The effect of non divergence-free velocity fields on field scale ground water solute transport

Abstract

Solute plume subjected to field scale hydraulic conductivity heterogeneity shows a large dispersion/macrodispersion, which is the manifestation of existing fields scale heterogeneity on the solute plume. On the other hand, due to the scarcity of hydraulic conductivity measurements at field scale, hydraulic conductivity heterogeneity can only be defined statistically, which makes the hydraulic conductivity a random variable/function. Random hydraulic conductivity as a parameter in flow equation makes the pore flow velocity also random and the ground water solute transport equation is a stochastic differential equation now. In this study, the ensemble average of stochastic ground water solute transport equation is taken by the cumulant expansion method in order to upscale the laboratory scale transport equation to field scale by assuming pore flow velocity is a non stationary, non divergence-free and unsteady random function of space and time. Besides the stochastic explanation of macrodispersion and the velocity correction term obtained by Kavvas and Karakas (J Hydrol 179:321–351, 1996) before a new velocity correction term, which is a function of mean pore flow velocity divergence, is obtained in this study due to strict second order cumulant expansion (without omitting any term after the expansion) performed. The significance of the new velocity correction term is investigated on a one dimensional transport problem driven by a density dependent flow field.

Appendices

Appendix A

The solution to Eq. 9 with the help of ordered exponential can be written as

$$H = {\left\lceil {\exp \left[\alpha {\int\limits_0^t {\mathcal{V}\,(\tau)\,\hbox{d}\tau }}\right]} \right\rceil}\,a$$

(28)

Here a stands for the non random/sure initial condition for the variable H. If we take ensemble average of the Eq. 28 we obtain

$$\langle H \rangle = {\left\lceil {\left\langle \exp \left[\alpha {\int\limits_0^t {\mathcal{V}\,(\tau)\,\hbox{d}\tau }}\right] \right\rangle} \right\rceil}\,\,a$$

(29)

Here we have followed van Kampen (1981) and used the commutativity of ensemble averaging operator with ordering. If we follow van Kampen (1981) again and treat the operators inside the ordering as if they commute we can write

$$\langle H \rangle = {\left\lceil {\exp \left[ \ll \exp \left[\alpha {\int\limits_0^t {\mathcal{V}(\tau)\,\hbox{d}\tau }}\right] - 1 \gg \right]} \right\rceil}\,a$$

(30)

which is also given by Kubo (1962). The symbol ≪ ≫ here stands for the cumulant operator. If we extend the second exponential inside and perform partial ordering we obtain

$$\langle H \rangle = {\left\lceil \begin{aligned} \exp &\left[\alpha {\int\limits_0^t {\ll \mathcal{V}(t_{1}) \gg \,\hbox{d}t_{1}}} + \alpha ^{2} {\int\limits_0^t {\hbox{d}t_{1} {\int\limits_0^{t_{1}} {\hbox{d}t_{2}}}\ll \mathcal{V}(t_{1}) \mathcal{V}(t_{2})}} \gg \right.\\ &\quad \left.+ \cdots + \alpha ^{m} {\int\limits_0^t {\hbox{d}t_{1} {\int\limits_0^{t_{1}} {\hbox{d}t_{2}}}\,\cdots\,{\int\limits_0^{t_{{m - 1}}} {\hbox{d}t_{m}}} \ll \mathcal{V}(t_{1})\,\mathcal{V}(t_{2})}}\,\cdots\,\mathcal{V}(t_{m})\right] \\ \end{aligned} \right\rceil}\,a\\ $$

(31)

Equation 31 at the second order is

$$\langle H \rangle = {\left\lceil {\exp \left[\alpha {\int\limits_0^t {\langle\mathcal{V}(t_{1}) \rangle\,\hbox{d}t_{1} + \alpha ^{2} {\int\limits_0^t {\hbox{d}t_{1} {\int\limits_0^{t_{1}} {\hbox{d}t_{2}}} \ll \mathcal{V}(t_{1}) \mathcal{V}(t_{2})}} \gg}}\right]} \right\rceil}\,a$$

(32)

Note that the cumulant operation is equal to the ensemble averaging when there is one argument inside the cumulant operator. Also it is important to recognize that the terms omitted are high order cumulants to reach the Eq. 32 and they are smaller than the low order cumulants kept in Eq. 32. If we write the differential equation for 〈H 〉 by using Eq. 32 we obtain Eq. 11.

Appendix B

Using Eqs. 7 and 10 in Eq. 11 we obtain

$$ \begin{aligned} & - {\left\lfloor {\exp {\left[ { - {\int\limits_0^t {A_{0} (\tau )\,{\text{d}}\,\tau } }} \right]}} \right\rfloor }A_{0} (t) < c > + {\left\lfloor {\exp \,{\left[ { - {\int\limits_0^t {A_{0} (\tau )\,{\text{d}}\,\tau } }} \right]}} \right\rfloor }\frac{{\partial < c > }} {{\partial t}} \\ & \quad = \alpha \,\, < {\left\lfloor {\exp {\left[ { - {\int\limits_0^t {A_{0} (\tau )\,\,{\text{d}}\tau } }} \right]}} \right\rfloor }\,\,A_{1} (t)\,\,\,\,{\left\lceil {\exp \,{\left[ {{\int\limits_0^t {A_{0} (\tau )\,{\text{d}}\,\tau } }} \right]}} \right\rceil } > \,\,{\left\lfloor {\exp {\left[ { - {\int\limits_0^t {A_{0} (\tau )\,{\text{d}}\,\tau } }} \right]}\,} \right\rfloor } < c\, > \\ & \,\,\,\,\,\,\,\,\,\,\, + \,\alpha ^{2} {\int\limits_0^t {{\left\{ {\,\,{\text{d}}s < < \,\,{\left\lfloor {\exp {\left[ { - {\int\limits_0^t {A_{0} (\tau )\,\,{\text{d}}\tau ]} }} \right]}\,} \right\rfloor }\,\,A_{1} (t)\,\,\,\,{\left\lceil {\exp \,{\left[ {{\int\limits_0^t {A_{0} (\tau )\,{\text{d}}\,\tau } }} \right]}} \right\rceil }{\left\lfloor {\exp {\left[ { - {\int\limits_0^{t - s} {A_{0} (\tau )\,\,{\text{d}}\tau } }} \right]}\,} \right\rfloor }\,\,A_{1} (t - s)\,\,\,\,{\left\lceil {\exp \,{\left[ {{\int\limits_0^{t - s} {A_{0} (\tau )\,{\text{d}}\tau } }} \right]}} \right\rceil } > > } \right\}}} }\,{\left\lfloor {\exp \,{\left[ { - {\int\limits_0^t {A_{0} (\tau )\,d\,\tau } }} \right]}} \right\rfloor } < c\, > \\ \end{aligned} $$

(33)

By using the following equality given by van Kampen (1981)

$${\left\lceil {\exp \left[{\int\limits_0^t {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rceil} = {\left\lceil {\exp \left[{\int\limits_{t_{1}}^t {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rceil}\,\,{\left\lceil {\exp \left[{\int\limits_0^{t_{1}} {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rceil}$$

(34)

we can obtain the following equalities

$${\left\lceil {\exp \left[{\int\limits_{t - s}^t {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rceil} = {\left\lceil {\exp \left[{\int\limits_0^t {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rceil}{\left\lfloor {\exp \left[ - {\int\limits_0^{t - s} {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rfloor}$$

(35)

$${\left\lfloor {\exp \left[ - {\int\limits_{t - s}^t {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rfloor} = {\left\lceil {\exp \left[{\int\limits_0^{t - s} {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rceil}\,\,{\left\lfloor {\exp \left[ - {\int\limits_0^t {A_{0} (\tau)\,\hbox{d}\,\tau }}\right]} \right\rfloor}$$

(36)

Using Eqs. 35 and 36 in Eq. 33 and performing left multiplication by the time ordered exponential of A ₀(t) we obtain the Eq. 12.

Appendix C

Although the following equalities, which are known as disentangling theorems, can be driven mathematically, we will just copy them from Wood and Kavvas (1999a) as

$$ \exp \left\lceil {\int\limits_{t - s}^t {{\text{d}}\tau \,\,\{ B_0 (\tau )\, + \,B_1 (\tau )\} } } \right\rceil = \exp \left\lceil {\int\limits_{t - s}^t {{\text{d}}\tau \,\,\,B_0 (\tau )\,\,} } \right\rceil \,\exp \left\lceil {\int\limits_{t - s}^t {{\text{d}}\tau \,\exp \left\lfloor { - \int\limits_{t - s}^\tau {{\text{d}}\eta \,B_0 (\eta )\,} } \right\rfloor \,B_1 (\tau )} \exp \left\lceil {\int\limits_{t - s}^\tau {{\text{d}}\eta \,B_0 (\eta )\,} } \right\rceil } \right\rceil$$

(37)

$$ \exp \left\lfloor { - \int\limits_{t - s}^t {{\text{d}}\tau \,\,\{ B_0 (\tau )\, + \,B_1 (\tau )\} } } \right\rfloor = \,\,\exp \left\lfloor { - \int\limits_{t - s}^t {{\text{d}}\tau \,\,\,\,\,\,\exp \left\lfloor { - \int\limits_{t - s}^\tau {{\text{d}}\eta \,\,\,\,B_0 (\eta )\,} } \right\rfloor \,\,\,\,\,B_1 (\tau )} \exp \left\lceil {\int\limits_{t - s}^\tau {{\text{d}}\eta \,\,\,B_0 (\eta )\,} } \right\rceil } \right\rfloor \,\exp \left\lfloor { - \int\limits_{t - s}^t {d\tau \,\,\,B_0 (\tau )\,\,} } \right\rfloor$$

(38)

If we separate our A ₀(t) operator as

$$\begin{aligned} B_{0}(t) &= - \langle v_{x} \rangle \frac{\partial}{{\partial \,x}} + D_{x} \frac{{\partial ^{2}}}{{\partial \,x^{2}}}\\ B_{1}(t) &= - \frac{{\partial \langle v_{x} \rangle}}{{\partial \,x}}\\ \end{aligned}$$

and use then in Eqs. 37 and 38 we obtain

$$ \exp \left\lceil {\int\limits_{t - s}^t {{\text{d}}\tau \,\,A_0 (\tau )\,} } \right\rceil = \exp \left\lceil {\int\limits_{t - s}^t {{\text{d}}\tau \,\left( { - \, < v_x > \frac{\partial } {{\partial \,x}}\,\, + \,\,D_x \frac{{\partial ^2 }} {{\partial \,x^2 }}} \right)} } \right\rceil \,\exp \left\lceil {\int\limits_{t - s}^t {{\text{d}}\tau \,\exp \left\lfloor { - \int\limits_{t - s}^\tau {{\text{d}}\eta \,\,\left( { - \, < v_x > \frac{\partial } {{\partial \,x}}\,\, + \,\,D_x \frac{{\partial ^2 }} {{\partial \,x^2 }}} \right)} } \right\rfloor } \left( { - \frac{{\partial \, < v_x \, > }} {{\partial \,x}}} \right)\exp \left\lceil {\int\limits_{t - s}^\tau {{\text{d}}\eta \,\,\left( { - \, < v_x > \frac{\partial } {{\partial \,x}}\,\, + \,\,D_x \frac{{\partial ^2 }} {{\partial \,x^2 }}} \right)} } \right\rceil } \right\rceil$$

(39)

$$ \exp \left\lfloor { - \int\limits_{t - s}^t {{\text{d}}\tau \,\,A_0 (\tau )\,} } \right\rfloor = \,\exp \left\lfloor { - \int\limits_{t - s}^t {{\text{d}}\tau \,\,\exp \left\lfloor { - \int\limits_{t - s}^\tau {{\text{d}}\eta \,\,\,\left( { - \, < v_x > \frac{\partial } {{\partial \,x}}\,\, + \,\,D_x \frac{{\partial ^2 }} {{\partial \,x^2 }}} \right)} } \right\rfloor } \left( { - \frac{{\partial \, < v_x \, > }} {{\partial \,x}}} \right)\,\exp \left\lceil {\int\limits_{t - s}^\tau {{\text{d}}\,\eta \,\,\left( { - \, < v_x > \frac{\partial } {{\partial \,x}}\,\, + \,\,D_x \frac{{\partial ^2 }} {{\partial \,x^2 }}} \right)} } \right\rceil } \right\rfloor \,\exp \left\lfloor { - \int\limits_{t - s}^t {{\text{d}}\tau \,\,\left( { - \, < v_x > \frac{\partial } {{\partial \,x}}\,\, + \,\,D_x \frac{{\partial ^2 }} {{\partial \,x^2 }}} \right)} } \right\rfloor$$

(40)

By using the following Lie operator properties below given by Grobner and Knapp (1967)

$$\begin{aligned} \exp ^{{\langle v \rangle \, \cdot \,\nabla}} F(x) &= F\,\left(\exp ^{{\langle v \rangle \cdot \nabla}} \,x\right),\\ \exp ^{{\, \langle v \rangle \, \cdot \,\nabla}} (\,F_{1} (x)\,F_{2} (x)\,) &= \left(\exp ^{{\langle v \rangle \cdot \nabla}} F_{1} (x)\right)\left(\exp ^{{\langle v \rangle \cdot \,\nabla}} F_{2} (x)\right) \end{aligned}$$

in Eq. 39 and 40 we obtain

$$\begin{aligned} \exp {\left\lceil {{\int\limits_{t - s}^t {\hbox{d}\tau A_{0} (\tau)}}} \right\rceil} &= \exp {\left\lceil {{\int\limits_{t - s}^t {\hbox{d}\tau \left( - \langle v_{x} \rangle \frac{\partial}{{\partial \,x}} + D_{x} \frac{{\partial ^{2}}}{{\partial \,x^{2}}}\right)}}} \right\rceil}\\ & \quad \times \exp {\left\lceil {{\int\limits_{t - s}^t {\hbox{d}\tau \left( - \frac{{\partial \langle v_{x} (x + \upsilon, \tau) \rangle}}{{\partial \,x}}\right)}}} \right\rceil} \\ \end{aligned}$$

(41)

$$\begin{aligned} \exp {\left\lfloor {- {\int\limits_{t - s}^t {\hbox{d}\tau A_{0} (\tau)}}} \right\rfloor} &= \exp {\left\lfloor {{\int\limits_{t - s}^t {\hbox{d}\tau \left(\frac{{\partial \langle v_{x} (x + \upsilon, \tau) \rangle}}{{\partial \,x}}\right)}}} \right\rfloor} \\ & \quad \times \exp {\left\lfloor {- {\int\limits_{t - s}^t {\hbox{d}\tau\left(- \, \langle v_{x} \rangle \frac{\partial}{{\partial \,x}} + D_{x} \frac{{\partial ^{2}}}{{\partial \,x^{2}}}\right)}}} \right\rfloor} \\ \end{aligned}$$

(42)

where

$$\upsilon = {\int\limits_{t - s}^\tau {\langle v(x,\eta) \rangle \,\hbox{d}\eta}}$$

If we write the simplified ordered exponentials of A ₀(t) (Eqs. 41, 42) in Eq. 13 our ensemble averaged equation becomes

$$\begin{aligned} \frac{{\partial \, \langle c \rangle}}{{\partial \,t}} &= A_{0} (t) \langle c \rangle \\ & \quad + \alpha ^{2} \,{\int\limits_0^t {\ll A_{1} (t)\,\exp {\left\lceil {{\int\limits_{t - s}^t {\hbox{d}\tau \left(- \langle v_{x} \rangle \frac{\partial}{{\partial \,x}} + D_{x} \frac{{\partial ^{2}}}{{\partial \,x^{2}}}\right)}}} \right\rceil}\exp {\left\lceil {{\int\limits_{t - s}^t {\hbox{d}\tau \left(\, - \frac{{\partial\langle v_{x} (x + \upsilon, \tau) \rangle}}{{\partial \,x}}\right)}}} \right\rceil}}} \\ & \quad \times A_{1} (t - s) \gg \exp {\left\lfloor {{\int\limits_{t - s}^t {\hbox{d}\tau \left(\frac{{\partial \, \langle v_{x} (x + \upsilon, \tau) \rangle}}{{\partial \,x}}\right)}}} \right\rfloor}\,\exp {\left\lfloor {- {\int\limits_{t - s}^t {\hbox{d}\tau \left(- \langle v_{x} \rangle \frac{\partial}{{\partial \,x}} + D_{x} \frac{{\partial ^{2}}}{{\partial \,x^{2}}}\right)}}} \right\rfloor} \\ & \quad \times \hbox{d}s\,\langle c \rangle \\ \end{aligned} $$

(43)

where

$$\upsilon = {\int\limits_{t - s}^\tau {\langle v(x,\eta) \rangle \,\hbox{d}\eta}}.$$

If we apply the Lie operator properties given above in Eq. 43 again, the Eq. 43 becomes

$$\begin{aligned} \frac{{\partial \langle c\rangle}}{{\partial \,t}} &= \,A_{0} (t) \langle c \rangle + \alpha ^{2} {\int\limits_0^t {\ll \,A_{1} (t) \exp {\left\lceil {{\int\limits_{t - s}^t {\hbox{d}\tau \left( - \frac{{\partial \langle v_{x} (x + \upsilon - \beta, \tau)\rangle}}{{\partial \,x}}\right)}}} \right\rceil}}} \\ &\quad \times A_{1} (x - \beta, t - s) \gg \,\exp {\left\lfloor {{\int\limits_{t - s}^t {\hbox{d}\tau \left(\frac{{\partial \langle v_{x} (x + \upsilon - \beta, \tau ) \rangle}}{{\partial \,x}}\right)}}} \right\rfloor}\,\hbox{d}s\, \langle c \rangle \\ \end{aligned} $$

(44)

where

$$\beta = {\int\limits_{t - s}^t {\langle v(x,\eta) \rangle \,\hbox{d}\eta}}, \quad \upsilon = {\int\limits_{t - s}^\tau {\langle v(x,\eta) \rangle\,\hbox{d}\eta}}$$

By writing α A ₁(t) operator in Eq. 44 we obtain

$$\begin{aligned} \frac{{\partial \langle c \rangle}}{{\partial \,t}} &= A_{0} (t) \langle c \rangle + {\int\limits_0^t {\ll \,\left(\frac{{\partial (- \overline{v} _{x} (x,t))}}{{\partial \,x}} + \left(- \overline{v} _{x} (x,t)\right)\frac{\partial}{{\partial \,x}}\right)}} \\ & \quad \times \exp {\left[ {{\int\limits_{t - s}^t {\hbox{d}\tau \left( - \frac{{\partial \langle v_{x} (x + \upsilon - \beta, \tau)\rangle}}{{\partial \,x}}\right)}}} \right]}\\ & \quad \times \left(\frac{{\partial \left(- \overline{v} _{x} (x - \beta, t - s)\right)}}{{\partial \,x}} + \left(- \overline{v} _{x} (x - \beta, t - s)\right)\frac{\partial}{{\partial \,x}}\right) \gg \\ & \quad \times \exp {\left[ {{\int\limits_{t - s}^t {\hbox{d}\tau \left(\frac{{\partial \langle v_{x} (x + \upsilon - \beta, \tau) \rangle}}{{\partial \,x}}\right)}}} \right]}\,\hbox{d}s\, \langle c \rangle\\ \end{aligned} $$

(45)

where

$$\overline{v} = v - \langle v \rangle, \quad \beta = {\int\limits_{t - s}^t {\langle v(x,\eta) \rangle \,\hbox{d}\eta,}} \quad \upsilon = {\int\limits_{t - s}^\tau {\langle v(x,\eta) \rangle \,\hbox{d} \eta}}$$

Note that ordered exponentials in Eq. 44 became regular exponentials since they contain only functions in them. After applying the differential operators in Eq. 45, we obtain the explicit ensemble averaged/field scale/up scaled solute transport equation as given in Eq. 14.