How do the big consulting firms get paid for benefits saved/the value of the solution for their clients

Hello,

Can anyone tell me how these big consulting firms get paid for the BENEFITS saved for their clients or the VALUE of the solution they provide? What I mean by that is I understand that consultants can get paid by the hour, the day or for a fixed duration for a number of different things including paid consulting, advisory, project managing changes, programme management, implementing a new technological solution, training staff etc etc etc but how to consulting firms get paid when they save a client say $5-10 million dollars and ask for 10-20% in return? HOW DO THEY MEASURE IT? I mean do they conduct some kind of (reverse) due diligence, do they send in auditors to look at the financials of the client, do they measure the value of new customers because of the work they have done? Do they cut costs and ask for the purchase invoices and work out 10%? I mean I work as an accountant an if you were to save a client $10m because of something you did, surely it would be open to interpretation. Both sides could discuss the interpret details for hours. Would they just send in the lawyers first to define what 'benefits saved' means to avoid questions later, how does it work?