A Note on Higher‐Order Corrections of the Head Covariances in Steady Aquifer Flow

Average uniform flow takes place in a heterogeneous aquifer of infinite extent. The input to the problem is the hydraulic conductivity, which is regarded as a random space function that is lognormal, stationary and statistically isotropic. The output is the water head field, which is a random function satisfying the equation of steady flow. The first‐order approximation of the head, in an asymptotic expansion for small log‐conductivity variance, is a normal function characterized completely by the head‐log‐conductivity cross‐covariance and the head covariance or variogram. These covariances are proportional to the log‐conductivity variance. By using spectral methods, second‐order corrections of the head covariances, proportional to the log‐conductivity variance squared, are derived explicitly. Detailed calculations are carried out for an exponential log‐conductivity covariance. The main finding of the note is that the first‐order approximation is very robust and even for a log‐conductivity variance equal to unity, the second‐order correction of the head variances is smaller than 10% of the first‐order approximation.