One-dimensional non-Darcy flow in a semi-infinite porous media: a multiphase implicit Stefan problem with phases divided by hydraulic gradients

Abstract

One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.

Appendices

Appendix A: Proof of the statements for $p= i+1$

Firstly, the auxiliary function $g_{i}(x)$ is defined using Eq. (50) with $p=i$+1, and the following recurrence formula can be deduced from the definition of $f_{i}(x)$

$$\begin{aligned} g_i (x)=K_i g_{i-1} (x)\exp [U_i (x)]. \end{aligned}$$

(A1)

The function $f_{i+1}(x)$ is then defined as

$$\begin{aligned}&\hbox {erf}[f_{i+1} (x)]=\hbox {erf}[\beta _i f_i (x)] +g_i (x)\exp [-\beta _i^2 f_i^2 (x)], x \in I_{i+1}.\nonumber \\ \end{aligned}$$

(A2)

Also the right hand side of Eq. (A2) should be less than 1, thus we have

$$\begin{aligned} \hbox {erfc}[\beta _i f_i (x)]\exp [\beta _i^2 f_i^2 (x)]>g_i (x). \end{aligned}$$

(A3)

The domain of definition $I_{i+1}$ is the non-negative solution of Eq. (A3). With x increasing from 0 to $y_{i-1}$, the left hand side of Eq. (A3) decreases from erfc(${\beta }_{i}E_{i})\hbox {exp}({\beta }_{i}^{2}E_{i}^{2})$ to 0, while the right hand side is an increasing function of x and always be positive (from the recurrence formula Eq. (A1)). Thus, there will be unique positive $y_{i}$ in $I_{i}$ which makes two sides of Eq. (A3) equal if Eq. (A4) is satisfied, and then the domain of definition $I_{i+1}$ is [0, $y_{i})$

$$\begin{aligned} g_i (0)<\hbox {erfc}\left( {\beta _i E_i } \right) \exp \left( {\beta _i^2 E_i^2 } \right) .\end{aligned}$$

(A4)

Using the recurrence formula Eq. (A1) repeatedly, Eq. (A4) can be transformed to the inequality presented in Eq. (48) with $p=i$+1.

The function $f_{i+1}(x)$ can then be rewritten as

$$\begin{aligned}&f_{i+1} (x)=\hbox {inverf}\{\hbox {erf}[\beta _i f_i (x)]+g_i (x)\exp [-\beta _i^2 f_i^2 (x)]\}, \nonumber \\&\quad x \in I_{i+1} = [0, y_{i}). \end{aligned}$$

(A5)

The auxiliary function $U_{i+1}(x)$ is still defined in the form of Eq. (49) with $p=i+1$, and it is obvious that $U_{i+1}(x)$ is positive. The derivative of $f_{i+1}(x)$ can be determined from its definition

$$\begin{aligned} {f}'_{i+1} =\exp (U_{i+1} )\left[ {\beta _i {f}'_i \left( {1-\sqrt{{\uppi }}\beta _i f_i g_i } \right) +\sqrt{{\uppi }}{g}'_i /2} \right] .\nonumber \\ \end{aligned}$$

(A6)

Multiplying both sides of Eq. (A3) with $\sqrt{{\uppi }}\beta _i f_i$, we can deduce $\sqrt{{\uppi }}\beta _i f_i g_i <1$; thus $f_{i+1}(x)$ is an increasing function of x, and it increases from $E_{i+1}$ to $+\infty$ with x increasing from 0 to $y_{i}$, where $E_{i+1}$ is given by Eq. (47) with $p=i+1$. From lemmas 3-4 in Ref. [37], $U_{i+1}(x)$ can also be proved as an increasing function of x. Thus all the statements for $p=i+1$ have been verified.

Appendix B: Discussion of the inequalities

Considering that the inequalities in Eq. (48) for $p \leqslant n+1$ are satisfied, then all the functions $f_{i}(x)$ and the values $E_{i}=f_{i}(0)$ are well defined. This Appendix investigates the monotonicity properties of the functions on two sides of Eq. (48).

The coefficients of consolidation $c_{\mathrm{v}i}$, the critical hydraulic gradients $J_{i}$ are all given constants, thus both sides of Eq. (48) are seemed as functions of a single augment $q_{\mathrm{c}}$, and we use $K_{1}$ as the augment instead for convenience.

The functions on the left hand side of Eq. (48) are increasing functions of $K_{1}$, while for the functions on the right hand side, the monotonicity is studied by an induction process.

The functions on the right hand side of Eq. (48) for $p = 2, 3$ are $R_{1}(K_{1})$ = 1 and

$$\begin{aligned} R_\mathrm{2} (K_1 )= & {} \frac{\hbox {erfc}(\beta _\mathrm{2} E_\mathrm{2} )}{\exp [E_\mathrm{2}^2 (1-\beta _\mathrm{2}^2 )]}\nonumber \\= & {} \frac{\hbox {erfc}(\beta _\mathrm{2} E_\mathrm{2} )\exp (\beta _\mathrm{2}^2 E_\mathrm{2}^2 )}{\exp (E_\mathrm{2}^2 )}. \end{aligned}$$

(B1)

Equation (B1) indicates that $R_{2}(K_{1})$ is a decreasing function of $E_{2}$. Since $E_{2 }= \hbox {inverf}(K_{1})$ is an increasing function of $K_{1}$, we know that $R_{2}(K_{1})$ is a decreasing function of $K_{1}$, and it can also be deduced from $E_{1} = 0$ that $E_{2}^{2}-{\beta }_{1}^{2}E_{1}^{2}$ is an increasing function of $K_{1}$.

Assuming for $s = 2,3,{\ldots },i$, we have already proved that $R_{s}(K_{1})$ is a decreasing function of $K_{1}$, and $E_{s}, E_{s}^{2}-{\beta }_{s-1}^{2}E_{s-1}^{2}$ are increasing functions of $K_{1}$.

For $s=i+1$, there is

$$\begin{aligned} \hbox {erf}(E_{i+1} )=\hbox {erf}(\beta _i E_i )+\prod _{m=1}^i {K_m \exp [E_m^2 (1-\beta _m^2 )]} . \end{aligned}$$

(B2)

Equation (B2) can be transformed to

$$\begin{aligned} \frac{\hbox {erfc}(E_{i+1} )}{\hbox {erfc}(\beta _i E_i )}=1-\frac{K_1 \mathop {\prod }\nolimits _{m=2}^i {K_m \exp (E_m^2 -\beta _{m-1}^2 E_{m-1}^2 )} }{\hbox {erfc}(\beta _i E_i )\exp (\beta _i^2 E_i^2 )}.\nonumber \\ \end{aligned}$$

(B3)

The right hand side of Eq. (B3) is a decreasing function of $K_{1}$, thus we have

$$\begin{aligned} \frac{\exp (-E_{i+1}^2 ){E}'_{i+{1}} }{\hbox {erfc}(E_{i+1} )}-\frac{\exp (-\beta _i^2 E_i^2 )\beta _i {E}'_i }{\hbox {erfc}(\beta _i E_i )}>0, \end{aligned}$$

(B4)

in which ${E}'_i$ represents $\hbox {d}E_{i}/\hbox {d}K_{1}$.

Equation (B4) indicates that ${E}'_{i+1} > 0$. Using lemma 3 in Ref. [37], we can deduce

$$\begin{aligned} E_{i+1} {E}'_{i+1} -\beta _i^2 E_i {E}'_i >0. \end{aligned}$$

(B5)

The right hand side of Eq. (48) for $p=i+2 (s=i+1)$ can be written as

$$\begin{aligned} R_{i+1} (K_1 )= & {} \frac{\hbox {erfc}(E_{i+1} \beta _{i+1} )}{\exp \left[ \mathop {\sum }\nolimits _{m=1}^{i+1} {E_m^2 } (1-\beta _m^2 )\right] }\nonumber \\= & {} \frac{\hbox {erfc}(E_{i+1} \beta _{i+1} )\exp (E_{i+1}^2 \beta _{i+1}^2 )}{\exp \left[ \mathop {\sum }\nolimits _{m=1}^i {(E_{m+1}^2 } -\beta _m^2 E_m^2 )\right] }. \end{aligned}$$

(B6)

From Eq. (B5), $E_{i+1}^{2}-{\beta }_{i}^{2}E_{i}^{2}$ is an increasing function of $K_{1}$. For the function on the rightmost hand side of Eq. (B6), the denominator is an increasing function of $K_{1}$, while the numerator is a decreasing function of $K_{1}$, therefore $R_{i+1}(K_{1})$ is a decreasing function of $K_{1}$. All the statements for $s=i+1$ have been verified. Through the induction process, we know that the functions on the right hand side of Eq. (48) are all decreasing functions of $K_{1}$.

Since $K_{1}$ is a decreasing function of $q_{\mathrm{c}}$, above discussions indicate that in order to satisfy the inequalities presented in Eq. (48), $q_{\mathrm{c}}$ should be large enough.