Methods Note/

Transient Divergent Flow and Transport in anInfinite Anisotropic Porous Formationby Simon A. Mathias

AbstractWhen seeking to predict plume geometry resulting from fluid injection through partially penetrating wells,

it is common to assume a steady-state spherically diverging flow field. In reality, the flow field is transient. Thesteady-flow assumption is likely to cause overestimation of injection plume radius since the accommodation offluid by increases in porosity and fluid density is ignored. In this paper, a transient solution is developed, resultingin a nonlinear ordinary differential equation expressing plume radius as a function of time. It is shown that theproblem can be fully described by one type curve. A critical time, tc, is identified at which the percentage errorof the steady-state flow solution compared to the fully dynamic problem is less than 1%. Only for large injectionrates and low permeabilities, does tc become greater than 1 h. Nevertheless, an improved approximate solutionis obtained by a simple linearization procedure. The critical time, tc for the new approximate solution is 0.3% ofthat required for the steady-state flow solution.

IntroductionFor many fluid injection scenarios, it is useful to

estimate anticipated plume geometries to inform optimaldesigns of injection strategies. For partially penetratingwells, this is often done by assuming a steady-state spher-ically diverging flow field (Bauer et al. 2001; Schroth andIstok 2005; Huang and Goltz 2006). This may be appro-priate for tracer tests where a quasi-steady flow regimeis developed prior to tracer injection. However, when thefluid to be tracked is injected from the start, a steady-flowassumption will be inaccurate during early times.

Anisotropic permeability is another factor that isoften ignored. In a recent paper, Schroth and Istok (2005)compared an approximate solution for the sphericallydiverging advection dispersion equation (using the methodof Gelhar and Collins 1971) to a numerical simulation ofa partially penetrating well problem. The approximation

Corresponding author: Department of Earth Sciences, DurhamUniversity, Durham DH1 3LE, UK; s.a.mathias@durham.ac.uk

Received July 2009, accepted October 2009.Copyright © 2009 The Author(s)Journal compilation ©2009NationalGroundWaterAssociation.doi: 10.1111/j.1745-6584.2009.00652.x

worked well where the length of the injection region wassmall compared to both aquifer thickness and maximumsolute frontal position, but broke down with increasinganisotropy. In isotropic formations where the ambientflow is negligible compared to the flow during injectionand the boundaries are sufficiently far away, the plumewill ultimately assume a spherical geometry. However,anisotropic permeability causes the plume to adopt anellipsoidal shape.

This paper presents the necessary governing equationsfor predicting injection fluid plume geometry due toinjection into an infinitesimal well point in an infiniteanisotropic porous formation. The resulting nonlinearordinary differential equation is integrated numerically.An approximate closed-form solution is then derivedby applying a simple linearization procedure. The workbuilds on that of Schroth and Istok (2005) by account-ing for anisotropic permeability and compressibility ofthe fluid and formation.

Similarity Solution for the Flow ModelConsider the governing equation for an infinitesimal

well point injecting at a constant rate, Q0 (L3T−1) into

438 Vol. 48, No. 3–GROUND WATER–May-June 2010 (pages 438–441) NGWA.org

an initially static, infinite homogenous anisotropic porousformation (e.g., Bear 1979)

S∂P

∂t= kx

μ

∂2P

∂x2+ ky

μ

∂2P

∂y2+ kz

μ

∂2P

∂z2+ Q0δ(x, y, z),

P (t = 0) = 0 (1)

where P (ML−1T−2) is pressure, t(T) is time,μ(ML−1T−1) is fluid viscosity, kx(L2), ky(L2), andkz(L2) are permeabilities in the principal directionsof anisotropy, x(L), y(L), and z(L) respectively, S =φe(cr + cf) (M−1LT2) is the storage coefficient, φe(−) isthe effective porosity, and cr(M−1LT2) and cf(M−1LT2)

are the formation and fluid compressibilities, respectively.The δ(x, y, z) (L−3) term denotes the Dirac delta function.

Following Bear and Dagan (1965), applying thetransformations:

y ′ =(

kx

ky

)1/2

y and z′ =(

kx

kz

)1/2

z (2)

the anisotropy is eliminated such that

S∂P

∂t= kx

μ

(∂2P

∂x2+ ∂2P

∂y′2 + ∂2P

∂z′2

)+ γ 1/2Q0δ(x, y

′, z

′)

(3)

where

γ = k2x

kykz

(4)

Introducing the radial distance

r2 = x2 + y′2 + z

′2 (5)

leads to the spherically divergent problem:

S∂P

∂t= − 1

4πr2

∂Q

∂r(6)

where Q(r, t) (L3T−1) is a flow rate (transformed withrespect to the anisotropy) found from

Q = −4πr2 kx

μ

dP

dr, Q(r = 0) = γ 1/2Q0 (7)

Application of the similarity transform

ψ = r2t−1 (8)

yields the initial value problem:

dQ

dψ= −SμQ

4kx

, limψ→0

Q = γ 1/2Q0 (9)

which has the analytical solution for the flow rate:

Q(ψ) = γ 1/2Q0 exp

(−Sμψ

4kx

)(10)

Substituting Equation 10 into Equation 7 and inte-grating with respect to r also leads to the equation for thepressure field (Kanwar et al. 1976)

P(r, t) = γ 1/2Q0μ

4πkx

{1

rexp

(−Sμr2

4kxt

)−

(πSμ

4kxt

)1/2

erfc

[(Sμr2

4kxt

)1/2]}

(11)

An Equation for the Plume FrontNow consider a solute, which has been mixed into the

injection fluid. The location of the plume front (ignoringdiffusion and dispersion) at time, t is denoted R = [X2 +(kx/ky)Y

2 + (kx/kz)Z2]1/2. The velocity of the front, v

(LT−1) is found from

dR

dt≡ v = Q(R, t)

4π(φk + ρbKd)R2(12)

where φk(−) is the kinematic porosity, ρb(ML−3) is thebulk density of the formation, and Kd(M−1L3) is the sorp-tion distribution coefficient (assuming linear equilibriumsorption).

Recalling Equation 10 then leads to the nonlinearordinary differential equation:

dR3

dt= 3γ 1/2Q0

4π(φk + ρbKd)exp

(−SμR2

4kxt

),

R(t = 0) = 0 (13)

Substituting the following expressions

t0 = Sμ, R0 =[

3γ 1/2Q0Sμ

4π(φk + ρbKd)

]1/3

, ε = R20

4kx

(14)

then yields:

dR3

dt= R3

0

t0exp

(−εt0R

2

R20 t

)(15)

from which it is seen that appropriate expressions fordimensionless time and dimensionless plume radius are

τ = t

ε3t0and ζ = R

εR0(16)

respectively, such that Equation 15 reduces to:

dζ 3

dτ= exp

(−ζ 2

τ

), ζ (τ = 0) = 0 (17)

The t0 (T) term is a characteristic time. Note that thestorage coefficient, S represents the bulk compressibilityof the system. Therefore, t0 compares how easily a mov-ing fluid compresses itself and the surrounding formationrock with how easily it moves, that is, its viscosity recip-rocal, μ−1. For an incompressible system, t0 is infinitesi-mally small. For a highly viscous fluid, t0 becomes large.For systems of low compressibility and/or low viscos-ity and/or high permeability, R0(L) is approximately theradius of the plume at time t = t0. The parameter, ε(–) isthe ratio of the square of this distance to permeability.

When the flow field is steady (i.e., S = 0)

ζ 0 ≡ limS→0

ζ = τ 1/3 (18)

To evaluate ζ for S > 0, Equation 17 must be solvednumerically.

NGWA.org S.A. Mathias GROUND WATER 48, no. 3: 438–441 439

Note that the expressions for τ and ζ further sim-plify to:

τ =[

4π(φk + ρbKd)

3γ 1/2Q0

]2 (4kx

Sμ

)3

t and

ζ =[

4π(φk + ρbKd)

3γ 1/2Q0

](4kx

Sμ

)R (19)

from which it is easily seen that

limS→0

R3 = 3γ 1/2Q0t

4π(φk + ρbKd)(20)

which, when γ = 1 (isotropic permeability), is identical tothat presented by Schroth and Istok (2005) for steady-statespherically diverging flow.

Simulation of the Plume FrontA question arises as to what value of τ is ζ ≈ ζ 0. To

this end, Equation 17 was solved using MATLAB’s ordi-nary differential equation solver, ODE23s. A comparisonplot of dimensionless plume radius against dimensionlesstime is presented in Figure 1. It can be seen that assumingsteady-state flow leads to an overestimate in plume radius,although this becomes less important with increasing time.This is due to the steady-flow assumption ignoring theaccommodation of fluid by increases in the porosity andfluid density associated with the formation and fluid com-pressibilities, respectively.

Figure 2 presents plots of normalized error betweenEquation 18 and the numerical solution of Equation 17,formally expressed as |ζ − ζ 0|/ζ . Here, it can be seen that

101

100

101

102

103

10 1

100

101

Dimensionless time, τ

Dim

ensi

onle

ss p

lum

e ra

dius

, ζ

ζζ

0

ζ1

ζ2

Figure 1. Plots of dimensionless plume radius againstdimensionless time with ζ (the fully dynamic problem)calculated from the numerical solution of Equation 17,ζ 0 (transient transport with steady flow) calculated fromEquation 18, ζ 1 (the linearized version of the fully dynamicproblem) calculated from Equation 22, and ζ 2 (the linearizedversion of the fully dynamic problem with the E1 termignored) calculated from Equation 23.

100

102

104

106

103

10 2

10 1

100

Dimensionless time, τ

Err

or

|ζ ζ0| / ζ

|ζ ζ1| / ζ

|ζ ζ2| / ζ

Figure 2. Plots of normalized error against dimensionlesstime associated with the three approximations: ζ 0 (transienttransport with steady flow), ζ 1 (the linearized version of thefully dynamic problem), and ζ 2 (the linearized version of thefully dynamic problem with the E1 term ignored).

the error does not get lower than 1% until τ ≥ 110,000.Although this maybe a useful range in many practicalscenarios, it would be convenient if a more accurateexpression was available. Note that the intrinsic errorassociated with the numerical solution is orders of mag-nitude smaller, as evidenced by the logarithmic decline inerror observed with increasing time (Figure 2).

An improved approximate solution, ζ 1 can be obta-ined by substituting ζ 0 into the right-hand side ofEquation 17, that is:

dζ 31

dτ= exp(−τ−1/3) (21)

which on integration yields:

ζ 1 =[(

τ − τ 2/3

2+ τ 1/3

2

)exp(−τ−1/3) − 1

2E1(τ

−1/3)

]1/3

(22)

where E1 denotes the exponential integral function. Asimilar linearization procedure was used by Wen et al.(2008) and Mathias et al. (2008) to obtain approximatesolutions for non-Darcian flow to wells.

A visual comparison of ζ , ζ 0, and ζ 1 is pre-sented in Figure 1. Indeed, it can be seen that ζ 1 morequickly approximates ζ than ζ 0. A plot of normalizederror between Equation 22 and the numerical solution ofEquation 17, formally expressed as |ζ − ζ 1|/ζ , is shownin Figure 2. Here, it can be seen that the error associ-ated with Equation 22 reduces to lower than 1% whenτ ≥ 413, substantially sooner than ζ 0.

The E1 term is relatively insignificant for times,τ ≥ 413 (at least seven orders of magnitude less thanthe exponential term). Therefore, a similarly reasonable

440 S.A. Mathias GROUND WATER 48, no. 3: 438–441 NGWA.org

10–1

100

101

10–8

10–6

10–4

10–2

100

102

104

Crit

ical

tim

e, t c (

hour

s)

Scaled injection rate, γ1/2 Q0 (m3/hour)

kx = 10–16 m2

kx = 10–15 m2

kx = 10–14 m2

Figure 3. Plots of critical time, tc at which the percentageerror of the steady-state flow solution compared to the fulltransient problem is less than 1% (i.e., τ = 110,000) for dif-ferent anisotropy-scaled flow rates and different permeabil-ities. Other parameters were set as follows: μ = 10−3 Pa s,S = 10−10 Pa−1, φk = 0.2, and Kd = 0.

approximation of ζ is obtained from:

ζ 2 =(

τ − τ 2/3

2+ τ 1/3

2

)1/3

exp

(−τ−1/3

3

)(23)

Surprisingly, a plot of normalized error betweenEquation 23 and the numerical solution of Equation 17,formally expressed as |ζ − ζ 2|/ζ , shown in Figure 2,reveals that the error associated with Equation 23 reducesto lower than 1% when τ is just 355.

At this stage, it is interesting to quantify dimensionaltimes at which the steady-state flow assumption becomesvalid, tc. This is achieved by inverting the expression for τ

(recall Equation 19) for time, t and setting τ = 110,000.Figure 3 shows plots of this critical time, tc against injec-tion rate (scaled for permeability anisotropy) for differ-ent permeabilities with μ = 10−3 Pa s, S = 10−10 Pa−1,φk = 0.2, and Kd = 0. Only for large injection rates andlow permeabilities, does tc become greater than an hour.Generally, the steady-state flow assumption should leadto reasonable plume radius estimates. However, tc willincrease with increasing anisotropy, γ .

Summary and ConclusionsIn this paper, the governing equations for the advec-

tive transport of a fluid injected, at a constant rate, viaan infinitesimal well point into an initially hydrostatic,homogenous, and anisotropic porous medium of infiniteextent are presented. Anisotropy was eliminated via atransformation of the spatial axes. The flow problemthen reduced to a linear ordinary differential equation byapplication of a similarity transform. This was solved toobtain an expression for flow rate as a function of radiusand time. A nonlinear ordinary differential equation was

then developed relating the injection plume radius withtime. For zero storage coefficient, the solution is trivialand analogous to the result one would get when assum-ing steady-state flow. For nonzero storage coefficients,the equation was solved numerically using MATLAB’sODE23s.

A critical time, tc was identified at which the percent-age error of the steady-state flow solution compared to thefully dynamic problem is less than 1%. Only for largeinjection rates and low permeabilities, does tc becomegreater than 1 h. Nevertheless, an improved approximatesolution was obtained by linearizing the nonlinear ordi-nary differential equation by replacing the radius term onthe right-hand side with that predicted assuming steadyflow. The critical time, tc for the new approximate solution(Equation 23) is 0.3% of that required for the steady-stateflow solution.

The above analysis strongly suggests that, for homo-genous systems with low levels of anisotropy, the com-monly used steady-state flow assumption should generallylead to a reasonable plume radius estimate. However, theextent to which this conclusion might be affected by thepresence of realistic heterogeneity is unclear and requiresfurther investigation.

AcknowledgmentsThis project was funded by the WorleyParsons Eco-

NomicsTM initiative, the National Grid Property Hold-ings Limited, and the UK Technology Strategy Board.Valuable suggestions by Junqi Huang, Ty Ferre, and anadditional anonymous reviewer were highly appreciated.

ReferencesBauer, P., S. Attinger, and W. Kinzelbach. 2001. Transport of

a decay chain in homogenous porous media: analyticalsolutions. Journal of Contaminant Hydrology 49, 217–239.

Bear, J. 1979. Hydraulics of Groundwater. New York: McGraw-Hill.

Bear, J., and G. Dagan. 1965. The relationship between solu-tions of flow problems in isotropic and anisotropic soils.Journal of Hydrology 3, 88–96.

Gelhar, A.D., and M.A. Collins. 1971. General analysis of lon-gitudinal dispersion in nonuniform flow. Water ResourcesResearch 7, no. 6: 1511–1521.

Huang, J., and M.N. Goltz. 2006. Analytical solutions for solutetransport in a spherically symmetric divergent flow field.Transport in Porous Media 63, 305–321.

Kanwar, R.S., S.D. Khepar, H.S. Chauhan, and G. Das. 1976.A simplified approach for the analysis of unsteady flow toa cavity well. Ground Water 14, no. 2: 101–104.

Mathias, S.A., A.P. Butler, and H. Zhan. 2008. Approximatesolutions for Forchheimer flow to a well. ASCE Journalof Hydraulic Engineering 134, no. 9: 1318–1325.

Schroth, M.H., and J.D. Istok. 2005. Approximate solutionfor solute transport during spherical-flow push-pull tests.Ground Water 43, no. 2: 280–284.

Wen, Z., G. Huang, and H. Zhan. 2008. An analytical solutionfor non-Darcian flow in a confined aquifer using the powerlaw function. Advances in Water Resources 31, 44–55.

NGWA.org S.A. Mathias GROUND WATER 48, no. 3: 438–441 441