R.L. Dabney explains the nature of atonement as satisfaction

The word which I should prefer to use, is one sanctioned by the constant usage of the Reformed theologians, "satisfaction." This expresses truly and specifically what Christ did for believers. It points explicitly to the divine law and perfection, whose demand for satisfaction constitute the great obstacles to pardon. It includes also, Christ’s perceptive, as well as His penal, compensation for our debt. We shall see that both Christ’s obedience to the perceptive law and His voluntary endurance of the penal sanction enter into His satisfaction, paid as our substitute. The established word, which has been deliberately attested and approved by the Church, is by all means to be retained. Atonement, or reconciliation is related to satisfaction, as effect to cause.

… The satisfaction of Christ is not idem facere; to do the identical thing required of the sinner, but satis facere; to do enough to be a just moral equivalent for what is due from the sinner. Hence, two consequences. Christ’s satisfaction cannot be forced on the divine Creditor as a legal tender; it does not free us ipso facto. And God, the Creditor, has an optional discretion to decline the proffer, if He chooses (before He is bound by His own covenant), or to accept it. Hence, the extent to which, and the terms on which Christ’s vicarious actions shall actually satisfy the law, depend simply on the stipulations made between Father and Son, in the covenant of redemption.

Yet, we shall by no means agree with the Scotists, and the early Remonstrants, that Christ did not make a real, and equivalent satisfaction for sinners debts. They say, that His sacrifice was not such, because He did not suffer really what sinners owed. He did not feel remorse, nor absolute despair, He did not suffer eternally; only His humanity suffered. But they suppose that the inadequate sufferings were taken as a ransom price, per account by a gracious waiver of God’s real claims of right. And they hold that any sacrifice, which God may please thus to receive, would be thereby made adequate. The difference between their view and the Reformed may be roughly, but fairly defined, by an illustration drawn from pecuniary obligations. A mechanic is justly indebted to a land owner in the sum of one hundred pounds and has no money wherewith to pay. Now, should a rich brother offer the landlord the full hundred pounds, in coin of the realm, this would be a legal tender. It would, ipso facto cancel the debt, even though the creditor captiously rejected it. Christ’s satisfaction is not ipso facto in this commercial sense. There is a second supposition, that the kind brother is not rich, but is himself an able mechanic, and seeing that the landlord is engaged in building, he proposes that he will work as a builder for him two hundred days, at ten shillings per diem (which is a fair price), to cancel his poor brother’s debt. This proposal, on the one hand, is not a "legal tender," and does not compel the creditor. He may say that he has already enough mechanics, who are paid in advance, so that he cannot take the proposal. But, if he judges it convenient to accept it, although he does not get the coin, he gets an actual equivalent for his claim, and a fair one. This is satisfact . The debtor may thus get a valid release on the terms freely covenanted between the surety and creditor. But there is a third plan. The kind brother has some "script" of the capital stock of some company, which, "by its face" amounts nominally to one hundred pounds, but all know that it is worth but little. Yet he goes to the creditor saying, "My brother and I have a pride about bearing the name of full payment of our debt. We propose that you take this "script" as one hundred pounds (which is its nominal amount), and give us a discharge, which shall state that you have payment in full." Now, if the creditor assents, this is payment per acceptilationem. Does Christ’s satisfaction amount to no more than this? We answer emphatically, it does amount to more. This disparaging conception is refuted by many scriptures, such as Isa. 13:21; 53:6. It is dishonorable to God, representing Him as conniving at a "legal fiction," and surrendering all standard of truth and justice to confusion. On this low scheme, it is impossible to see how any real necessity for satisfaction could exist.

… A stick of wood, and an ingot of gold are subjected to the same fire. The wood is permanently consumed, the gold is only melted, because it is a precious metal, incapable of natural oxidation, and it is gathered, undiminished, from the ashes of the furnace. But the fire was the same! And then, the infinite dignity of Christ’s person gives to His temporal sufferings a moral value equal to the weight of all the guilt of the world.

Christ, or His work, is also called lutron, ransom price; and the transaction an apolutrwsi, or redeeming. The obvious idea here, is that of purchase, by a price, or equivalent, out of bondage. He is also our ilasmo, or exilasmo, making for us propitiation, ilasthrion. Expiation is the sacrificial and satisfactory action, making the offended Judge propitious to the transgressor. These terms applied to Christ’s suffering work, justify us in describing His sacrifice, as His vicarious suffering of the penalties due our sins, to satisfy God’s justice and thus reconcile Him to us. (R.L. Dabney, The Nature of Christ's Sacrifice)