## 1 Introduction
On the supply side, or profile requirement, there is a lack of spatial resolution and predictive ecological information to reveal key impacts that allow the external constraints on the ecosystem services to be checked in a region in the sustainability context [43; 44]. On the other hand, when multiple dimensions (technological, ecological, economic) and multiple periods (years, decades) and scales (national, regional, local) are considered in the assessment of sustainability for a process, city, or region and its WEF nexus, such as the one performed by MuSIASEM, it is possible to obtain a comprehensive view [6]. In this way, it is possible to assess the biophysical constraints on the supply side and the environmental impact on the sink side in socio-ecological systems (SES) through the addition of the flow-fund (FF) model [14], relational analysis [45; 46], and the non-equilibrium thermodynamics of the social-ecological system [15], the pillars of MuSIASEM. Hence, a different approach needs to be developed to connect regionalized inventories, i.e., on the supply side, with impact assessment results to adapt LCA for the evaluation of the SES nexus and its external and internal constraints. The LCA results of the studies conducted thus far do not provide adequate information on the required pattern of supply flows and environmental load per day or year for a process or a PS. Moreover, there is no connection with external constraints and with the continuous pressure on the environment of a place and time. In this sense, it is recognized that the processes operate from the technosphere in LCA, and the supply ecosystem services for the biosphere are two separate processes. Hence, this research proposes establishing this inextricably metabolic relation between them, considering the MuSIASEM as an analytical framework. Mainly, three elements are considered to build an analytical framework and operationalize LCA coupled to the EFs: (i) the FF model, (ii) the time-scale, and (iii) spatial analysis, are used to establish this bridge to LCA improvement. The objective of this study is to present a methodological proposal on the flows-EFs nexus into the LCI and life cycle environmental impact (LCIA) based on the FF model and spatiotemporal scale, incorporating the perspective of the socio-ecological metabolism [47; 48] from a LCA study [3] related to CHP plants without (w/o) and with (w) post-carbon capture (CHP wPCC). Coupling the FF model with the LCA allows for a FF-LCA integrated approach that incorporates the building requirements of the flow patterns (input/output) and EFs (e.g., catchment area, forest land, and stocks-hydrocarbon reserves) rather than only flows to characterize the flow-fund ratios considering a spatiotemporal scale of a one year of operation of a CHP plant to analyze the biophysical constraints from a bottom-up causality point of view. In contrast, the FU model used has been addressed without considering the quantification of the interaction between the demand impact (flows) and ecosystems (ecological funds) for the processes or PS as CHP plant on the micro time scale (seconds). The results can be used to (i) show the implications of a PS considering the biophysical constraints identified from the supply side and the environmental effects on the sink capacity in a territory to implement mitigation plans, (ii) to make decisions related to the viability of the business or future investments at the local or regional level, and (iii) to build regional scenarios by using water, the PES (natural gas), and CO\({}_{2}\) emissions. To guide this study, the content of this paper is organized as follows: Section 2 provides an introduction regarding MuSIASEM principles (Section 2.1) and additional subsection (Section 2.2) to understand the research design, highlighting the importance of the FF model to build a methodological FF-LCA integrating approach. The data LCA case study of a CHP w/o and w PCC [3] was used to show the integrated approach (Section 2.3).