第 1 建模基本方法论

本章极少数值方法建模的基本方法论，涉及一些基础的建模思路和数学基础。

1.1 建模基本思路

我称之为CLAMS方法，包含以下步骤：

1. Conceptual Model - 描述物理过程，形成概念模型（或认知模型）
2. Laws of Physics - 使用物理规律
3. Assumptions - 列出合理假设，简化问题
4. Math equations - 使用数学公式表达物理规律和假设
5. Solver - 求解数学公式

求解数学公式的过程，可以尝试寻找其解析解(Analytical solution)，也可以使用数值方法求得数值解(Numerical Solution)。

数值方法本质上是对离散（非连续）时空模型中因变量(Dependant variable)分布和变化的数学近似描述，从理论的解析解到数值解虽然损失了精度，但解析解通常无法求得，而数值方法可给出误差可接受的近似解。

案例：自由落体运动

描述 问题描述下图。

自由落体运动示意图

问题：任意\(t>0\)时刻的速度，即\(v(t) = ?\)。

建模步骤：

1. 认知模型：

   自由落体运动

2. 物理定律：

   牛顿定律： \(F = ma\)。

3. 假设：

   \(v(t=0) = 0\)

   且

   \(F_{drag}(t) \propto v(t)\)，即\(F_{drag} = cv\)。

4. 数学公式：

   由\(F = ma\)和\(\frac{dv}{dt} = a\)可得：

   \[\tag{1} \frac{dv}{dt} = a = \frac{F}{m}\]

   根据物体受力分析, 其受到向下的重力\(F_{g} = mg\)和向上的空气阻力\(F_{drag} = cv\)，空气阻力在此假设与物体运动速度成正比关系。则其受力平衡公式为： \[\tag{2} F = F_{g} - F_{drag} = mg - cv\]

   综合公式（1）和（2），则得到： \[\tag{3} \frac{dv}{dt} = g - \frac{c}{m} v\]

   

   自由落体运动的受力分析

5. 公式求解：

   初始条件：\(v(0) = 0\)

   积分求解(解析解)： \[v(t) = \frac {mg}{c}\left[ 1- exp(-\frac{c}{m}t) \right]\]

6. 结果绘图：

c = 15 # drag coeefficient
g = 9.8 # Gravity
m = 150  # Mass in kg
x = seq(0,100, 1)  # Time
y = m*g/c *(1 - exp(-1 * c / m * x))  # Vecocity
plot(x, y, type='l', xlab='Time (s)', ylab='Velocity (m/s)', col=2, lwd=2); 
grid()

变量表:

• \(v(t)\) - 随时间变化的物体速度
• \(m\) - 物体质量
• \(g\) - 重力加速度
• \(a\) - 物体运动的加速度
• \(c\) - 空气阻力系数
• \(F\) - 物体所受的力
• \(F_{g}\) - 重力
• \(F_{drag}\) - 空气阻力

1.2 典型控制方程

1.2.1 一维承压地下水运动

Sure, let’s derive the governing equation for one-dimensional confined aquifer groundwater flow step by step. We will use Darcy’s Law and the principle of mass conservation.

1.2.1.1 Step 1: Darcy’s Law

Darcy’s Law describes the flow of groundwater through porous media. It states that the discharge per unit area (specific discharge or Darcy velocity, \(q\)) is proportional to the hydraulic gradient: \[q = -K \frac{\partial h}{\partial x}\] where: - \(q\) is the specific discharge (Darcy velocity) [L/T]. - \(K\) is the hydraulic conductivity of the aquifer [L/T]. - \(h\) is the hydraulic head [L]. - \(x\) is the spatial coordinate in the direction of flow [L].

1.2.1.2 Step 2: Conservation of Mass

Consider a control volume of length \(\Delta x\), cross-sectional area \(A\), and located at position \(x\) along the direction of flow within a confined aquifer.

1.2.1.2.1 Inflow and Outflow

• The rate of inflow at \(x\): \(q(x) \cdot A\).
• The rate of outflow at $ x + x $: \(q(x + \Delta x) \cdot A\).

Using a Taylor series expansion for \(q(x + \Delta x)\): \[q(x + \Delta x) \approx q(x) + \left( \frac{\partial q}{\partial x} \right) \Delta x\]

1.2.1.2.2 Net Flow

The net rate of flow into the control volume is: \[q(x) \cdot A - \left( q(x) + \left( \frac{\partial q}{\partial x} \right) \Delta x \right) \cdot A\] \[= -A \left( \frac{\partial q}{\partial x} \right) \Delta x\]

1.2.1.3 Step 3: Storage in the Aquifer

The change in storage within the control volume over a time interval \(\Delta t\) can be expressed using the specific storage \(S_s\), which is the amount of water per unit volume of the aquifer that is stored or released from storage per unit change in hydraulic head: \[\Delta S = S_s \cdot A \cdot \Delta x \cdot \frac{\partial h}{\partial t} \cdot \Delta t\]

1.2.1.4 Step 4: Applying Conservation of Mass

According to the conservation of mass principle, the rate of change of storage in the control volume must equal the net rate of flow into the control volume: \[-A \left( \frac{\partial q}{\partial x} \right) \Delta x = S_s \cdot A \cdot \Delta x \cdot \frac{\partial h}{\partial t}\]

1.2.1.5 Step 5: Substituting Darcy’s Law

Substitute \(q = -K \frac{\partial h}{\partial x}\) into the equation: \[-A \left( \frac{\partial}{\partial x} \left( -K \frac{\partial h}{\partial x} \right) \right) \Delta x = S_s \cdot A \cdot \Delta x \cdot \frac{\partial h}{\partial t}\]

Simplify the equation: \[A \left( K \frac{\partial^2 h}{\partial x^2} \right) \Delta x = S_s \cdot A \cdot \Delta x \cdot \frac{\partial h}{\partial t}\]

#### Step 6: Simplifying and Rearranging Cancel out the common terms \(A\) and \(\Delta x\): \[K \frac{\partial^2 h}{\partial x^2} = S_s \frac{\partial h}{\partial t}\]

#### Final Governing Equation The one-dimensional groundwater flow equation for a confined aquifer is: \[\frac{\partial h}{\partial t} = \frac{K}{S_s} \frac{\partial^2 h}{\partial x^2}\]

Define the hydraulic diffusivity \(D\) as: \[D = \frac{K}{S_s}\]

Thus, the governing equation can also be written as: \[\frac{\partial h}{\partial t} = D \frac{\partial^2 h}{\partial x^2}\]

This partial differential equation describes how the hydraulic head \(h\) varies with time \(t\) and position \(x\) within the confined aquifer.

1.2.2 二维承压地下水运动

Sure, let’s derive the governing equation for two-dimensional confined aquifer groundwater flow step by step using Darcy’s Law and the principle of mass conservation.

1.2.2.1 Step 1: Darcy’s Law

In two dimensions, Darcy’s Law describes the flow of groundwater through porous media. It states that the discharge per unit area (specific discharge or Darcy velocity, \(\mathbf{q}\)) is proportional to the hydraulic gradient: \[\mathbf{q} = -K \nabla h\] where: - \(\mathbf{q}\) is the specific discharge (Darcy velocity) vector [L/T]. - \(K\) is the hydraulic conductivity of the aquifer [L/T]. - \(h\) is the hydraulic head [L]. - \(\nabla h\) is the hydraulic gradient, which in two dimensions can be written as: \[\nabla h = \left( \frac{\partial h}{\partial x}, \frac{\partial h}{\partial y} \right)\]

1.2.2.2 Step 2: Conservation of Mass

Consider a differential control volume in the aquifer with dimensions \(dx\) by \(dy\) and thickness \(b\).

1.2.2.2.1 Inflow and Outflow

For simplicity, let’s assume the flow is in the \(x\)- and \(y\)-directions.

• The rate of inflow in the \(x\)-direction at \(x\): $ q_x(x) b dy $

  • The rate of inflow in the \(y\)-direction at \(y\): $ q_y(y) b dx $

  The rate of outflow in the \(x\)-direction at \(x + dx\): \[q_x(x + dx) \cdot b \cdot dy \approx \left( q_x(x) + \frac{\partial q_x}{\partial x} dx \right) \cdot b \cdot dy\]

  The rate of outflow in the \(y\)-direction at \(y + dy\): \[q_y(y + dy) \cdot b \cdot dx \approx \left( q_y(y) + \frac{\partial q_y}{\partial y} dy \right) \cdot b \cdot dx\]

  Net Flow

  The net rate of flow into the control volume is: \[\text{Net inflow in } x\text{-direction} = \left[ q_x(x) \cdot b \cdot dy \right] - \left[ \left( q_x(x) + \frac{\partial q_x}{\partial x} dx \right) \cdot b \cdot dy \right]\] \[= -b \cdot dy \cdot \frac{\partial q_x}{\partial x} dx\]

\[\text{Net inflow in } y\text{-direction} = \left[ q_y(y) \cdot b \cdot dx \right] - \left[ \left( q_y(y) + \frac{\partial q_y}{\partial y} dy \right) \cdot b \cdot dx \right]\] \[= -b \cdot dx \cdot \frac{\partial q_y}{\partial y} dy\]

The total net inflow into the control volume is: \[-b \left( \frac{\partial q_x}{\partial x} dx \cdot dy + \frac{\partial q_y}{\partial y} dy \cdot dx \right)\] \[= -b \left( \frac{\partial q_x}{\partial x} + \frac{\partial q_y}{\partial y} \right) dx \cdot dy\]

#### Step 3: Storage in the Aquifer The change in storage within the control volume over a time interval \(\Delta t\) can be expressed using the specific storage \(S_s\): \[\Delta S = S_s \cdot b \cdot dx \cdot dy \cdot \frac{\partial h}{\partial t} \cdot \Delta t\]

#### Step 4: Applying Conservation of Mass According to the conservation of mass principle, the rate of change of storage in the control volume must equal the net rate of flow into the control volume: \[-b \left( \frac{\partial q_x}{\partial x} + \frac{\partial q_y}{\partial y} \right) dx \cdot dy = S_s \cdot b \cdot dx \cdot dy \cdot \frac{\partial h}{\partial t}\]

#### Step 5: Substituting Darcy’s Law Substitute \(q_x = -K \frac{\partial h}{\partial x}\) and \(q_y = -K \frac{\partial h}{\partial y}\) into the equation: \[-b \left( \frac{\partial}{\partial x} \left( -K \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( -K \frac{\partial h}{\partial y} \right) \right) dx \cdot dy = S_s \cdot b \cdot dx \cdot dy \cdot \frac{\partial h}{\partial t}\]

Simplify the equation: \[b \left( K \frac{\partial^2 h}{\partial x^2} + K \frac{\partial^2 h}{\partial y^2} \right) dx \cdot dy = S_s \cdot b \cdot dx \cdot dy \cdot \frac{\partial h}{\partial t}\]

#### Step 6: Simplifying and Rearranging Cancel out the common terms \(b\), \(dx\), and \(dy\): \[K \left( \frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} \right) = S_s \frac{\partial h}{\partial t}\]

#### Final Governing Equation The two-dimensional groundwater flow equation for a confined aquifer is: \[\frac{\partial h}{\partial t} = \frac{K}{S_s} \left( \frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} \right)\]

Define the hydraulic diffusivity \(D\) as: \[D = \frac{K}{S_s}\]

Thus, the governing equation can also be written as: \[\frac{\partial h}{\partial t} = D \left( \frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} \right)\]

This partial differential equation describes how the hydraulic head \(h\) varies with time \(t\) and position \((x, y)\) within the confined aquifer.

1.2.3 一维非承压地下水运动

Let’s go through the derivation for one-dimensional unconfined aquifer groundwater flow more carefully, considering the variation in aquifer thickness due to changes in the water table.

1.2.3.1 Step-by-Step Derivation for One-Dimensional Unconfined Aquifer

1.2.3.2 Step 1: Darcy’s Law

For unconfined groundwater flow, Darcy’s Law in one dimension is: \[q = -K \frac{\partial h}{\partial x}\] where: - \(q\) is the specific discharge (Darcy velocity) [L/T]. - \(K\) is the hydraulic conductivity of the aquifer [L/T]. - \(h\) is the hydraulic head [L]. - \(x\) is the spatial coordinate in the direction of flow [L].

1.2.3.3 Step 2: Volumetric Flow Rate

The volumetric flow rate \(Q\) at a point \(x\) for an unconfined aquifer with variable saturated thickness \(h\) is given by: \[Q = q \cdot b \cdot h\] where \(b\) is the aquifer width perpendicular to the flow direction. For simplicity, we assume $ b = 1 $ unit width, leading to: \[Q = q \cdot h = -K h \frac{\partial h}{\partial x}\]

1.2.3.4 Step 3: Conservation of Mass

Consider a differential control volume in the unconfined aquifer of width \(dx\) and saturated thickness $ h(x) $.

1.2.3.4.1 Inflow and Outflow

• Inflow at \(x\): \(Q(x) = -K h \frac{\partial h}{\partial x}\)
• Outflow at \(x + dx\): \(Q(x + dx) = -K h \frac{\partial h}{\partial x} + \left( \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) \right) dx\)

1.2.3.4.2 Net Flow

The net inflow into the control volume is: \[Q(x) - Q(x + dx) = -K h \frac{\partial h}{\partial x} - \left( -K h \frac{\partial h}{\partial x} + \left( \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) \right) dx \right)\] \[= - \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) dx\]

1.2.3.5 Step 4: Change in Storage

The change in storage within the control volume $ dx h $ over a time interval $ t $ can be expressed using the specific yield \(S_y\), which measures the volume of water released from storage per unit decline in the water table: \[\Delta S = S_y \cdot dx \cdot \frac{\partial h}{\partial t} \cdot \Delta t\]

1.2.3.6 Step 5: Applying Conservation of Mass

According to the conservation of mass principle, the rate of change of storage in the control volume must equal the net rate of flow into the control volume: \[- \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) dx = S_y \cdot dx \cdot \frac{\partial h}{\partial t}\]

1.2.3.7 Step 6: Simplifying and Rearranging

Simplify the equation: \[\frac{\partial}{\partial x} \left( K h \frac{\partial h}{\partial x} \right) = S_y \frac{\partial h}{\partial t}\]

1.2.3.8 Final Governing Equation

The one-dimensional groundwater flow equation for an unconfined aquifer is: \[S_y \frac{\partial h}{\partial t} = \frac{\partial}{\partial x} \left( K h \frac{\partial h}{\partial x} \right)\]

This is the governing equation for transient groundwater flow in an unconfined aquifer. It accounts for changes in the saturated thickness \(h\) due to fluctuations in the water table.

1.2.4 二维非承压地下水运动

Sure, let’s derive the governing equation for two-dimensional unconfined aquifer groundwater flow step by step. We’ll again use Darcy’s Law, the principle of mass conservation, and the Dupuit assumption, which simplifies the analysis by assuming horizontal flow and a vertical hydraulic gradient.

1.2.4.1 Step-by-Step Derivation for Two-Dimensional Unconfined Aquifer

1.2.4.2 Step 1: Darcy’s Law

In two dimensions, Darcy’s Law for an unconfined aquifer can be written as:

\[\mathbf{q} = -K \nabla h\]

where: - \(\mathbf{q}\) is the specific discharge (Darcy velocity) vector [L/T]. - \(K\) is the hydraulic conductivity of the aquifer [L/T]. - \(h\) is the hydraulic head [L]. - \(\nabla h\) is the hydraulic gradient, which in two dimensions can be written as: \[\nabla h = \left( \frac{\partial h}{\partial x}, \frac{\partial h}{\partial y} \right)\]

1.2.4.3 Step 2: Volumetric Flow Rate

For a control volume in the unconfined aquifer, the volumetric flow rate \(Q\) at a point in two dimensions is given by:

\[Q_x = q_x \cdot h = -K h \frac{\partial h}{\partial x}\] \[Q_y = q_y \cdot h = -K h \frac{\partial h}{\partial y}\]

where \(h\) is the saturated thickness of the aquifer.

1.2.4.4 Step 3: Conservation of Mass

Consider a differential control volume in the unconfined aquifer with dimensions \(dx\) by \(dy\) and saturated thickness \(h\).

1.2.4.4.1 Inflow and Outflow

• Inflow in the \(x\)-direction at \(x\): \(Q_x(x) = -K h \frac{\partial h}{\partial x}\)

• Outflow in the \(x\)-direction at \(x + dx\): \(Q_x(x + dx) = -K h \frac{\partial h}{\partial x} + \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) dx\)

• Inflow in the \(y\)-direction at \(y\): \(Q_y(y) = -K h \frac{\partial h}{\partial y}\)

• Outflow in the \(y\)-direction at \(y + dy\): \(Q_y(y + dy) = -K h \frac{\partial h}{\partial y} + \frac{\partial}{\partial y} \left( -K h \frac{\partial h}{\partial y} \right) dy\)

1.2.4.4.2 Net Flow

The net inflow into the control volume is:

\[\text{Net inflow in } x\text{-direction} = Q_x(x) - Q_x(x + dx)\] \[= -K h \frac{\partial h}{\partial x} - \left( -K h \frac{\partial h}{\partial x} + \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) dx \right)\] \[= - \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) dx\]

\[\text{Net inflow in } y\text{-direction} = Q_y(y) - Q_y(y + dy)\] \[= -K h \frac{\partial h}{\partial y} - \left( -K h \frac{\partial h}{\partial y} + \frac{\partial}{\partial y} \left( -K h \frac{\partial h}{\partial y} \right) dy \right)\] \[= - \frac{\partial}{\partial y} \left( -K h \frac{\partial h}{\partial y} \right) dy\]

The total net inflow into the control volume is:

\[\text{Net inflow} = - \left( \frac{\partial}{\partial x} \left( -K h \frac{\partial h}{\partial x} \right) dx + \frac{\partial}{\partial y} \left( -K h \frac{\partial h}{\partial y} \right) dy \right)\] \[= \left( \frac{\partial}{\partial x} \left( K h \frac{\partial h}{\partial x} \right) dx + \frac{\partial}{\partial y} \left( K h \frac{\partial h}{\partial y} \right) dy \right)\]

1.2.4.5 Step 4: Change in Storage

The change in storage within the control volume over a time interval \(\Delta t\) can be expressed using the specific yield \(S_y\), which is the volume of water released from storage per unit surface area per unit decline in the water table:

\[\Delta S = S_y \cdot dx \cdot dy \cdot \frac{\partial h}{\partial t} \cdot \Delta t\]

1.2.4.6 Step 5: Applying Conservation of Mass

According to the conservation of mass principle, the rate of change of storage in the control volume must equal the net rate of flow into the control volume:

\[\frac{\partial}{\partial x} \left( K h \frac{\partial h}{\partial x} \right) dx + \frac{\partial}{\partial y} \left( K h \frac{\partial h}{\partial y} \right) dy = S_y \cdot dx \cdot dy \cdot \frac{\partial h}{\partial t}\]

1.2.4.7 Step 6: Simplifying and Rearranging

Divide through by $ dx dy $:

\[\frac{\partial}{\partial x} \left( K h \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( K h \frac{\partial h}{\partial y} \right) = S_y \frac{\partial h}{\partial t}\]

1.2.4.8 Final Governing Equation

The two-dimensional groundwater flow equation for an unconfined aquifer is:

\[S_y \frac{\partial h}{\partial t} = \frac{\partial}{\partial x} \left( K h \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( K h \frac{\partial h}{\partial y} \right)\]

This is the governing equation for transient groundwater flow in a two-dimensional unconfined aquifer. It accounts for the variation in the saturated thickness \(h\) due to fluctuations in the water table and assumes horizontal flow and a vertical hydraulic gradient, as per the Dupuit approximation.

1.2.5 三维非承压地下水运动

You are right. The Dupuit assumption is primarily applicable to simplifying two-dimensional flow problems in unconfined aquifers by assuming horizontal flow and neglecting vertical gradients. For a three-dimensional unconfined aquifer groundwater flow, we need to take a different approach that does not rely on the Dupuit assumption.

Let’s derive the governing equation for three-dimensional unconfined aquifer groundwater flow without using the Dupuit assumption, considering the changes in storage and flow in all three dimensions properly.

1.2.5.1 Step-by-Step Derivation for Three-Dimensional Unconfined Aquifer

1.2.5.2 Step 1: Darcy’s Law

In three dimensions, Darcy’s Law for groundwater flow in an unconfined aquifer is given by:

\[\mathbf{q} = -K \nabla h\]

where: - \(\mathbf{q}\) is the specific discharge (Darcy velocity) vector [L/T]. - \(K\) is the hydraulic conductivity of the aquifer [L/T]. - \(h\) is the hydraulic head [L]. - \(\nabla h\) is the hydraulic gradient, which in three dimensions is written as: \[\nabla h = \left( \frac{\partial h}{\partial x}, \frac{\partial h}{\partial y}, \frac{\partial h}{\partial z} \right)\]

1.2.5.3 Step 2: Conservation of Mass

Consider a differential control volume in the unconfined aquifer with dimensions \(dx\), \(dy\), and \(dz\), where \(z\) is the vertical direction.

1.2.5.3.1 Inflow and Outflow

• Inflow in the \(x\)-direction at \(x\): \(Q_x(x) = q_x A_x = -K \frac{\partial h}{\partial x} A_x\), where \(A_x = dy \cdot dz\) is the cross-sectional area perpendicular to the \(x\)-direction.
• Outflow in the \(x\)-direction at \(x + dx\): \(Q_x(x + dx) = \left( q_x + \frac{\partial q_x}{\partial x} dx \right) A_x\)

Similarly, for the \(y\)- and \(z\)-directions:

• Inflow in the \(y\)-direction at \(y\): \(Q_y(y) = q_y A_y = -K \frac{\partial h}{\partial y} A_y\), where \(A_y = dx \cdot dz\).

• Outflow in the \(y\)-direction at \(y + dy\): \(Q_y(y + dy) = \left( q_y + \frac{\partial q_y}{\partial y} dy \right) A_y\)

• Inflow in the \(z\)-direction at \(z\): \(Q_z(z) = q_z A_z = -K \frac{\partial h}{\partial z} A_z\), where \(A_z = dx \cdot dy\).

• Outflow in the \(z\)-direction at \(z + dz\): \(Q_z(z + dz) = \left( q_z + \frac{\partial q_z}{\partial z} dz \right) A_z\)

1.2.5.3.2 Net Flow

The net inflow into the control volume is the sum of the net inflows in each direction:

\[\text{Net inflow in } x\text{-direction} = Q_x(x) - Q_x(x + dx)\] \[= \left( -K \frac{\partial h}{\partial x} \right) A_x - \left( -K \frac{\partial h}{\partial x} + \frac{\partial}{\partial x} \left( -K \frac{\partial h}{\partial x} \right) dx \right) A_x\] \[= - A_x \frac{\partial}{\partial x} \left( -K \frac{\partial h}{\partial x} \right) dx\]

Similarly,

\[\text{Net inflow in } y\text{-direction} = - A_y \frac{\partial}{\partial y} \left( -K \frac{\partial h}{\partial y} \right) dy\] \[\text{Net inflow in } z\text{-direction} = - A_z \frac{\partial}{\partial z} \left( -K \frac{\partial h}{\partial z} \right) dz\]

The total net inflow into the control volume is:

\[\text{Net inflow} = - \left( \frac{\partial}{\partial x} \left( -K \frac{\partial h}{\partial x} \right) dx \cdot dy \cdot dz + \frac{\partial}{\partial y} \left( -K \frac{\partial h}{\partial y} \right) dx \cdot dy \cdot dz + \frac{\partial}{\partial z} \left( -K \frac{\partial h}{\partial z} \right) dx \cdot dy \cdot dz \right)\] \[= \left( \frac{\partial}{\partial x} \left( K \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( K \frac{\partial h}{\partial y} \right) + \frac{\partial}{\partial z} \left( K \frac{\partial h}{\partial z} \right) \right) dx \cdot dy \cdot dz\]

1.2.5.4 Step 3: Change in Storage

The change in storage within the control volume over a time interval \(\Delta t\) can be expressed using the specific yield \(S_y\), which is the volume of water released from storage per unit surface area per unit decline in the water table:

\[\Delta S = S_y \cdot dx \cdot dy \cdot dz \cdot \frac{\partial h}{\partial t} \cdot \Delta t\]

1.2.5.5 Step 4: Applying Conservation of Mass

According to the conservation of mass principle, the rate of change of storage in the control volume must equal the net rate of flow into the control volume:

\[S_y \frac{\partial h}{\partial t} \cdot dx \cdot dy \cdot dz = \left( \frac{\partial}{\partial x} \left( K \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( K \frac{\partial h}{\partial y} \right) + \frac{\partial}{\partial z} \left( K \frac{\partial h}{\partial z} \right) \right) dx \cdot dy \cdot dz\]

1.2.5.6 Step 5: Simplifying and Rearranging

Divide through by $ dx dy dz $:

\[S_y \frac{\partial h}{\partial t} = \frac{\partial}{\partial x} \left( K \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( K \frac{\partial h}{\partial y} \right) + \frac{\partial}{\partial z} \left( K \frac{\partial h}{\partial z} \right)\]

1.2.5.7 Final Governing Equation

The three-dimensional groundwater flow equation for an unconfined aquifer is:

\[S_y \frac{\partial h}{\partial t} = \frac{\partial}{\partial x} \left( K \frac{\partial h}{\partial x} \right) + \frac{\partial}{\partial y} \left( K \frac{\partial h}{\partial y} \right) + \frac{\partial}{\partial z} \left( K \frac{\partial h}{\partial z} \right)\]

This equation describes the transient groundwater flow in a three-dimensional unconfined aquifer, accounting for variations in the hydraulic head in all three spatial dimensions. This derivation does not rely on the Dupuit assumption, making it suitable for three-dimensional flow problems.

1.2.6 一维热传导方程

Let’s go through the step-by-step derivation of the one-dimensional heat conduction equation based on Fourier’s law in a structured manner.

1.2.6.1 Step 1: Fourier’s Law of Heat Conduction

Fourier’s law states that the heat flux \(q\) (amount of heat per unit area per unit time) is proportional to the negative gradient of the temperature \(T\):

\[q = -k \frac{\partial T}{\partial x}\] where:

• \(q\)is the heat flux [W/m\(^2\)].
• \(k\)is the thermal conductivity of the material [W/(m·K)].
• \(\frac{\partial T}{\partial x}\)is the temperature gradient in the\(x\)-direction [K/m].

1.2.6.2 Step 2: Energy Conservation in a Differential Element

Consider a small differential control volume of length\(dx\), cross-sectional area\(A\), and located at position \(x\) along the rod’s length.

1.2.6.2.1 Heat Entering and Leaving the Control Volume

• The rate of heat entering at \(x\): \(q(x) \cdot A\)
• The rate of heat leaving at \(x + dx\): \(q(x + dx) \cdot A\)

Using a Taylor series expansion for \(q(x + dx)\):

\[q(x + dx) \approx q(x) + \left( \frac{\partial q}{\partial x} \right) dx\]

1.2.6.2.2 Net Heat Flow

The net rate of heat entering the differential element is: \[q(x) \cdot A - \left( q(x) + \left( \frac{\partial q}{\partial x} \right) dx \right) \cdot A\]

\[= -A \left( \frac{\partial q}{\partial x} \right) dx\]

1.2.6.3 Step 3: Heat Storage in the Differential Element

The change in internal energy (\(\Delta U\)) within the differential element over a time interval \(\Delta t\) can be expressed using the specific heat capacity \(c\) and density \(\rho\) of the material: \[\Delta U = \rho \cdot c \cdot A \cdot dx \cdot \frac{\partial T}{\partial t} \cdot \Delta t\]

1.2.6.4 Step 4: Applying Conservation of Energy

Assuming no internal heat generation and applying the conservation of energy principle, the rate of heat entering the control volume must equal the rate of energy storage within the control volume: \[-A \left( \frac{\partial q}{\partial x} \right) dx = \rho \cdot c \cdot A \cdot dx \cdot \frac{\partial T}{\partial t}\]

1.2.6.5 Step 5: Substituting Fourier’s Law

Substitute\(q = -k \frac{\partial T}{\partial x}\)into the equation: \[-A \left( \frac{\partial}{\partial x} \left( -k \frac{\partial T}{\partial x} \right) \right) dx = \rho \cdot c \cdot A \cdot dx \cdot \frac{\partial T}{\partial t}\]

Simplify the equation: \[A \left( k \frac{\partial^2 T}{\partial x^2} \right) dx = \rho \cdot c \cdot A \cdot dx \cdot \frac{\partial T}{\partial t}\]

1.2.6.6 Step 6: Simplifying and Rearranging

Cancel out the common terms\(A\)and\(dx\): \[k \frac{\partial^2 T}{\partial x^2} = \rho c \frac{\partial T}{\partial t}\]

Divide both sides by\(\rho c\): \[\frac{\partial T}{\partial t} = \frac{k}{\rho c} \frac{\partial^2 T}{\partial x^2}\]

Define the thermal diffusivity\(\alpha\)as: \[\alpha = \frac{k}{\rho c}\]

1.2.6.7 Final Governing Equation

The one-dimensional heat conduction equation (also called the heat diffusion equation) is: \[\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\]

This partial differential equation describes how the temperature \(T\) varies with time \(t\) and position \(x\) within the material.

1.2.7 一维溶质运移-扩散方程

Sure, let’s derive the governing equation for a one-dimensional solute advection-diffusion problem step by step. This equation describes how the solute concentration changes over time due to both advection (transport by the flow of the water) and diffusion (spreading due to concentration gradients).

1.2.7.1 Step-by-Step Derivation for One-Dimensional Solute Advection-Diffusion

1.2.7.2 Step 1: Define Variables

• \(c(x, t)\): Solute concentration [M/L³].
• \(u\): Velocity of the fluid in the x-direction [L/T].
• \(D\): Diffusion coefficient [L²/T].
• \(x\): Spatial coordinate in the x-direction [L].
• \(t\): Time [T].

1.2.7.3 Step 2: Conservation of Mass (Continuity Equation)

Consider a differential control volume of length \(dx\) in the x-direction.

1.2.7.3.1 Inflow and Outflow by Advection

• Inflow of solute by advection at position \(x\): \[ J_{\text{adv,in}} = u c(x, t) \]

• Outflow of solute by advection at position $ x + dx $: \[ J_{\text{adv,out}} = u c(x + dx, t) \approx u \left( c(x, t) + \frac{\partial c}{\partial x} dx \right) \]

1.2.7.3.2 Inflow and Outflow by Diffusion

• Inflow of solute by diffusion at position \(x\): \[ J_{\text{diff,in}} = -D \frac{\partial c}{\partial x} \]

• Outflow of solute by diffusion at position $ x + dx $: \[ J_{\text{diff,out}} = -D \frac{\partial c}{\partial x} \Bigg|_{x+dx} \approx -D \left( \frac{\partial c}{\partial x} + \frac{\partial}{\partial x} \left( \frac{\partial c}{\partial x} \right) dx \right) \]

1.2.7.4 Step 3: Net Flow

The net inflow of solute into the control volume is the difference between the inflow and outflow due to both advection and diffusion.

1.2.7.4.1 Net Advection Flow

\[ \text{Net advective flow} = J_{\text{adv,in}} - J_{\text{adv,out}} = u c(x, t) - u \left( c(x, t) + \frac{\partial c}{\partial x} dx \right) = -u \frac{\partial c}{\partial x} dx \]

1.2.7.4.2 Net Diffusion Flow

\[ \text{Net diffusive flow} = J_{\text{diff,in}} - J_{\text{diff,out}} = -D \frac{\partial c}{\partial x} - \left( -D \left( \frac{\partial c}{\partial x} + \frac{\partial}{\partial x} \left( \frac{\partial c}{\partial x} \right) dx \right) \right) = D \frac{\partial^2 c}{\partial x^2} dx \]

1.2.7.5 Step 4: Change in Storage

The change in solute mass within the control volume over a time interval \(\Delta t\) is:

\[ \Delta S = \frac{\partial c}{\partial t} dx \Delta t \]

1.2.7.6 Step 5: Applying Conservation of Mass

According to the conservation of mass principle, the rate of change of solute storage in the control volume must equal the net rate of solute flow into the control volume:

\[ \frac{\partial c}{\partial t} dx = -u \frac{\partial c}{\partial x} dx + D \frac{\partial^2 c}{\partial x^2} dx \]

1.2.7.7 Step 6: Simplifying and Rearranging

Divide through by \(dx\):

\[ \frac{\partial c}{\partial t} = -u \frac{\partial c}{\partial x} + D \frac{\partial^2 c}{\partial x^2} \]

1.2.7.8 Final Governing Equation

The final governing equation for the one-dimensional solute advection-diffusion problem is:

\[ \frac{\partial c}{\partial t} + u \frac{\partial c}{\partial x} = D \frac{\partial^2 c}{\partial x^2} \]

This partial differential equation describes how the solute concentration \(c\) varies with time \(t\) and position \(x\) due to advection by the flow field \(u\) and diffusion characterized by the diffusion coefficient \(D\).

1.2.8 二维溶质运移-扩散方程

Let’s derive the governing equation for a two-dimensional solute advection-diffusion problem step by step. This equation describes how a solute concentration changes over time due to both advection (transport by the flow of the water) and diffusion (spreading due to concentration gradients).

1.2.8.1 Step-by-Step Derivation for Two-Dimensional Solute Advection-Diffusion

1.2.8.2 Step 1: Define Variables

• $ c(x, y, t) $: solute concentration [M/L³].
• \(u\): velocity component in the x-direction [L/T].
• \(v\): velocity component in the y-direction [L/T].
• \(D_x\): diffusion coefficient in the x-direction [L²/T].
• \(D_y\): diffusion coefficient in the y-direction [L²/T].

1.2.8.3 Step 2: Conservation of Mass (Continuity Equation)

Consider a differential control volume in the \(x\)-\(y\) plane with dimensions \(dx\) and \(dy\).

1.2.8.3.1 Inflow and Outflow

• Inflow of solute by advection in the \(x\)-direction at \(x\): \[u(x) c(x) \cdot dy\]

• Outflow of solute by advection in the \(x\)-direction at $ x + dx $: \[u(x + dx) c(x + dx) \cdot dy \approx (u(x) c(x) + \frac{\partial}{\partial x} (u c) dx) \cdot dy\]

• Inflow of solute by advection in the \(y\)-direction at \(y\): \[v(y) c(y) \cdot dx\]

• Outflow of solute by advection in the \(y\)-direction at $ y + dy $: \[v(y + dy) c(y + dy) \cdot dx \approx (v(y) c(y) + \frac{\partial}{\partial y} (v c) dy) \cdot dx\]

• Inflow of solute by diffusion in the \(x\)-direction at \(x\): \[-D_x \frac{\partial c}{\partial x} \cdot dy\]

• Outflow of solute by diffusion in the \(x\)-direction at $ x + dx $: \[-D_x \frac{\partial c}{\partial x} \cdot dy - \frac{\partial}{\partial x} \left( -D_x \frac{\partial c}{\partial x} \right) dx \cdot dy\]

• Inflow of solute by diffusion in the \(y\)-direction at \(y\): \[-D_y \frac{\partial c}{\partial y} \cdot dx\]

• Outflow of solute by diffusion in the \(y\)-direction at $ y + dy $: \[-D_y \frac{\partial c}{\partial y} \cdot dx - \frac{\partial}{\partial y} \left( -D_y \frac{\partial c}{\partial y} \right) dy \cdot dx\]

1.2.8.3.2 Net Flow

The net inflow of solute into the control volume due to advection and diffusion is:

\[\text{Net inflow in } x\text{-direction} = \left( u(x) c(x) \cdot dy - (u(x + dx) c(x + dx) \cdot dy) \right) + \left( -D_x \frac{\partial c}{\partial x} \cdot dy - \left( -D_x \frac{\partial c}{\partial x} - \frac{\partial}{\partial x} \left( -D_x \frac{\partial c}{\partial x} \right) dx \right) dy \right)\] \[= - \frac{\partial}{\partial x} \left( u c \right) dx \cdot dy + \frac{\partial}{\partial x} \left( D_x \frac{\partial c}{\partial x} \right) dx \cdot dy\]

Similarly, for the \(y\)-direction:

\[\text{Net inflow in } y\text{-direction} = - \frac{\partial}{\partial y} \left( v c \right) dy \cdot dx + \frac{\partial}{\partial y} \left( D_y \frac{\partial c}{\partial y} \right) dy \cdot dx\]

The total net inflow of solute into the control volume is:

\[\text{Total net inflow} = \left( \frac{\partial}{\partial x} \left( D_x \frac{\partial c}{\partial x} \right) - \frac{\partial}{\partial x} \left( u c \right) \right) dx \cdot dy + \left( \frac{\partial}{\partial y} \left( D_y \frac{\partial c}{\partial y} \right) - \frac{\partial}{\partial y} \left( v c \right) \right) dy \cdot dx\]

1.2.8.4 Step 3: Change in Storage

The change in solute mass within the control volume over a time interval $ t $ is:

\[\Delta S = \frac{\partial c}{\partial t} \cdot dx \cdot dy \cdot \Delta t\]

1.2.8.5 Step 4: Applying Conservation of Mass

According to the conservation of mass principle, the rate of change of solute storage in the control volume must equal the net rate of solute flow into the control volume:

\[\frac{\partial c}{\partial t} = \left( \frac{\partial}{\partial x} \left( D_x \frac{\partial c}{\partial x} \right) - \frac{\partial}{\partial x} \left( u c \right) \right) + \left( \frac{\partial}{\partial y} \left( D_y \frac{\partial c}{\partial y} \right) - \frac{\partial}{\partial y} \left( v c \right) \right)\]

1.2.8.6 Step 5: Simplifying and Rearranging

Combine the terms and rearrange:

\[\frac{\partial c}{\partial t} = D_x \frac{\partial^2 c}{\partial x^2} + D_y \frac{\partial^2 c}{\partial y^2} - \frac{\partial}{\partial x} \left( u c \right) - \frac{\partial}{\partial y} \left( v c \right)\]

1.2.8.7 Final Governing Equation

The final governing equation for the two-dimensional solute advection-diffusion problem is:

\[\frac{\partial c}{\partial t} + \frac{\partial}{\partial x} (u c) + \frac{\partial}{\partial y} (v c) = D_x \frac{\partial^2 c}{\partial x^2} + D_y \frac{\partial^2 c}{\partial y^2}\]

This partial differential equation describes how the solute concentration \(c\) varies with time \(t\) and position $ (x, y) $ due to advection by the flow field $ (u, v) $ and diffusion characterized by the diffusion coefficients \(D_x\) and \(D_y\).