Interaction Between Biofilm Growth and Fluid Flow in Seafloor Hydrothermal Systems

Abstract

We adapt a set of differential equations commonly used in the mathematical biofilm literature to study the interaction between biofilm growth and fluid flow in seafloor hydrothermal systems. The resulting equations admit a steady state solution; phase space analysis indicates that the system will converge to this point from a wide variety of initial conditions. We present numerical solutions of the biofilm equations in which detachment processes dominate at the base of the system and a biofilm-rich zone propagates upward, halting before reaching the surface. The maximum height of this zone depends on the vigor of fluid flow and the concentration of limiting nutrient. We find no solutions resulting in cyclically varying rock permeability. There is, however, a characteristic time scale for the zone of high biofilm concentration or lowered permeability to reach minimum subsurface depth.

7 Appendix

1.1 7.1 Leibniz formula

Assume that we have a function of the form f(x, y, t) and let $\zeta$ equal y or t. If we integrate the derivative of this function with respect to $\zeta$ from $x=0$ to $x=g(y,t)$, we obtain

$$\begin{aligned} \frac{\partial }{\partial \zeta }\int _0^g f dx = \int _0^g \frac{\partial f}{\partial \zeta }dx + f_g\frac{\partial g}{\partial \zeta }. \end{aligned}$$

(99)

1.2 7.2 Alternative derivation of (70)

As before we begin with $\int _0^w\phi Cdx=\phi _b L {\bar{C}}+\int _L^w Cdx$. But now, with the idea that the integral is the area under the curve in mind (see Fig. 6), we approximate the integral on the right hand side with $(w-L){\bar{C}}$. In this way, we obtain

$$\begin{aligned} \int _0^w\phi C dx=\phi _b L{\bar{C}}+w{\bar{C}}-L{\bar{C}}=(\phi _b-1)L{\bar{C}}+w{\bar{C}}. \end{aligned}$$

(100)

Hence,

$$\begin{aligned} \frac{\partial }{\partial t}\int _0^w \phi C dt = (\phi _b-1)L\frac{\partial {\bar{C}}}{\partial t}+(\phi _b-1){\bar{C}}\frac{\partial L}{\partial t}+w\frac{\partial {\bar{C}}}{\partial t}, \end{aligned}$$

(101)

as before.

Fig. 6

We represent the chemical concentration in the channel to reflect assumption 3, and approximate the area under the curve between $x=L$ and $x=w$ with the shaded area, $(w-L){\bar{C}}$