Forward Problem

Key Takeaways

• Seismic forward modeling (forward problem in seismic acquisition and processing) encompasses several levels of physical approximation depending on the computational resources available and the level of accuracy required: ray-based forward modeling (using geometrical ray tracing through a velocity model, computing traveltimes and amplitudes along the ray paths from the source to each receiver) is computationally fast (seconds to minutes for a 3D velocity model) and provides adequate accuracy for seismic migration velocity model building and structural interpretation in most geological settings, but fails in regions of strong velocity gradients, shadow zones, or diffractions that violate the high-frequency ray approximation; finite-difference or finite-element wave equation modeling (computing the full seismic wavefield by numerically solving the acoustic or elastic wave equation on a discretized grid of the subsurface) provides exact physics within the discrete approximation and correctly models diffractions, multiples, and mode conversions, but requires computation times of hours to days for realistic 3D models on CPU clusters, or minutes to hours on GPU-accelerated hardware; the choice of forward model fidelity is driven by the inversion objective: full-waveform inversion (FWI) requires elastic wave equation forward modeling for accuracy, while migration velocity analysis can use ray-based tomographic forward models; reservoir history matching requires multiphase flow simulation as the forward model, which itself may range from simple material balance calculations (seconds) to full-field compositional simulation (hours to days).
• Gravity and magnetic forward modeling is computationally straightforward compared to seismic because the governing equations (Poisson's equation for gravity, the magnetic scalar potential equation for magnetics) involve no wave propagation and can be solved by direct integration or Green's function summation over the source body: the gravitational attraction of a subsurface density anomaly (a salt body, a dense igneous intrusion, a sedimentary basin filled with low-density sediment) is computed by dividing the body into elementary mass elements (prisms, spheres, or tesseroids in 3D) and summing the vertical component of gravitational attraction at each observation point on the surface; the forward problem for a single 3D density distribution (with 10^6 to 10^8 elementary elements) can be computed in minutes on a modern workstation using fast Fourier transform (FFT) acceleration or Barnes-Hut tree-code approximations; the simplicity of the gravity forward problem makes it a powerful tool for basin modeling (computing the predicted Bouguer anomaly for a structural model and comparing to observed gravity to test the model), salt body geometry inference (determining the extent and shape of a salt canopy or diapir that is consistent with both the observed gravity and the seismic interpretation), and deep crustal structure analysis (where the isostatic compensation of mountain belts and ocean basins can be tested by computing the predicted gravity signature of proposed crustal thickness models).
• Well log forward modeling (synthetic log generation) is the forward problem for petrophysical log interpretation: given a model of the formation (lithology, porosity, fluid saturation, clay content, and formation water salinity at each depth), the tool response equations predict the log readings that each logging tool should record in those formation conditions; the density log response is predicted as rho_bulk = rho_matrix*(1-phi) + rho_fluid*phi (where rho_matrix depends on the mineral composition and rho_fluid depends on the fluid type and saturation); the neutron log response involves more complex corrections for tool standoff, borehole fluid, and formation geometry; the resistivity log response involves the solution of an electrical current flow problem in the invaded and uninvaded formation zones around the borehole; the comparison of synthetic log responses computed from the petrophysical model with the measured log responses at the same depth identifies intervals where the model is inconsistent with the data (indicating that the assumed lithology, fluid type, or saturation is incorrect), guiding an iterative refinement of the model until the forward-calculated logs match all observed log curves simultaneously; this iterative forward problem approach to log interpretation is the basis of simultaneous petrophysical inversion (SPI), where all available log curves are matched simultaneously by optimizing a single consistent petrophysical model at each depth level.
• Basin modeling forward problems compute the burial, temperature, and maturation history of a sedimentary basin from a geological model of the basin structure, stratigraphy, and heat flow, predicting the present-day distributions of source rock maturity (vitrinite reflectance, transformation ratio), oil and gas generation (volume and composition of generated hydrocarbons), and fluid overpressure (from compaction disequilibrium, hydrocarbon generation, or lateral pressure transfer): given the input geological model (stratigraphic column with deposition ages and thicknesses, erosion events, paleowater depths for compaction history, and basal heat flux from the lithosphere), the 1D or 3D forward basin model integrates the differential equations of compaction (Athy's law or a calibrated porosity-depth relationship), heat conduction (Fourier's law with time-varying boundary conditions), kinetic models of kerogen cracking (Arrhenius equations for hydrocarbon generation as a function of time-temperature history), and fluid flow (Darcy's law for pore fluid pressure evolution); the forward model output (maturity maps, generation histograms, pressure profiles) is compared to observed data (measured vitrinite reflectance from well samples, measured formation pressures from DST or MDT tools, oil and gas geochemistry from produced fluids) to calibrate the model parameters (heat flow history, erosion amounts, paleo-water depths) through the inverse problem; successful calibration of the forward basin model provides the confidence basis for predicting source rock maturity and hydrocarbon generation in undrilled areas of the basin.
• The relationship between forward problem accuracy and inverse problem solution quality is fundamental to all quantitative geoscientific interpretation: if the forward model physics is oversimplified (using acoustic rather than elastic wave propagation in FWI of converted wave data, or using 1D piston-drive rather than 3D multiphase simulation in history matching), the inversion will converge to a model that matches the observed data but is not the correct geological model -- the forward model error is absorbed into apparent model parameter changes that are artifacts rather than real geological features; conversely, using unnecessarily accurate (and computationally expensive) forward models when simpler approximations would suffice wastes computation and may prevent the inversion from being run with enough iterations or enough realizations to adequately sample the model uncertainty; the forward problem accuracy required is determined by the noise level in the observed data (which sets a lower bound on the misfit that needs to be achieved) and by the sensitivity of the observable to the parameters of interest (which determines whether the simplified physics captures the relevant physics of the measurement).