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Transp Porous Med (2008) 73:120DOI 10.1007/s11242-007-9156-x

Steady Periodic Gas Flow Around a Well of a CAES

Plant

Roy Kushnir Amos Ullmann Abraham Dayan

Received: 11 April 2007 / Accepted: 6 July 2007 / Published online: 14 August 2007 Springer Science+Business Media B.V. 2007

Abstract The design of a Compressed Air Energy Storage (CAES) plant requiresknowledge of the pressure and temperature variations within the reservoir, forexpected sets of plant operation. In the current work, a closed form approximateanalytical solution for the pressure variations, in porous media reservoirs, was derivedfor conditions of steady periodic isothermal radial gas flow. Two different expressionsfor the pressure variation were obtained, one as an infinite series and the other as

an integral, where the latter is the computationally preferred solution. In order toevaluate the model accuracy, a finite difference numerical solution of the full non-linear problem was developed. The accuracy of the analytical solution was confirmedthrough, both, error analysis and comparison against the numerical calculations. Theanalytical solution can be used to calculate the well pressure variations and the ra-dius of the active region around the well. Examples of calculations are provided, anda parametric study is presented to demonstrate the sensitivity of the well pressureto pertinent parameters. The model could eventually yield improved CAES plantdesigns.

Keywords Compressed air energy storage (CAES) Gas storage Compressible gasflow Porous reservoirs Nonlinear diffusion equation Periodic boundary condition

Nomenclature

C Constant,C = e = 1.781072 . . .CD Charging discharging time ratiof PorosityF(t) Dimensionless well mass flow rateFB Function defined in Eq.28

R. Kushnir A. Ullmann (B) A. DayanDepartment of Fluid Mechanics and Heat transfer, School of Mechanical Engineering,Tel Aviv University, Tel Aviv 69978, Israele-mail: [email protected]

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2 R. Kushnir et al.

h Well screen lengthk Permeabilitymc Compressor mass flow rate

mc Dimensionless compressor mass flow rate,mcZRT

hkP20

N (x) Function defined in Eq.20p PressureP0 Reservoir initial pressurep Dimensionless pressure,p/P0r Radial coordinaterw Well radiusr Dimensionless radial coordinate,r/rwR Specific gas constantR Penetration radius

R Dimensionless penetration radius,R/rwt Timetp Cycle time periodt Dimensionless time,t/tpti i = 1, 2, 3, dimensionless times, see Fig.1T Temperaturevr Gas radial superficial velocityZ Gas compressibility factor

Greek Symbol

p Pneumatic diffusivity,kP0/f Stretching parameter (x) Function defined in Eq.20 Modified pressure, defined in Eq.13 Dimensionless modified pressure,/P0 Dimensionless coordinate, see Eq.41 Gas viscosityn Lag anglen Defined in Eq.20 Gas density

Dimensionless cycle time period,tpp/r2w

Subscript

s Steady periodic condition

1 Introduction

The electric power consumption undergoes significant variations. It reaches its peakduring daylight and drops to its trough at nighttime. Storage of excess power capacityduring off peak hours is desirable for energy conservation and environmental protec-

tion purposes. It is also economical, since the stored energy is provided at marginalcosts. In principle, inexpensive off peak excess electrical energy is stored for subse-quent use during hours of peak demand. In this respect, the Compressed Air EnergyStorage (CAES) facility is a highly attractive venue for having a well-managed powergeneration. In a CAES plant, air is compressed into an underground reservoir throughthe consumption of inexpensive excess electrical power. During peak hours, the

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Steady Periodic Gas Flow 3

compressed air is fired to expand in a gas turbine for electrical power generation.Three geologically different types of underground reservoirs are feasible: salt cav-erns, hard rock caverns, and porous reservoirs (such as aquifers or depleted reser-voirs). Previous investigations of CAES plants thermodynamic performance were

based on the assumption of constant pressure storage (e.g.,Vadasz et al. 1989;Najjarand Zaamout 1998). This assumption is viable for storage in hard rock caverns withhydraulic compensation and, for certain conditions of porous media storage. The cur-rent work is focused on porous reservoirs, and among others, examines the conditionsthat conform to the assumption of constant pressure distribution. It is important tonote that experimental testing proved the technology viability without identificationof any contamination of the reservoir and stored air (Allen et al. 1984;ANR StorageCompany 1986).

Models of gas transport in porous reservoirs have been largely based on the Darcylaw. Accordingly, the isothermal transient compressible gas flow in porous media is

described by a nonlinear partial differential equation (Muskat 1937). A self-similarsolution of the equation exists for an infinite reservoir around a zero radius well(Barenblatt et al. 1990). The solution is for a uniform initial pressure distributionsubject to a constant gas flow rate at the well. This solution, in some cases, could bealso applicable for non-zero well radius and bounded reservoirs. However, for timedependent boundary conditions (such as in a CAES plant) a similarity solution doesnot exist, and solution of the equation is likely to rely on approximate analyticalmethods or numerical schemes.

A numerical study of the behavior and suitability of an aquifer based CAES was

conducted byAyers (1982).The solution for the air pressure distribution around thewell was obtained by a numerical integration of the governing dimensional-differentialequation. Two models of isothermal airflow have been developed. The first is a one-dimensional radial flow around a single well and the second is a two-dimensionalhorizontal flow around a multiple-well system. These models were used to design awell-field system for a 1000 MW 10-h CAES plant, for several potential sites.

Braester and Bear (1984)developed a two-dimensional isothermal gas flow modelfor a partially penetrating single well subject to a daily periodic mass flow rate.The model accounts for the location variation of the aquifer air-water interface.The solution was obtained by the Galerkin finite-element method. In the range of

the parameters studied, the variations in the air-water interface were small. Whereas,the pressure fluctuations at the well were pronounced in cases of short well penetration(less than 20% of the gas layer height).

An approximate one-dimensional analytical solution for the isothermal radial flowaround a well was obtained by perturbation methods(Shnaid and Olek 1995). Thegas pressure was assumed as a sum of spatial averaged pressure and small pressureperturbations. This assumption led to a linear equation of the pressure perturbation,which was subsequently solved by an eigenfunction expansion. The results indicatedthat the pressure transient triggered by the initial well flow quickly disappear, leaving

subsequently a stabilized gas flow regime which is governed by the Poisson equationfor the pressure perturbation. Separately,Olek (1998)used an eigenfunction expan-sion to reduce the nonlinear one-dimensional isothermal radial flow equation to aninfinite system of initial value first-order ordinary differential equations. The lattercan be solved by standard numerical codes for various boundary conditions.

Sakakura (1953)analytically studied the transient behavior of radial gas flow, sub-ject to a constant well pressure. Through a variable transformation, the nonlinear

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4 R. Kushnir et al.

equation was reduced to a heat conduction equation with a variable diffusivity. Asolution was derived for constant diffusivity, and by using minimal and maximal dif-fusivities the nonlinear effects on the transformed variable were found negligible.Ritchie and Sakakura (1956) extended the solution to account for non-isothermal gas

flow and prescribed constant well mass flux. A similar approach was conducted in thecurrent work for periodic boundary conditions.

Inspection of the aforementioned models clearly indicates that a need for explicitanalytic formulae exists. Such formulae could be used to identify the influence of,both, the reservoir characteristics and CAES plant operating conditions on the reser-voir pressure. In order to address that need, the current work was undertaken withthe purpose of providing an analytical tool to calculate the steady periodic pressuredistribution around a well during CAES plant operations. An approximate analyti-cal solution was developed for typical operating conditions; namely, two periods ofconstant well mass flow rate for the charging and discharging phases and no flow

in between. Additionally, the operation is usually characterized by small pressurefluctuations relative to the reservoir average pressure.

2 Problem Formulation

Consider a fully penetrated well located in a porous reservoir. During a CAES plantoperation air flows into and out of the reservoir.It was shown that for typical reservoirs,the momentum equation reduces to the Darcy law equation(Shnaid and Olek 1995).

Hence, for radial gas flow, the continuity and momentum equations, subject to thegeneralized gas state equation, are:

(f)

t + 1

r

(rvr)

r = 0 (1)

vr= k

p

r (2)

=p

ZRT

(3)

where vris the gas radial superficial velocity,fand k are the medium porosity and per-meability, respectively. The remaining symbols are consistent with common notations.The model is based on the assumption that the reservoir can adequately be repre-sented as an isotropic and homogeneous porous space with, both, constant effectiveporosity and permeability.

The airflow is essentially isothermal owing to, both, the air cooling immediatelyafter the compression stage, and the immense thermal inertia of the porous medium(as compared to that of the gas). Indeed, field test data (Allen et al. 1984)revealed

that air temperature variations in the reservoir are minor when a compressor aftercooler is used. From the isothermal flow assumptions, it is also sensible to considerthe fluid viscosity as constant. Hence, substitution of Eqs. 2and 3into Eq.1yields thefollowing nonlinear partial differential equation

p

t= k

2f

2p2

r2 + 1

r

p2

r

(4)

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The pertinent initial and boundary conditions are

att = 0 p = P0 (5)

atr = rw vr(2 rh) =mcF(t) (6)

atr p P0 (7)

where h is the well screen length (here, equal to the gas layer height), and the productmcF(t)represents the gas mass flow rate at the well during plant operation (mcis thecompressor mass flow rate, andF(t)is a dimensionless periodic function with a time

period tp). Initially the reservoir gas pressure is uniformly P0. Given that the pressurefluctuations are smaller thanP0, the compressibility factor is approximately constantand evaluated according to Z = Z(T, P0). In contrast to previous investigationsthat addressed closed finite reservoirs (e.g,Shnaid and Olek 1995), an unboundedreservoir is studied. The latter produces simpler form solutions that apply to boundedwells provided that the reservoir radius is larger than the fluctuating gas penetrationradii (the effective region for gas storage around the well).

The dimensionless form of Eqs. 47 when P0, rwand tpare the pressure, length andtime scales, respectively, is

pt

= 2

2p2r2

+ 1r

p2r

(8)

att= 0 p= 1 (9)

atr= 1 p2

r = mc F(t) (10)

atr p 1 (11)

where

p pP0

, r rrw

, t ttp

, tpkP0r2wf

tppr2w

, mcmcZRT

hkP20(12)

Equations 811 could be reduced to a simpler form through introduction of a modifiedpressure , according to

p2 = P20 + P0 i.e. p2 = 1 + (13)

Such a variable change is commonly used in non linear heat conduction problems(Carslaw and Jaeger 1959). In effect is a mass flux potential. Substitution ofEq.13into Eqs.811,yields

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6 R. Kushnir et al.

t = p

2

r2+ 1

r

r

(14)

at t= 0 = 0 (15)

at r= 1

r= mc F(t) (16)

at r 0 (17)In terms of the modified pressure, the nonlinearity is represented solely by thecoefficientp. As aforementioned, realistically, the pressure fluctuations within thereservoir are smaller thanP0. Thus, as a first approximation, Eq.14can be linearizedby substitutingp= 1. Indeed, for typical operating conditions and reservoir charac-teristics, the nonlinearity effects on Eq.14can be shown to be small (see AppendixA). Therefore, for practical conditions, the solution of the linearized equation is suf-ficiently accurate to represent the full non-linear equation solution.

3 Analytical Solution

3.1 Fourier Series Method

A straightforward method to obtain the steady periodic solution of the linearized Eqs.

1417is through a Fourier series representation. First, the equations are solved for amass flow rate that is expressed as a harmonic function of time. This is accomplishedby the complex combination method(Arpaci 1966).Therefore, when

F(t) = sin(2 nt + n) (18)the steady periodic solution of Eqs.1417(withp= 1) was found to be

s=mcn

N0(nr)

N1(n)sin

2 nt + n + 0

nr 1(n) 3

4

(19)

where

n=

2 n

(20)

N (x)=

Ker2 (x) + Kei2 (x), (x) = arg (Ker (x) + Kei (x)i)Ker and Kei are the Kelvin functions of order . The subscript s in

s indicates

steady periodic solution that satisfies, both, the differential equation and boundaryconditions, but not the initial condition.

Equation19 represents a pressure wave which oscillates everywhere at the same

frequency as the disturbance at the well. The amplitude of oscillation, appearing asan expression of Kelvin functions, diminishes as the radial distance from the wellincreases. The dimensionless radius R at which the pressure amplitude reduces to afraction from that at the well, satisfies the equation

N0

2 n

R

N0

2 n

= 0 (21)

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It follows from Eq.21that higher disturbances frequency (larger n) produce shorterpenetration radiusR.

The solution for a harmonic mass flow rate is a building block for construction ofany periodic mass flow function. This is accomplished by a purely sine Fourier series

representation ofF(t), whichin effect is a superposition of harmonic forcing functionsfor which the solution (19) applies. For a CAES plant operating at a compressor andturbine constant mass flow rates, the actual flow rate at the well is (see Fig.1)

F= 1, m< t

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8 R. Kushnir et al.

3.2 Laplace Transform Method

In order to circumvent the aforementioned convergence problem of the pressure nearthe well, an alternative solution is proposed (seeJaeger 1953). For that, the Laplace

transform method is incorporated. For the present problem, the corresponding solu-tion for the well pressure can be easily evaluated.

Applying the Laplace transform to Eqs. 1417(with p= 1), subject to F(t) asexpressed by (22), yields the solution

(r,s) =mc

1 est1 + CD(est3 est2 )

(1 es)s

s1

K0

s1r

K1

s1 (25)

where(r,s)denotes the Laplace transform of(r, t), and K0 and K1 are themodified Bessel functions of the second kind of order zero and one, respectively. Theinversion theorem for the Laplace transform states that

(r, t) = 12 i

+ii

(r,s)est

ds (26)

whereis a positive constant. The integrand of (26) has a branch point ats= 0 andsimple poles ats = 2 ni, n = 1,2, . . .. Applying the Residue Theorem to the integral(26) for the contour seen in Fig.2, yields

(r, t) = 2mc

0

1 e2t1+ CD(e2t3 e2t2 )(1 e2 )2e2t

FB

r, 1/2

d+ s (r, t)

(27)where

FB

r, 1/2

= J0

1/2r

Y1

1/2 Y0 1/2rJ1 1/2

J21

1/2+ Y21 1/2 (28)

Jand Yare the Bessel functions of the first and second kind of order

, respectively.The first term on the right hand side of (27) stands for the transient part of the solution,which vanishes astprogresses. It is produced from the integrals on the contours FEand DC. The integrals over the arcs AF and CB vanish as the contour radius tends toinfinity. Likewise, the integral over the small circle around the origin vanishes as thecircle radius tends to zero. The second term, of(27), represents the steady periodicpart of the solution. It can be evaluated by calculating the sum of the residues at thepoles. However, that approach would produce the Fourier series of (23). Therefore,s is evaluated differently.

Equation 27 is a general solution, valid at all times. Hence, the steady periodic term

can be found by equating Eq.27with the values of during the first period. In thiscontext, the solution of for 0 t t1 is well known (Carslaw and Jaeger 1959,p. 338)

(r, t) 1(r, t) =2mc

0

e

2t 1

FBd

2 0 t t1 (29)

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Fig. 2 The contour for theintegration of Eq.26

A

B

C D

EF

x

y

The values ofin the subsequent times can be found easily by

(r, t) 2(r, t) = 1(r, t) 1(r, t t1 ) t1

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10 R. Kushnir et al.

3.3 Asymptotic Formulae

Consider a set of CAES plant operating conditions and reservoir characteristics thatcorrespond to a large. For such sets, the solutions (23) and (31) can be significantly

simplified by using their asymptotic approximation. The asymptotic representation ofN1(x)and 1(x), subject tox

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s= mc2

Ei

r24t

Ei r2

4 (t t1)

CDEi r2

4 (t

t2 )

+CDEi r2

4 (t

t3 )

2I(r, t) t3 1 for t1 < t t2 , etc. In effect, these conditionsstate that the time elapsed from the beginning of each period should be substantiallylarger than the characteristic time r2w/p. Consequently and as opposed to the Fourierseries, the maximum and minimum pressures can be easily calculated by the above

equations, each for its indicated time. Furthermore, it turns out that Eqs. 37 and 39provide accurate pressure predictions for the entire range of realistic conditions, ascompared to those obtained by the exact expression.

Based on the linearity of the modified pressure representation, the solution of asingle well can be extended to a field of multiple wells by superposition. The pressureat any point is obtained from the summation of the neighboring wells contributions,through the use of Eqs. 37 and 39.

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4 Numerical Solution

In order to support the analytical results, a numerical solution of the nonlinearEqs. 811was developed. Using central differences for the spatial derivatives, the

partial differential equation and boundary conditions were converted to a systemof initial value ordinary differential equations. As aforementioned, high-frequencycomponents of the pressure wave vanish rapidly as the distance from the well in-creases. Therefore, to conduct an effective numerical computation of the differentialequation, the grid points were arranged in increasing intervals as the distance fromthe well increases. In order to develop a central difference numerical scheme, the r

coordinate is transformed to a uniform grid size variable. The transformation used forthat purpose is(Tannehill et al. 1997)

= 1 ln + 1

r1R

1/ 1 +

r1R

1

ln

+11

1< < (41)

It transforms the physical domain r [1, R], with its clustered grid points near thewell, into a uniform grid computational domain [0,1]. The stretching parameter,, clusters more points near the well (within therdomain) as 1.

Applying the transformation to Eqs.811,yields

p

t=

2

2

2p2

2 +

+

r

p2

(42)

at t= 0 p= 1 (43)

at = 0 p2

= mc F(t) (44)

at = 1 p

= 0 (45)

where and are the first and second derivatives with respect to r. For the numerical

computation of unbounded reservoir, the boundary condition (11) is assigned to abounded reservoir with an external radius equal to the penetration radius, R.

For a central difference derivatives representation, the semi discrete form ofEq.42is

pjt

= 2pj

pj+1 2pj+pj1

2 + 2

pj+1 pj1

2

2

+ +

rp

j

pj+1 pj1

2 r=rjj= 1,2, . . . , N 1 (46)

where

= 1N

, j= j, rj= 1+(R 1) + 1 ( 1)

+111j

1 +

+111j j= 0,1, . . . , N (47)

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Note that Eq. 46 is not the only possible finite difference scheme available. Alternativeschemes could be developed, for instance, with discretization ofp2, or, through a semidiscrete form of Eqs.1417(generated for). Nonetheless, Eq.46was found to bethe most stable scheme for numerical computations.

Discretization of the boundary conditions and application of the semi discrete formof Eq.42,yields the boundary grid points equations

p0t

=

22p0p1 p0

2 +mc F(t)

2p0

2+

2

1

r

mc F(t)2

|r=r0

(48)

pN t

=

22pNp

N1 pN2

|r=rN

(49)

Equations4649comprise a complete set. They are solved for a uniform initial con-

dition of unity.The solutions were computed with the problem solving environment Maple

(Maplesoft 2003), based on the default stiff method, which is an implicit Rosen-brock third-fourth order Runge-Kutta method (see Shampine and Corless 2000). Thelocation of the external boundary,R, is determined by solving Eq.21for = 0.5%(n= 1); where, as shown subsequently, the pressure oscillations are negligible. Theset of equations were solved forN= 20. The stretching parameter , was chosen suchthat a relative error of less than 0.1% is achieved (between the exact and Numericalsolutions of the linearized problem, at the well maximum and minimum pressures).

Like the penetration radiusR

, the stretching parameter

, depends solely on theparameter . Finally, the numerical computation was verified through a successfulreproduction ofAyers (1982)results, as seen in Fig. 3.The radial pressure distribu-tions at the end of each of the four periods of a single cycle are shown in the figure.The results are for boundary conditions of two constant dimensionless well pressures(rather than constant mass flow rates), which are: 1.2 for the charging phase and 0.8for the discharging phase.

Fig. 3 Numerical calculateddimensionless pressure as a

function of dimensionlessradius, for an initial pressure of1, a loading pressure of 1.2from 0 to 6 h, an extractionpressure of 0.8 from 10 to 14 h,and zero flow for all otherperiods. (R= 3378.4,= 1.12 107)

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Table 1 Representative ranges of reservoir characteristics, operating conditions and their corre-sponding dimensionless parameters

Variable Definitions Minimum value Maximum value Units

f Porosity 0.05 0.35 unitlessk Permeability 100 5000 mdh Layer height 5 25 mrw Well radius 0.05 0.6 mP0 Initial air pressure 2 7 MPamc Compressor flow rate 1 50 kg/sT Air temperature 300 400 K

Air viscosity 1.8 105 2.4 105 kg/m stp Time period 24 24 hourZ Compressibility factor 0.99 1.01 unitless

p Pneumatic diffusivity 2 102 40 m2/sr2w/p Characteristic time 7

105 15 s

tpp/r2w 5 103 1 109mc mcZRT/( h k P20) 8 105 20t1

t1/tp 6/24 12/24

t2 t1 (t2 t1)/tp 2/24 8/24t3 t2 (t3 t2)/tp 2/24 10/24

5 Results and Discussion

5.1 General Considerations

The analytical analysis reveals the dimensionless parameters affecting the time andspatial pressure distribution within the reservoir. These parameters are , mc , andthe dimensionless time intervals: the charging time t1 , the storage time t

2 t1 , and

the power generation time t3 t2 . Representative ranges of reservoir characteristic,operating conditions and their corresponding dimensionless parameters are listedin Table 1. As seen in the table, both and mc (which represent all the physicalproperties) have a wide range of applicable values. The remaining dimensionlessparameters (the time intervals) are determined so as to meet the local power demand

and production capacity.Analytically and numerically calculated reservoir pressures for a cycle period at

different radii (for steady periodic conditions), are illustrated in Fig. 4a (for theindicated set of operating conditions). The curves reveal a diminution in amplitudeand a progressive phase lag (though small) as rincreases. Additionally, it is apparentthat higher-pressure harmonics fluctuations disappear at increased distance from thewell. The pressure dependence on r, at the end of each time interval, is seen in Fig. 4b.Due to a small phase lag, the curves for t= t1 andt= t3 approximately representthe pressure envelop, which expectedly decreases as rincreases. It is clearly observed

that the penetration of a well pressure fluctuation into the reservoir does not exceeda certain distance; beyond that distance the pressure oscillations are negligible. Notethat the pressure distribution at the end of each storage time (t= t2 , t= 1) is nearlyuniform.

A good agreement exists between the analytical and numerical solutions. Asexpected (see Appendix A), the analytical solution produces somewhat smaller pres-sures. The deviations att= t3 (the highest fluctuation) is smaller than 1%. Further-

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0.68

0.76

0.84

0.92

1

1.08

1.16

0 0.2 0.4 0.6 0.8 1

t*

ps*

NumericalAnalytical

Discharging

r*=1

22.25

256.35

951.65

Charging Storage

Storage

t 1*

t2* t3

*

0.68

0.76

0.84

0.92

1

1.08

1.16

0 0.5 1 1.5 2 2.5 3

log(r*)

ps*

NumericalAnalyticalt

*=t 1

*

1

t2*

t3*

(a) (b)

Fig. 4 Steady periodic dimensionless pressure oscillations for = 5 105, mc= 0.05, t1= 7/24,

t2= 14/24 andt3= 18/24. (a) versus dimensionless time at different dimensionless radii; (b) versusdimensionless radius at different dimensionless times

Fig. 5 Nonlinear effects onthe well pressure at differentpressure fluctuations

0.68

0.76

0.84

0.92

1

1.08

1.16

0 0.2 0.4 0.6 0.8 1

t*

ps

*

NonLinear

Linear

mc*=0.01

.

0.03

0.05

= 5105

t1*=7/24

t2*=14/24

t3*=18/24

more, for the indicated value of, calculated pressures by the asymptotic formulae,Eqs. 37 and 39, coincide with those of the exact expressions.

As seen in Fig.4a, the well pressure undergoes rapid changes at the beginning ofeach time interval (owing to the flow variations). Following those sharp variations,the pressure change rate moderates as it approaches stable conditions. The well pres-sure reaches its crest at the end of the charging stage and drops to its trough at theend of the power generation stage. In order to examine the difference between thecurrent solutions to a solution of the classical linearized equation (of the pressure),

calculations were conducted for several pressure fluctuations. As seen in Fig. 5,non-linear effects of Eq.8,for large pressure fluctuations, produce substantially smallerpressures than those of the linear solution. It stems from the fact that compressiblegas flow acts as a spring rather than a rigid non-compressible flow.

As aforementioned, the fluctuating gas penetration radius can be calculated byconsidering only the fundamental harmonic term of s . The dimensionless radiusR, at which the fundamental harmonic amplitude decreases by a factor is plotted

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Fig. 6 The dimensionless radiusR at which the fundamental harmonic amplitude decreases by afactor

in Fig. 6. According to Eq. 21, for a given , the dimensionless radius R depend

only on . In order to identify the maximum value of that ascertains negligiblepressure oscillations beyond the corresponding R, Eqs. 1418 were solved once more.Subsequently, the boundary condition (17) was replaced by a closed reservoir with anexternal radius equal to the penetration radius, R. The solution was than compared tothat of the unbounded reservoir, Eq.19. Indeed asdecreases the solutions coincide.It turns out that already at = 1% the deviation in amplitude and phase of the wellpressure are negligible. As seen in Fig.6,Ris nearly proportional to the square rootof, and for = 1% is approximately equal to1/2.

5.2 Parametric Study

For the design of a CAES plant, predictions of the well pressure oscillations arerequired, and in particular their maximum and minimum values. The maximum pres-sure is one constraint that the compressor train must work against. The minimumpressure is essentially the turbine inlet pressure (excluding losses). Therefore, theexploration of the well pressure sensitivity to operational parameters is of essence.

The dimensionless parametermcand can vary within several orders of magnitude.The way in which these parameters affect the well pressure oscillations is illustrated inFig.7a and b. As seen in the figures, the pressure oscillation amplitude increases with

bothmc and. From these two parameters, mc is more influencing. That dominancecan also be observed from inspection of Eq. 39, where the modified pressure, s , isdirectly proportional tomc , whileappears as a logarithmical argument. In order tospecifically demonstrate that point, the range of pressure oscillation ps= ps max ps minis plotted in Fig.8a and b as a function ofandmc , respectively. As seen,psis moderately dependent on and strongly onmc . By using several wells, it is possibleto reducemc with its associated pressure fluctuations.

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0.68

0.76

0.84

0.92

1

1.08

1.16

0 0.2 0.4 0.6 0.8 1

t*

ps

*

NumericalAnalytical

mc*=0.03

t 1*=7/24

t2*=14/24

t3*=18/24

5105

5108

=5103

.

0.68

0.76

0.84

0.92

1

1.08

1.16

0 0.2 0.4 0.6 0.8 1

t*

ps

*

NumericalAnalytical

mc*=0.02

.

0.05

0.08

=5103

t 1*=7/24

t2*=14/24

t3*=18/24

(a) (b)

Fig. 7 Steady periodic dimensionless well pressure during a cycle. (a) at differents; (b) at different

m

cs

0.01

0.1

1

2 4 6 8 10

log( )

ps

*

p=

s*

xam

p-

s*

nim

NumericalAnalytical

mc*=0.04

0.02

0.01

.

t 1*=7/24

t2*=14/24

t3*=18/24

0.01

0.1

1

-3 -2.5 -2 -1.5 -1 -0.5

log(m c)

ps

*

p=

s*

xam

p-

s*

nim

NumericalAnalytical

.

=109

105

103

t 1*=7/24

t2*=14/24

t3*=18/24

*

(a) (b)

Fig. 8 Ranges of dimensionless pressure variations. (a) versus at different mc s; (b) versus mc atdifferents

As previously illustrated, Figs. 7and8substantiate the agreement between theanalytical and numerical solutions. Within the range of parameter shown in the fig-

ures, the deviations of the predicted well pressure from the numerical results at t= t3are smaller than 3%. As expected (see Appendix A), for the same pressure fluctua-tions the discrepancy decreases as increases. In effect, for the entire range of therelevant parameters, the asymptotic formulae could be used to calculate the reservoirpressure oscillations. It could be seen from Table1,that the values of the character-istic timer2w/pare significantly smaller than the time intervals of each period. Evenfor the maximum value ofr2w/p(of 15 s) the conditions for the asymptotic solutionsapplicability are fulfilled already within minutes from the beginning of each period.

As previously discussed, all three: the charging time, the power generation time,

and the resulting storage time depend on the local power demand and productioncapacity. Realistic bounds for these time intervals are shown in Table1.Essentially, alarger charging time, or power generation time, produce higher-pressure fluctuations.However, for a given mass of stored air (i.e. mc t1= const), it is expected that largercharging time spans, or power generation time spans, produce smaller pressure fluctu-ations. These assertions are seen in Fig.9,when the modified pressure oscillations atthe well are plotted for different charging time spans (a) and power generation spans

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-40

-30

-20

-10

0

10

20

0 0.2 0.4 0.6 0.8 1

t*

s

*

mc*t 1

*

t 1*=7/24

9/2412/24

.

r*=1

=5105

t2*=14/24

t3*=18/24

-40

-30

-20

-10

0

10

20

0 0.2 0.4 0.6 0.8 1

t*

s

*

t2*=12/24

t3*=22/24

13/24

20/24

14/24

18/24

r*=1

=5105

t 1*=7/24

mc*t 1

*.

(a) (b)

Fig. 9 The variation of s /mc t1 during a cycle for a fixed mass stored (mc t

1= const). (a) for

different charging time spans; (b) for different power generation time spans

(b). Evidently, it is preferred to expand the compression and power generation timespans as much as possible to mitigate the pressure fluctuations, and associated losses.

6 Conclusions

A combined analytical and numerical investigation of the steady periodic gas flowaround a well in porous reservoirs for a CAES plant operation was conducted. Auseful closed form solution for the reservoir pressure distribution was developed.It was shown that, for practical conditions, this solution is sufficiently accurate torepresent the full non-linear equation solution. Moreover, the analysis provides simpleand useful asymptotic expressions. The following conclusions were drawn from theinvestigation:

By incorporating a pertinent variable substitution, it is possible to obtain an equa-tion that is insensitive to non linear effects caused by the gas compressibility,provided that the time is substantially larger than the characteristic time r2w/p(see also Appendix A).

For a given set of charging, storage and production periods, the dimensionlesspressure fluctuations were found to depend on the dimensionless parameters andmc . From the two,mc is the more influencing parameter.

It is desirable to expand the compression and power generation time spans, asmuch as possible, to mitigate the pressure fluctuations and associated losses, for agiven air mass.

As it turns out, the dimensionless distance R, which characterizes the effectivestorage radius, depends only on , and is close in value to 1/2.

The analytical solution can be used to construct a solution for multiple well systems.It provides an important tool that could eventually support compressed air storageoptimization analyses. Field test data are needed for model validation, and for possiblemodel adjustments, to account for real reservoir conditions.

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Appendix A: Error Analysis

The solution for the modified pressure was obtained through the linearization ofthe governing differential equation. In order to estimate the errors entailed by the

solution method, a simple analysis is presented (seeAmetov and Danielyan 1973).Consider the problem for the modified pressure at the first time interval 0 t t1 .

During that interval, the differential equation and, both, the initial and boundaryconditions are

t = p

2

r2+ 1

r

r

=

1 +

2

r2+ 1

r

r

(A.1)

(r, 0) = 0,

r|r=1= mc , (, t) = 0 (A.2)

Now, let the functions1and2satisfy the conditions (A.2) and the equations

1t

= pmin

21r2

+ 1r

1r

,

2 t

= pmax

22r2

+ 1r

2r

(A.3)

where pmin and pmax represent the upper and lower limits of the reservoir dimen-

sionless pressure. Clearly, pmin=1, while the upper limit remains unknown. Since/t 0, the following inequalities hold true

1 t 1 + 1

21r2 +

1

r

1r , 2

t 1 + 222r2 +

1

r

2r

(A.4)It follows from (A.4) that(Ametov and Danielyan 1973)

1 2 (A.5)hence,1and

2are the lower and upper bounds of the unknown exact solution

.When t >> 1, the solutions of Eqs. A.3under conditions(A.2) are(Ritchie andSakakura 1956)

1= mc2 Eir

2

4t

, 2= mc2 Ei r

2

4pmaxt

(A.6)

Particularly at the well

1=mc2 ln

4t

C

, 2=

mc2 ln

4pmaxt

C

(A.7)

with a relative error

RE = 2 1

1= lnpmax

ln

4t

C

(A.8)

It is seen that the error depends on pmax and t. Obviously, larger values ofpmax

entails greater errors. However, larger values oft produces lower errors. For thetypical values of= 5 105, t= t1= 9/24, and anmc s that correspond withpmax=1.2, 1.6, and 2; one gets relative errors of 1.41%, 3.63%, and 5.35%, respectively. Asseen, even if the well pressure is doubled, the relative error between the upper andlower reservoir modified pressure limits is merely 5%.

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The same analysis can be applied to the withdrawal of air from the reservoir. It canbe done by replacingmc with mc in (A.2). For this case, pmax= 1 and pmin is theunknown. However, since /t 0, 2 represents the lower bound and 1 theupper bound of the unknown exact solution . Again, if the well pressure is reduced

by a half, the error between the limits would be about 5%.The comparison with the numerical solution validates the conclusions drawn from

the simple analysis. First, it shows that the analytical solution predicts somewhat lowerpressures. Second, it shows that for identical extreme pressure values, the relative errordecreases as increases. Finally, within the range of parameter checked, the deviationsof the predicted well pressure from the numerical results, at the maximum fluctuationpoints, were smaller than 3%. It is therefore concluded that for the present problem,where the values of the characteristic time r2w/p are significantly smaller than thetimet(t >>1) and the pressure fluctuations are moderate, the nonlinearity effectsof Eq.A.1are negligible.

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