*4.3. Unit Determination*

According to the above-mentioned assumptions and principles of the mixed-unit method, the unconfined aquifer of the Aksu River Basin was divided into seven small units (a, b, c, d, e, f, and g), and its confined aquifer is divided into five small units (C, D, E, F, and G), as shown in Figure 6.

**Figure 6.** Mixed-unit division in the confined aquifer.

#### *4.4. Calculation of Aquifer Recharging and Discharging*

In the mixed-unit model, the unconfined aquifer and confined aquifer are divided into a finite number of discrete small units, which are discrete at an interval of Δt. Their solutes are fully mixed, and the components of each solute are evenly distributed in all small units. Therefore, the equilibrium equation of water in a small unit within the period of Δt can be expressed as:

$$\mathbf{Q\_n} - \mathbf{W\_n} + \sum\_{i=1}^{\text{ln}} \mathbf{q\_{in}} - \sum\_{i=1}^{\text{ln}} \mathbf{q\_{rj}} = \mathbf{e\_n} \tag{1}$$

According to the assumption on the water balance of each small unit, the mass balance equation of the dissolved component k in unit n can be obtained as:

$$\mathbf{C\_{nk}}\mathbf{Q\_n} - \mathbf{C\_{nk}} \left[ \mathbf{W\_n} + \sum\_{j=1}^{\text{ln}} \mathbf{q\_{nj}} \right] + \sum\_{i=1}^{\text{ln}} \mathbf{q\_{in}}\mathbf{C\_{irk}} = \mathbf{e\_{nk}} \text{ k} = 1, \ 2, \ \dots, \ \text{k} \tag{2}$$

where, Qn represents the time average flow value into unit n, Wn is the average value of the flow out from unit n, qin represents the average flow from unit i into n, en is the deviation of water balance caused by various errors from the flow entering or exiting the unit n, k is the average concentration of the tracer k in one unit, and Cnk is the average concentration of the trace k in the k in unit n.

After Equations (1) and (2) are combined into a rectangular matrix of known concentrations in unit n, in which the first row represents the water balance and the other rows represent the solute mass conservation balance, the Equation (3) can be obtained with any unit n:

$$\mathbf{C\_n q\_n + D\_n = E\_n} \tag{3}$$

where, qn represents the flow through the boundary of small unit n:

$$\mathbf{q}\_{\mathbf{n}} = [\mathbf{q}\_{1\mathbf{n}} \mathbf{q}\_{2\mathbf{n}} \dots \mathbf{q}\_{\mathbf{in}} \mathbf{q}\_{\mathbf{n}1} \mathbf{q}\_{\mathbf{n}2} \dots \mathbf{q}\_{\mathbf{n}} \mathbf{j}\_{\mathbf{n}}] (\mathbf{I}\mathbf{n} + \mathbf{J}\mathbf{n}) \times \mathbf{1} \tag{4}$$

Dn is the measurable and quantifiable known items in unit n (such as the known outflow and pumping volumes), and En represents the unknown error vector in the unit as,

$$\mathbf{E}\_n = [\mathbf{e}\_n \mathbf{e}\_{n1} \mathbf{e}\_{n1} \dots \mathbf{e}\_{nk}](1+\mathbb{K}) \times 1\tag{5}$$

According to Equation (3) (Adar (1988)), through the minimization of the sum function J of square error and evaluation of the sum of square error of all units, the flow composition of the aquifer can be obtained as,

$$\mathbf{J} = \sum\_{1}^{N} [\mathbf{E}\_{\mathbf{n}}^{T} \mathbf{W} \mathbf{E}\_{\mathbf{n}}] = \sum\_{1}^{N} (\mathbf{c}\_{\mathbf{n}} \mathbf{q}\_{\mathbf{n}} + \mathbf{D}\_{\mathbf{n}})^{T} \mathbf{W} (\mathbf{c}\_{\mathbf{n}} \mathbf{q}\_{\mathbf{n}} + \mathbf{D}\_{\mathbf{n}}) \tag{6}$$
