To ensure and to demonstrate the mechanical integrity of a welded structure, precise ultrasonic testing (UT) is often mandatory. The importance of the link between nondestructive testing (NDT) and the assessment of structural integrity is recalled. However, it is difficult to achieve great efficiency as the welding of thick and heavy structural part produces heterogeneous material. Heterogeneity results from the welding process itself as well as from the material solidification laws. For thick components, several welding passes are deposited, and temperature gradients create material grain elongation and/or size variations. In many cases, the welded material is also anisotropic, this anisotropy being due to the metal used, for example, austenitic stainless steel. At the early stages of ultrasonic testing, this kind of welded material was considered too unpredictable, and thus too difficult to be tested by ultrasounds without possible diagnosis errors and misunderstandings. At the end of the 1990s, an algorithmic solution to predict the material organisation began to be developed using data included in the welding notebook. This algorithm or modelling solution was called MINA. This present work recalls, in a synthetic form, the path followed to create this algorithm combining the use of solidification laws and the knowledge of the order of passes in the case of shielded metal arc welding (SMAW). This work describes and questions the simplifications used to produce a robust algorithm able to give a digital description of the material for wave simulation code. Step by step, advances and demonstrations are described as well as the limitations, and ways to progress are sketched. Recent developments are then explained and discussed for modelling in the case of gas tungsten arc welding (GTAW), in addition to discussions about 3D modelling for the future. The discussion includes alternative ways to represent the welded material and challenges to continue to produce more and more convincing weld material model to qualify and to make use of UT methods.