This may be a simple question, but I decided to post it here (not just on MSE) because it is very related to a research topic: capillary surfaces.
Let $(M^3,g)$ be a Riemannian $3$-manifold with boundary. Fix an embedding $\varphi: \Sigma \to M$ of a compact surface $\Sigma$ with boundary such that $\varphi(\Sigma) \cap \partial M = \varphi(\partial \Sigma)$, and let $\Phi : \Sigma \times (-\varepsilon, \varepsilon) \to M$ be a variation of $\varphi$ by embeddings that also satisfy this last condition. Also let $\Sigma_t = \Phi(\Sigma \times \{t\})$.
When dealing with capillary surfaces, one often considers the following function, called the wetting area, associated with the previous data: $W : (-\varepsilon, \varepsilon) \to \mathbb{R}$ given by
$$W(t) = \int_{\partial \Sigma \times [0,t]} \Phi^*(\operatorname{d}A_{\partial M}),$$
where $\operatorname{d}A_{\partial M}$ denotes the area element of $\partial M$. Suppose that for some $t$ (positive or negative) the boundary $\partial \Sigma_t$ lies on one side of $\partial \Sigma$. Is it true that $W(t) > 0$ in this case? Could you help me to ellucidate this definition, please?