## 1 Introduction  
As an essential resource for human development, water is required throughout the life-cycle processes of all of society. In the context of unprecedented economic development and rising urbanization, water usage (i.e., withdrawal) by humans has increased from 500 km\({}^{3}\)yr\({}^{-1}\) to nearly 4000 km\({}^{3}\)yr\({}^{-1}\) over the last century, with an annual increase rate of 1.5% between 1960 and 2010 [1, 2]. This increasing water usage has aggravated water scarcity, affecting more than 2 million people globally. In addition, it is predicted that more than half of the global population will live in regions suffering from at least moderate water shortage by 2050 [3, 4]. Among global water usage, the principal user of water is in the agriculture sector, accounting for 70% of total water usage, with the remaining part being attributable to the industrial sector and domestic sectors [5]. The global domestic water demand is projected to see a 130% increase by 2050, which is much faster compared to other water sectors [6]. Therefore, with economic development, domestic water will play a major role in total water usage in the near future. A long-term evaluation of domestic water usage trends could provide references for policymakers, and is emerging as a paramount issue for efficient and sustainable management of water resources [7]. A series of studies has been conducted on the evaluation of water usage trends, based on different theories and methodologies. These published studies were performed based on two integrated criteria: drivers and approaches. The first one investigated water-usage drivers in a short timeframe. Domestic water usage is considered to be related to a series of drivers, including economic, climate, population, water price, and policies. [PERSON] et al. quantified socioeconomic drivers, such as urban population and service GVA, to investigate the key drivers of water changes [7, 8]. [PERSON] et al. considered climate change as a factor affecting domestic water [9, 10]. [PERSON] et al. proved the significant relationship between regional GDP, population, and water consumption [11]. [PERSON] modeled the linear relationship between water usage and water price [12]. [PERSON] et al. described the GDP per capita, population, and water withdrawal efficient as synthetic factors for domestic water change [13]. Among these factors, the investigation of long-term drivers was constrained by the lack of continuous data, except for the economic drivers, which could be traced back to 1900 by the World Bank [14]. The second one is the investigation of approaches. These approaches can be divided into two types, namely, single-equation models and hybrid models. Single-equation models include the linear regression model [10], whale optimization algorithm [15], artificial neural network [16], pseudo-panel approach [8], and so on. Hybrid models are driven by macroscale socioeconomic activity to simulate water use in specific regions. Hybrid models include IMAGE (Integrated Model to Assess the Global Environment) in IMAGE regions [17], QUAIDS (Quadratic Almost Ideal Demand System) models in Spanish [12], and IUWM (Integrated Urban Water Management) in Australia [18]. Therefore, a universal and long-term evaluation of water-usage trends is still missing, and a new approach should be introduced. In this research, we adopted the expanded S-curve model to assess the long-term relationship between water usage and economics, which are presented by domestic water usage per capita (DWPC) and GDP per capita, respectively. The S-curve model has been widely used in mineral and energy resource evaluation. The S-curve pattern was first proposed by French mathematician [PERSON] in 1838 for the description of biological population [19], and was employed by [PERSON] et al. [20] and [PERSON] et al. [21] to quantify the relationships between economics and energy and steel, respectively. Besides, in the water-usage studies by [PERSON] [22] and [PERSON] et al.