## 1. Introduction

[2] Starting with the pioneering work of *Alemi et al*. [1976], centrifuges have been a convenient tool to measure quickly soil properties. Effectively increasing the effect of gravity shortens the duration of experiments, although as a consequence, care must be taken so that measured capillary pressures have their static values [*Oung et al*., 2005]. Most experiments have been carried out under steady state conditions for simplicity and reliability. Nimmo and coworkers [*Nimmo et al*., 1987; *Nimmo*, 1990; *Simunek and Nimmo*, 2005; *Caputo and Nimmo*, 2005] adapted the steady state results to interpret transient experiments as well. Some applications are ideally suited for centrifuge, e.g., flow in fractures [*Levy et al*., 2002]; colloids transport in porous media [*Sharma et al*., 2008]; air sparging [*Marulanda et al*., 2000]; geo-environmental problems [*Savvidou and Culligan*, 1998]. There have been many other important contributions to the field which are described in the careful review of *van den Berg et al*. [2009].

[3] To transfer results from the centrifuge to the prototype, scaling laws are required [*Culligan and Barry*, 1998; *Barry et al*., 2001]. Interpretation of data is not easy and requires very careful numerical simulations [*Ataie-Ashtiani et al*., 2003]. *Basha and Mina* [1999] pointed out the great advantage of analytical solutions, when attainable, because of their simplicity and transparency, and also if they can be used as a check of the accuracy of the numerical solutions. *Basha and Mina* [1999] then offered an analytical approximation to be used for steady state measurements of unsaturated hydraulic conductivity with a centrifuge. This case is obviously the most fundamental and they knew full well that their solution was only a first step as it had only a 10% precision on the average and it required two different approximations to cover the whole range of properties. *Parlange et al*. [2001] suggested some minor improvement of the solution with further insight provided by *Basha* [2001]. However, the accuracy, though improved, was still not outstanding with a maximum error around 10%. In this paper, after several years, we are finally able to cover the whole range of conditions with a maximum error of less than 1%.

[4] Following *Basha and Mina* [1999], we write the steady state centrifuge equation as

for a *Gardner* [1958] type of soil water conductivity, *k*,

[5] We changed the signs of the constants A, C, and D so they are positive here. is the pressure relative to the pressure at the bottom of the column, so at *R* is the distance from the axis of the centrifuge measured in units of the length *L* of the column so that the top of the column is closer than the bottom to the axis of rotation, i.e., and *k _{o}* is a characteristic conductivity value. With

*w*the angular velocity and

*q*the flux density,

[6] Note that if the *D* term in equation (1) is equal to zero, if the *D* term can always be absorbed in the *AR* term by changing the position of *R =* 0. In the following, we drop the *D* term without any loss of generality.

[7] in equation (3) represents the relative importance of centrifugal and capillary forces, whereas shows the impact of the flux density, i.e., with the equilibrium pressure is obtained when centrifugal and capillary forces balance each other with no flow.

[8] A first important step is to reduce the number of parameters from three (*R _{b}, A*, and

*C*) to two by relinquishing the condition that the length of the column be taken as unit of length. To do so, we change variables taking:

with

with boundary condition,

and are now the only two parameters entering the problem.

[9] We take the examples of *Basha and Mina* [1999], which cover a wide range of conditions, i.e., *A =* 1 and 3; *C =* 5 and 0.5; with *n =* 2 and 5, eight cases altogether. Their boundary condition, equation (3), was for Table 1 gives the corresponding values of and as well as which is the top of the column at

n = 2 | |||||||
---|---|---|---|---|---|---|---|

C | A | f_{1} | r_{1} | Ψ_{1} | r_{2} | ||

0.5 | 1 | 1.854 | 3.175 | 0.630 | 2.381 | 0.493 | |

0.5 | 3 | 2.192 | 4.579 | 0.437 | 3.434 | 0.667 | |

5 | 1 | 2.651 | 9.864 | 2.028 | 7.398 | – | |

5 | 3 | 3.165 | 6.840 | 2.924 | 5.130 | 0.219 |

n = 5 | |||||||||
---|---|---|---|---|---|---|---|---|---|

^{a}For the two cases above, the asymptotes are for,
| |||||||||

C | A | β^{−1} | λ | f_{1} | r_{1} | Ψ_{1} | r_{2} | ||

0.5 | 1 | 0.367 | 79.055 | 1.304 | 3.704 | 0.857 | 2.777 | 24.43 | |

0.5 | 3 | 0.393 | 58.917 | 1.435 | 6.035 | 0.759 | 4.526 | 56.55 | |

5 | 1 | 1.371 | 7.794 | 1.266 | 5.846 | – | |||

5 | 3 | 0.339 | 114.267 | 1.509 | 4.783 | 1.430 | 3.588 | 5.718 |

[10] To solve equation (6), we have to consider two regions separately, an upper and lower region. Those two regions are separated by a boundary where *f* still obeys equation (6) but satisfies the condition

[11] For *r* large, and using this estimate to calculate we obtain to the first order

and to the second order, using the first order to calculate

[12] Higher-order terms could easily be calculated, but will not be found necessary. Clearly, equations (9) and (10) should be accurate when *r* is large; however, we would like to find an accurate down to Using this value of as a check, we can find and Table 2 gives those values for as well as the value obtained numerically. We find that is always too small and too large, suggesting that some “average” would be more accurate. In equation (11), the geometric average of the second terms in equations (9) and (10) were used, giving the value shown in equation (12)

yielding,

n | f_{20} | f_{10} | Numerics | Equation (12) |
---|---|---|---|---|

^{a}The corresponding numerical results and the predictions of equation (12) are also given.
| ||||

2 | 0.630 | 0.794 | 0.7290 | 0.7309 |

3 | 0.644 | 0.803 | 0.7521 | 0.7519 |

4 | 0.673 | 0.820 | 0.7793 | 0.7785 |

5 | 0.699 | 0.836 | 0.8018 | 0.8008 |

[13] Note that in general, are physically positive; hence the negative solution of equation (11) can only have a mathematical meaning when *n* is a positive integer. Table 2 shows the excellent accuracy of equation (12). Note that for the two limits, equation (12) predicts the exact value of With the example of we shall discuss the negative branch later, again for mathematical interest. The value of *n* can only be known approximately so if it were to change from an even integer value to an infinitesimally close value, the negative branch would suddenly disappear, confirming that the negative branch is not relevant physically.

[14] To solve equation (6), either above or below the boundary, we consider the case when part of is close to For that case, we rewrite equation (6) as

and linearize that equation to obtain

where are between to be chosen later. The solution of equation (14) can be written as

[15] No lower limit was put in the integral as any change could always be absorbed by a new constant We now choose by a simple interpolation between e.g.,

where are constants to be obtained later. Using from equation (16) in equation (15) yields

where we used i.e., the asymptotic approximation for large *r* when can differ the most.

[16] To estimate we first look at the zeros of the denominator in equation (17), where Equation (6) shows that when this happens, around then behaves like which is only possible if and equation (17) gives for

where we used Hence, or

[17] Using now equation (9) to calculate *dr* in the integral in equation (17) (higher-order terms could also be used) yields

with

where is the value of

[18] Since the solution is only physical for equation (20) applies to the upper region, i.e., above the boundary given by

[19] If the boundary condition is below that boundary, no asymptote is available to find As a result, determination of the solution is more difficult to obtain. We take the boundary condition as

and define *g*, following equation (21) as

where and is a constant, with As in the case above the boundary, we could try

[20] Note we change the signs of the as can be zero but not infinite in that region. However, is finite and nonzero so equation (24) can apply only if we keep in the denominator only, then

with and value of at To satisfy the derivative condition at requires at once,

which gives quite easily once is known. Imposing that, equation (25) satisfies the derivative of gives

[21] Starting at equation (11) yields ; then equations (26), (27), and (25) at relate the three unknowns: (note that is irrelevant and could be taken equal to 1 without loss of generality). Note also that if by chance i.e., then equation (27) reduces to equation (26) and we are short one equation. In that case, we would impose that the second derivative is satisfied at or

[22] Obviously, equation (28) would be far easier to use than equation (27) but being a second derivative condition, it is less accurate than a first derivative condition when