Metastable SiGe films were grown by MBE on Si (001) substrates and annealed to promote varying degrees of partial relaxation. X-ray diffraction reciprocal-space analysis was then used to monitor the structural evolution of the displacement fields of the dislocation array with increasing misfit density. The diffuse-x-ray-scattering patterns of the dislocated heterolayers were compared with lineal-misfit densities determined by defect etching, leading us to develop a geometric model which provides a framework for understanding the early-stage evolution of the displacement fields of the dislocation array, and which also explicitly links diffuse x-ray intensity to misfit density. At low misfit density, the diffuse intensity arises from two-dimensional displacement fields associated with single-nonoverlapping dislocations. As misfit density increases, the displacement fields of individual dislocations increasingly overlap producing three-dimensional displacements. The evolving diffuse intensity reflects the transition from 2-D to 3-D displacement fields. Finally, it is demonstrated that the diffuse x-ray intensity of the strained epilayer can be used to accurately measure lineal misfit-dislocation densities from 400 to 20,000 lines/cm.