We focus on the so-called Warburg effect and its mathematical modelling by means of the acid-mediated invasion hypothesis, namely the typical strategy of acidity increasing against the environment operated by tumours to regulate their growth, which is translated into a system of reaction-diffusion equations starting from the seminal work by Gatenby-Gawlinsky [Cancer Research 56(24) 1996] The main theoretical issues are the existence of travelling wave solutions and the numerical simulation of experimental results. We make a novel assumption, namely that acidification caused by lactic acid production proceeds over a heterogeneous micro-environment which is compatible with physical tissues, and that is advantageous for the spread of cancer cells, whilst it destroys healthy cells. Hence, the original modelling is modified by taking into account heterogeneous (possibly periodic) diffusion coefficients, and we illustrate the passages leading to an approximation of travelling wave solutions for various biological regimes. Then, we introduce the numerical algorithm, together with its consistency and convergence analysis, and we simulate many different cases depending on the biological context. Finally, we make considerations on a possible homogenisation phenomenon of travelling fronts in the long run dynamics and we provide a comparison between analytical and numerical approximations.