We restrict examples to the two-dimensional disk, corresponding to axial flow in circular pipes, to allow direct comparison with the weighting functions of Shercliff and Bevir, presented above. However, the methods presented here easily extend to general flows in 3-dimensional domains, with the only change required is using the appropriate Neumann Green's function. We develop a dipole form of the forward map from flow to boundary potential, to examine the theoretical limits of flowmeters, with potentially multiple point electrodes around the boundary of the pipe. The singular value decomposition (SVD) of the forward map then demonstrates the capability of flowmeters to imaging the spatial flow distribution and a simple reconstruction of a few velocity fields is presented. Our motivation for this work started with the goal of developing accurate inversion for an application in measuring flow in subsurface rivers, or aquifers, in Aoteoroa New Zealand, within a multidisciplinary team developing instrumentation and processing. Measurements in a lab-scale aquifer simulator built for this project are reported in [22] showing correlation with measured bulk flow after interference effects are removed. Presently, there is no efficient way of measuring the natural ground-water flow distribution. Electromagnetic flowmeters seem a promising technology as the principle is non-invasive, and the flowmeter responds to bulk or averaged properties, thereby allowing remote measurement of bulk flow, at least in principle. However, there are many practical obstacles to developing such an instrument for measuring subsurface flow in the natural environment. In contrast to industrial settings, the region of flow is unconstrained while flow direction and velocity is likely to be somewhat random. Electromagnetic flowmeters for open channels are reported in [17] showing some success, though the application to underground sensing is more challenging as the region of flow is essentially unbounded. Further, sensing electrodes are likely to be placed within the flow (and non-zero magnetic field) leading to potentially unbounded sensitivity to local flow around the electrode and to details of electrode geometry. Since electrodes will need to be pushed into the ground for subsurface applications, electrode geometry is unlikely to be precisely known. In contrast, almost all analyses of electromagnetic flowmeters assume zero flow at electrodes, for example due to boundary effects [15], so ignore flow at electrodes and also details of electrode geometry. We note that [PERSON]'s successful measurement [2] employed electrodes in mud banks, so were in regions of zero flow. We analyze the response of flowmeters to flow at electrodes using a novel dipole form in Section 2, and further discuss this issue in Section 4.3. As noted above, the analysis presented here can be generalized to include all these aspects of underground flow sensing. Practical instrumentation also suffers from interference from unwanted magnetic and electric field coupling, and electro-chemical effects at electrodes (see [17; 22] and reference therein). A further potential difficulty is that the conductivity underground may not be spatially homogeneous, so not satisfying the assumptions required for the flowmeter Equation (1) to hold. These last two difficulties are outside the scope of this paper. This paper is structured as follows: The Neumann Green's function in the disk is presented and expanded upon in Section 2, that is then developed to give the dipole form of the forward map for EM flowmeters. The relationship to Shercliff's and Bevir's weight functions is explained. Section 3 gives details required for discrete approximation of the forward map for computational purposes, then uses the computational form to calculate the singular value decomposition (SVD) of the forward map that displays sensitivities of EM flowmeters to spatial flow distribution.