INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2013; 37:1908–1910Published online 5 February 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2174

Forchheimer and added mass effects in the flow towards a well

José Jorge Nader*,†

Department of Structural and Geotechnical Engineering, Polytechnic School, University of São Paulo, 05508-900, SãoPaulo, Brazil

SUMMARY

In this note, we examine the flow towards a well in a confined aquifer in the presence of an interaction forcedefined by the sum of three terms, namely, a Darcy term (linear in the velocity), a Forchheimer term(quadratic in the velocity), and an added-mass term (linear in the acceleration). We obtain the exact dynamicsolution for the piezometric head distribution around the well and investigate the relative importance of thenon-Darcian terms. Copyright © 2013 John Wiley & Sons, Ltd.

Received 27 August 2012; Revised 26 September 2012; Accepted 16 December 2012

KEY WORDS: soil hydrodynamics; seepage; groundwater; high velocity flow; Forchheimer; added mass

In higher velocity flows, differently from usual seepage problems, the force exerted by the soil particleson the moving water is not well described by a linear function of the velocity (e.g., [1–5]). In order tomodel such flows, more complex expressions for the interaction force have been proposed as, forinstance, the following quite general law [6], obtained from the traditional linear Darcy force by theaddition of a Forchheimer term (quadratic in the velocity) and an added mass term (linear in theacceleration):

f ¼ �av� b vk kv� la; (1)

valid only for incompressible fluids. In the aforementioned equation, f is the interaction force per unitvolume, v is the velocity (‖v‖ denotes the norm of v), a is the acceleration, and a, b, and l are positiveconstants (only homogeneous soils will be treated here). In the Darcy term, a is related to the porosityn, the water unit weight g, and the permeability coefficient k through a = n g/k.

The aim of this short note is to investigate the importance of the two non-Darcian terms in the steadyflow of water towards a well in a confined aquifer. For that purpose, we derive the dynamicsolution (i.e., with the acceleration taken into account in the equation of motion) for the piezometrichead field around the well in the presence of an interaction force given by Equation 1.

Consider first the differential equation of motion: - grad p� gez + f= ra, which, after theintroduction of the piezometric head, H= z + p/g, becomes

-g gradH þ f ¼ ra: (2)

In the aforementioned equations, p is the pressure, ez is the unit vector pointing upward (associatedto the vertical coordinate z: ez = grad z) and r is the water density.

*Correspondence to: José Jorge Nader, Department of Structural and Geotechnical Engineering, Polytechnic School,University of São Paulo, 05508-900, São Paulo, Brazil.†E-mail: jjnader@usp.br

Copyright © 2013 John Wiley & Sons, Ltd.

FORCHHEIMER AND ADDED MASS EFFECTS IN THE FLOW TOWARDS A WELL 1909

To study the radial flow to a well, it is convenient to employ cylindrical coordinates (r, θ, z), withbasis vectors (er,eθ,ez). We search for the distribution of H corresponding to a radial steady flowwith velocity field v(r) =� v(r)er. The acceleration field is, therefore, expressed by a(r) =� [v2(r)/r]er.

Consider now the continuity equation: div(nv) = 0. For constant n, it reduces to d(rv (r))/dr= 0, sothe product rv(r) is constant, say, rv(r) = w. Hence v=� (w/r)er and a =� (w2/r3)er. If D is the aquiferthickness, the flow rate is Q= 2prDnv(r) = 2pDnw. The interaction force (Equation 1), in turn, readsf= (aw/r +bw2/r2 + lw2/r3) er. From Equation (2), we obtain

dH

drrð Þ ¼ 1

gawrþ bw2

r2þ rþ lð Þw2

r3

� �; (3)

which, after integration, yields

Figure 1. A, B, and C as a function of the distance r from the well axis. (ΔH=A+B +C).

Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2013; 37:1908–1910DOI: 10.1002/nag

1910 J. J. NADER

ΔH ¼ awg

lnr

r0þ bw2

g1r0

� 1r

� �þ rþ lð Þ w2

2g1r20

� 1r2

� �; (4)

where ΔH=H(r)�H(r0) is the piezometric head difference between a point located at a distance rfrom the well axis and a point at the well (r0 is the well radius). For the sake of simplicity, the threeterms on the right side of Equation 4 will be indicated by A, B, and C, respectively. In thetraditional quasi-static solution for Darcian flows (e.g., [7]), obtained with a= 0, a 6¼ 0, b= l = 0,only A is present. In the dynamic solution for Darcian flows [8], obtained with a 6¼ 0, a 6¼ 0,b= l = 0, only A and part of C (i.e., C with l = 0) are present.

In Equation (4), note first that A is a linear function of the flow rate Q (recall that Q= 2pDnw),whereas B and C vary as Q2. All three terms start from 0 (at r = r0) and increase monotonically withr (at a decreasing rate). But, while A increases without bound, B and C are limited, tendingrespectively to B1= bw2/gr0 and C1 ¼ rþ lð Þw2=2gr20, as r!1.

Let us now evaluate the significance of B and C in relation to A. The ratios

b ¼ B

A¼ bw

ar�10 � r�1

lnr=r0; (5)

c ¼ C

A¼ rþ lð Þw

2ar�20 � r�2

lnr=r0(6)

are decreasing functions of r and so the role of B and C, in comparison with A, is more important nearthe well. An upper bound for the values of b and c can be obtained by computing the limits as r tends tor0: b and c tend to b0 = bw/ar0 and c0 ¼ rþ lð Þw=ar20, respectively.

The existence of upper bounds for B, C, b and c is an interesting property of the solution. Theyare useful for assessing the relevance of the non-Darcian terms. Clearly, B and C are more importantfor higher flow rates and smaller well radii. If Q is doubled, b0 and c0 are also doubled,whereas B1 and C1 are multiplied by four. If r0 is halved, b0 and B1 are doubled, whereas c0and C1 are multiplied by four.

In order to illustrate the magnitude of these numbers in a particular case, consider the followingvalues: r0 = 0.10 m, Q/D= 10� 2m3/s/m, n = 0. 3, r= 103kg/m3, g = 9.81� 103N/m3, and, for a coarsesand [6, 9]: a = 2� 104 Pa m� 2s, b= 3� 105 Pa m� 3s2, l = 2� 103Pa m� 2s2. In this case:B1ffi 8.6� 10� 3m, C1ffi 4.3� 10� 4m, b0ffi 8� 10� 1, c0 = 8� 10� 2. For the adopted values,Figure 1 shows graphs of A, B, and C as a function of r.

Finally, for a direct comparison of the terms B and C, note that the ratio d ¼ C=B ¼rþ lð Þ r�1

0 þ r�1� �

=2b does not depend on Q and decreases with r. At r= r0, d equals d0 = (r+ l)/br0 (upper bound); and as r!1, d! d1= (r+ l)/2br0 (lower bound). Both vary inversely as r0.With the numerical values presented earlier: d0ffi 10� 1, d1ffi 5� 10� 2.

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Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2013; 37:1908–1910DOI: 10.1002/nag