Porous media pressure distribution in centrifugal fields



[1] The simplest use of centrifuges to measure soil properties relies on steady state conditions. Analytical solutions, especially if they are simple, make interpretation of data more direct and transparent. Previous approximations are simplified and have a greatly improved accuracy. Using previous examples as a test, the error on pressure is always less than 1%, compared to about 10% with previous approximations.

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

display math(1)

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

display math(2)

[5] We changed the signs of the constants A, C, and D so they are positive here. inline image is the pressure relative to the pressure at the bottom of the column, inline image so inline image at inline image 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., inline image and ko is a characteristic conductivity value. With w the angular velocity and q the flux density,

display math(3)

[6] Note that if inline image the D term in equation (1) is equal to zero, if inline image 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]  inline image in equation (3) represents the relative importance of centrifugal and capillary forces, whereas inline image shows the impact of the flux density, i.e., with inline image 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 (Rb, 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:

display math(4)


display math(5)
display math(6)

with boundary condition,

display math(7)

inline image and inline imageare 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 inline image Table 1 gives the corresponding values of inline image and inline image as well as inline image which is the top of the column at inline image

Table 1. Parameters Necessary to Plot the Analytical Results for Basha and Mina [1999], Examples in Figure 3 for inline image and Figures 5 and 6 for inline image With Three Below the Asymptote, inline image and One Above in Each Casea
n = 2
 CAf1r1Ψ1r2 inline image
inline image0.511.8543.1750.6302.3810.493
inline image0.532.1924.5790.4373.4340.667
inline image512.6519.8642.0287.398
inline image533.1656.8402.9245.1300.219
n = 5
  1. a

    For the two cases above, the asymptotes are for, inline image

 CAβ−1λf1r1Ψ1r2 inline image
inline image0.510.36779.0551.3043.7040.8572.77724.43
inline image0.530.39358.9171.4356.0350.7594.52656.55
inline image51  1.3717.7941.2665.846
inline image530.339114.2671.5094.7831.4303.5885.718

[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 inline image where f still obeys equation (6) but satisfies the condition

display math(8)

[11] For r large, inline imageand using this estimate to calculate inline imagewe obtain to the first order

display math(9)

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

display math(10)

[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 inline image down to inline image Using this value of inline image as a check, we can find inline image and inline image Table 2 gives those values for inline image as well as the value obtained numerically. We find that inline image is always too small and inline image 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 inline image shown in equation (12)

display math(11)


display math(12)
Table 2. Values of inline image From Equations (10) and (9) for Various na
nf20f10NumericsEquation (12)
  1. a

    The corresponding numerical results and the predictions of equation (12) are also given.


[13] Note that inline image 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, inline image equation (12) predicts the exact value of inline image With the example of inline image 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, inline image we consider the case when part of inline image is close to inline image For that case, we rewrite equation (6) as

display math(13)

and linearize that equation to obtain

display math(14)

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

display math(15)

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

display math(16)

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

display math(17)

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

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

display math(18)

where we used inline image Hence, inline image or

display math(19)

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

display math(20)


display math(21)

where inline image is the value of inline image

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

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

display math(22)

and define g, following equation (21) as

display math(23)

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

display math(24)

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

display math(25)

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

display math(26)

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

display math(27)

[21] Starting at inline image equation (11) yields inline image; then equations (26), (27), and (25) at inline image relate the three unknowns: inline image (note that inline image is irrelevant and could be taken equal to 1 without loss of generality). Note also that if by chance inline image i.e., inline image 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 inline image or

display math(28)

[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 inline image

2. Application to the Examples of Basha and Mina

[23] Examples are for n = 2, about the minimum value for a clay, and n = 5, typical value for a sand [Basha and Mina, 1999]. As explained earlier, integer values, especially even ones, give negative branches, inline image which are not physical, but will be touched upon here for mathematical completeness. Among even integers, n = 2 has an exact solution expressible in terms of Airy functions. Others values of n yielding exact analytical solutions are n = 0; ½ and 1 which are not considered here, as they are too small to have physical meaning.

[24] For inline image we can write exactly,

display math(29)

[25]  inline image being the two Airy functions, with

display math(30)

[26] Note that here the subscript “0” refers here to values at r = 0, and not to values at inline image

[27] The case n = 2 is also unique as equations (26) and (27), together with equation (25) at inline image yield inline image which is also in agreement with equation (28).

[28] Figure 1 shows a variety of curves for n = 2, differing from their starting value at r = 0; from the top (as indicated on the figure) with inline image inline image Note that the curves inline image are such that the product of their inline image is equal to inline image In that case, according to equations (20) and (25), inline image and hence inline image should be the same as long as they are large enough for our asymptotic calculations to hold. Clearly, this is true when inline image but not for inline image as expected. Figure 2 compares numerical and analytical solutions for the Basha and Mina cases for inline image (each curve is identified by the value of inline image). When equation (6) is used, the comparison includes the nonphysical region of inline image with essentially no discrepancy. Figure 3 repeats the comparison with equation (1), and inline image showing more details; of course, the agreement is excellent.

Figure 1.

Exact solutions inline image for inline image at different starting points at inline image with inline image Only values for inline image have physical meaning.

Figure 2.

Four cases for inline image following the example of Basha and Mina [1999]. Numbers for each curve identify the starting values inline image see Table 1. Solid lines are the numerical results and the dots are the analytical results. The two asymptotes labeled inline image correspond to inline image and inline image in equation (29). Although the agreement of numerics and analysis is excellent, between the two asymptotes, only for inline image are the results physically meaningful.

Figure 3.

Details of the examples of Basha and Mina [1999] using the variables of equation (1), with inline image The solid lines are the numerical results and dots the analytical results.

[29] Figure 4 shows the general mathematical case for inline image including inline image which, again, would not be present if n was not an integer. The figure is much simpler than the corresponding one for inline image because only even integers have a solution inline image Figure 5 shows the comparison between the numerics and the analysis using equation (1) when equation (28) rather than equation (27) is applied which greatly simplifies the calculation. The figure shows that for C small, the maximum error is around 3%, more than the chosen threshold of 1%. The difficulty of taking either equation (27) or (28) to estimate inline image did not appear for inline image as both gave at once inline image Figure 6 shows that when equation (27) is applied, the error disappears, which is natural since the derivative condition is applied where the boundary condition is used rather than a curvature condition at inline image

Figure 4.

Sketch of two curves for inline image identified by the value of inline image one slightly above inline image one slightly below (for this last one, only the part with inline image is physically meaningful). In this case, inline image being an odd integer, there is only one asymptote inline image starting at inline image

Figure 5.

Details of Basha and Mina's [1999] cases for inline image when the simple equation (28) is used, showing the significant error when inline image is small. The analysis is shown by dots and the numerics by solid lines.

Figure 6.

Same cases as in Figure 5 using equation (27), rather than equation (28). The errors for the cases with inline image small have disappeared. The analysis is shown by the dots and the numerics by the solid lines.

[30] In all cases, equation (11) is used to obtain inline image for a given inline image However, for a given inline image to obtain inline image we used an iterative procedure. We start with inline image and use this value to obtain an estimate of the term in the inline image bracket in equation (11) and use the new estimate of inline image thus obtained to repeat the procedure. Numerically, equation (6), with inline image is integrated using a Runge-Kutta procedure, starting with inline image where inline image is very large, larger than any inline image of interest, e.g., inline image Integrating backward, the asymptote is approached very quickly, yielding a stable solution. Forward integration, starting at a point very close to the asymptote, yields an unstable solution which eventually diverges from the asymptote. This is clearly seen in Figure 1, where the curves starting at inline image with inline image equal to inline image and inline image which are close to the exact value of inline image still diverge at the short distance when inline image

[31] The values of inline image in Table 1, i.e., the asymptotes when inline image are obtained starting from the boundary condition inline image As explained above, inline image is then calculated and inline image is obtained from equation (20). Using those values n equation (21) yields inline image from equation (11).

3. Conclusion

[32] We have obtained an extremely accurate approximation to predict pressure in a centrifuge for steady state conditions when conductivity is a power law of pressure. The accuracy makes the use of the solution quite reliable to predict soil water properties. The two difficulties in previously available approximations, i.e., using two different approximations depending on soil water properties, and limited accuracy, have been resolved. Here, the form of the approximation depends only on whether inline image is greater or less than inline image