METHOD FOR QUICKLY ACQUIRING SOIL HYDRAULIC PROPERTIES IN SITU BASED ON PONDED INFILTRATION EXPERIMENT

Abstract

A method for quickly acquiring soil hydraulic properties in situ based on a ponded infiltration experiment is provided. The method adopts a novel derivation method, and is based on a Richards' equation and a Brooks-Corey model to derive an analytical solution that accurately describes one-dimensional infiltration into homogeneous soil under ponded conditions. The method, for the first time, provides a detailed description of a developing saturated zone in the soil water profile, which is the most vital infiltration characteristic during ponded infiltration. The method proposes an optimized estimation method for parameters based on an inverse process of the analytical solution. The method can quickly acquire soil hydraulic properties in situ field by measuring a cumulative infiltration amount and a length of a wetting front over time during one-dimensional ponded infiltration experiment through a time-domain reflectometer (TDR).

Claims

1. A method for quickly acquiring soil hydraulic properties in situ based on ponded infiltration experiment, comprising the following steps: (1) measuring a saturated water content ?.sub.s, an initial water content ?.sub.i, and a residual water content ?.sub.r of soil; (2) monitoring variations of a cumulative infiltration amount I and a length z.sub.f of a wetting front over an infiltration time t during an in situ field one-dimensional ponded infiltration experiment; and (3) calculating target parameters n, h.sub.d, and K.sub.s with an optimized calculation method according to an objective function (9): $\begin{array}{cc}\min Q=\underset{i=1}{\overset{N}{.Math.}}{\left({\hat{I}}_i\left(n,h_d,K_s\right)-I_i\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{\left({\hat{z}}_{f,i}\left(n,h_d,K_s\right)-z_{f,i}\right)}^2& \left(9\right)\end{array}$ wherein, I.sub.i and ?.sub.i respectively denote an observed value of the cumulative infiltration amount and a predicted value of the cumulative infiltration amount calculated by an analytical model; z.sub.f,i and {circumflex over (z)}.sub.f,i respectively denote an observed value of the length of the wetting front and a predicted value of the length of the wetting front calculated by the analytical model; and N denotes a total number of observed values.

2. The method for quickly acquiring soil hydraulic properties in situ based on ponded infiltration experiment according to claim 1, wherein step (3) further comprises: optimizing the objective function (9) through a Levenberg-Marquardt algorithm to estimate correct values of the target parameters n, h.sub.d, and K.sub.s to minimize the objective function (9); and calculating estimated values ?(n, h.sub.d, K.sub.s) and (n, h.sub.d, K.sub.s) as follows: calculating, by a Newton-Simpson method, an estimated value (n.sub.e, h.sub.d,e, K.sub.s,e) of a length z.sub.s of a saturated zone at any infiltration time t, given a set of estimation parameters n.sub.e, h.sub.d,e, and K.sub.s,e, according to Equation (5): $\begin{array}{cc}{z}_s-\left(B_3+B_5\right)\ln \left(1+\frac{z_s}{B_3}\right)+B_5\ln \left(1+\frac{B_2{z}_s}{B_3}\right)+\frac{B_1}{\left(1-B_2\right)B_2}\left(1-\frac{B_3}{\left(B_2{z}_s+B_3\right)}\right)=B_{4}t& \left(5\right)\end{array}$ calculating, based on the estimated value (n.sub.e, h.sub.d,e, K.sub.s,e), an estimated value ?(n.sub.e, h.sub.d,e, K.sub.s,e) of the cumulative infiltration amount I according to Equation (6): $\begin{array}{cc}I=\left({?}_s-{?}_i\right)\left(z_s+\frac{b_1{z}_s}{B_2{z}_s+B_3}\right)+K_{i}t& \left(6\right)\end{array}$ calculating an estimated value (n.sub.e, h.sub.d,e, K.sub.s,e) of the length z.sub.r of the wetting front according to Equation (7): $\begin{array}{cc}{z}_f=z_s+\frac{B_1{z}_s}{B_0\left(B_2{z}_s+B_3\right)}& \left(7\right)\end{array}$ wherein, unknown parameters in the above equations are defined as follows: $\begin{array}{cc}\{\begin{array}{c}{S}_i=\frac{{?}_i-{?}_r}{{?}_s-{?}_r}\\ a=\frac{n}{2n+2}\\ b=1-S_i^{\frac{1}{a}}\\ B_0=\left({?}_s-{?}_r\right)\frac{1-\left(1+\mathrm{ab}\right)S_i}{b\left(a+1\right)}\\ B_1=\frac{1-S_i^{1+1/n}}{3n+1}\\ B_2=1-\frac{\left(a+1\right)\left(1-S_i\right)-\left(a\left(a+2\right)b-\left(1-b\right)\left(1-S_i\right)\right)S_i^{3+2/n}}{\left(1-\left(1+\mathrm{ab}\right)S_i\right)\left(a+2\right)}\\ B_3=h_p+h_d\\ B_4=\frac{K_s}{{?}_s-{?}_i}\\ B_5=\frac{B_1}{{\left(1-B_2\right)}^2}\end{array}.& \left(8\right)\end{array}$

3. The method for quickly acquiring soil hydraulic properties in situ based on ponded infiltration experiment according to claim 1, wherein steps (1) and (2) comprise a parameter acquisition method as follows: inserting, before infiltration, a probe of a time-domain reflectometer (TDR) into target soil for an one-dimensional ponded infiltration experiment with a ponded depth of h.sub.p; starting acquiring a time-domain reflection signal from the probe of the TDR before infiltration; calculating the initial water content ?.sub.i and the saturated water content ?.sub.s of the soil; sampling the soil with a cutting ring; and measuring the residual water content ?.sub.r of the soil; and acquiring time series data z.sub.f-t of the length of the wetting front and time series data I-t of the cumulative infiltration amount in real-time by interpreting a reflection waveform of the TDR during one-dimensional ponded infiltration.

4. The method for quickly acquiring soil hydraulic properties in situ based on ponded infiltration experiment according to claim 3, wherein steps (1) and (2) further comprise a parameter acquisition method as follows: (A) inserting, before infiltration, the probe of the TDR with a length of L vertically and wholly into the target soil until a head of the probe is flush with the soil surface; and acquiring, by the TDR, a reflection signal from the probe of the TDR continuously at a predetermined frequency before infiltration, until a wetting front exceeds the length of the probe and a soil layer monitored by probe of the TDR reaches a saturated state; (B) determining that there is no wetting front within a measurement range of the TDR before and at an end of infiltration; extracting apparent positions of the head and a tail of the probe before infiltration from the reflection waveform of the TDR, and acquiring an initial apparent length L.sub.ad of a soil layer monitored by the probe of the TDR; extracting apparent positions of the head and the tail of the probe at the end of infiltration, and acquiring a saturated apparent length L.sub.as of the soil layer monitored by the probe of the TDR; calculating, based on an electromagnetic wave transmission theory, an initial average dielectric constant K.sub.ad=(L.sub.ad/L).sup.2 and a saturated average dielectric constant K.sub.as=(L.sub.as/L).sup.2 of the monitored soil layer; and converting, based on a relationship between a soil water content and a dielectric constant K.sub.ad and K.sub.as, into the initial water content ?.sub.i and the saturated water content ?.sub.s of the soil, respectively; and (C) determining that there is a wetting front within the measurement range of the TDR after infiltration begins; extracting, at each moment, an apparent length L.sub.a of the soil layer monitored by the probe of the TDR and an apparent length L.sub.aw of the wetting front from the reflection waveform of the TDR before the wetting front reaches the tail of the probe; calculating an average dielectric constant K.sub.a=(L.sub.a/L).sup.2 of the monitored soil layer based on the apparent length L.sub.a; converting, based on the relationship between the soil water content and the dielectric constant, the average dielectric constant K.sub.a of the soil layer monitored by the probe of the TDR into a real-time average water content ?; calculating the cumulative infiltration amount of the soil, I=L*(???.sub.i); acquiring the time series data I-1 of the cumulative infiltration amount and the length of the wetting front, z.sub.f=L*(L.sub.ad?L.sub.a+L.sub.aw)/L.sub.ad; and acquiring the time series data z.sub.f-t of the length of the wetting front.

5. The method for quickly acquiring soil hydraulic properties in situ based on ponded infiltration experiment according to claim 4, wherein step (C) further comprises: extracting, by a second-derivative method, apparent positions of the wetting front and the tail of the probe from the reflection waveform data of the TDR: acquiring first-derivative and second-derivative of waveform data; determining a first zero point of the first-derivative of the waveform data after occurrence of a maximum value of the first-derivative as an apparent position X.sub.h corresponding to the head of the probe of the TDR; determining positions where second and third local maximum values of the second-derivative appear after X.sub.h as an apparent position X.sub.f corresponding to the wetting front and an apparent position X.sub.e corresponding to the tail of the probe of the TDR, respectively; determining the apparent length L.sub.a of the monitored soil layer as a distance X.sub.e-X.sub.n between the apparent positions of the head and the tail of the probe; and determining the apparent length L.sub.aw of the wetting front as a distance X.sub.f-X.sub.h between the apparent positions of the head and the tail of the probe.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0025] FIG. 1 is a flowchart of a method for quickly acquiring soil hydraulic properties in situ based on a one-dimensional ponded infiltration experiment according to the present disclosure;

[0026] FIG. 2 shows a comparison between variations of a length of a wetting front observed by a TDR method and a conventional method;

[0027] FIG. 3 shows a comparison between variations of a cumulative infiltration amount simulated by an analytical solution and observed in a test; and

[0028] FIG. 4 shows a comparison between a soil water retention curve predicted by the method of the present disclosure and a measurement result of a pressure membrane method.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0029] The content of the present disclosure will be further described below in conjunction with the drawings, but they should not be construed as limiting the present disclosure. Modifications and substitutions made to methods, steps, or conditions of the present disclosure without departing from the spirit and essence of the present disclosure fall within the scope of the present disclosure. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art. FIG. 1 shows a calculation process according to the present disclosure.

Embodiment 1

[0030] The technical principles of the present disclosure are as follows. Based on a Richards' equation, a Brooks-Corey model, soil water flux-concentration relationship, and mean value theorem of integrals, the present disclosure acquires a profile equation describing water infiltration into soil. The present disclosure derives a complete analytical model for describing one-dimensional infiltration into homogeneous soil under ponded conditions based on the principle of mass conservation and the profile equation. Specifically:

[0031] The Richards' equation and its initial and boundary conditions are expressed as follows:

[00006]$\begin{array}{cc}\{\begin{array}{c}\frac{??}{?t}=\frac{?}{?z}\left[K\left(h\right)\frac{?h}{?z}+K\left(h\right)\right]\\ ?\left(z,0\right)={?}_i\\ ?\left(?,t\right)={?}_i\\ h\left(0,t\right)=h_p\end{array}& \left(1\right)\end{array}$

[0032] ? and ?.sub.i respectively denote a soil water content and an initial soil water content (cm.sup.3 cm.sup.?3); t denotes an infiltration time (min); z denotes a soil depth (cm); K denotes a soil hydraulic conductivity (cm min.sup.?1); h denotes a soil matrix potential (cm); and h.sub.p denotes a ponding depth at the soil surface (cm).

[0033] The Brooks-Corey model describes the relationship between a soil unsaturated hydraulic conductivity, a soil matrix potential, and a soil water content as follows:

[00007]$\begin{array}{cc}S=\frac{?-{?}_r}{{?}_s-{?}_r}=\{\begin{array}{cc}\frac{?-{?}_r}{{?}_s-{?}_r}={.Math.\frac{h_d}{h}.Math.}^n,& h?-h_d\\ 1,& h>-h_d\end{array}& \left(2\right)\\ K\left(h\right)=K_s{.Math.\frac{h_d}{h}.Math.}^m={K_s\left(\frac{?-{?}_r}{{?}_s-{?}_r}\right)}^{\frac{m}{n}}& \left(3\right)\end{array}$

[0034] ?.sub.s and ?.sub.r respectively denote a saturated water content and a residual water content of the soil (cm.sup.3 cm.sup.?3); K.sub.s denotes a saturated hydraulic conductivity of the soil (cm min.sup.?1); n denotes a distribution index of a soil pore size; and ?h.sub.d denotes a water-entry suction of the soil (cm), m=3n+2.

[0035] Equation (4) describes a variation relationship of effective soil water saturation S with soil depth z (cm) in a water profile of one-dimensional ponded infiltration. In Equation (4), z.sub.f denotes a length of a wetting front (cm); and z.sub.s denotes a length of a saturated zone (cm).

[00008]$\begin{array}{cc}S=\frac{?-{?}_r}{{?}_s-{?}_r}=\{\begin{array}{cc}{\left(1-b\frac{z-z_s}{z_f-z_s}\right)}^a+{?}_r,& z?z_s\\ 1,& z<z_s\end{array}& \left(4\right)\end{array}$

[0036] Equation (5) describes a quantitative relationship between the length z.sub.s of the saturated zone in the soil water profile and the infiltration time t (min) during ponded infiltration.

[00009]$\begin{array}{cc}{z}_s-\left(B_3+B_5\right)\ln \left(1+\frac{z_s}{B_3}\right)+B_5\ln \left(1+\frac{B_2{z}_s}{B_3}\right)+\frac{B_1}{\left(1-B_2\right)B_2}\left(1-\frac{B_3}{\left(B_2{z}_s+B_3\right)}\right)=B_{4}t& \left(5\right)\end{array}$

[0037] Equation (6) describes a quantitative relationship between the length z.sub.s (cm) of the saturated zone in the soil water profile and the length z.sub.f (cm) of the wetting front during ponded infiltration.

[00010]$\begin{array}{cc}{z}_f=z_s+\frac{B_1{z}_s}{B_0\left(B_2{z}_s+B_3\right)}& \left(6\right)\end{array}$

[0038] Equation (7) describes a quantitative relationship between the length of the saturated zone z.sub.s (cm) in the soil water profile and cumulative infiltration amount I (cm) during the infiltration process.

[00011]$\begin{array}{cc}I=\left({?}_s-{?}_i\right)\left(z_s+\frac{b_1{z}_s}{B_2{z}_s+B_3}\right)+K_{i}t& \left(7\right)\end{array}$

where K.sub.i denotes a soil hydraulic conductivity at the initial water content (cm min.sup.?1). The remaining unknown parameters are defined as follows.

[00012]$\begin{array}{cc}\{\begin{array}{c}{S}_i=\frac{{?}_i-{?}_r}{{?}_s-{?}_r}\\ a=\frac{n}{2n+2}\\ b=1-S_i^{\frac{1}{a}}\\ B_0=\left({?}_s-{?}_r\right)\frac{1-\left(1+\mathrm{ab}\right)S_i}{b\left(a+1\right)}\\ B_1=\frac{1-S_i^{1+1/n}}{3n+1}\\ B_2=1-\frac{\left(a+1\right)\left(1-S_i\right)-\left(a\left(a+2\right)b-\left(1-b\right)\left(1-S_i\right)\right)S_i^{3+2/n}}{\left(1-\left(1+\mathrm{ab}\right)S_i\right)\left(a+2\right)}\\ B_3=h_p+h_d\\ B_4=\frac{K_s}{{?}_s-{?}_i}\\ B_5=\frac{B_1}{{\left(1-B_2\right)}^2}\end{array}& \left(8\right)\end{array}$

[0039] Through the above inverse process of the analytical model, an optimized calculation method for soil hydraulic properties (n, h.sub.d, and K.sub.s) is acquired. The method includes the following steps.

[0040] Step 1. Variations of cumulative infiltration amount I and length z.sub.f of a wetting front over infiltration time t during an in-situ field one-dimensional infiltration under ponded conditions of ponded depth h.sub.p are derived.

[0041] Step 2. Saturated water content ?.sub.s, initial water content ?.sub.i, and residual water content ?.sub.r of soil are measured. Based on these parameters and datasets I-t and z.sub.f-t, target parameters n, h.sub.d, and K.sub.s are calculated with an optimized calculation method according to objective function (9).

[00013]$\begin{array}{cc}\min Q=\underset{i=1}{\overset{N}{.Math.}}{\left({\hat{I}}_i\left(n,h_d,K_s\right)-I_i\right)}^2+\underset{i=1}{\overset{N}{.Math.}}{\left({\hat{z}}_{f,i}\left(n,h_d,K_s\right)-z_{f,i}\right)}^2& \left(9\right)\end{array}$

[0042] ? and I.sub.i respectively denote an observed value of the cumulative infiltration amount and a predicted value of the cumulative infiltration amount calculated by an analytical model; {circumflex over (z)}.sub.f,i and z.sub.f,i respectively denote an observed value of the length of the wetting front and a predicted value of length of the wetting front calculated by the analytical model; and N denotes a total number of observed values.

[0043] The objective function (9) is optimized through a Levenberg-Marquardt algorithm, with the idea of estimating correct values of the parameters (n, h.sub.d, and K.sub.s) to minimize the value of objective function (9). Estimated values ?(n, h.sub.d, K.sub.s) and (n, h.sub.d, K.sub.s) are calculated as follows. Given a set of estimation parameters (n.sub.e, h.sub.d,e, and K.sub.s,e), an estimated value (n.sub.e, h.sub.d,e, K.sub.s,e) of length z.sub.s of a saturated zone at any infiltration time I is calculated by a Newton-Simpson method according to Equation (5).

[00014]$\begin{array}{cc}{z}_s-\left(B_3+B_5\right)\ln \left(1+\frac{z_s}{B_3}\right)+B_5\ln \left(1+\frac{B_2{z}_s}{B_3}\right)+\frac{B_1}{\left(1-B_2\right)B_2}\left(1-\frac{B_3}{\left(B_2{z}_s+B_3\right)}\right)=B_{4}t& \left(5\right)\end{array}$

[0044] Based on the (n.sub.e, h.sub.d,e, K.sub.s,e), estimated value ? (n.sub.e, h.sub.d,e, K.sub.s,e) of the cumulative infiltration amount I is calculated according to Equation (7):

[00015]$\begin{array}{cc}I=\left({?}_s-{?}_i\right)\left(z_s+\frac{B_1{z}_s}{B_2{z}_s+B_3}\right)+K_{i}t& \left(7\right)\end{array}$

[0045] Estimated value (n.sub.e, h.sub.d,e, K.sub.s,e) of the length z.sub.f of the wetting front is calculated according to Equation (6):

[00016]$\begin{array}{cc}{z}_f=z_s+\frac{B_1{z}_s}{B_0\left(B_2{z}_s+B_3\right)}& \left(6\right)\end{array}$

[0046] The parameter definition is the same as above.

Embodiment 2

[0047] Step 1. Red soil was sampled from Yingtan, Jiangxi Province, China (28.202942?N, 116.948483?E, 20.9% sand, 34.9% silt, and 44.2% clay). The soil sample was air-dried, crushed, and passed through a 2 mm sieve. Then the soil was loaded into an organic glass column with an inner diameter of 19 cm and a length of 30 cm at a unit weight of 1.29 g cm.sup.?3. Before an infiltration experiment began, a 30 cm long probe of a TDR was inserted into a soil surface to measure an initial water content of the soil, ?.sub.i=0.058 cm.sup.3 cm.sup.?3. Due to the low initial water content, a residual water content was temporarily assumed to be approximately the initial water content, that is, ?.sub.r=0.058 cm.sup.3 cm.sup.?3. In practical applications, if a more accurate residual water content is required, a soil sample is taken with a cutting ring within 0.5 m of a study area and is brought back to a laboratory to measure the residual water content with a pressure membrane.

[0048] Step 2. Constant-pressure water was supplied through a commonly used Mariotte bottle in the industry to conduct a one-dimensional ponded infiltration experiment with a ponded depth of h.sub.p=1 cm. An automatic water level monitoring and acquisition system was provided. During an ponded infiltration process, a real-time reflection waveform signal of the measured soil was acquired through the probe of the TDR. Data series I-t and z.sub.f-f (observed values shown in FIGS. 2 and 3) of a cumulative infiltration amount and a length of a wetting front over time were acquired through waveform analysis. After the infiltration experiment was completed, the water supply was continued until the soil column was saturated. The saturated water content of the soil was measured through the probe of the TDR, ?.sub.s=0.513 cm.sup.3 cm.sup.?3.

[0049] Step 3. Based on known parameters (?.sub.i, ?.sub.s, ?.sub.r, and h.sub.p) and the time series data I-1 of the cumulative infiltration amount and the time series data z.sub.f-t of the wetting front, a Levenberg-Marquardt algorithm was introduced to optimize Equation (9) to estimate parameters n, h.sub.d, and K.sub.s. The estimated results and the measured values acquired through the standard pressure membrane method and the constant-head method are shown in Table 1 and FIG. 4. It can be seen that the method for estimating the soil hydraulic parameters provided by the present disclosure is an accurate method.

TABLE-US-00001 TABLE 1 Comparison between estimated values, acquired by the present disclosure, and measured values of soil hydraulic parameters n h.sub.d (cm) K.sub.s (cm .Math. min.sup.?1) Measured values 0.17 1.2 0.079 Estimated values 0.1965 2.588 0.081

[0050] It should be noted that the above embodiments are only intended to explain, rather than to limit the technical solutions of the present disclosure. Although the present disclosure is described in detail with reference to embodiments, those skilled in the art should understand that modifications or equivalent substitutions may be made to the technical solutions of the present disclosure without departing from the spirit and scope of the technical solutions of the present disclosure, and such modifications or equivalent substitutions should be included within the scope of the claims of the present disclosure.