< r < ,

z = 0. (30)

In fact, Eq. 30can also be derived from Eq.28,following perturbation solution method tech-

niques (based on the first expansion term, while neglecting the interface movement).

From the pressure distribution solution, the shape of the interfacez = (r, t)during

discharge can be determined from Eq. 29.However, under certain conditions of flow in the

air zone, the interface becomes unstable and water could be sucked into the well. In order

to avoid such occurrence, the conditions that ascertain the airwater interface stability are

subsequently explored.

3 Pressure Distribution

Evidently, from the imposed periodic boundary condition, the solution for the modified

pressure can be constructed from two separate parts (transient and steady periodic). The

transient part, which is affected by the initial condition, vanishes practically within a few

cycles. Therefore, the conditions within the reservoir are determined by the steady periodic

solution.

3.1 Fourier Series Method

A straightforward method is used to obtain the steady periodic solution of Eqs. 2227and

30,through a Fourier series representation of F(t). First, the equations are solved for the

mass flow rate expressed as a harmonic function of time, namely,

F(t) = sin(2 nt + n ). (31)

It is convenient to introduce the steady periodic modified pressure distribution for an

unbounded reservoir caused by a periodic source element distribution at z = along thewell (Carslaw and Jaeger 1959, p. 263)

ds =mc d

2he

nz

(r2+(z)2)r2 + (z )2

sin

2 nt + n

n

z

r2 + (z )2

, (32)

where mc d/2h is the element strength amplitude with dbeing the element dimensionless

length. In order to satisfy theboundary conditions (24) and(30), a corresponding set of source

elements along the z axis at points (0, 2m ), m = 0,1,2, . . .must be accounted for.

The solution can then be obtained by integrating the derived expression for all s from 0 toh. Consequently, the steady periodic solution of Eqs. 2227and30(with p = 1), subject

to F(t)as expressed in(31), is

s =mc

2h

m=

z2m+hz2mh

e

nz

(r2+2)r2 + 2

sin

2 nt + n

n

z

r2 + 2

d. (33)

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In the derivation, for simplicity, the variable transformation z 2m = was incor-

porated. Generally, the solution is a fast converging series. This is especially valid for smaller

zwhere the contributions of remote elements become negligible.

The solution for a harmonic mass flow rate is a building block for the construction of any

periodic mass flow function. This is accomplished with a sine Fourier series ofF(t

), whicheffectively is a mere superposition of harmonic forcing functions for which the solution (33)

applies. The solution of the actual mass flow rate (see Fig.2), obtained through the Fourier

series application, is therefore

s =mc

h

n=1

Cn

n

m=

z2m+hz2mh

e

nz

(r2+2)r2 + 2

sin2 nt + n nz

(r2 + 2)

d, (34)where

Cn =

A2n + B

2n , n = ar g(Bn + An i ),

An =cos

nt1

sin

nt1

CD cos

n(t3 + t2 )

sin

n(t3 t2 )

, and (35)

Bn =sin2

nt1

CD sin

n(t3 + t2 )

sin

n(t3 t2 )

Equation34is an exact steady periodic solution to the linearized problem. However, in the

time vicinity of a flow step change (see Fig.2), the solution series (for n) converges extremelyslowly for small values ofr. Hence, the Fourier series solution does not reveal the maximum

and minimum well pressures, occurring at t1andt3 , respectively. Nevertheless, the solution

is useful for all other times, and it also serves as a validation tool of the Laplace transform

solution, subsequently presented.

Although the plant operating conditions are determined by the steady periodic solution,

published numerical studies usually focused only on the first cycle solution. Hence, it is of

interest to obtain the first cycle solution and compare the limiting conditions of both solu-

tions. The solution for the charging stage of the first cycle (when F(t) = 1) can be obtained

in a similar way as applied to the steady periodic solution method, where the steady periodic

point source is replaced by a continuous source. The modified pressure for 0 t t1 istherefore

f =mc

2h

m=

z2m+hz2mh

1r2 + 2

erfc

r2 + 2

4z t

d 0 t t1 . (36)

The values off in the subsequent times can be readily found by superposition

f2(r,z, t) = f(r

,z, t) f(r,z, t t1 ) t

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3.2 Laplace Transform Method

In order to circumvent the slow convergence of the Fourier series, an alternative solution

based on the Laplace transform is incorporated. The method provides a steady periodic

pressure solution that can be easily evaluated for all values ofr

. Applying, first, the Laplacetransform to Eqs. 2227and30(with p =1), and subsequently, the finite Fourier cosine

transform, yields the solution

(r,z, s) = mcF(s)K0

s

zr

+2mc

F(s)

h

m=1

K0

(m )2 +

s

zr

sin(m h) cos(mz)

m, (38)

where (r,z, s)and F(s)denotes the Laplace transform of(r, z, t)and F(t),

respectively, and K0is the modified Bessel function of the second kind of order zero. The

first term on the RHS of (38), represents the solution of a fully penetrating well radial flow

(h = 1). The second term describes the vertical pressure variations around a partially

penetrating well.

The first cycle compression solution can be found by the substitution F(s) =1/s(i.e.,

F(t) = 1) in Eq. 38.The expression obtained is then integrated, according to the inversion

theorem. Consequently, the modified pressure for 0 t t1is

f

= mc

2Ei r2

4z t + 2mc

h

m=1

K0(mr)

em2 2z t

0

J0(r)e

2z t

2 + (m )2 d

sin(m h)cos(mz)

m, 0 t t1 (39)

where Ei, is the exponential integral, and J0is the Bessel function of the first kind of order

zero. The values offin subsequent times can be found by Eqs. 37ac. Equation 39, as

it stands, is merely an equivalent form of Eq. 36, and has no advantage on the latter. Actu-

ally, it carries a disadvantage, since as r 0, the series containingK0(mr)requires an

extremely large numberof terms beforeconverging(and is singular at r

= 0). Asshownsub-sequently, that problem can be avoided by replacing the Bessel series with its corresponding

power series representation in terms ofr.

The superiority of the Laplace transform method is clearly apparent when considering the

steady periodic solution. The procedure used to obtain the steady periodic solution (follow-

ing the inversion theorem incorporation) is described byKushnir et al.(2008). Applying that

procedure on Eq. 38yields

s = f + m

cI0 +

2mc

h

m=1Im e

m2 2z tsin(m h) cos(mz)

m0 t 1, (40)

where

Im =

0

1 e(2+m2 2z )t

1 + CD

e(

2+m2 2z )t3 e(

2+m2 2z )t2

(1 e(

2+m22z ))(2 + m2 2z )J0

r

1/2

z

e

2t d

m = 0, 1, 2,... (41)

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Equation40is an exact steady periodic solution of the linearized partially penetrating well

representation. For all relevant periods, the solution is a fast converging series, and the

integrals Im can be easily calculated. Consequently, the steady periodic modified pressure

expressed by Eq. 40is far more suitable for pressure computations at small values ofr than

its equivalent Fourier series form (Eq. 34). Furthermore, inspection of the Laplace transformsolutions reveals that for 2z t

>>1 (where t is the dimensionless elapsed time from the

beginning of each stage), transient effects in the vertical direction are practically negligible.

For such times, the solutions could be significantly simplified by neglecting those transient

effects. Accordingly, the steady periodic solution is reduced to

s = s,r +

2mc F(t)

h

m=1

K0

mr sin(m h) cos(mz)

m, (42)

wheres,ris the steady periodic solution of a fully penetrating well given by

2s,r

mc= Ei

r2

4z t

+ 2I0 0< t

t1 (43a)

2s,r

mc= Ei

r2

4z t

+ Ei

r2

4z (t t1 )

+ 2I0 t

1

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For such times, pressure transient effects in the vertical direction are negligible, and the

vertical pressure changes are of a steady periodic nature. Notice that the first cycle solution

can be calculated by Eq. 42, simply from the omission of the 2I0(or I0a )term. Thus, the

Laplace transform method also exposes the difference between the first cycle and the steady

periodic cycle solutions.As seen from Eq. 42, the modified pressure is practically independent ofz for rs closer

or larger than 2. Effects of the partial penetration of the well are therefore confined to a

dimensionless radial distance smaller than 2. Beyond that distance, the flow is essentially

radial, and the modified pressure can be calculated bys,r. Up to a distance of 2, the vertical

flow is prominent, as obtained from the summation term of(42). For smaller rs, more

terms are needed to calculate the Bessel series. In particular, as r approaches zero, the series

requires an extremely large number of terms to calculate. For small values ofr, it is more

advantageous to address the two equivalent solutions, Eqs. 36and39.On comparing those

solutions, subject toz t , and by expressing(r2 + 2)1/2 in terms of a power series

ofr, the following representation emerges

2

h

m=1

K0(mr)

sin(m h) cos(mz)

m

= lnr

4

1

2h

ln

2 ((z + h) /2) sin ( (z + h) /2)

2 ((z h) /2) sin ( (z h) /2)

ln

z h

z + h

z + h +

r2 + (z + h)2

z h +

r2 + (z h)2

+r2

16

2, 1

z

2

h

2

2, 1

z

2+

h

2

+

2,

z h

2

2,

z + h

2

+

2

z + h

2

2

z h

2

3r

4

512

4, 1 z

2 h

2

4, 1 z

2+ h

2

+

4, z

h

2

4,

z + h

2

+

2

z + h

4

2

z h

4+

, (46)

where is the Hurwitz Zeta function. Excluding extreme cases, the value ofr at the well

is small enough so that only the logarithmic terms, on the RHS of(46), must be retained to

calculate the well pressure. Furthermore, the term ln(r

/4) is the only singular function atr = 0. As expected, it cancels out when combined with the singular term ofs,r(see Eqs.

44a, 44c). Consequently, a finite value of the pressure beneath the well (r =0,z > h)

can be calculated. The pressure variations beneath the well are of particular significance, by

providing the basis for the calculation of the highest interface rise, which occurs atr = 0.

As aforementioned, the maximum and minimum reservoir pressures are of practical inter-

est. These pressures occur at the top of the well (r = rw, z = 0) at times t1 and t

3 ,

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(a) (b)

Fig. 3 First period air pressure variations at the well (r = rw , z = 0): comparison between Eq. 42and the

exact solution for mc = 0.005, rw = 0.014, z = 100, h

= 0.5, t1 = 7/24, t2

= 14/24, andt3 = 18/24.

aCharging and subsequent storage andbdischarge and subsequent storage

respectively. Upon substituting the indicated location and times in Eq. 42and46one obtains,

2smax

mc=ln

z t1

4C+ I0a +

1

h

ln

r2w + h

2 + h

2 (h/2)

r2w + h2 h

2 (h/2)

+ O(r2w )

2

sminmc

=ln t

3t3 t

1

CD ln z

t

3 t

2

4C

+I0a CD

h

ln

r2w + h

2 + h

2 (h/2)r2w + h

2 h

2 (h/2)+ O(r2w )

, (47)

where I0ais calculated at the appropriate time (t1 for maximum andt

3 for minimum). The

first cycle extreme pressures can be calculated simply from the omission ofI0a . As seen, the

Laplace transform method provides simple and useful expressions of the well pressure.

The above expressions were obtained for conditions of negligible transient effects in thevertical direction. In order to assess if those transient effects are indeed negligible based

on their duration, well pressure variations during the first cycle were calculated for typical

operating conditions and reservoir characteristics, as seen in Fig. 3a and b. The figures pres-

ent a comparison between the exact solution (of the linear problem, Eqs. 36or39) and the

simplified solution (Eq. 42,omitting I0a ). All the stages were shifted in time to represent

them on a single time scale (t). As it turns out, already at 2z t =2, the solutions are

essentially identical (with proximity smaller than 0.02%). That is, for a cycle time of 24h

andz = 100, 3 min after the beginning of each stage, vertical pressure transients terminate.

In order to estimate the error entailed by the linearization of Eq. 22(obtained by settingp =1), the analysis described in the appendix ofKushnir et al.(2008) was incorporated.

The results confirm that the nonlinearity effects of Eq. 22are indeed negligible. It is worth

noting that the linearized modified pressure representation allows the extension of the sin-

gle well solution to solutions of multiple well fields, simply by superposition. Likewise, for

finite reservoir boundaries, solutions can be derived from the superposition of appropriate

well images.

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4 Interface Location and Stability

In principle, knowledge of the reservoir pressure distribution enables the calculation of the

interface curvez = (r, t)during discharge based on Eq.29.The highest interface rise

occurs right beneath the well, approximately at the end of the withdrawal stage (r

= 0, t

=t3 ). Based on the pressure distribution solution, the modified pressure beneath the well at

that time is

2smc

r = 0t = t3

=lnt3

t3 t1

CD lnz

t3 t2

4C

+I0a CD

h ln

2 ((z h) /2) sin ((z h) /2)

2 ((z + h) /2) sin ((z + h) /2). (48)

The substitution of this expression into Eq. 29yields an implicit equation from which theextent of the highest interface rise can be determined (subject to the set of pertinent param-

eters: mc , h, g, z , t

1 , t

2 , t

3 ).

In order to demonstrate how the location and stability of the interface can be determined,

the air pressure distribution beneath the well at the end of the first withdrawal stage is plotted

in Fig. 4a and b, for different values ofmc s and penetration depths, respectively. The linear

dotted line in the figure refers to the water hydrostatic pressure distribution beneath the well.

The intersection points of the dotted line with the air pressure curves represent the solutions

of Eq. 29. As seen from Fig. 4a, for mc =0.006, there are two solutions (points A and B).

At point B, the pressure gradient in the air zone exceeds the hydrostatic gradient in the water

zone. Hence, the height at B is unstable, and water could be drawn into the well. Conversely,point A pertains to a physically stable height, as the air pressure gradient is smaller than that

of the water zone. As mcis increased, the interface height rises up to a critical height (point

C). At that point, the pressure gradient at both sides of the interface is identical, and any

increase in mcwill drive the water into the well. Therefore, the interface height at point C is

defined as critical, and the corresponding value ofmc(0.01255) is the critical suction rate.

Beyond that critical rate, a stable interface height cannot exist, as seen from the inspection of

the curve mc = 0.024. Similarly, for any given value ofmc , there exists a critical penetration

depth, as seen in Fig. 4b.

Following the above, for any given compressed air pressure distribution, the airwaterinterface stability can be determined. Evidently, the critical mass flow rate and interface rise

are defined by two conditions stating that both the pressure and the vertical pressure gradi-

ent on each side of the interface are identical. Hence, the equations determining the critical

conditions are

r = 0,z = , t = t31 +

1/2= 1 g

1 z

(49)

r = 0,z = , t = t3

z = 2g

1 g

1 z

. (50)

Given a set of parameters (h

, g

, z , t

1 , t

2 , t

3 ), the critical dimensionless mass flow rate,mccr, and the critical interface rise can thereby be determined.

In reality, critical conditions can involve appreciable interface rises, whereas the derived

solutions are for relatively small interface fluctuations. Nonetheless,Muskat(1937) showed

that such an approximation will only affect the quantitative details of the results. It does

not invalidate the general conclusions relating to the existence of critical conditions and

the dependence of the critical flow on its pertinent parameters. Indeed, comparison of the

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(a) (b)

Fig. 4 Air and water pressure distributions beneath the well at the termination of the first discharge stage

(r = 0, t = t3 ) forz = 78.75, g = 0.03924, t1 = 10/24, t2 = 12/24, andt3 = 22/24.aAt differentmc s andbat differenth

s

Muskat and Wyckoff(1935) approximate solution with solutions that addresses the interface

shape (Wheatley 1985; Hoyland et al. 1989), reveals that the approximation entails somewhat

higher critical flow rates but represents quite accurately the influence of the relevant param-

eters (well screen length, layer height, etc.) on the critical conditions. Furthermore, from

a quantitative point of view, the approximate critical flow rate is sufficient to estimate the

number of wells required for any desired application, and thereby determines the economicfeasibility.

5 Results and Discussion

5.1 General Considerations

Table1provides representative ranges of the studied reservoir characteristics, operating con-

ditions, and their corresponding dimensionless parameters. As seen in the table, the values

ofrware indeed small enough to justify the omission of terms containingr2w, and terms ofhigher power, in Eq. 47.These terms can be significant only ifkzis much larger thankr(the

maximum value ofrwin the table is based on a permeability ratio of unity). In addition, it was

shown in Fig.3 that vertical pressure transients terminate when the time elapsed at each stage,

from its initiation, is approximately twice the vertical characteristic time, H2/(z 2). That

is, even for the maximum vertical characteristic time of 2,700 s (Table1), vertical pressure

transients terminate way before the end of each stage.

In order to closely inspect the nature of the pressure distribution, calculated steady peri-

odic pressure distributions for a cycle period at different radii and elevations are illustrated

in Fig. 5a and b, respectively (for the indicated set of operating conditions). As seen, the wellpressure undergoes rapid changes at the beginning of each time interval (owing to the flow

variations). Following those sharp responses, the rate of change of the pressure moderates, as

it approaches stable conditions. The well pressure reaches its crest at the end of the charging

stage and drops to its troughat the termination of the power generationstage.Figure5a reveals

a diminution of the pressure amplitude with a slightly progressive phase lag asr increases.

As observed, the penetration of a well pressure fluctuation into the reservoir does not exceed

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Table 1 Representative ranges of reservoir characteristics, operating conditions, and their corresponding

dimensionless parameters (The maximal value ofrw is based on the maximal permeability ratio of 1, and

separately, the specific gas constant is R = 0.287kJ/kg K)

Variable Definitions Minimum value Maximum value Units

f Porosity 0.05 0.35 unitless

kr Radial permeability 100 5,000 md

kz Vertical permeability 100 5,000 md

H Layer height 5 25 m

rw Well radius 0.05 0.6 m

P0 Initial air pressure 2 7 MPa

mc Compressor flow rate 1 50 kg/s

T Air temperature 300 400 K

Air viscosity 1.8 105 2.4 105 kg/m s

w Water density 930 1,000 kg/m3

tp Time period 24 24 h

Z Air compressibility factor 0.99 1.01 unitlessr, z Pneumatic diffusivities 0.02 40 m

2/s

H2/(z 2) Vertical characteristic time 0.07 2.7 103 s

z tp z /H2 3 1.3 105

rw (kz / kr)1/2rw/H 3 10

4 0.12

mc mcZRT/(H krP20) 8 10

5 20

g w g H/P0 7 103 0.12

t1

t1/tp 6/24 12/24

t2

t1

(t2t1)/tp 2/24 8/24

t3

t2 (t3t2)/tp 2/24 10/24

a certain distance. Beyond that distance, the pressure oscillations vanish. Hence, the infinite

reservoir approach is valid when the effective reservoir radius exceeds the fluctuating gas

penetration radius. For a fully penetrating well, the penetration radius is approximately equal

to (tpr)1/2 (Kushnir et al. 2008). For a partially penetrating well, the ratio of the maximum

pressure (at any radius) versus the maximum well pressure is likely to drop within a shorter

radius than the fully penetrating well (owning to a higher maximum well pressure). As seen

in Fig. 5b, proceeding vertically downward, the pressure amplitude decreases along the well,

and the phase lag is practically negligible. Based on the latter, calculations of the critical

conditions can be performed at the end of each withdrawal stage, representing the time ofminimal pressure in the gas layer below the well.

Since previous investigations addressed only the first cycle solution, it is of essence to

compare the extreme conditions of the first cycle and periodically steady-state solutions.

For that purpose, the relative error(pf ps )/p

s was calculated and plotted (including the

pressure) against mc , for the maximum and minimum pressure values, as seen in Fig. 6a

and b, respectively. For the particular calculations shown in Fig. 6,the relative error is indeed

negligible for moderate pressure fluctuations. It becomes perceptible only in cases of imprac-

tical high pressure fluctuations. It is important to note that applicable ranges of operating

conditions are to avoid interface instability, andthus arebounded by m

ccr(here, in particular,mccr =0.0163 for g =0.12). Similar calculations for the entire range of the parameters

appearing in Table 1lead to the same conclusion. Therefore, it seems that for all practical

conditions, subject to the critical conditions(mc mccr)and within the limitations of mod-

erate pressure fluctuations (psmin 0.75), the relative error can be considered negligible. In

conclusion, for normal operating conditions, the first cycle solution adequately represents the

steady periodic solution. Hence, to obtain a cycle period pressure distribution, it is sufficient

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(a) (b)

Fig. 5 Steady periodic dimensionless pressure oscillations for t1

=7/24, t2

=14/24, and t3

=18/24.

aAt different dimensionless radii andbat different dimensionless depths

to know the solution for a constant well mass flow rate. Any first cycle pressure distribution

(for a daily or a weekly cycle) can be constructed from that solution.

5.2 Parametric Study

For the design of a CAES plant, predictions of the air pressure oscillations in the reservoir

during operation are of interest, in particular, as it pertains to the oscillations extreme values.

In general, an increase in pressure oscillations entails larger compression and lower turbineoutput, both, reducing the plant efficiency. Clearly, to mitigate pressure fluctuations and

associated losses, the well screen length must be extended as much as possible. On the other

hand, larger penetrations run the risk of undesired water suction. In order to demonstrate the

effects of the penetration of the well on pressure fluctuations, the range of pressure oscillation

ps = psmax p

sminis plotted as a function ofh

in Fig.7ac, for several values ofmc , rw,

andz . It is seen that ps is highly dependent onh

for a range of small well penetrations.

This range widens along with larger ps , as mc increases or r

wdecreases. At longer well

penetrations, theps variation moderates.

Expectedly, Fig. 7ac reveals thatpsincreases with both mcandz , and decreases with

rw. From these parameters, mc is the most influencing. Its dominance can be observed inEq.47as well, since the modified pressure is directly proportional to mc . The parameterzappears in a logarithmic term and therefore affects mildlyps . As seen from, both, Fig. 7b

and Eq. 47,the influence ofrwon ps is strongly dependent on the well penetration. At

short penetrations,psis strongly effected byrw, while at deep penetrations, p

sbecomes

independent ofrw. This assertion stems from the fact that at high well penetrations, the flow

is radial and thus depends on z /r2w ( r), while at short well penetration, vertical flow is

appreciable and varies with rw. Notice that the plots do not address instability limitations

of the flow, and as such are representative only for the corresponding ranges of stable flow

conditions.Beyond theconsiderationofpressurefluctuations, theCAES plantmust bedesignedwithin

the limitation of critical flow rates. As aforementioned, the critical dimensionless mass flow

rate and interface rise arecalculated by Eqs. 49 and 50. Results of such calculations areseen in

Fig.8, showing the critical conditions dependence on the relative well penetration for various

values ofg (8a) andz(8b). The predicted trends agree with the results of previous inves-

tigations of steady oil wells critical conditions (e.g.,Wheatley 1985;Hoyland et al. 1989).

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(a)

(b)

Fig. 6 Relative error (between the steady periodic and first period pressures) as a function ofm

cfor different

values ofrw .aAt the cycle maximum pressure and bat the cycle minimum pressure

Evidently, critical flow rates are smaller for deeper well penetrations, with a markedly higher

sensitivity at deep penetrations. On the other hand, the relative interface critical rise is larger

at deeper well penetrations (the relative interface rise is defined as y = (H )/(H h)).

It is seen that mccrincreases with g and decreases with z . As it turns out, the critical

flow rate strongly depends on the gas layer height, where a mild extension ofHentails a

significant increase ofmccr . In addition, the critical flow rate is directly proportional to the

horizontal permeability and approximately inversely proportional to the temperature (effectsof viscosity and compressibility owing to temperature changes are negligible). This is not a

consequence of the linearization and applies also to the exact solution of Eqs. 12through18

(sincekrand T appear only in mc ). The vertical permeability and the porosity, which appear

solely inz , affect little the critical flow rate.

In general, the variables that can mitigate the pressure fluctuations also stabilize the inter-

face. Excluded from such variables are the well screen length and the vertical permeability.

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(a) (b)

(c)

Fig. 7 Ranges of dimensionless pressure variations versus relative well penetrations.aAt different mc s,

bat differentrws, andcat differentzs

(a) (b)

Fig. 8 Critical dimensionless flow rates and relative interface rises versus relative well penetrations.aAt

differentg s andbat differentz s

This is manifested in Figs. 7and8where larger well penetrations reduce both the pressure

fluctuations and the critical flow rate (notice that an increase in vertical permeability can be

interpreted as a well penetration elongation). A decrease in pressure oscillations is desirable

because it reduces the compression work and enlarges the turbine output. However, lower

critical flow rates would require more wells. In order to elucidate these opposing effects, the

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Fig. 9 Effects of the well penetration on the critical dimensionless flow rate and the corresponding range of

pressure variations

critical flow rate and the corresponding range of pressure oscillations (calculated at mccr)

are plotted in Fig. 9. Obviously, the choice of the preferred screen length must be strongly

based on economics.

It is warranted to extend the discussion and address quantitatively real critical flow rates.Referring to the Huntorf CAES plant, where the compressor works 12h at 108kg/s and the

turbine works 3h (producing 290MW) with an inlet pressure of 43 bar(Crotogino et al.

2001). Though the plant operates with salt cavern reservoirs, the same operating conditions

are subsequently used to determine how many wells are required to perform similarly with

an aquifer reservoir. The maximum allowable flow rate, per well, is therefore calculated

for t1 = 12/24, t2 = 18/24, t

3 = 21/24, and P0 = 45 bar. Assuming an aquifer with

f =0.2, kr =500 md, kz =400 md, and T=310K (Z = 0.9946, =1.964 105 Pas,

w = 995kg/m3), and then forH = 25m, the highest critical flow rate (h = 0) is 2.34 kg/s.

It means that for a total of 108kg/s, at least 46 wells are needed (excluding well interference).

Accordingly, for H =15 m, mccr =0.724 kg/s (149 wells), and for H =5 m, mccr =0.0627 kg/s (1,722 wells). The calculation demonstrates that water coning could constitute

a severe limitation, especially for large scale plants. Consequently, it is apparent that CAES

designs must give preference to the smallest practical mc(e.g., larger H, andkror smaller

T) for the reduction of both number of wells and pressure fluctuations.

All the above results are based on fixed time intervals (charging, storage, and power gener-

ation). Realistic bounds for these intervals are shown in Table 1.Basically, the time intervals

are determined so as to meet the local power demand and production capacity. Effects of

the duration period of the power generation on the critical flow rate and the corresponding

range of minimum pressure are shown in Fig. 10.As seen, prolonging the duration of powergeneration can significantly increase the well storage capacity. Moreover, the corresponding

minimum pressure is even slightly higher owing to smaller discharge flow rates. As for the

compression phase, referring to the bounds shown in Table 1,prolongation of its duration

period would not affect the well storage capacity (i.e., mccrt1 const), but would reduce

pmaxowing to a corresponding smaller flow rate. Consequently, it is desirable to expand the

duration periods of compression and power generation, as much as feasible.

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Braester, C., Bear, J.: Some hydrodynamics aspects of compressed-air energy storage in aquifers. J. Hydrol.

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Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn. Oxford University Press, Oxford (1959)

Crotogino, F., Mohmeyer, K.U., Scharf, R.: Huntorf CAES: more than 20 years of successful operation. SMRI

Spring Meeting, Orlando, USA, pp.351357 (2001)

Hoyland, L.A., Papatzacos, P., Skjaeveland, S.M.: Critical rate for water coning: correlation and analyticalsolution. SPE Reserv. Eng.4(4), 495502 (1989). doi:10.2118/15855-PA

Kushnir, R., Ullmann, A., Dayan, A.: Steady periodic gas flow around a well of a CAES plant. Transp. Porous

Media73(1), 120 (2008). doi:10.1007/s11242-007-9156-x

Meiri, D., Karadi, G.M.: Simulation of air storage aquifer by finite element model. Int. J. Numer. Anal. Meth.

Geomech. 6(3), 339351 (1982). doi:10.1002/nag.1610060306

Muskat, M.: TheFlow of Homogeneous Fluid through PorousMedia. 1st edn. McGraw-Hill, New York(1937)

Muskat, M., Wyckoff, R.D.: An approximate theory of water coning in oil production. Trans. AIME114, 144

163 (1935)

Ruud, N.C., Kabala, Z.J.: Response of a partially penetrating well in a heterogeneous aquifer: integrated well-

face flux vs. uniform well-face flux boundary conditions. J. Hydrol. (Amst.)194, 7694 (1997). doi:10.

1016/S0022-1694(96)03217-9

Wheatley, M.J.: An approximate theory of oil/water coning. Paper SPE 14210, SPE 60th annual technicalconference and exhibition, Las Vegas, USA (1985)

Wiles, L.E.: Numerical analysis of temperature and flow effects in a dry, two-dimensional, porous media res-

ervoir used for compressed air energy storage, technical report PNL-3047, Pacific Northwest Laboratory

(1979)

Wiles, L.E., McCann, R.A.: Water coning in porous media reservoirs for compressed air energy storage,

technical report PNL-3470, Pacific Northwest Laboratory (1981)
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