## Achievements

# STUDY ON THE NUMERICAL SIMULATION OF MOISTURE BEHAVIOR IN FIXED DUNE UNDER WATER EXTRACTION BY PLANT ROOTS

Qian Ju^{1}，Long Xunjian^{1}，Ma Jinzhu^{1}，Zhang Huichang^{1}*，Yi Lixing^{2}

1.Key Laboratory of Western China’s Environmental Systems, MOE, Lanzhou University, Lanzhou 730000, China;

2.College of Environmental Science and Engineer, Nankai University, Tianjin 300071,China;*Author for correspondence

**Abstract:** Water uptake by roots in fixed dune was discussed in this paper. The field experiments had been daily carried out on time within the 0cm～175cm depth of the fixed dune planted *Nitraria tangutorum* which is located in the north of Dengmaying basin at the southwestern edge of Tengger desert in Shiyanghe river basin in the northwestern China’s Gansu province to measure moisture content and moisture potential with sand layers by Neutron moisture meter and Tensionmeter. Based on the measured data in the field, the parameters of dune moisture transport, such as unsaturated conductivity, moisture diffusivity etc., were made using mathematical statistics method. Then water uptake model by *Nitraria tangutorum* roots which was related to the rate of evaporation measured by evaporator and dune moisture contents was established at the macroscopic angle. In addition, the numerical simulations of moisture behavior in fixed dune during the growth of *Nitraria tangutorum* and rainless period in June and August were conducted. The relative errors between simulated data and measured ones were -21.97%～5.67%, mainly -0.016%～3.56%, which was indicated in the numerical simulation of dune moisture at the depth of 25cm, 55cm and 100cm on June 5th, 10th and 12th. Meantime, the predicted curves of moisture distribution and the measured ones in the dune profile in autumn were compared. The results from the verification and analysis of dune moisture simulated data showed that the precision of root uptake rate was higher, and it was able to indicate the regularity of water uptake during the rainless period, the predicted moisture distribution curve in the dune profile during *Nitraria tangutorum*growth season by the use of mathematical model was basically consistent with the measured one.

**Key words:** fixed dune; water uptake by plant roots; numerical simulation; moisture prediction

**Introduction**

Limited precipitation is the main source for savageness vegetation existent of no irrigation in desert environment. The plants which grow in desert absorb precipitation mainly depended on dune moisture. Root system is the plants apparatus for absorbing, and it happens every minute. Even though lamina stoma closed, and without transgress of rising moisture during night, the root system are still absorbing little moisture by one certain speed, in order to replenish overfull loss of plants in the daytime, and modulate moisture balance of plants (Kang et al., 1994). It is important part of moisture recycle in groundwater-dune-plant-atmosphere continuum (GSPAC), and is also one of the main contents of plant and moisture connection study (Shi et al., 2003; Zhao et al., 2002). The GSPAC environment factors including dune temperature, moisture content, water solution concentration, aerate status and weather condition would affect the water uptake by plant roots that grow in desert (Zhao et al., 2002; Ji et al., 2006). When the dune containing water in section is different, it affects the distributing of boot system absorbing in section. For example, the main bibulous region appears in shallow dune after precipitation, while sand bed becoming dryer, water uptake zone moves to down direction (Kang et al., 1994).

**1 General Situation of Study Area**

The study plot lies in the north of Dengmaying Lake（103°15′E，38°7′N）which is the largest lake bottomland at the southwestern edge of Tengeer desert in Shiyanghe river basin of Gansu province in northwestern China, and its weather belongs to typical aridity hungriness climate. Mean annual precipitation is 188.0mm, and fasten from June to August, rainfall occupy 82.4% of the whole year. Mean annual evaporation of water surface is 1880.9mm. Normal annual temperature is 7.8℃, and frost-free period is 150d. Ordinary year prevail northwester, mean annual sandstorm day is 120d, and mainly from March to May. Endow with groundwater belongs to unconfined water in desert; the lithology of containing water layer is Upper Permian graining sand and silt sand. Groundwater level is 270cm, and submit period change, the changing range is small, average value is 36 cm (Qian et al., 1999, Yi et al., 1997).

Setting observation point on the elliptic dune (research dune) covered with *Nitraria tangutorum* bush of study plot that is 5m high and wide, 20m long. Using the ZG-1 Neutron moisture meter and WM-1 mercury Tensionmeter regular timing observe moisture content and moist potential at the depth of 0～175cm day by day at the same time. And there is atmosphere observation station in open place that is apart from the dune100m away, which observes precipitation, water surface evaporation and temperature.

Research dune is mainly made up of middle sand and graining sand, which occupied about 90.74%, and the particle diameter smaller than 0.05mm is just 0.03%. Main moist physics constants are field moisture capacity4.51%, maximum hygroscopicity0.17%. Capillarity water rise height is 57cm, total porosity is 41.93%. Cover degree of *Nitraria tangutorum* bush that develop well in nature on dune is upper, and boot system is up growth.

**2 General Resume of the Water Uptake Models by Plant Roots**

The intensity of root system water absorption is showed by root system water absorptivity, that means per volume soil of root system absorbed moist volume in unit period (Shao, 1996; Lei et al., 1988). The research of root system absorption rule and water absorptivity shows, the studies are most based on root system absorption models. And totally including three types: (1) experience model, just as simple root absorption models put forward by Gardner (Gardner,1960), wheat、corn root system absorption models established by Yao Jianwen (Yao,1989). (2) semi-theory and semi-experience model, Molz-Remson、Nimah-Hanks models etc.(Molz F. J. et al., 1970; Nimah M. N. et al., 1973). (3) Theory model, for example, Hillel lodged the model on the basis of simulate water movement by electricity (Kang et al., 1994; Hillel D., 1976; Zhao et al.,1999)^{.}And many root system models were established successfully. The representative models（calculated by root system water absorption）include: (1) Assuming root system absorptivity pro rata with soil-water potential or soil containing moisture quantity. (2) Using transpiration consume water capacity in root system to estimating root system absorptivity (Kang et al., 1994). The research of root system that begin with the end of 50 decades and early 60decades in 20 century suffer from develop modeling, rework and perfect, and have gained quite systemic study production as yet, but researching of configuration, absorption mechanism of root system, specialty application and process visual and etc. are still in investigating (Zhao et al., 1999).

The plant root system absorption models in existence mostly include parameters difficult to assured outside such as density, resistance or perviousness and etc., and some difficulties in actually application. But whether found root system absorption model, or analysis distributing rule of root system water absorptivity, both need to take the root system water absorptivity dynamic data. So, we can consider the root system as a whole one, confirm root section moist movement equation from microscope angle, and estimate root system water absorptivity by observed data in fields (Kang et al., 1994).

**3 Numerical Simulation of Moisture Behavior in Dune under Water Extraction by Plant Roots**

**3.1Confirming of dune moisture dynamic parameters**

Determination of unsaturated coefficient of conductivity, dune moisture diffusivity in field is important content for dune moisture dynamics research, and it is also crucial precondition of dune moisture dynamic simulation in root system absorption condition.

3.1.1 Dune moisture characteristics curve

It means the relation curve that dune moisture substratum potential *j*_{m} or suction s change with volume moisture content*q*,* *and shows the relation between dune moisture energy and quality, also important curve that response the basic characteristics of dune moisture (Lei et al., 1988).

This paper chose observed *j*_{m}-*q* data in filed, using elements of least-squares method and experience function *s*=*ab** ^{q}*, and by regression analysis gets the expression of dune moisture characteristic curve and it can be written as:

* s*=61.7946´0.8314^{100}^{q}* *

*r*=0.9228 （1）

3.1.2 Unsaturated transmissibility

Basis on Darcy’s law and moisture balance theory corresponding unsaturated transmissibility could be calculated due to *q*. Through observed *j*_{m} and *q* data, regression by least-squares method obtains as following (Ma et al.,1998; Zhang et al.,1998):

*K*（*q*）=3.5157´10^{-8}´1.6375^{100}^{q}

*r*=0.9259 （2）

Where *K*（*q*）is the unsaturated transmissibility (cm/s).

3.1.3 Diffusivity of dune moisture

Calculating from expressions of diffusivity *D*（*q*）and *s*（*q*）,i.e.:

*D*（*q*）=*K*（*q*）/*C*（*q*）=4.0904´10^{-4}´1.3614^{100}* ^{q}*（3）

Where *D*（*q*）is diffusivity of dune moisture（cm^{2}/s）, *C*（*q*）is specific capacity（cm^{-1}）,that is named *d**q*/*d**j*_{m.}

**3.2 Establishment of water uptake model by plant roots**

Single root experience model is one of the sublimate model, it is propitious to analyze root system absorption mechanism, relation of root-moist potential and soil-moisture potential, but inapplicable to dynamic dune base moist modeling (Kang et al.,1994; Zhao et al.,1999). Therefore, it should resolve actual problems of dune moisture movement based on macroscopic level.

Study of root system absorption on macroscopic points is adding root absorption item S on the right of dune moisture movement equation, when vertical is one-dimensional flow, the mathematics equation is given as (Kang et al.,1994; Liu et al., 2004):

（4）

Where t is time, z is depth of sand bed, and downwards is positive; *D*_{T} is moist diffusivity effected by temperature, *T *is temperature, *S*（z, t）root absorption item, named root water absorption.

Other denotation meaning is the same as frontal.

When dune moisture content change little, temperature dispatch direct evocable moisture movement is small, can dismission, and also ignore dune moisture movement parameter infected by temperature challenge, and formula（4）is transformed into:

（5）

（6）

Formula（6）is nonlinear, non-homogeneous partial differential equation, adopting finite-difference equation evaluates the closest result.

Change formula（6）into difference equation by implicit difference equation, that is using the difference coefficient instead partial differential equation of the equation and then gets the result:

（7）

Where *i* is special step serial number, *i *=0, 1, …, n; j is time step serial number,* j*=0，1，…，*m*；。

Each root system water absorption of dune section in facultative period of time can calculate by substituting parameter *K*（*q*）, *D*（*q*）to formula（7）. The calculation depth *L *= 175cm, calculation period is during* Nitraria tangutorum* growth period (from May to September), special step is:

cm，period step D*t*=1*d*.

Adopting nonlinear regress analysis method gets research region root system water adsorption *S *changing with moisture content *q*, and the experience equation is:

*S *(*z*，*t*)= *E*_{0 }(t) *AB*^{100}^{q}^{（}^{z}^{，}^{t}^{）}

*r* = 0.9492 （8）

Where *E*_{0 }(t) is water surface evaporation in t period, *A*=1.9728´10^{-4}； *B*=1.1858；*q*（*z*，*t*）is average volume water content of dune depth z in period t（%）.

**3.3 Numerical simulation of moisture behavior in dune**

3.3.1 Elementary condition

Distributing of dune elementary moisture content has already knew, thus:

（9）

3.3.2 Boundary condition

The precipitation of proving ground is relatively little, so dune surface didn’t form water and run-flow. The simulation period during continuum evaporation after secondary precipitation was selected. The upper boundary of evaporation control condition is^{ }(lei et al.,1988):

（10）

Where *E*_{s }(t) is dune surface evaporation intensity in t period, it is calculated by regression Stat. relation of formula（11） (Yi et al.,1997; Ma et al.,1998; Zhang et al.,1998).

（11）

Where *q*_{r }is average volume water content of 25cm depth of dune surface layer（%）.

The lower boundary condition takes the first kind edition by the button measure point of neutron instrument（*z*=175cm）:

（12）

3.3.3 Numerical simulation

The modeling calculation region is dune aeration zone, the depth is 175cm. Express by equation*q*, mathematics positive value questions considering vertical one-dimensional moisture movement of plant root system absorption is:

（13）

Where the symbols are the same as forward.

Formula（13）is two- dimensional nonlinear partial differential equation, evaluation by numerical value method. Researching adopt unit horizon area and height for calculation sand column. By two group perpendicular with one another liner divided calculation region into rectangular gridding, deviding depth into n（*n*=10） unit from z direction,* *and node number is *j*=0，1，2，…，*m*，special step is D*z*（the same as front）, and provide time into m period, node number is *j*=0，1，2，…，*m*，period step is D*t*=1*d*. each node （including inside node and outside node）adopt implicit iterative method separately, and get the quantity of water equilibrium equation:

（14）

Where *b*_{0}，*c*_{0}，*d*_{0}，*d*_{n-1}，*a*_{i}，*b*_{i}，*c*_{i}，*d*_{i }are calculated by formula（15）(Kang et al., 1994).

（15）

Where *r*_{1}=D*t*/（D*z*）^{2}，*r*_{2 }=D*t*/2（D*z*）， is average evaporation intensity of period, , are sand pillar *z*_{i-1/2}、*z*_{i+1/2 }section moisture diffuse ratio in *j*+1 period respectively, , are sand pillar *z*_{i-1/2}、*z*_{i+1/2 }section unsaturated conductivity in *j*+1 period respectively, , are D（）and K（）,* **q*^{j}_{i} is point *i* in period *j* volume water content, _{ }is root system water adsorption of point *i*in period *j*+1/2.

Formula （14）is nonlinear three diagonal equation, evaluation by pursuit method after iterative linear, the evaluation progress includes elimination and back substitution. While numerical value calculating, *K*（*q*）,*D*（*q*）are presented by working experience equation, and other value of parameters are calculated by geometry average method.

**4 Results Verification and Analysis of Numerical Simulation of Moisture Behavior in Dune**

Modeling the dune moisture dynamic state during dry period of *Nitraria tangutorum* growth time in June and August by mathematic model, and contrasted with observed data. Total modeling time is 27d, and chosen time of June 5, June 10, and June12 simulation data of dune water content compare with observed data（table 1）, relative error is between -21.97%～5.67%, and most between -0.016%～3.56%. And also present dune water content distributing simulation section of June 22, August 24 comparison progress with observed section（fig.1）

June 22 August 24

**Fig.1 Comparison of dune moisture simulation values****（****a****）****with observed values****（****b****）****during ***Nitraria tangutorum*** growth period**

** **

**Table 1 Moisture simulation and observed values at different depths in dune in June**（cm，%）

Date | | 25 | | | 55 | | | 100 | |

day/month | Observed value | Simulationvalue | Relative error | Observed value | Simulationvalue | Relative error | Observed value | Simulationvalue | Relative error |

5/6 | 2.385 | 2.470 | 3.56 | 4.982 | 4.720 | -5.26 | 6.089 | 6.434 | 5.67 |

10/6 | 1.925 | 1.502 | -21.97 | 4.862 | 4.975 | 2.32 | 5.647 | 5.756 | 1.93 |

12/6 | 1.856 | 1.603 | -13.63 | 5.109 | 5.221 | 2.19 | 6.433 | 6.432 | -0.016 |

In order to verify the reliability of model, moisture behavior in dune was predicted during *Nitraria tangutorum* growth later period in autumn.

**Fig.2 Comparison of dune moisture forecast values****（****a****）****with observed values****（****b****）****on Sep. 2**

** **

**5 Conclusions and Discussion**

The root system absorption model in this paper can correctly reflect root system absorption rule in dry spell, dynamic numerical value modeling result is testing well with observed value, and this model also could forecast dune section water content distributing during plant growing period（dry period）. However, the key of exactly build up of full purport root system model is accurate determinate dune water content and moisture potential, improve dune moisture movement parameter precision, check out distributing characteristic of root system structure, extend depth, root system density and etc, perfect dune moisture movement elements researching. Dynamic numerical value modeling and forecast dependability depend on mathematics assure problem correction, modeling period special step and period step confirm reasonably. In a word, these problems still need more discussion, resolving and perfection.

**Acknowledgements**

This study was funded by the National Natural Science Foundation of China (No. 49272143) and the National Key Development and Programming Project for Basic Research of China (No. G2000048701).

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