CN105259088A

Movatterモバイル変換

Info

Publication number: CN105259088A
Application number: CN201510736421.5A
Authority: CN
Inventors: 王世梅; 刘佳龙; 向玲; 李正辉; 鲁芃
Original assignee: China Three Gorges University CTGU
Current assignee: China Three Gorges University CTGU
Priority date: 2015-11-03
Filing date: 2015-11-03
Publication date: 2016-01-20

Abstract

一种用于快速确定非饱和土渗透性函数的方法及装置。运用前向欧拉法和后向欧拉法，对土壤水分运动基本方程Richards偏微分方程进行离散化和反演求解，得到两种非饱和土渗透系数表达式，然后通过定水头一维垂直非饱和入渗试验测得试样中Δz处的体积含水量θ_t及张力h_t随时间变化的一系列数据，将实验数据代入所得的两种表达式，联合二种表达式的计算结果可快速精确地获得非饱和土的渗透性函数。实验装置包括试样筒体、量测系统、供水系统和集水系统，供水系统用溢流方式提供试验所需的定水头水源，集水系统用于收集并量测出水量，量测系统包括用于量测体积含水量θ_t和张力h_t的传感器以及用于数据采集的自动采集装置，传感器埋设于试样筒内部一定位置Δz处。

A method and device for quickly determining the permeability function of unsaturated soil. Using the forward Euler method and the backward Euler method, the basic equation of soil moisture movement, Richards partial differential equation, is discretized and inversely solved, and two expressions of the unsaturated soil permeability coefficient are obtained, and then the one-dimensional vertical nonlinear A series of data of the volumetric water content θ_t and tension h_t at Δz in the sample measured by the saturated infiltration test, and the experimental data are substituted into the two expressions obtained, and the calculation results combined with the two expressions can be quickly Accurately obtain the permeability function of unsaturated soils. The experimental device includes a sample cylinder, a measurement system, a water supply system and a water collection system. The water supply system uses an overflow method to provide the constant head water source required for the test. The water collection system is used to collect and measure the water output. The measurement system includes The sensor used to measure the volumetric water content θ_t and the tension h_t and the automatic acquisition device used for data collection, the sensor is buried at a certain position Δz inside the sample cylinder.

Description

Translated fromChinese

一种用于快速确定非饱和土渗透性函数的方法及装置A method and device for quickly determining the permeability function of unsaturated soil

技术领域technical field

本发明涉及非饱和土渗透系数实验测定领域，特别是一种用于快速确定非饱和土渗透性函数的方法及装置。The invention relates to the field of experimental measurement of unsaturated soil permeability coefficient, in particular to a method and device for quickly determining the permeability function of unsaturated soil.

背景技术Background technique

非饱和渗透系数与土体含水量之间的函数关系称为渗透性函数。地球表面覆盖着的土体大部分处于非饱和状态，随着饱和度的减小，土的非饱和渗透系数减小的幅度可以产生几个甚至更多数量级的变化。因此，如何确定土的非饱和渗透系数，则成为常见的边坡工程、基础工程、大坝等岩土工程以及污染物迁移等非饱和渗流分析中的一个关键问题。The functional relationship between the unsaturated permeability coefficient and the soil water content is called the permeability function. Most of the soil covered on the earth's surface is in an unsaturated state. With the decrease of saturation, the degree of reduction of the unsaturated permeability coefficient of the soil can produce changes of several or more orders of magnitude. Therefore, how to determine the unsaturated permeability coefficient of soil has become a key issue in the unsaturated seepage analysis of common slope engineering, foundation engineering, dam and other geotechnical engineering and pollutant migration.

目前，土的非饱和渗透性函数的获取主要通过经验公式(间接方法)和渗透试验(直接方法)确定。由于非饱和土渗透性试验对于试验仪器、试验过程及试验周期都要求很高，实际应用中几乎都采用非饱和土的土水特征曲线进行预测，这种半经验的确定方法并不具有普适性，常常出现这种半经验的非饱和土渗透性函数被大量滥用，而导致计算分析结果与实际工程相差甚远。因此，非饱和土的渗透性函数最好通过试验测出。At present, the unsaturated permeability function of soil is mainly determined by empirical formula (indirect method) and permeability test (direct method). Since the unsaturated soil permeability test has high requirements on the test equipment, test process and test period, the soil-water characteristic curve of unsaturated soil is almost always used for prediction in practical applications. This semi-empirical determination method is not universal. However, it often occurs that this semi-empirical permeability function of unsaturated soil is widely abused, which leads to a large difference between the calculation and analysis results and the actual engineering. Therefore, the permeability function of unsaturated soils is best determined experimentally.

常用的试验方法分为稳态试验方法(即流速不随时间变化)和非稳态试验方法(即流速随时间变化)。由于非饱和土在高基质吸力作用下的低渗透性，使得稳态试验方法极其耗时，因此非稳态试验方法更受青睐，其中最常用的为瞬态剖面法，在试验中量测非稳态水流中土样的张力及体积含水量分布及其随时间的变化，试验过程中水力梯度和流速都随时间变化，然后利用体积含水量计算流速，由流速和水力梯度之比给出渗透系数。瞬态剖面法试验过程十分复杂，要控制流量边界条件，要防止试样的任何部位有明显的浸润面或饱和，时间周期也较长。相对于流量边界条件，水头边界条件要容易控制和稳定得多，因此研发一种通过控制水头边界条件、能快速进行非饱和土渗透性试验并能获得非饱和土渗透性函数的实验装置及方法，对于将非饱和土理论应用于实际工程意义重大。Commonly used test methods are divided into steady-state test methods (that is, the flow rate does not change with time) and unsteady-state test methods (that is, the flow rate changes with time). Due to the low permeability of unsaturated soil under high matrix suction, the steady-state test method is extremely time-consuming, so the unsteady-state test method is more popular, and the most commonly used is the transient profile method. Tension and volumetric water content distribution of soil samples in steady-state water flow and their changes with time. During the test, the hydraulic gradient and flow velocity both change with time. Then, the volumetric water content is used to calculate the flow velocity, and the permeability is given by the ratio of flow velocity and hydraulic gradient. coefficient. The test process of the transient section method is very complicated. It is necessary to control the flow boundary conditions and prevent any part of the sample from having an obvious wetting surface or saturation, and the time period is also long. Compared with the flow boundary conditions, the hydraulic head boundary conditions are much easier to control and stable. Therefore, an experimental device and method that can quickly conduct unsaturated soil permeability tests and obtain unsaturated soil permeability functions by controlling the hydraulic head boundary conditions , which is of great significance for applying unsaturated soil theory to practical engineering.

通过对土壤水分运动基本控制方程的偏微分形式进行离散，获得非饱和土渗透系数的数值求解式，根据数值求解式，结合简单定水头一维垂直非饱和入渗试验来快速精确地确定非饱和土渗透性函数，是本发明的基本思想。By discretizing the partial differential form of the basic control equation of soil water movement, the numerical solution formula of the unsaturated soil permeability coefficient is obtained. According to the numerical solution formula, combined with a simple one-dimensional vertical unsaturated infiltration test with constant water head, the unsaturated soil can be quickly and accurately determined. Soil permeability function is the basic idea of the present invention.

发明内容Contents of the invention

本发明所要解决的技术问题是提供一种用于快速确定非饱和土渗透性函数的方法及装置，利用前向欧拉法和后向欧拉法通过对土壤水分运动基本方程Richards偏微分方程进行离散化和反演求解，得到两种非饱和土渗透系数表达式并研发定水头一维垂直非饱和入渗实验装置；通过定水头一维垂直非饱和入渗试验测得试样中Δz处的体积含水量θ_t及张力ht随时间变化的一系列数据，将实验数据代入所得的两种渗透系数表达式，联合二种表达式的计算结果可快速精确地获得非饱和土的渗透性函数。The technical problem to be solved by this invention is to provide a method and device for quickly determining the permeability function of unsaturated soil, using the forward Euler method and the backward Euler method to carry out the basic equation of soil moisture movement Richards partial differential equation Discretization and inversion solution to obtain two expressions of unsaturated soil permeability coefficient and develop a constant head one-dimensional vertical unsaturated infiltration experimental device; through the constant water head one-dimensional vertical unsaturated infiltration test to measure the A series of data of volumetric water content θ_t and tension ht changing with time, substituting the experimental data into the obtained two kinds of permeability coefficient expressions, combined with the calculation results of the two expressions can quickly and accurately obtain the permeability function of unsaturated soil.

为解决上述技术问题，本发明所采用的技术方案如下：In order to solve the problems of the technologies described above, the technical scheme adopted in the present invention is as follows:

一种用于快速确定非饱和土渗透性函数的方法，其特征是包括以下步骤：A method for quickly determining the permeability function of unsaturated soil, characterized in that it comprises the following steps:

1)利用前向欧拉法和后向欧拉法通过对土壤水分运动基本方程Richards偏微分方程进行离散化和反演求解，获得了两种非饱和渗透系数表达式：1) Using the forward Euler method and the backward Euler method to discretize and invert the Richards partial differential equation, the basic equation of soil moisture movement, to obtain two expressions of the unsaturated permeability coefficient:

${K K}_{t t} = = \frac{{K K}_{s the s}}{11 - - {h h}^{' '} (({θ θ}_{t t})) \cdot &Center Dot; \frac{{θ θ}_{t t} - - {θ θ}_{s the s}}{Δ Δ z z}} - - - - - - ((I I))$

${K K}_{t t} = = \frac{\frac{{θ θ}_{t t + + 11} - - {θ θ}_{t t}}{Δ Δ t t} \cdot &Center Dot; Δ Δ z z - - {K K}_{s the s}}{{h h}^{' '} (({θ θ}_{t t})) \cdot \cdot \frac{{θ θ}_{t t} - - {θ θ}_{s the s}}{Δ Δ z z} - - 11} - - - - - - ((I I I I))$

式中：In the formula:

K_t表示体积含水量θ_t对应的渗透系数；K_t represents the permeability coefficient corresponding to the volumetric water content θ_t ;

K_s表示饱和渗透系数；K_s represents the saturated permeability coefficient;

表示基质势h对于体积含水量θ求导； Indicates the derivative of the matrix potential h for the volumetric water content θ;

Δz表示测定体积含水量θ_t随时间t变化测点处距离水土接触面的距离；Δz represents the distance between the measuring point and the water-soil contact surface when the volumetric water content θ_t changes with time t;

θ_s表示饱和体积含水量；θ_s represents the saturated volumetric water content;

式(I)含有体积含水量θ和空间距离Δz两个变量，式(II)含有体积含水量θ、时间Δt、及空间距离Δz三个变量；Formula (I) contains two variables of volumetric water content θ and spatial distance Δz, and formula (II) contains three variables of volumetric water content θ, time Δt, and spatial distance Δz;

2)利用非饱和土渗透实验装置，通过一维垂直非饱和入渗试验测得土样测试点Δz处的体积含水量θ_t及张力ht随时间变化的一系列数据，并将所得到的数据代入式(I)及式(II)，快速确定非饱和土的渗透性函数。2) Using the unsaturated soil infiltration test device, through a one-dimensional vertical unsaturated infiltration test to measure a series of data of the volumetric water content θ_t and tension ht at the soil sample test point Δz with time, and the obtained data Substitute into formula (I) and formula (II), quickly determine the permeability function of unsaturated soil.

一种用于快速确定非饱和土渗透性函数的实验装置，包括试样筒体、量测系统、集水系统和供水系统，其特征是：An experimental device for quickly determining the permeability function of unsaturated soil, including a sample cylinder, a measurement system, a water collection system and a water supply system, is characterized by:

量测系统包括用于量测体积含水量θ_t的土壤水分传感器和用于测量张力ht的土壤张力传感器，以及用于数据采集的数据采集装置，土壤水分传感器和土壤张力传感器测试端埋设于试样筒内部Δz处，另一端与数据采集装置连接；The measurement system includes a soil moisture sensor for measuring the volumetric water content θt, a soil tension sensor for measuring the tension_ht , and a data acquisition device for data collection. The test ends of the soil moisture sensor and the soil tension sensor are embedded in the test At Δz inside the sample cylinder, the other end is connected to the data acquisition device;

试样筒体内设有上过滤层和下过滤层，上过滤层由上至下分别为上过滤板、上钢丝网、上滤纸，下过滤层由上至下分别为下滤纸、下钢丝网、下过滤板，上过滤层和下过滤层之间的空间内用于填充土样，在试样筒体上反滤层下方一段距离上开有两个开口，用于设置土壤水分传感器和土壤张力传感器；There are an upper filter layer and a lower filter layer in the sample cylinder. The upper filter layer is the upper filter plate, the upper steel wire mesh, and the upper filter paper from top to bottom. The lower filter layer is the lower filter paper, the lower steel wire mesh, The lower filter plate, the space between the upper filter layer and the lower filter layer is used to fill the soil sample, and there are two openings at a distance below the reverse filter layer on the upper filter layer of the sample cylinder, which are used to set the soil moisture sensor and soil tension sensor;

供水系统包括供水桶、潜水泵、进水管和溢流管，潜水泵放置在供水桶内，潜水泵通过进水管与试样筒体连接，进水管一端与潜水泵连接，另一端位于试样筒体内壁的上过滤层上方，溢流管一端与试样筒体连接，连接位置位于进水管与试样筒体连接位置的下方，溢流管另一端位于供水桶内；The water supply system includes a water supply bucket, a submersible pump, a water inlet pipe and an overflow pipe. The submersible pump is placed in the water supply bucket. The submersible pump is connected to the sample cylinder through the water inlet pipe. Above the upper filter layer of the inner wall, one end of the overflow pipe is connected to the sample cylinder, the connection position is located below the connection position between the water inlet pipe and the sample cylinder, and the other end of the overflow pipe is located in the water supply bucket;

集水系统包括排水管、蓄水瓶和天平，天平放置在集水系统底座上，蓄水瓶放置在天平上，蓄水瓶用于收集试验流出的积水，排水管一端置于试样筒体内下过滤层的下方，另一端伸入蓄水瓶内。The water collection system includes a drainage pipe, a water storage bottle and a balance. The balance is placed on the base of the water collection system. The water storage bottle is placed on the balance. The water storage bottle is used to collect the accumulated water flowing out of the test. Below the lower filter layer in the body, the other end stretches into the water storage bottle.

优选的方案中，所述的溢流管与试样筒的连接位置位于上过滤层上方5cm处，用于提供5cm的定水头边界条件。In a preferred solution, the connection position of the overflow pipe and the sample cylinder is located 5 cm above the upper filter layer, so as to provide a 5 cm constant head boundary condition.

优选的方案中，所述的试样筒体下方设有试验柜台，试样筒体放置在试验柜台上方，集水系统放置在试验柜台下面。In a preferred solution, a test counter is provided under the sample cylinder, the sample cylinder is placed above the test counter, and the water collection system is placed under the test counter.

优选的方案中，所述的蓄水瓶上设有蓄水瓶盖，排水管穿过蓄水瓶盖伸入蓄水瓶中，蓄水瓶盖与蓄水瓶、排水管紧密贴合保证蓄水瓶的密封。In a preferred solution, the water storage bottle is provided with a water storage bottle cap, and the drain pipe extends into the water storage bottle through the water storage bottle cap, and the water storage bottle cap is closely fitted with the water storage bottle and the drain pipe to ensure that the Water bottle seal.

优选的方案中，所述的试样筒体上方设有顶盖，顶盖能够开启或关闭。In a preferred solution, a top cover is provided above the sample cylinder, and the top cover can be opened or closed.

优选的方案中，所述的进水管位于试样筒体内上过滤层的上方的端头上设有水喷头。In a preferred solution, a water nozzle is provided on the end of the water inlet pipe located above the upper filter layer in the sample cylinder.

本发明提供的一种用于快速确定非饱和土渗透系数的方法及装置，通过采用上述方法及装置结构，具有以下有益效果：A method and device for quickly determining the permeability coefficient of unsaturated soil provided by the present invention, by adopting the above method and device structure, has the following beneficial effects:

(1)利用前向欧拉法和后向欧拉法对土壤水分运动基本方程进行离散和反演求解，获得了两种非饱和土渗透性函数表达式，直接由得出的表达式结合简单一维入渗试验便可计算得出非饱和土渗透系数，与瞬态剖面法通过实时计算变动的流速及水力梯度获得渗透系数相比，方法更加简单，函数表达式意义更加清楚；(1) Using forward Euler method and backward Euler method to discretize and invert the basic equation of soil moisture movement, two expressions of unsaturated soil permeability functions are obtained, which are directly combined with simple The one-dimensional infiltration test can be used to calculate the unsaturated soil permeability coefficient. Compared with the transient profile method which obtains the permeability coefficient by calculating the changing flow velocity and hydraulic gradient in real time, the method is simpler and the meaning of the function expression is clearer;

(2)只需要知道固定位置Δz处体积含水量θ和张力h_t随时间t的关系，代入上述非饱和土渗透性函数表达式便可以求得土的非饱和渗透性函数，将两个计算表达式在较低体积含水量与较高体积含水量下的计算结果相结合便可以获得较准确的土的非饱和渗透性函数；(2) It is only necessary to know the relationship between the volumetric water content θ at the fixed position Δz and the tension h_t with time t, and then substitute the above unsaturated soil permeability function expression to obtain the soil unsaturated permeability function. The two calculations A more accurate unsaturated permeability function of soil can be obtained by combining the calculation results of the expression at lower volumetric water content and higher volumetric water content;

(3)将供水系统设计成一种恒定水头的方法，利用潜水泵将不锈钢圆桶中的无气水通过进水管末端的喷头，均匀喷到过滤板上面，通过过滤板将水均匀分布渗入圆形土体中，随着水位的升高，当水位达到溢水口高度时，多余的水将从出水口溢出，从而到达控制恒定水头压力的目的，为试验提供稳定的边界条件。(3) Design the water supply system as a method of constant water head, use the submersible pump to pass the airless water in the stainless steel drum through the nozzle at the end of the water inlet pipe, and evenly spray it on the filter plate, and evenly distribute the water through the filter plate. In the soil, as the water level rises, when the water level reaches the height of the overflow, the excess water will overflow from the outlet, so as to achieve the purpose of controlling the constant head pressure and provide stable boundary conditions for the test.

附图说明Description of drawings

下面结合附图和实施例对本发明作进一步说明：Below in conjunction with accompanying drawing and embodiment the present invention will be further described:

图1为本发明的装置整体结构示意图。Figure 1 is a schematic diagram of the overall structure of the device of the present invention.

图2本发明的简化入渗模型示意图。Fig. 2 is a schematic diagram of a simplified infiltration model of the present invention.

图中：顶盖1，水喷头2，进水管3，试样筒体4，水5，上过滤板6，上钢丝网7，上滤纸8，土壤水分传感器9，土壤张力传感器10，土样11，下滤纸12，下钢丝网13，下过滤板14，开口15，排水管16，蓄水瓶盖17，蓄水瓶18，积水19，天平20，试验台21，集水系统底座22，溢流管23，供水桶24，潜水泵25,水土接触面26，数据采集装置27。In the figure: top cover 1, water nozzle 2, water inlet pipe 3, sample cylinder 4, water 5, upper filter plate 6, upper steel wire mesh 7, upper filter paper 8, soil moisture sensor 9, soil tension sensor 10, soil sample 11. Lower filter paper 12, lower steel mesh 13, lower filter plate 14, opening 15, drain pipe 16, water storage bottle cap 17, water storage bottle 18, accumulated water 19, balance 20, test bench 21, water collection system base 22 , Overflow pipe 23, water supply bucket 24, submersible pump 25, water-soil contact surface 26, data acquisition device 27.

具体实施方式detailed description

关于非饱和土渗透系数表达式的推导：The derivation of the expression of unsaturated soil permeability coefficient:

1.土壤水分运动基本控制方程1. Basic governing equations of soil moisture movement

土壤水分运动基本方程的一维垂直入渗形式如下：The one-dimensional vertical infiltration form of the basic equation of soil water movement is as follows:

$\frac{\partial \partial θ θ}{\partial \partial t t} = = \frac{\partial \partial}{\partial \partial z z} [[K K ((θ θ)) \cdot &Center Dot; \frac{\partial \partial h h}{\partial \partial z z}]] - - \frac{\partial \partial K K ((θ θ))}{\partial \partial z z} - - - - - - ((I I I I I I))$

式中：In the formula:

θ表示体积含水量；θ represents volume water content;

t表示时间；t means time;

K(θ)表示对应于θ的渗透系数；K(θ) represents the permeability coefficient corresponding to θ;

h表示基质势。h represents the matrix potential.

令则式(III)可以变化为如下形式：make Then formula (III) can be changed into the following form:

$\frac{\partial \partial θ θ}{\partial \partial t t} = = \frac{\partial \partial}{\partial \partial z z} [[K K ((θ θ)) \cdot &Center Dot; \frac{\partial \partial h h}{\partial \partial θ θ} \cdot &Center Dot; \frac{\partial \partial θ θ}{\partial \partial z z}]] - - \frac{\partial \partial K K ((θ θ))}{\partial \partial z z} - - - - - - ((I I V V))$

2.土壤水分运动基本控制方程的离散和反演求解2. Discretization and inversion of the basic governing equations of soil moisture movement

在一维非饱和入渗模型中，设i代表空间位置，j代表时间位置，时间间隔为Δt，空间间隔为Δz。可分别采用向前欧拉法和向后欧拉法对方程(IV)进行离散和反演求解，得到两种渗透系数K(θ)表达式。In the one-dimensional unsaturated infiltration model, let i represent the space position, j represent the time position, the time interval is Δt, and the space interval is Δz. The forward Euler method and the backward Euler method can be used to solve equation (IV) discretely and inversely, and two expressions of the permeability coefficient K(θ) can be obtained.

建立如附图2所示的简化一维垂直入渗离散模型，其中蓝色部分表示土样顶部极薄的充水层，即透水石部分，土黄色部分代表土样。在土柱竖直方向设立0、1、2三个位置点，0和1为水土界面附近非常接近的两点，可认为该两点的含水量始终等于饱和含水量，不随时间变化。A simplified one-dimensional vertical infiltration discrete model is established as shown in Figure 2, in which the blue part represents the extremely thin water-filled layer on the top of the soil sample, that is, the part of the permeable stone, and the khaki part represents the soil sample. Three points 0, 1, and 2 are set up in the vertical direction of the soil column. 0 and 1 are two points very close to the water-soil interface. It can be considered that the water content of these two points is always equal to the saturated water content and does not change with time.

向前欧拉法和向后欧拉法对对方程(IV)进行离散和反演求解过程如下：The forward Euler method and the backward Euler method discretize and inversely solve the equation (IV) as follows:

2.1向前欧拉法对方程(IV)进行离散和反演求解：2.1 Forward Euler method to solve equation (IV) discretely and inversely:

对于 $\frac{\partial θ}{\partial t} = \frac{\partial}{\partial z} [K (θ) \cdot \frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z}] - \frac{\partial K (θ)}{\partial z} = \frac{\partial K \cdot (\frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z} - 1)}{\partial z}$ for $\frac{\partial θ}{\partial t} = \frac{\partial}{\partial z} [K (θ) &Center Dot; \frac{\partial h}{\partial θ} &Center Dot; \frac{\partial θ}{\partial z}] - \frac{\partial K (θ)}{\partial z} = \frac{\partial K &Center Dot; (\frac{\partial h}{\partial θ} &Center Dot; \frac{\partial θ}{\partial z} - 1)}{\partial z}$

设 $h^{'} (θ) = \frac{\partial h}{\partial θ}, f = K \cdot \frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z} - 1 = K \cdot (h^{'} (θ) \cdot \frac{\partial θ}{\partial z} - 1),$ Assume $h^{'} (θ) = \frac{\partial h}{\partial θ}, f = K \cdot \frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z} - 1 = K \cdot (h^{'} (θ) &Center Dot; \frac{\partial θ}{\partial z} - 1),$

则：but:

$\frac{\partial \partial θ θ}{\partial \partial t t} = = \frac{\partial \partial f f}{\partial \partial z z} - - - - - - ((V V))$

运用前向欧拉法对时间和空间进行离散，Using forward Euler method to discretize time and space,

左边 $\frac{\partial θ}{\partial t} = \frac{θ_{i}^{j + 1} - θ_{i}^{j}}{Δ t}$ left $\frac{\partial θ}{\partial t} = \frac{θ_{i}^{j + 1} - θ_{i}^{j}}{Δ t}$

右边 $\frac{\partial f}{\partial z} = \frac{f_{i + 1}^{j} - f_{i}^{j}}{Δ z}$ right $\frac{\partial f}{\partial z} = \frac{f_{i + 1}^{j} - f_{i}^{j}}{Δ z}$

${f f}_{i i + + 11}^{j j} = = {K K}_{i i + + 11}^{j j} [[{h h}^{' '} (({θ θ}_{i i + + 11}^{j j})) \cdot &Center Dot; {((\frac{\partial \partial θ θ}{\partial \partial z z}))}_{i i + + 11}^{j j} - - 11]] = = {K K}_{i i + + 11}^{j j} [[{h h}^{' '} (({θ θ}_{i i + + 11}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{i i + + 11}^{j j} - - {θ θ}_{i i}^{j j}}{Δ Δ z z} - - 11]]$

${f f}_{i i}^{j j} = = {K K}_{i i}^{j j} [[{h h}^{' '} (({θ θ}_{i i}^{j j})) \cdot &Center Dot; {((\frac{\partial \partial θ θ}{\partial \partial z z}))}_{i i}^{j j} - - 11]] = = {K K}_{i i}^{j j} [[{h h}^{' '} (({θ θ}_{i i}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{i i}^{j j} - - {θ θ}_{i i - - 11}^{j j}}{Δ Δ z z} - - 11]]$

将上述两式代入式(V)得到最终离散形式如下：Substituting the above two formulas into formula (V), the final discrete form is as follows:

$\frac{{θ θ}_{i i}^{j j + + 11} - - {θ θ}_{i i}^{j j}}{Δ Δ t t} = = \frac{11}{Δ Δ z z} {{{K K}_{i i + + 11}^{j j} [[{h h}^{' '} (({θ θ}_{i i + + 11}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{i i + + 11}^{j j} - - {θ θ}_{i i}^{j j}}{Δ Δ z z} - - 11]] - - {K K}_{i i}^{j j} [[{h h}^{' '} (({θ θ}_{i i}^{j j})) \cdot \cdot \frac{{θ θ}_{i i}^{j j} - - {θ θ}_{i i - - 11}^{j j}}{Δ Δ z z} - - 11]]}}$

简化的入渗模型示意图如附图2所示，当i＝1时，上述方程表述如下：The schematic diagram of the simplified infiltration model is shown in Figure 2. When i=1, the above equation is expressed as follows:

$\frac{{θ θ}_{11}^{j j + + 11} - - {θ θ}_{11}^{j j}}{Δ Δ t t} = = \frac{11}{{Δz Δz}_{22}} {{{K K}_{22}^{j j} [[{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot \cdot \frac{{θ θ}_{22}^{j j} - - {θ θ}_{11}^{j j}}{{Δz Δz}_{22}} - - 11]] - - {K K}_{11}^{j j} [[{h h}^{' '} (({θ θ}_{11}^{j j})) \cdot \cdot \frac{{θ θ}_{11}^{j j} - - {θ θ}_{00}^{j j}}{{Δz Δz}_{11}} - - 11]]}} - - - - - - ((V V I I))$

位置i＝0和i＝1为水土接触面26上、下相邻的两点，故和不随时间而变，为饱和含水量θ_s，为饱和渗透系数K_s，则式(VI)变化为：The positions i=0 and i=1 are two adjacent points above and below the water-soil contact surface 26, so and does not change with time, is the saturated water content θ_s , is the saturated permeability coefficient K_s , then formula (VI) changes to:

$\frac{{θ θ}_{s the s} - - {θ θ}_{s the s}}{Δ Δ t t} = = \frac{11}{{Δz Δz}_{22}} {{{K K}_{22}^{j j} [[{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{22}^{j j} - - {θ θ}_{s the s}}{{Δz Δz}_{22}} - - 11]] - - {K K}_{11}^{j j} [[{h h}^{' '} (({θ θ}_{s the s})) \cdot \cdot \frac{{θ θ}_{s the s} - - {θ θ}_{s the s}}{{Δz Δz}_{11}} - - 11]]}}$

$00 = = \frac{11}{{Δz Δz}_{22}} {{{K K}_{22}^{j j} [[{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{22}^{j j} - - {θ θ}_{s the s}}{{Δz Δz}_{22}} - - 11]] + + {K K}_{s the s}}}$

则：but:

${K K}_{22}^{j j} = = \frac{{K K}_{s the s}}{11 - - {h h}^{' '} (({θ θ}_{22}^{j j})) \cdot \cdot \frac{{θ θ}_{22}^{j j} - - {θ θ}_{s the s}}{{Δz Δz}_{22}}} - - - - - - ((V V I I I I))$

从式(VII)可以看出，推导后的渗透系数表达式是以含水量传感器到水土接触面的间距Δz以及体积含水量θ为因变量，与时间t没有关系。因此，只需通过一维非饱和垂直入渗试验在测点Δz处测得任意j时刻的土样体积含水量θ和张力h，就可以通过式(VII)计算出相应的渗透系数，将不同时刻的试验数据分别带入式(VII)计算便可获得一系列不同体积含水量所对应的渗透系数，进而此得到非饱和渗透性函数K(θ)。From formula (VII), it can be seen that the derived permeability coefficient expression takes the distance Δz from the water content sensor to the water-soil contact surface and the volumetric water content θ as dependent variables, and has nothing to do with time t. Therefore, it is only necessary to measure the volumetric water content θ and tension h of the soil sample at any j moment at the measuring point Δz through the one-dimensional unsaturated vertical infiltration test, and the corresponding permeability coefficient can be calculated by formula (VII). The test data at each moment are brought into formula (VII) for calculation to obtain a series of permeability coefficients corresponding to different volumetric water contents, and then obtain the unsaturated permeability function K(θ).

2.2运用向后欧拉法对方程(IV)进行离散和反演求解：2.2 Use the backward Euler method to solve equation (IV) discretely and inversely:

对于 $\frac{\partial θ}{\partial t} = \frac{\partial}{\partial z} [K (θ) \cdot \frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z}] - \frac{\partial K (θ)}{\partial z} = \frac{\partial K \cdot (\frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z} - 1)}{\partial z}$ for $\frac{\partial θ}{\partial t} = \frac{\partial}{\partial z} [K (θ) &Center Dot; \frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z}] - \frac{\partial K (θ)}{\partial z} = \frac{\partial K &Center Dot; (\frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z} - 1)}{\partial z}$

设 $h^{'} (θ) = \frac{\partial h}{\partial θ}, f = K \cdot \frac{\partial h}{\partial θ} \cdot \frac{\partial θ}{\partial z} - 1 = K \cdot (h^{'} (θ) \cdot \frac{\partial θ}{\partial z} - 1),$ Assume $h^{'} (θ) = \frac{\partial h}{\partial θ}, f = K &Center Dot; \frac{\partial h}{\partial θ} &Center Dot; \frac{\partial θ}{\partial z} - 1 = K &Center Dot; (h^{'} (θ) &Center Dot; \frac{\partial θ}{\partial z} - 1),$

则：but:

$\frac{\partial \partial θ θ}{\partial \partial t t} = = \frac{\partial \partial f f}{\partial \partial z z} - - - - - - ((V V I I I I I I))$

运用前向欧拉法对时间和空间进行离散Discretization of time and space using forward Euler method

右边 $\frac{\partial f}{\partial z} = \frac{f_{i}^{j} - f_{i - 1}^{j}}{Δ z}$ right $\frac{\partial f}{\partial z} = \frac{f_{i}^{j} - f_{i - 1}^{j}}{Δ z}$

${f f}_{i i - - 11}^{j j} = = {K K}_{i i - - 11}^{j j} [[{h h}^{' '} (({θ θ}_{i i - - 11}^{j j})) \cdot &Center Dot; {((\frac{\partial \partial θ θ}{\partial \partial z z}))}_{i i - - 11}^{j j} - - 11]] = = {K K}_{i i - - 11}^{j j} [[{h h}^{' '} (({θ θ}_{i i - - 11}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{i i - - 11}^{j j} - - {θ θ}_{i i - - 22}^{j j}}{Δ Δ z z} - - 11]]$

将上述两式代入式(VIII)得到最终离散形式如下：Substituting the above two formulas into formula (VIII), the final discrete form is as follows:

$\frac{{θ θ}_{i i}^{j j + + 11} - - {θ θ}_{i i}^{j j}}{Δ Δ t t} = = \frac{11}{Δ Δ z z} {{{K K}_{i i}^{j j} [[{h h}^{' '} (({θ θ}_{i i}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{i i}^{j j} - - {θ θ}_{i i - - 11}^{j j}}{Δ Δ z z} - - 11]] - - {K K}_{i i - - 11}^{j j} [[{h h}^{' '} (({θ θ}_{i i - - 11}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{i i - - 11}^{j j} - - {θ θ}_{i i - - 22}^{j j}}{Δ Δ z z} - - 11]]}} - - - - - - ((I I X x))$

简化的入渗模型示意图如附图2所示，当i＝2时，上述方程表述如下：The schematic diagram of the simplified infiltration model is shown in Figure 2. When i=2, the above equation is expressed as follows:

$\frac{{θ θ}_{22}^{j j + + 11} - - {θ θ}_{22}^{j j}}{Δ Δ t t} = = \frac{11}{{Δz Δz}_{22}} {{{K K}_{22}^{j j} [[{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot &Center Dot; \frac{{θ θ}_{22}^{j j} - - {θ θ}_{11}^{j j}}{{Δz Δz}_{22}} - - 11]] - - {K K}_{11}^{j j} [[{h h}^{' '} (({θ θ}_{11}^{j j})) \cdot \cdot \frac{{θ θ}_{11}^{j j} - - {θ θ}_{00}^{j j}}{{Δz Δz}_{11}} - - 11]]}} - - - - - - ((X x))$

位置i＝0和i＝1为水土接触面26上、下相邻的两点，故和不随时间而变，为饱和含水量θ_s，为饱和渗透系数K_s，则式(X)变化为：The positions i=0 and i=1 are two adjacent points above and below the water-soil contact surface 26, so and does not change with time, is the saturated water content θ_s , is the saturation permeability coefficient K_s , then formula (X) changes to:

$\frac{{θ θ}_{22}^{j j + + 11} - - {θ θ}_{22}^{j j}}{Δ Δ t t} = = \frac{11}{{Δz Δz}_{22}} {{{K K}_{22}^{j j} [[{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot \cdot \frac{{θ θ}_{22}^{j j} - - {θ θ}_{s the s}}{{Δz Δz}_{22}} - - 11]] - - {K K}_{11}^{j j} [[{h h}^{' '} (({θ θ}_{s the s})) \cdot \cdot \frac{{θ θ}_{s the s} - - {θ θ}_{s the s}}{{Δz Δz}_{11}} - - 11]]}}$

$\frac{{θ θ}_{22}^{j j + + 11} - - {θ θ}_{22}^{j j}}{Δ Δ t t} = = \frac{11}{{Δz Δz}_{22}} {{{K K}_{22}^{j j} [[{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot \cdot \frac{{θ θ}_{22}^{j j} - - {θ θ}_{s the s}}{{Δz Δz}_{22}} - - 11]] - - {K K}_{s the s}}}$

则：but:

${K K}_{22}^{j j} = = \frac{\frac{{θ θ}_{22}^{j j + + 11} - - {θ θ}_{22}^{j j}}{Δ Δ t t} \cdot \cdot {Δz Δz}_{22} - - {K K}_{s the s}}{{h h}^{' '} (({θ θ}_{22}^{j j})) \cdot \cdot \frac{{θ θ}_{22}^{j j} - - {θ θ}_{s the s}}{{Δz Δz}_{22}} - - 11} - - - - - - ((X x I I))$

从式(XI)可以看出，推导后的渗透系数表达式是以时间t和含水量传感器到水土接触面的间距Δz以及体积含水量θ为因变量。因此，只需通过一维非饱和垂直入渗试验在测点Δz处测得任意j时刻的土样体积含水量θ和张力h，就可以通过式(XI)计算出相应的渗透系数，将不同时刻的试验数据分别带入式(XI)计算便可获得一系列不同体积含水量所对应的渗透系数，进而此得到非饱和渗透性函数K(θ)。It can be seen from formula (XI) that the derived permeability coefficient expression takes the time t, the distance Δz from the water content sensor to the water-soil contact surface, and the volumetric water content θ as dependent variables. Therefore, it is only necessary to measure the volumetric water content θ and the tension h of the soil sample at any j moment at the measuring point Δz through the one-dimensional unsaturated vertical infiltration test, and the corresponding permeability coefficient can be calculated by formula (XI). The test data at each moment are brought into formula (XI) for calculation to obtain a series of permeability coefficients corresponding to different volumetric water contents, and then obtain the unsaturated permeability function K(θ).

综上所述，通过对一维垂直入渗方程的推导得到了两种非饱和渗透系数的表达式，之后通过对一维入渗物理模型的简化处理，得出了一个点位控制下的非饱和渗透系数表达式的两种形式：To sum up, through the derivation of the one-dimensional vertical infiltration equation, two expressions of unsaturated permeability coefficients are obtained, and then through the simplified treatment of the one-dimensional infiltration physical model, a non-saturated permeability under the control of point position is obtained. Two forms of saturated permeability coefficient expression:

${K K}_{t t} = = \frac{\frac{{θ θ}_{t t + + 11} - - {θ θ}_{t t}}{Δ Δ t t} \cdot &Center Dot; Δ Δ z z - - {K K}_{s the s}}{{h h}^{' '} (({θ θ}_{t t})) \cdot &Center Dot; \frac{{θ θ}_{t t} - - {θ θ}_{s the s}}{Δ Δ z z} - - 11} - - - - - - ((I I I I))$

式中：In the formula:

表示基质势对于体积含水量求导； Indicates the derivative of the matrix potential with respect to the volumetric water content;

θ_s表示饱和体积含水量。θ_s represents the saturated volumetric water content.

通过数值试验方法对式(I)和式(II)进行可靠性验证，发现不含时间t的非饱和渗透性函数表达式(I)可以较好的预测较低含水量条件下的非饱和渗透系数，含有时间t的表达式(II)可以更为精确的计算高含水量条件下的非饱和渗透性函数。最终决定联合两个表达式的计算结果，即将式(I)在较低体积含水量与式(II)在较高体积含水量下的计算结果相结合以获得较准确连续的非饱和土渗透系数K(θ)，进而得到非饱和土渗透性函数。The reliability of formulas (I) and (II) was verified by numerical experiments, and it was found that the unsaturated permeability function expression (I) without time t can better predict the unsaturated permeability under the condition of lower water content Coefficient, expression (II) including time t can more accurately calculate the unsaturated permeability function under the condition of high water content. Finally, it was decided to combine the calculation results of the two expressions, that is, to combine the calculation results of formula (I) at a lower volumetric water content with the calculation results of formula (II) at a higher volumetric water content to obtain a more accurate and continuous unsaturated soil permeability coefficient K(θ), and then get the unsaturated soil permeability function.