Restoring Environmental Justice: On the Coupled Dynamical Analysis of Lake Powell and Lake Mead

A new omen of the current climate crisis is surfacing, threatening lives relying on Lake Powell and Lake Mead. As the largest reservoirs in Colorado River and its tributary system, the two lakes are su ﬀ ering from structural water deﬁciency caused by persistent global warming and human related environmental destruction. This study proposes threshold values of water recycling and water saving by applying mathematical models within reality-based circumstances. The dynamic nature of environmental factors has been elucidated by applying a system of ordinary di ﬀ erential equations (ODE). In addition, water level predictions under various scenarios are obtained from the ODE system solutions. Our study provides a clear and simulated roadmap reﬂective of the scale and urgency of restoring climate justice in Lake Powell and Lake Mead.


Introduction
Water is the driving force of all lives.Ecosystems and mankind cannot survive without it.Hoover Dam, built on the Colorado River and its tributaries, is one of the iconic marvels of modern engineering and has played a crucial role in the health, well-being and economic prosperity of the Southwest in the United States.Today, the Bureau of Reclamation estimates around 40 million people are reliant on the Colorado River Basin (Baculi et. al, 2022) (see Figure 1).
When water allocations were decided by the 1922 Colorado River Compact, the Colorado River Basin was going through a period of unusually wet years (Barrett et. al, 2008).As a result, the average annual flow of the Colorado River was overshot, at 17.5 MAF.(Baculi et. al, 2022).This Compact, and later on, the Law of the River decided the allotment of the Upper and Lower Basins: each would receive 7.5 MAF annually.The current burden of Colorado River also includes the annual 1.5 MAF apportionment to Mexico (Bureau of Reclamation) and approximately 0.9 MAF of evaporation (Cerveny et. al, 2022).However, recent assessment reveals that the actual long term average natural flow of the Colorado River is 14.8 MAF -about 16% lower than what was assumed in the 1922 Compact allocations (Baculi et. al, 2022).The hardest working river in the Southwest is overexploited.
In addition to the enduring and parched drought overcasting the Basin since 1998, increasing demand on water for economic growth has further aggravated the situation.As a result, the water levels of Lake Powell and Lake Mead keep dropping.Lake Powell is just under twenty seven percent full and Lake Mead is now only about a quarter full (Barrett et al, 2008).The severe water depletion threatens access to clean water for drinking for millions of Americans.It severely impacts agricultural yields, biodiversity, and the environment of the surrounding ecosystems.Moreover, it limits the abilities of industries to operate as there is not enough water flow to produce hydroelectric power (Xu and Ramanathan 2017).In addition to electricity generation, the two lakes serve as a major source of grid resilience when full, backing up energy where solar or wind power is on shortage (Bureau of Reclamation).This function is at risk.Clearly, the long-term sustainability of Lake Powell and Lake Mead and the implications for millions people living in the Southwest is a national concern.
Built on extent literature and publicly available data, this study focuses on a unique aspect of the current crisis C the overuse of the Colorado River.Prior studies indicate that water levels failed to recover even after years of above-average snowfall occurring upstream in the River, implying human related environmental destruction as a striking impetus among other culprits.Assuming the climate trend continues, a more careful recalculation of partition of use for the agriculture, residential and industry sectors is necessary.
Grounded on prior findings, this study models the dynamic relation among the following factors: precipitation amount, evaporation rate, groundwater refill rate, water recycling rate and apportionment between stakeholders.Differently from to optimize the allotment among constituents and outflows from Lake Powell to Lake Mead.Specifically, our analyses adopts the realistic amount of total inflow of 14.8 MAF in most recent 5 years as a base.The analyses also count an additional 0.77 MAF of inflow into Lake Mead from other tributaries.The binding conditions for the ODE system are: a) at any moment, water levels have to maintain equal to or greater than the last 3-year average values b) a minimum of 1.6 MAF flow to Mexico is guaranteed.As a result, the projected water levels from my models meet the USBRs environmental sustainability goals, reflective of pragmatic and attainable solutions to various reality-based scenarios that could incur during our global climate plight.This study contributes to dynamic modeling skills in environmental research as well as serves as a roadmap to recover water levels for the two lakes.

Method
The water volume evolution of Lake Powell can be modeled through where V p is the volume of Lake Powell and dt is the rate of change of V p with respect to time t, f p is the rate of outflow from Lake Powell into Lake Mead, S p is the total surface area of Lake Powell, and α p S p is the net rate of loss of water volume due to the combined influence of evaporation and precipitation through S p .The coefficient α p is the difference of rate of evaporation and precipitation per unit area per unit time subject to Lake Powell.I p is the net rate of water inflow to Lake Powell from Colorado River.
With the apportionment of water usage of the Upper Basin states, the inflow I p can be further decomposed as ( where I p0 is the rate of inflow to Lake Powell from the upstream Colorado River and various tributaries.Without human Introducing equation (2) into equation ( 1), we arrive at the governing equation for Lake Powell, Similarly, the water volume of Lake Mead is modeled through where f o + f MEX is the outflow of Lake Mead into the Colorado River that arrives in Mexico (including the water evaporated along the flow), from which f o is the outflow that eventually arrives in the ocean, and f MEX is the water outflow allocated to Mexico.S m is the surface area of Lake Mead, α p is the net rate of water loss through S m per unit area (caused by precipitation and evaporation).f p is the outflow of Lake Powell which is assumed to have completely reached Lake Mead.I m0 is the inflow coming from tributaries between Lake Powell and Lake Mead.U m is the rate of water use related to Lake Mead allocated to the Lower Basin states.The proportion of the Lower Basins water use that can be recycled is denoted γ m U m , where γ m is a proportionality factor.k m U m (t − ∆t) is the amount of water that leaves Lake Powell to refill groundwater reservoirs, where k m is the proportionality coefficient.Equations ( 4) and (3) jointly determine the dynamics of the water volume for the system of reservoirs of Lake Mead and Lake Powell.
We are interested in constructing a system of equations from which the water height of Lake Mead and Lake Powell can be modeled.In order to convert the volume and the surface area in equations ( 3) and (4) into functions of height, we make considerations as follows.
Let ∆h = h − h be the maximum depth of the lake, where h is the surface elevation and h is the elevation of the bottom, for either Lake Mead or Lake Powell.Although the 3D shape of the lake is irregular, it can be simplified, to the leading order, as a right cone.
As shown in Fig.
(2), a lakes volume can be modeled as where θ is half the opening angle according to Fig.
(2), and the lakes surface area is Time differentiating equation ( 5) yields Substituting equations ( 7) and (6) into equations ( 3) and (4), denoting h p and h m for the lake surface elevation of Lake Powell and Lake Mead to replace h, respectively, and denoting h p and h m the elevation for the bottom of the Lake Powell and Lake Mead, respectively, to replace h , we obtain Figure 2. (a) Simplified 3D shape of the lake and its cross-section.The surface elevation is h and the elevation of the bottom of the lake is h.The maximum depth of the lake is ∆h = hCh.(b) The vertical cross section of the lake where the radius of the surface area is approximated as ∆h tan θ were we have introduced f m = f o + f MEX as the total outflow from Lake Mead to Mexico.Equations ( 8) and ( 9) are the governing system of ordinary differential equations for the elevations of Lake Powell and Lake Mead.We want to find the proper set of parameters ( f p , γ p , U p , γ m , U m ) on the right hand side of the two equations such that the elevations, h p and h m , as the solutions to the system, may be sustainably obtained above the safe threshold for peoples living water supply.

Simulations and Analysis
Based on historical data, the annual net water evaporation (evaporation minus precipitation) per square feet can be estimated as α p = 5.7 feet for Lake Powell (Dally 2008) and α m = 6.2 feet for Lake Mead (Dally 2008).According to the Bureau of Reclamation, the bottom elevation of Lake Powell is h p = 3117 and h m = 650 feet for Lake Mead.The total water inflow from Colorado River upstream is approximated as I p0 = 14.8 MAF per year.The water inflow from Colorado tributaries between Lake Powell and Lake Mead is estimated I m0 = 0.77 MAF per year.The tanθ of the lake opening angle θ as shown in Fig. ( 2b) is estimated to be tanθ = 6 for both lakes (Release U.S. Geological Survey Open File Report 2003).The coupled system of equations ( 8) and ( 9) can be solved numerically with finite difference by Euler method, in which we approximate the time derivative of the water height as where h can be either h p or h m , ∆t is a fixed finite difference time increment.The solutions of water height is viewed as functions of time and a set of human related factors: The notation h(t|) emphasizes that G = { f p , U p , γ p , k p , U m , γ m , k m } generates the time-varying solution of water height.Different solutions may be obtained once scenarios of different parameter settings of G are raised.The annual inflow of water to Lake Mead I m0 + f p can be estimated at 9.00 MAF, 8.23 MAF of which can be attributed to the outflow of Lake Powell (Western Resource Advocates), f p = 8.23 MAF, and the rest, I m0 = 0.77 MAF, from downstream tributaries.Other major adjustabe factors are the Upper Basin states water apportionment U p = 7.5 MAF/year, the Lower Basin states water apportionment U m = 7.5MAF/year, and the net outflow of Lake Mead to Mexico f m = 1.5 + 0.6 = 2.1 MAF/year (0.6 MAF is the estimated evaporation before the outflow reaches Mexico and 1.5 MAF is the actual apportionment arrives at Mexico).These numbers of U p , U m , and f m were determined based on an over-estimated inflow of the Colorado River I p0 into Lake Powell historically observed over an abnormal wet period in the 1920s.Modern estimates ascribe the inflow I p0 to a value of 14.8 MAF/year, significantly lower than what was originally estimated.The resulting overshot U p , U m , and f m are the main reason behind the current water crisis.Our simulation (Fig. 3) reveals that should Lake Powell and Lake Mead continue their current water allocations to the Upper Basin and Lower Basin states, they would reach dead pool in 2−3 months from their current elevation of 3560 ft for Lake Powell and 1090 ft for Lake Mead, where dead pool is reached at ff 3530 ft for Lake Powell and 1065 ft for Lake Mead (Bureau of Reclamation).The simulation assumes the water recycling rate for Lake Powell and Lake Mead γ p = γ m = 15%, and the water refilling rate k p = k m = 10% (Castle et al. 2014).We set the finite difference time increment ∆t = 1 month (Leake et al. 2013).All annual inflow/outflow and apportionment are averaged to monthly amount by dividing with 12.
From 1995 to 2015, the percentage of industry and mining usage of water remain relatively small and takes only 1% -2% of the total usage (Tang et al. 2009).We may ignore the impact of competing interests of water availability coming from industry and mining over residential and agriculture.The residential living water usage has grown from 12% in 1995 to 19% in 2015 (Tang et al. 2009).
At the same time, the irrigated agriculture, although declined from 86% to 77%, remains by the largest category of the If we uplift the water recycling rate from 15% to 20%, while keeping the apportionment fixed (that is U p = U m = 7.5MAF f m = 2.1 MAF and f p = 8.1 MAF), the water elevation for Lake Mead will gradually rise (Fig. 4a) as opposed to quickly declining to the elevation of a dead pool (Fig. 3).However, at this recycling rate, the Lake Powell remains the same fate with only slightly longer lifetime extension to 7 months.If we continue to increase the recycling rate by as little as 1%.That is γ p = γ m = 21%.Both Lake Mead and Lake Powell will gradually regain their water elevation over a period of 36 months and the elevation will continue to rise if the recycling rate is kept at this rate (Fig. 4).Our simulations, as shown in figures (4a) and (4) suggest that as small as 1% fluctuation of γ influences the resulting water level significantly.Given the current technology status, reaching to a recycling rate of 21% is infeasible in common situations.In reality, the water recycling rate can achieve no more than 15% and the apportionment for U p , U m , f m , and the outflow of Lake Powell (also inflow to Lake Mead) f p must be adjusted, in order to maintain Lake Mead and Lake Power to their operational level in the next few decades.Sustainable water levels can be obtained from solutions to the system of ordinary equations (8)

Figure 1 .
Figure 1.Colorado River Basin.Lake Powell holds the outflow from the Colorado River Upper Basin states, and Lake Mead is the main reservoir formed by the Hoover Dam on the border between Southern Nevada and Northwestern Arizona

Figure 3 .
Figure 3. Evolution of water height for Lake Powell (top) and Lake Mead (bottom) reach the dead pool level if the annual apportionment to the Upper Basin states, Lower Basin states, and Mexico remain unchanged: U p = U m = 7.5 MAF, f m = 2.1 MAF (a) Elevation prediction at γ p = γ m = 20%, U p = U m = 7.5 MAF/year, f m = 2.1 MAF/year.(b) Elevation prediction at γ p = γ m = 21%, U p = U m = 7.5 MAF/year, f m = 2.1 MAF/year.

Figure 4 .
Figure 4. Prediction of water elevation for Lake Powell and Lake Mead using slightly different recycling rate at a) γ p = γ m = 20% and b) γ p = γ m = 21% while the annual apportionment is kept unchanged U p = U m = 7.5 MAF, f m = 2.1 MAF