An Inverse Method to Estimate the Root Water Uptake Source-Sink Term in Soil Water Transport Equation under the Effect of Superabsorbent Polymer

<div><p>The widespread use of superabsorbent polymers (SAPs) in arid regions improves the efficiency of local land and water use. However, SAPs’ repeated absorption and release of water has periodic and unstable effects on both soil’s physical and chemical properties and on the growth of plant roots, which complicates modeling of water movement in SAP-treated soils. In this paper, we proposea model of soil water movement for SAP-treated soils. The residence time of SAP in the soil and the duration of the experiment were considered as the same parameter <i>t</i>. This simplifies previously proposed models in which the residence time of SAP in the soil and the experiment’s duration were considered as two independent parameters. Numerical testing was carried out on the inverse method of estimating the source/sink term of root water uptake in the model of soil water movement under the effect of SAP. The test results show that time interval, hydraulic parameters, test error, and instrument precision had a significant influence on the stability of the inverse method, while time step, layering of soil, and boundary conditions had relatively smaller effects. A comprehensive analysis of the method’s stability, calculation, and accuracy suggests that the proposed inverse method applies if the following conditions are satisfied: the time interval is between 5 d and 17 d; the time step is between 1000 and 10000; the test error is ≥ 0.9; the instrument precision is ≤ 0.03; and the rate of soil surface evaporation is ≤ 0.6 mm/d.</p></div

Calculated water uptake distribution in the soil profile at different levels of unsaturated diffusivity.

<p>Use inverse method to calculate distribution of water uptake in the soil profile at different unsaturated diffusivity (100<i>D</i>, 10<i>D</i>, 5<i>D</i>, 0.5<i>D</i>, 0.1<i>D)</i>, 100<i>D</i> means the value of unsaturated diffusivity is increased by 100 times than the normal <i>D</i>, and 0.01<i>D</i> means the value of unsaturated diffusivity is reduced by 100 times than the normal <i>D</i>, Theoretical values are calculated from the root water uptake model developed by Shao AJ et al [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p

An Inverse Method to Estimate the Root Water Uptake Source-Sink Term in Soil Water Transport Equation under the Effect of Superabsorbent Polymer - Fig 8

<p>Theoretical moisture and calculated water uptake distribution in the soil profile at different rates of soil surface evaporation (a) Theoretical moisture distribution (b) Calculated water uptake distribution. Use inverse method to calculate distribution of water uptake in the soil profile at different rates of soil surface evaporation (<i>E</i> = 0.03, <i>E</i> = 0.1, <i>E</i> = 0.3, <i>E</i> = 0.6), The “<i>E</i>” stands for evaporation. All the distribution of water uptake in the soil profile are calculated from applying SAP 3d to 15d. Theoretical distribution of moisture and water uptake are calculated from the root water uptake model developed by Shao AJ et al[<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p

Selection of parameters for numerical testing.

<p>Selection of parameters for numerical testing.</p

Theoretical moisture and calculated water uptake distribution in the profile of layered soil.

<p>(a) Theoretical moisture distribution; (b) Calculated water uptake distribution. Use inverse method to calculate distribution of water uptake in the profile of layered soil (sandy loam+silty soil), Theoretical distribution of moisture and water uptake are calculated from the root water uptake model developed by Shao AJ et al [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p

Theoretical moisture and calculated water uptake distribution in the soil profile at different times.

<p>(a) Theoretical moisture distribution; (b) Calculated water uptake distribution. Use inverse method to calculate distribution of moisture (apply SAP after 0, 1, 3, 5, 8, 10, 13, 15, 17, 20d) and water uptake (from apply SAP 3d to 5d, 3d to 8d, 3d to 10d, 3d to 13d, 3d to 15d, 3d to 17d, 3d to 20d) in the soil profile at different times. Theoretical values are calculated from the root water uptake model developed by Shao AJ et al [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p

Calculated water uptake distribution in the soil profile at different instrument precisions.

<p>Use inverse method to calculate distribution of water uptake in the soil profile at different instrument precisions (<i>w</i> = 0.01, <i>w</i> = 0.03, <i>w</i> = 0.05, <i>w</i> = 0.1), The bigger the value of <i>w</i>, the smaller the instrument precisions. Theoretical values are calculated from the root water uptake model developed by Shao AJ et al[<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p

Calculated water uptake distribution in the soil profile at different test errors.

<p>Use inverse method to calculate distribution of water uptake in the soil profile at different test errors (<i>per</i> = 0.95, <i>per</i> = 0.90, <i>per</i> = 0.85, <i>per</i> = 0.80, <i>per</i> = 0.70), The bigger the value of <i>per</i>, the smaller the test error. Theoretical values are calculated from the root water uptake model developed by Shao AJ et al[<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p

Calculated water uptake distribution in the soil profile at different levels of unsaturated hydraulic conductivity.

<p>Use inverse method to calculate distribution of water uptake in the soil profile at different unsaturated hydraulic conductivity (100<i>K</i>, 10<i>K</i>, 5<i>K</i>, 0.5<i>K</i>, 0.1<i>K</i>, 0.01<i>K</i>, 0.001<i>K</i>). 100<i>K</i> means the value of unsaturated hydraulic conductivity is increased by 100 times than the normal <i>K</i>, and 0.01<i>K</i> means the value of unsaturated hydraulic conductivity is reduced by 100 times than the normal <i>K</i>, Theoretical values are calculated from the root water uptake model developed by Shao AJ et al [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref033" target="_blank">33</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0159936#pone.0159936.ref034" target="_blank">34</a>] (<i>S</i> = <i>ET</i>×<i>A</i>×[<i>e^-B</i>(<i>lnZ-C</i>)²]/Z).</p