We introduce a family of abelian sandpile models with two parameters n, m ∈ N defined on finite lattices on d-dimensional torus. Sites with 2dn + m or more grains of sand are unstable and topple, and in each toppling m grains dissipate from the system. Because of dissipation in bulk, the models are well-defined on the shift-invariant lattices and the infinite-volume limit of systems can be taken. From the determinantal expressions, we obtain the asymptotic forms of the avalanche propagators and the height-(0, 0) correlations of sandpiles for large distances in the infinite-volume limit in any dimensions d ≥ 2. We show that both of them decay exponentially with the correlation length ξ(d, a) = (√<d> sinh^<−1>√<a(a + 2)> )^<−1>, if the dissipation rate a = m/(2dn) is positive. Considering a series of models with increasing n, we discuss the limit a ↓ 0 and the critical exponent defined by ν_a = −< lim>___<a↓0> (log ξ(d, a)) / (log a) is determined as ν_a = 1 / 2 for all d ≥ 2. Comparison with the q ↓ 0 limit of q-state Potts model in external magnetic field is discussed.