We will in this section analyze four variants of the capillary tube model (Scheidegger 1953, 1974). In each case, we calculate the co-moving velocity \(v_m\) and then use it to demonstrate the consistency of our theory. We also use this section to clarify the physical meaning of the area derivatives introduced in Sect. 3.
Parallel Capillaries Filled with Either Fluid
In this simplest case, we envision a bundle of N parallel capillaries, all equal. Each tube has a cross-sectional inner (i.e., pore) area \(a_p\). \(N_w\) of these capillaries are filled with the wetting fluid, and \(N_n\) are filled with the non-wetting fluid, so that \(N_w+N_n=N\). Let us assume that the seepage velocity in the capillaries filled with wetting fluid is \(v_w\) and the seepage velocity in the capillaries filled with non-wetting fluid is \(v_n\). The total volumetric flow rate is then
$$\begin{aligned} Q=N_w a_p v_w + N_n a_p v_n\;. \end{aligned}$$
(49)
The wetting saturation is given by
$$\begin{aligned} S_w=\frac{A_w}{A_p}=\frac{N_w}{N}\;. \end{aligned}$$
(50)
We calculate the derivative of Q with respect to \(A_w\) to determine the thermodynamic velocity \({\hat{v}}_w\) defined in Eq. (18) together with Eq. (49),
$$\begin{aligned} {\hat{v}}_w=\left( \frac{\partial Q}{\partial A_w}\right) _{A_n}= \frac{1}{a_p\delta N_w}\ \left[ Q(N_w+\delta N_w,N_n)-Q(N_w,N_n)\right] =v_w\;. \end{aligned}$$
(51)
The derivative is thus performed changing the number of capillaries from N to \(N+\delta N\) and letting all the added capillaries contain the wetting fluid so that \(N_w\rightarrow N_w+\delta N_w=N_w+\delta N\). Likewise, we find using Eq. (19)
$$\begin{aligned} {\hat{v}}_n=\left( \frac{\partial Q}{\partial A_n}\right) _{A_w}= \frac{1}{a_p\delta N_n}\ \left[ Q(N_w,N_n)-Q(N_w,N_n+\delta N_n)\right] =v_n\;. \end{aligned}$$
(52)
From Eqs. (33) and (34), we find that the co-moving velocity is zero,
$$\begin{aligned} v_m=0\; \end{aligned}$$
(53)
for this system. Hence, we have that the seepage velocities \(v_w\) and \(v_n\) are equal to the thermodynamic velocities \({\hat{v}}_w\) and \({\hat{v}}_n\).
Parallel Capillaries with Bubbles
Suppose that we have N parallel capillaries as shown in Fig. 3. Each tube has a length L and an average inner area \(a_p\). The diameter of each capillary varies along the long axis. Each capillary is filled with a bubble train of wetting or non-wetting fluid. The capillary forces due to the interfaces vary as the bubble train moves due to the varying diameter. We furthermore assume that the wetting fluid, or more precisely the more wetting fluid (in contrast to the non-wetting fluid which is the less wetting fluid) does not form films along the pore walls so that the fluids do not pass each other; see Sinha et al. (2013) for details. The volume of the wetting fluid in each tube is \(L_w a_p\), and the volume of the non-wetting fluid is \(L_n a_p\). Hence, the saturations are \(S_w=L_w/L\) and \(S_n=L_n/L\) for each tube.
Suppose now that the seepage velocity in each tube is v when averaged over time. Both the wetting and non-wetting seepage velocities must be equal to the average seepage velocity since the bubbles do not pass each other:
$$\begin{aligned} v_w=v_n=v\;. \end{aligned}$$
(54)
We now make an imaginary cut through the capillaries orthogonal to the flow direction as shown in Fig. 3. There will be a number \(N_w\) capillaries where the cut passes through the wetting fluid and a number \(N_n\) capillaries where the cut passes through the non-wetting fluid. Averaging over time, we must have
$$\begin{aligned} \frac{\langle N_w\rangle }{N} = \frac{L_w}{L}=S_w \end{aligned}$$
(55)
and
$$\begin{aligned} \frac{\langle N_n\rangle }{N} = \frac{L_n}{L}=S_n\;. \end{aligned}$$
(56)
Hence, the wetting and non-wetting areas defined in Sect. 2 are
$$\begin{aligned} A_w=\langle N_w\rangle a_p \end{aligned}$$
(57)
and
$$\begin{aligned} A_n=\langle N_n\rangle \ a_p;. \end{aligned}$$
(58)
We also have that
$$\begin{aligned} A_p=N a_p\;. \end{aligned}$$
(59)
We will now calculate derivative (18) defining the thermodynamic velocity \({\hat{v}}_w\). We do this by changing the pore area \(A_p\rightarrow A_p+\delta A_p=Na_p+a_p \delta N\). We wish to change \(A_w\) while keeping \(A_n\) fixed. This can only be done by adjusting \(S_w\) while changing \(\delta N\). This leads to the two equations
$$\begin{aligned} \delta \langle N_w\rangle =\delta [N S_w]=N\delta S_w+S_w\delta N=\delta N\;, \end{aligned}$$
(60)
and
$$\begin{aligned} \delta \langle N_n\rangle =\delta [N (1-S_w)]=-N\delta S_w+(1-S_w)\delta N=0\;. \end{aligned}$$
(61)
We solve either (60) or (61) for \(\delta S_w\) (they contain the same information), finding
$$\begin{aligned} \delta S_w=\frac{(1-S_w)}{N}\ \delta N\;. \end{aligned}$$
(62)
Hence, we have
$$\begin{aligned} {\hat{v}}_w= & {} \left( \frac{\partial Q}{\partial A_w}\right) _{A_n} =\frac{1}{\delta A_p}\ \left[ (A_p+\delta A_p) v(S_w+\delta S_w)-A_pv(S_w)\right] \nonumber \\= & {} \frac{1}{\delta N}\ \left[ (N+\delta N) v(S_w+\delta S_w)-Nv(S_w)\right] \nonumber \\= & {} \frac{1}{\delta N}\ \left[ (N+\delta N) [v(S_w)+(\mathrm{d}v/\mathrm{d}S_w)\delta S_w]-Nv(S_w)\right] \nonumber \\= & {} \frac{1}{\delta N}\ \left[ (N+\delta N) [v(S_w)+(1-S_w)(\mathrm{d}v/\mathrm{d}S_w)(\delta N/N)]-Nv(S_w)\right] \nonumber \\= & {} v(S_w)+S_n\ \frac{\mathrm{d}v(S_w)}{\mathrm{d}S_w}\;. \end{aligned}$$
(63)
We recognize that this equation is Eq. (27). Hence, we could have taken this equation as the starting point of discussing this model. However, doing the derivative explicitly demonstrates its operational meaning.
Rather than performing a similar calculation for \({\hat{v}}_n\) as in (63) for \({\hat{v}}_w\), we use Eq. (29) and have
$$\begin{aligned} {\hat{v}}_n=\left( \frac{\partial Q}{\partial A_n}\right) _{A_w}=v(S_w)-S_w\ \frac{\mathrm{d}v(S_w)}{\mathrm{d}S_w}\;. \end{aligned}$$
(64)
We now combine Eqs. (54), (63) and (64) with (33) and (34) and read off
$$\begin{aligned} v_m=\frac{\mathrm{d}v}{\mathrm{d}S_w}\;. \end{aligned}$$
(65)
We see that the expression for \(v_m\) is independent of the constitutive equation that relates flow rate to the driving forces. It expresses that both fluid species are forced to move with the same seepage velocity, whereas the thermodynamic velocities remain different for the two fluids.
Combining Eq. (65) for \(v_m\) with Eqs. (33) and (34) gives Eq. (54) as it must.
This result gives us an opportunity to clarify the physical meaning of the co-moving velocity \(v_m\). We have assumed that the bubbles cannot pass each other as shown in Fig. 3. Equation (65) gives upon substitution in Eqs. (33) and (34),
$$\begin{aligned} v_w={\hat{v}}_w-S_n\frac{\mathrm{d}v}{\mathrm{d}S_w}\;, \end{aligned}$$
(66)
and
$$\begin{aligned} v_n={\hat{v}}_n+S_w\frac{\mathrm{d}v}{\mathrm{d}S_w}\;. \end{aligned}$$
(67)
We see that these two equations compensate exactly for the two Eqs. (27) and (29),
$$\begin{aligned} {\hat{v}}_w= & {} v+S_n\frac{\mathrm{d}v}{\mathrm{d}S_w}\;,\\ {\hat{v}}_n= & {} v-S_w\frac{\mathrm{d}v}{\mathrm{d}S_w}\;,\\ \end{aligned}$$
that give the thermodynamic velocities resulting in \(v_w=v_n=v\).
Parallel Capillaries with a Subset of Smaller Ones
We now turn to the third example. There are N parallel capillaries of length L. A fraction of these capillaries has an inner area \(a_s\), whereas the rest has an inner area \(a_l\). The total pore area carried by the small capillaries is \(A_s\) and the total pore area carried by the larger capillaries is \(A_l\) so that
$$\begin{aligned} A_p=A_s+A_l\;, \end{aligned}$$
(68)
We define the fraction of small capillaries to be \(S_{w,i}\), so that
$$\begin{aligned} A_s=S_{w,i} A_p\;. \end{aligned}$$
(69)
We now assume that the small capillaries are so narrow that only the wetting fluid enters them. These pores constitute the irreducible wetting fluid contents of the model in that we cannot go below this saturation. However, they still contribute to the flow. Hence, at a saturation \(S_w\) which we assume to be larger than or equal to \(S_{w,i}\) which is then the irreducible wetting fluid saturation, we have that the wetting pore area is
$$\begin{aligned} A_w=S_{w,i} A_p+(S_w-S_{w,i})A_p=A_s+A_{lw}\;, \end{aligned}$$
(70)
where we have defined \(A_{lw}=(S_w-S_{w,i})A_p\).
We assume there is a seepage velocity \(v_{sw}\) in the small capillaries, a seepage velocity \(v_{lw}\) in the larger capillaries filled with wetting fluid and a seepage velocity \(v_n\) in the larger filled with non-wetting fluid. The velocities \(v_{sw}\), \(v_{lw}\) and \(v_n\) are independent of each other. The total volumetric flow rate is then given by
$$\begin{aligned} Q=A_s v_{sw}+A_{lw} v_{lw}+A_n v_{n}\;, \end{aligned}$$
(71)
or in terms of average seepage velocity
$$\begin{aligned} v=S_{w,i} v_{sw}+(S_w-S_{w,i}) v_{lw}+S_n v_{n}\;. \end{aligned}$$
(72)
We will now calculate derivative (18) defining the thermodynamic velocity \({\hat{v}}_w\). We could have done this using Eq. (27). However, it is instructive to perform the derivative yet again through area differentials so that their meaning becomes clear.
In order to calculate derivative (19), we change the pore area \(A_p\rightarrow A_p+\delta A_p\) so that \(\delta A_w=\delta A_p\) and \(\delta A_n=0\). In addition, we have that \(\delta A_s=S_{w,i} \delta A_p\), so that
$$\begin{aligned} \delta A_w=\delta A_s+\delta A_{lw}=S_{w,i} \delta A_p+\delta A_{lw}=\delta A_p\;. \end{aligned}$$
(73)
Hence, we have
$$\begin{aligned} \delta A_{lw}=(1-S_{w,i})\delta A_p\;. \end{aligned}$$
(74)
We now combine this expression with Eq. (71) for the total volumetric flow Q to find
$$\begin{aligned} {\hat{v}}_w= & {} \left( \frac{\partial Q}{\partial A_w}\right) _{A_n} =\frac{1}{\delta A_p}\ \left[ \delta A_s v_{sw} + \delta A_{lw} v_{lw} \right] \nonumber \\= & {} v_{lw}+S_{w,i} (v_{sw}-v_{lw}). \end{aligned}$$
(75)
We could have found \({\hat{v}}_n\) in the same way. However, using (29) combined with (72) gives
$$\begin{aligned} {\hat{v}}_n=v_{n}+S_{w,i}(v_{sw}-v_{lw})\;. \end{aligned}$$
(76)
We now use Eq. (34) to find
$$\begin{aligned} {\hat{v}}_n-v_n= - v_m S_w = S_{w,i}(v_{sw}-v_{lw})=-\left[ \frac{S_{w,i}}{S_w}(v_{lw}-v_{sw})\right] S_w\;, \end{aligned}$$
(77)
so that
$$\begin{aligned} v_m=\frac{S_{w,i}}{S_w}\left( v_{lw}-v_{sw}\right) \;. \end{aligned}$$
(78)
As a check, we now use the co-moving velocity found in (78) together with Eqs. (75) and (33) to calculate \(v_w\). We find
$$\begin{aligned} v_w={\hat{v}}_w-v_m S_n=\frac{1}{S_w}\left[ S_{w,i} v_{sw}+(S_w-S_{w,i}) v_{lw}\right] \;, \end{aligned}$$
(79)
which is the expected result; see Eq. (72).
Large Capillaries with Bubbles and Small Capillaries with Wetting Fluid only
The fourth example that we consider is a combination of the two previous models discussed in Sects. 7.2 and 7.3, namely a set of N capillary tubes. A fraction \(S_{w,i}\) of these capillaries has an inner area \(a_s\), whereas the rest has an inner area \(a_l\). The capillaries with the larger area contain bubbles as in Sect. 7.2. The smaller capillaries contain only wetting liquid as in Sect. 7.3. The fluid contents in the small capillaries are irreducible. The seepage velocity in the smaller capillaries is \(v_{sw}\). The wetting and non-wetting seepage velocities in the larger capillaries are the same,
$$\begin{aligned} v_n=v_{lw}=v_l\;. \end{aligned}$$
(80)
Hence, the seepage velocity is then
$$\begin{aligned} v=S_{w,i} v_{sw}+(1-S_{w,i})v_l\;. \end{aligned}$$
(81)
We have that
$$\begin{aligned} \frac{\mathrm{d}v}{\mathrm{d}S_w}=(1-S_{w,i})\frac{\mathrm{d}v_l}{\mathrm{d}S_w}\;, \end{aligned}$$
(82)
since \(v_{sw}\) is independent of \(S_w\). Equation (29) gives
$$\begin{aligned} {\hat{v}}_n=S_{w,i} v_{sw}+\left( 1-S_{w,i}\right) \left( v_l-S_w\frac{\mathrm{d}v_l}{\mathrm{d}S_w}\right) \;, \end{aligned}$$
(83)
where we have used (81). We now use Eqs. (34) and (80) to find
$$\begin{aligned} {\hat{v}}_n-v_n={\hat{v}}_n-v_l=-S_w v_m\;. \end{aligned}$$
(84)
Combining Eqs. (82), (83) and (84) gives
$$\begin{aligned} v_m=\frac{S_{w,i}}{S_w}\left( v_l-v_{sw}\right) +\frac{\mathrm{d}v}{\mathrm{d}S_w}\;. \end{aligned}$$
(85)
Hence, the co-moving velocity is a sum of the co-moving velocities of the two previous sections; see (65) and (78).
We check the consistency of this calculation by using the co-moving velocity found in (85) together with Eq. (42) to calculate \(v_w\). We find as expected
$$\begin{aligned} v_w=\frac{1}{S_w}\left[ S_{w,i} v_{sw}+(S_w-S_{w,i}) v_{l}\right] \;. \end{aligned}$$
(86)
These four analytically tractable examples have allowed us to demonstrate in detail how the thermodynamic formalism that we have introduced in this paper works. We have in particular calculated the co-moving velocity \(v_m\) in all four cases. As is clear from these examples, it does not depend on the constitutive equation or any other equation except through v and \(\mathrm{d}v/\mathrm{d}S_w\).
Hence, the discussion of these four models has not included the constitutive Eq. (48). In order to have the seepage velocities \(v_w\) and \(v_n\) as functions of the driving forces, we would need to include the constitutive equation and the incompressibility condition (46) in the analysis.